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+Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+Ultra-Low-Loss Silicon Nitride Photonics Based on 
+Deposited Films Compatible with Foundries 
+ 
+Xingchen Ji,1,3,* Yoshitomo Okawachi,2 Andres Gil-Molina,1 Mateus Corato-Zanarella,1 
+Samantha Roberts,1 Alexander L. Gaeta,2 and Michal Lipson1,* 
+ 
+1Department of Electrical Engineering, Columbia University, New York, NY, 10027, USA  
+2Department of Applied Physics and Applied Mathematics, Columbia University, New York, 
+NY, 10027, USA 
+3Currently at John Hopcroft Center for Computer Science, School of Electronic Information and 
+Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China 
+*Corresponding Author: E-mail: xingchenji@sjtu.edu.cn and ml3745@columbia.edu 
+ 
+Abstract:  
+ 
+The fabrication processes of silicon nitride (Si3N4) photonic devices used in foundries require low 
+temperature deposition, which typically leads to high propagation losses. Here, we show that 
+propagation loss as low as 0.42 dB/cm can be achieved using foundry compatible processes by 
+solely reducing waveguide surface roughness. By post-processing the fabricated devices using 
+rapid thermal anneal (RTA) and furnace anneal, we achieve propagation losses down to 0.28 
+dB/cm and 0.06 dB/cm, respectively. These low losses are comparable to the conventional devices 
+using high temperature, high-stress LPCVD films. We also tune the dispersion of the devices, and 
+proved that these devices can be used for linear and nonlinear applications.  Low threshold 
+parametric oscillation, broadband frequency combs and narrow-linewidth laser are demonstrated. 
+Our work demonstrates the feasibility of scalable photonic systems based on foundries. 
+ 
+
+Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+1. Introduction 
+To date, ultra-low-loss silicon nitride (Si3N4) waveguides and resonators have been demonstrated 
+almost exclusively using films deposited at high temperature, while foundries mostly rely on Si3N4 
+films deposited at low temperature. The high temperature deposition  uses low-pressure chemical 
+vapor deposition (LPCVD), while low temperature deposition uses plasma-enhanced chemical 
+vapor deposition (PECVD). PECVD Si3N4 is the most commonly used thin film in foundries as an 
+insulator or a chemical barrier layer, however, the high propagation losses in these films limit their 
+applications in photonics. LPCVD Si3N4 is not used in foundries due to the high temperature 
+required and high film stress. Therefore, reducing losses in PECVD Si3N4 photonic devices is 
+critical for integrating photonics devices with electronics, which could be used to realize high 
+performance, scalable systems and realize system-level innovation[1]. 
+Previously, there have been efforts to reduce losses in PECVD Si3N4 films by chemically 
+changing the film composition[2–5]. By lowering the ammonium concentration during the 
+deposition, losses down to 1.5 dB/cm have been shown[2]. However, these losses remain too high 
+for most photonic applications. Researchers have also substituted conventional precursors with 
+deuterated ones to reduce the losses of the film, losses down to 0.3 dB/cm have been shown[6]. 
+However, these methods require special precursors and deposition tools, which are not commonly 
+available in foundries. 
+ 
+2. Film deposition and waveguide fabrication 
+Here we show that low-loss can be achieved in a standard PECVD process by physically reducing 
+waveguide surface roughness. The fabrication process is schematically shown in Figure 1. We 
+deposit Si3N4 using PECVD at 350 °C in a single step onto a thermally oxidized 4-inch silicon 
+
+Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+wafer. The gases used for deposition are a mixture of silane (SiH4: 20 sccm) diluted by nitrogen 
+(N2: 1425 sccm) and pure ammonia (NH3: 30 sccm), with a process pressure of 1900 mTorr. The 
+plasma frequencies alternate between a high frequency (13.56 MHz) with a power of 200 W and 
+a low frequency (100 kHz) with a power of 160 W. The time duration for the two frequencies is 8 
+seconds and 12 seconds, respectively. The above parameters ensure that the deposition of Si3N4 
+film has very low film stress and high uniformity. The measured stress for the Si3N4 film on a test 
+wafer is 93.4 MPa and tensile, which is more than an order of magnitude lower than LPCVD Si3N4 
+films deposited at high temperature. The low stress allows us to deposit thicker films without any 
+cracking.  
+ 
+Figure 1. Schematic of our low-temperature PECVD Si3N4 fabrication processes. 
+The process steps here are fully compatible with CMOS electronics. 
+ 
+We design high confinement waveguides based on the deposited PECVD films allows for 
+strong dispersion engineering. One can see in Figure 2, the strong mode overlaps with the top 
+surface that can exhibit a roughness of several nanometers for PECVD films[7,8]. 
+
+Si,N4
+SiO2
+Si,N4
+SiO2
+SisN4
+SiO2
+SiO2
+SiO2
+Resist
+Resist
+SiO2
+SiO,
+Sio,
+SigN4
+SisN4
+SisN4
+ SiO2
+SiO2
+SiO2
+SiO2
+SisN4
+SiO2Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+ 
+Figure 2. Mode simulation and microscope images of fabricated devices. (a) Mode 
+simulation of 730 nm tall and 1500 nm wide waveguide showing that the mode is 
+highly confined in the geometry we have chosen. (b) Top view optical microscope 
+image of a 115 µm radius ring resonator. 
+To reduce scattering from the top surface of PECVD Si3N4, we use chemical mechanical 
+planarization (CMP) to smooth the surface, as roughness traditionally leads to a high loss. We 
+show the atomic-force microscopy (AFM) scans before and after the polishing step in Figure 3. 
+The root-mean-squared (RMS) roughness is decreased from 1.36 nm before polishing to 0.20 nm 
+after polishing. In order to reduce the roughness from the sidewalls and protect the polished top 
+surface, we use a SiO2 hard mask deposited using PECVD after CMP and use a dry etching process 
+with a much higher oxygen flow. This etching process has been proved to substantially reduce the 
+polymerization process during etching and decreases the roughness[9]. We pattern our devices with 
+electron beam lithography using ma-N 2403 resist and use multipass writing algorithms to further 
+reduce sidewall roughness caused by the lithography itself[9,10]. Finally, we clad the devices with 
+2 μm of SiO2 deposited using PECVD for waveguide protection. The fabricated devices consist of 
+resonators with a radius of 115 μm, a height of 730 nm and a width of 1500 nm, which are coupled 
+to a waveguide of the same width and height. These dimensions ensure high confinement.  
+
+730nm
+1500 nm
+100μmPublished in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+ 
+ 
+Figure 3. AFM measurement of the top surface of PECVD Si3N4. (a)  3D AFM scan 
+of the top surface before CMP with RMS roughness of 1.36 nm and a correlation 
+length of 27.6 nm. (b) 2D image of Si3N4 top surface before CMP and scaled to -
+5.0 – 5.0 nm with RMS roughness of 1.36 nm. (c) 3D image of Si3N4 top surface 
+after CMP with RMS roughness of 0.20 nm and a correlation length of 2.96 nm. (d) 
+2D image of Si3N4 top surface after CMP and scaled to -1.0 – 1.0 nm with RMS 
+roughness of 0.20 nm. Note the different scale bars on (a) and (c). 
+ 
+3. Fundamental loss extraction and discussion 
+The quality factor is a measure of the sharpness of the resonance relative to its central frequency. 
+It represents how well the resonator can store energy and can be written as[11,12]:  
+                              
+                          (1) 
+The quality factor defined in Equation 1 is the loaded quality factor. The intrinsic quality factor 
+of the cavity which is directly related to the propagation losses can be written as[13,14]:  
+                        
+                    (2) 
+0
+L
+Q
+w
+w
+= D
+min
+2
+1
+L
+i
+Q
+Q
+T
+=
+±
+
+5.0
+5.0 nm
+300 nm
+300 nm
+100 nm
+100 nm
+100 nm
+-5.0
+5
+.0 nm
+300 nm
+300 nm
+100 nm
+100 nm
+-1.0
+100 nmPublished in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+Tmin is the on-resonance normalized transmission minimum, 
+sign is corresponding to 
+undercoupled and overcoupled condition. The schematic of the experimental setup for quality 
+factor measurement and frequency comb generation is shown in Figure 4. The resonators we 
+fabricated and measured here have a height of 730 nm, a width of 1500 nm and a bending radius 
+of 115 µm. We measure an intrinsic quality factor of 724,000, corresponding to a propagation loss 
+of 0.42 dB/cm. In Figure 5(a), we show the measured resonance and normalized transmission 
+spectrum over a broad wavelength range. To the best of our knowledge, this is the lowest 
+propagation loss reported to date in a standard PECVD film compatible with foundries.  
+ 
+Figure 4. Schematic of the experimental setup for measuring transmission spectra 
+and resonator linewidth to characterize the quality factor and generate frequency 
+combs. FPC: fiber polarization controller; PD: photodetector; and OSA: optical 
+spectrum analyzer. Note that amplifier is not needed for transmission measurement. 
+ 
+To minimize both surface scattering losses, as well as bulk loss, we post-process the films with 
+a rapid thermal anneal (RTA). With RTA, we achieve an even higher intrinsic quality factor of 
+more than 1 million, corresponding to a propagation loss of 0.28 dB/cm. RTA has been 
+successfully applied in the microelectronics industry and it has particular relevance for CMOS 
+technology, specifically in steps such as implant annealing, oxidation, and source and drain contact 
+junctions[15,16]. The process reduces loss by driving out the non-bonded atomic and molecular 
+hydrogen trapped in microvoids of the structure and further densifies the films[17,18]. We apply 
+RTA at 800 °C for 5 mins to the cladded devices. In Figure 5(b), we show the measured resonance 
+±
+
+000
+Laser
+Amplifier
+ChipPublished in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+and normalized transmission spectrum over a broad wavelength range. The thermal budget is 
+below the tolerance of most CMOS electronics and can be used to further reduce losses for devices 
+with microheaters or dopants.  
+We show that by post-processing foundry-compatible devices with furnace anneal (appropriate 
+for devices with high thermal budget), the propagation loss can be comparable to those fabricated 
+using high temperature, high-stress LPCVD films. Furnace anneal differs from RTA, with higher 
+temperatures (above 1000 °C [19–24]) and longer anneal times (several hours). We anneal cladded 
+devices at 1150 °C in a nitrogen atmosphere for 3 hours and no defects or cracks were observed. 
+We achieve a quality factor of 4.7 million, which corresponds to a propagation loss of 0.06 dB/cm. 
+In Figure 5(c), we show the measured resonance and normalized transmission spectrum over a 
+broad wavelength range.  
+
+Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+  
+Figure 5. (a) Device without annealing shows a measured full width half maximum 
+(FWHM) of 595 MHz around 1600 nm and measured normalized transmission 
+spectrum over a broad wavelength range. (b) Device after rapid thermal anneal 
+shows a measured full width half maximum (FWHM) of 423 MHz around 1600 nm 
+and measured normalized transmission spectrum over a broad wavelength range. 
+(c) Device after furnace anneal shows a measured full width half maximum 
+(FWHM) of 52 MHz around 1600 nm and measured normalized transmission 
+spectrum over a broad wavelength range. 
+ 
+We show that for as-fabricated devices, the bulk losses dominate over the surface scattering 
+loss, and can be as low as 0.33 dB/cm, while for post-fabrication annealed devices, the bulk losses 
+
+No anneal
+(a)
+1.0
+0.8
+0.8
+0.6
+ed
+Normalized
+0.4
+595 MHz
+0.2
+Qi = 0.72 million
+0.2
+ION
+0.0
+1320 1530 1540 1550 1560 1570 1580 1590 1600 1610 1620
+-2000
+-1000
+0
+1000
+2000
+Wavelength (nm)
+ Frequency (MHz)
+(b)
+Rapidthermalanneal
+1.0
+0.8
+0.6
+Normalized
+0.4
+423 MHz
+0.2
+Qi = 1.1 million
+0.2
+0.0
+-2000
+-1000
+0
+1000
+2000
+Wavelength (nm)
+△Frequency (MHz)
+(c)
+Furnace anneal
+1.0
+0.8
+0.8
+0.6
+52 MHz
+ed
+0.4
+Qi= 4.7million
+0.2
+Nor
+0.0
+1320 1530 1540 1550 1560 1570 1580 1590 1600 1610 1620
+-400
+-200
+0
+200
+400
+Wavelength (nm)
+△Frequency(MHz)Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+are comparable to the surface scattering loss, and can be as low as 0.04 dB/cm. We extract the loss 
+contributions by comparing the losses between two different structures with different mode 
+overlap with the interfaces. 
+,
+,
+ are the overlap of the optical field with the waveguide core, 
+the top and bottom surfaces, and sidewalls respectively for the two different waveguide widths[25]. 
+These parameters are calculated using FEM simulations (performed with COMSOL). We also use 
+the Payne-Lacey model[26] to relate scattering loss to the surface’s RMS roughness (σ) and the 
+correlation length (Lc), both extracted from the AFM measurements. The method used here to 
+extract the loss contributions is similar to the one used in ref[9]. We find that for complete overlap 
+of the mode with the interfaces, the scattering losses are
+~ 0.0002 dB/cm and 
+~ 0.0024 dB/cm at the SiO2/Si3N4 top interface and Si3N4/SiO2 bottom interface, respectively. The 
+estimated surface scattering and bulk loss contributions for different thermal treatments (shown in 
+Table 1) are extracted from Equation 3 and Equation 4 below:  
+ 
+                    (3) 
+      (4) 
+We find that both bulk loss and surface scattering losses are reduced after RTA and furnace 
+anneal, which indicates that the chemical and physical properties of the films are improved by 
+thermal treatment. From Table 1 and Equation 3, if the surface scattering loss were eliminated, 
+one could reduce the propagation loss down to 0.33 dB/cm. By post-processing with RTA at 800 
+°C, one could reduce the propagation loss to 0.23 dB/cm. The propagation loss can be further 
+reduced if RTA were performed at a higher temperature to break down bonded hydrogen. By post-
+processing with furnace anneal, one could reduce the propagation loss in these devices to 0.04 
+dB/cm if the surface scattering loss were eliminated.  
+1
+h
+2
+h
+3
+h
+_
+top scatter
+a
+_
+bottom scatter
+a
+1
+_
+_
+_
+_
+ring
+bulk
+loss
+top
+scatter
+bottom
+scatter
+sidewalls
+scatter
+a
+a
+a
+a
+a
+=
++
++
++
+2
+1
+_
+2
+_
+_
+3
+_
+ring
+bulk
+loss
+top
+scatter
+bottom
+scatter
+sidewalls
+scatter
+a
+h a
+h
+a
+a
+h a
+=
++
++
++
+（
+）
+
+Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+Table 1. The extracted surface scattering and bulk loss contribution in PECVD film. 
+ 
+Bulk Loss 
+Surface Scattering Loss  Total Loss 
+No Anneal 
+0.33 dB/cm 
+0.09 dB/cm 
+ 0.42 dB/cm 
+Rapid Thermal Anneal 0.23 dB/cm 
+0.05 dB/cm 
+ 0.28 dB/cm 
+Furnace Anneal 
+0.04 dB/cm 
+0.02 dB/cm 
+ 0.06 dB/cm 
+ 
+The structure fabricated without any post-fabrication thermal treatment exhibits a high 
+confinement of 87% and a low propagation loss of 0.42 dB/cm. High confinement is necessary for 
+tailoring the waveguide dispersion to achieve phase matching in nonlinear processes as well as for 
+tighter bends, thus allowing small footprints required in large-scale photonic systems. We compare 
+the confinement factor and propagation loss achieved in this work with other state-of-the-art works 
+realized in foundry compatible PECVD platform without any thermal treatment in Figure 6[2,3,5,27–
+30].  
+ 
+Figure 6. Loss and confinement achieved in this work compared with other state-
+of-the-art works based on PECVD platform. All points including this work are for 
+devices fabricated without any thermal treatment[2,3,5,27–30]. 
+
+10
+(This work)
+1/Loss (cm)
+5
+Y. Huang, et al (2014)
++N. Sherwood-Droz, et al (2011)
+C. Lacava et al, (2017)
+E. A. Douglas et al, (2016)
+s. Mao et al, (2008)
+L.Wang, et al (2018)
++K. Ikeda, et al (2008)
+0
+40
+50
+60
+70
+80
+90
+100
+Confinement Factor (%)Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+4. Dispersion engineering 
+We show the dispersion of the devices can be tuned by post-processing with furnace anneal. In 
+order to engineer the dispersion, we derive the Sellmeier equations for PECVD Si3N4 films from 
+ellipsometry performed over 200–1690 nm and 1.7–34 μm wavelength ranges using J.A. Woollam 
+M-2000 and IR-VASE instruments. We show the measured spectra from 200-1750 nm before and 
+after annealing in Figure 7(a) and Figure 7(b). We fit the spectra over the wavelength range 300–
+2000 nm to obtain the following Sellmeier equations for Si3N4 before and after furnace anneal.  
+ 
+ 
+ 
+ 
+𝜆 is in units of nanometer. We show the simulated dispersions based on the Sellmeier equations 
+for silicon nitride resonators with a cross section of 730 nm x 1500 nm and a bending radius of 
+115 µm before and after annealing in Figure 7(c). The dashed line separates the anomalous group-
+velocity dispersion (GVD) regime and the normal GVD regime. One can see that the device with 
+the same cross section of 730 nm x 1500 nm exhibits normal GVD before anneal and anomalous 
+GVD after anneal. 
+3
+4
+2
+9
+2
+2
+2
+2
+2
+8
+2
+2.61
+1.11 10
+(
+_
+)
+1
+139.77
+2.51 10
+Si N
+n
+before
+anneal
+l
+l
+l
+l
+´
+= +
++
+-
+-
+´
+（
+）
+3
+4
+2
+9
+2
+2
+2
+2
+2
+8
+2
+2.97
+1.57 10
+(
+_
+)
+1
+-144.86
+- 3.80 10
+Si N
+n
+after
+anneal
+l
+l
+l
+l
+´
+= +
++
+´
+（
+）
+
+Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+  
+ 
+Figure 7. (a) Refractive index n and extinction coefficient k for the wavelength 
+range 200–1750 nm before annealing. (b) Refractive index n and extinction 
+coefficient k for the wavelength range 200-1750 nm after annealing. (c) Dispersion 
+simulations for fundamental TE mode of a silicon nitride ring resonator with a cross 
+section of 730 nm ´ 1500 nm and a bending radius of 115 µm before and after 
+annealing. The dashed line separates the anomalous group-velocity dispersion 
+regime and the normal group-velocity dispersion regime. 
+ 
+ 
+Before annealing 
+After annealing 
+
+(a)
+2.5
+0.4
+Extinction (
+n
+2.4
+Refraction,
+0.3
+2.3
+n
+2.2
+Coefficient k
+0.2
+2.1
+of
+2
+0.1
+1.8
+0
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+Wavelength (nm)
+(b)
+2.5
+0.2
+n
+2.4
+n
+0.15
+2.3
+Refrac
+-k
+2.2
+0.1
+Coefficie
+R
+xepul
+2.1
+0.05
+ient k
+2
+1.9
+0
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+Wavelength (nm)
+(c)
+100
+Before annealing
+(wy/wu/sd) 
+-After annealing
+50
+ispersion (
+-50
+-100
+D
+-150
+200
+1000
+1200
+1400
+1600
+1800
+2000
+Wavelength (nm)Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+5. Linear and nonlinear applications 
+We demonstrate low threshold parametric oscillation and frequency combs generation using 
+foundry compatible devices post-processed with furnace anneal leveraging our ability to engineer 
+the dispersion. We show the evolution of the comb generation process and observe transitions into 
+various comb states in Figure 8 using a pump wavelength of 1550 nm. As the power in the 
+resonator builds, we see the primary sidebands form at the parametric gain peak due to degenerate 
+four-wave mixing as shown in Figure 8(a). We show the transition into the mini-combs in Figure 
+8(b) and eventually the broadband frequency combs with an on-chip pump power of 202 mW in 
+Figure 8(c). The parametric oscillation threshold is measured as low as 3 mW, which is close to 
+the theoretical limit of 2.7 mW. 
+
+Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+ 
+Figure 8. Evolution of the frequency comb generation process. (a) Primary 
+sidebands form at the parametric gain peak due to degenerate four-wave mixing. 
+(b) The mini-comb formation. (c) Broadband Kerr frequency comb with an on-chip 
+pump power of 202 mW. 
+ 
+We demonstrate that modal-collapse of a multimode Fabry-Perot laser diode (FPL) can be 
+realized by using the same device. Therefore, we obtain a single-wavelength emission laser thanks 
+to the increased robustness to coupling loss of  a FPL[31] and strong feedback of the high quality 
+factor resonator.  The system is composed of a commercial single transverse-mode FPL (Thorlabs 
+FPL1001C) and the high quality resonator as shown in Figure 9.  
+
+(a)
+10
+0
+10
+Power (dBm)
+20
+-30
+40
+50
+-60
+-70
+1450
+1500
+1550
+1600
+1650
+1700
+(b)
+Wavelength (nm)
+10
+0
+10
+(dBm)
+20
+30
+-40
+-50
+60
+-70
+1450
+1500
+1550
+1600
+1650
+1700
+(c)
+Wavelength (nm)
+10
+0
+-10
+Power (dBm)
+-20
+-30
+40
+50
+60
+-70
+1450
+1500
+1550
+1600
+1650
+1700
+Wavelength (nm)Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+ 
+Figure 9. Schematic of the experimental setup for lasing measurement. A 
+commercial single transverse-mode Fabry-Perot Laser Diode (Thorlabs 
+FPL1001C) is coupled to the high quality factor resonator. The spectrum of the 
+laser is measured with an optical spectrum analyzer (OSA). 
+ 
+A feedback signal from the high quality factor resonator leads to self-injection locking of the 
+FPL laser resulting in a locked laser with single longitudinal-mode emission and narrow-linewidth. 
+The spectrum of the unlocked free-running laser and the locked laser are shown in Figure 10. The 
+side-mode suppression ratio (SMSR) is at least 29 dB and the linewidth is measured below 
+resolution limit of the optical spectrum analyzer. We have calculated the intrinsic linewidth to be 
+in the range of 1 - 10 kHz. For this calculation we have considered the Schawlow–Townes 
+linewidth of the free-running laser and the linewidth reduction due to self-injection locking 
+following a similar procedure as explained in Ref [31]. The coupling structure for our device here 
+is inverse taper and it could be optimized for coupling to FPL, so better SMSRs and even narrower 
+linewidths can be achieved with improved coupling. 
+       
+Figure 10. (a) Optical spectra of the unlocked free-running laser. (b) Optical 
+spectra of the locked narrow-linewidth laser to the ring resonator. Side-mode 
+suppression ratio (SMSR) is at least 29 dB. 
+
+Chip
+Fabry-Perot Laser Diode(a)
+Free Running
+Locked
+Power (10 dB/div.)
+Power (10 dB/div.)
+29 dB
+1520
+1524
+1528
+1532
+1536
+1540
+1520
+1524
+1528
+1532
+1536
+1540
+Wavelength (nm)
+Wavelength (nm)Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+6. Conclusion and Discussion 
+Our work demonstrates the feasibility of obtaining ultra-low loss devices directly from foundries. 
+We show that these foundry compatible devices with or without a simple post-processing step can 
+be used for linear and nonlinear applications where ultra-low loss and dispersion are required. Low 
+threshold parametric oscillation, broadband frequency combs and narrow-linewidth laser are 
+demonstrated. The fundamental limit of loss in our devices is extracted and proved to be 
+comparable with the loss achieved in LPCVD films. Our work provides a promising path for 
+scalable photonic systems based on foundries.   
+Recently, reactive sputtering silicon nitride films annealed at 400℃ in ambient atmosphere 
+have been shown to achieve propagation losses down to 0.54 dB/cm[32]. Optical frequency 
+combs[32] and hybrid integration with lithium niobate on insulator platforms[33,34] have been 
+successfully demonstrated, which makes the reactive sputtering another promising method for 
+producing low-loss silicon nitride films. Since the losses in reactive sputtering devices are 
+currently limited by scattering from the sidewall roughness rather than H-bond absorption losses[35], 
+these devices could further benefit from the processes and techniques we developed here. 
+ 
+Acknowledgements 
+The authors would like to acknowledge Ron Synowicki from J.A. Woollam Co., the leading 
+manufacturer of spectroscopic ellipsometers for optical properties measurements. Research 
+reported in this work was performed in part at the Cornell NanoScale Science & Technology 
+Facility (CNF), a member of the National Nanotechnology Coordinated Infrastructure (NNCI) 
+supported by National Science Foundation (Grant NNCI-2025233). The authors acknowledge 
+support from the PIPES program funded by DARPA (HR0011-19-2-0014), the PINE program 
+funded by the ARPA-E (DE-AR0000843), and the AFOSR STTR program (FA9550-20-1-0297). 
+ 
+
+Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+References 
+[1] 
+A. H. Atabaki, S. Moazeni, F. Pavanello, H. Gevorgyan, J. Notaros, L. Alloatti, M. T. 
+Wade, C. Sun, S. A. Kruger, H. Meng, K. Al Qubaisi, I. Wang, B. Zhang, A. Khilo, C. V. 
+Baiocco, M. A. Popović, V. M. Stojanović, R. J. Ram, Nature 2018, 556, 349. 
+[2] 
+E. A. Douglas, P. Mahony, A. Starbuck, A. Pomerene, D. C. Trotter, C. T. DeRose, 
+Optical Materials Express 2016, 6, 2892. 
+[3] 
+S. C. Mao, S. H. Tao, Y. L. Xu, X. W. Sun, M. B. Yu, G. Q. Lo, D. L. Kwong, Optics 
+Express 2008, 16, 20809. 
+[4] 
+T. Domínguez Bucio, A. Z. Khokhar, C. Lacava, S. Stankovic, G. Z. Mashanovich, P. 
+Petropoulos, F. Y. Gardes, Journal of Physics D: Applied Physics 2017, 50, 025106. 
+[5] 
+C. Lacava, S. Stankovic, A. Z. Khokhar, T. D. Bucio, F. Y. Gardes, G. T. Reed, D. J. 
+Richardson, P. Petropoulos, Scientific Reports 2017, 7, DOI 10.1038/s41598-017-00062-6. 
+[6] 
+J. Chiles, N. Nader, D. D. Hickstein, S. P. Yu, T. C. Briles, D. Carlson, H. Jung, J. M. 
+Shainline, S. Diddams, S. B. Papp, S. W. Nam, R. P. Mirin, Optics Letters 2018, 43, 1527. 
+[7] 
+G.-R. Yang, Y.-P. Zhao, Y. Z. Hu, T. Paul Chow, R. J. Gutmann, Thin Solid Films 1998, 
+333, 219. 
+[8] 
+H. Huang, K. J. Winchester, A. Suvorova, B. R. Lawn, Y. Liu, X. Z. Hu, J. M. Dell, L. 
+Faraone, Materials Science and Engineering: A 2006, 435–436, 453. 
+[9] 
+X. Ji, F. A. S. Barbosa, S. P. Roberts, A. Dutt, J. Cardenas, Y. Okawachi, A. Bryant, A. 
+L. Gaeta, M. Lipson, Optica 2017, 4, 619. 
+[10] 
+R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, Y. Aida, Journal of Vacuum 
+Science & Technology B: Microelectronics and Nanometer Structures 2011, 29, 06F309. 
+[11] 
+B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.- Laine, Journal of Lightwave Technology 
+1997, 15, 998. 
+[12] 
+W. Bogaerts, P. D. Heyn, T. V. Vaerenbergh, K. D. Vos, S. K. Selvaraja, T. Claes, P. 
+Dumon, P. Bienstman, D. V. Thourhout, R. Baets, Laser & Photonics Reviews 2012, 6, 47. 
+[13] 
+A. Yariv, Electronics Letters 2000, 36, 321. 
+[14] 
+A. Yariv, IEEE Photonics Technology Letters 2002, 14, 483. 
+[15] 
+M. J. Hart, A. G. R. Evans, Semiconductor Science and Technology 1988, 3, 421. 
+[16] 
+V. Trivedi, S. J. Pearton, Solid-State Electronics 2002, 46, 777. 
+[17] 
+F. L. Martínez, A. del Prado, I. Mártil, G. González-Diaz, W. Bohne, W. Fuhs, J. 
+Röhrich, B. Selle, I. Sieber, Phys. Rev. B 2001, 63, 245320. 
+[18] 
+R. Bousbih, W. Dimassi, I. Haddadi, H. Ezzaouia, Phys. Status Solidi C 2012, 9, 2189. 
+[19] 
+M. J. Shaw, J. Guo, G. A. Vawter, S. Habermehl, C. T. Sullivan, in Proc.SPIE (Eds.: E. 
+G. Johnson, G. P. Nordin, T. J. Suleski), Proc.SPIE, San Jose, CA, 2005. 
+[20] 
+Y. Xuan, Y. Liu, L. T. Varghese, A. J. Metcalf, X. Xue, P.-H. Wang, K. Han, J. A. 
+Jaramillo-Villegas, A. Al Noman, C. Wang, S. Kim, M. Teng, Y. J. Lee, B. Niu, L. Fan, J. 
+Wang, D. E. Leaird, A. M. Weiner, M. Qi, Optica 2016, 3, 1171. 
+[21] 
+K. Luke, Y. Okawachi, M. R. E. Lamont, A. L. Gaeta, M. Lipson, Optics Letters 2015, 
+40, 4823. 
+[22] 
+M. H. P. Pfeiffer, J. Liu, A. S. Raja, T. Morais, B. Ghadiani, T. J. Kippenberg, Optica, 
+OPTICA 2018, 5, 884. 
+[23] 
+X. Ji, X. Yao, A. Klenner, Y. Gan, A. L. Gaeta, C. P. Hendon, M. Lipson, Opt. Express, 
+OE 2019, 27, 19896. 
+[24] 
+T. Gu, M. Yu, D.-L. Kwong, C. W. Wong, Optics Express 2014, 22, 18412. 
+
+Published in Laser & Photonics Reviews. DOI: 10.1002/lpor.202200544 (2022). 
+ 
+ 
+ 
+[25] 
+T. Barwicz, H. A. Haus, Journal of Lightwave Technology 2005, 23, 2719. 
+[26] 
+F. P. Payne, J. P. R. Lacey, Optical and Quantum Electronics 1994, 26, 977. 
+[27] 
+K. Ikeda, R. E. Saperstein, N. Alic, Y. Fainman, Optics Express 2008, 16, 12987. 
+[28] 
+N. Sherwood-Droz, M. Lipson, Opt. Express, OE 2011, 19, 17758. 
+[29] 
+Y. Huang, J. Song, X. Luo, T.-Y. Liow, G.-Q. Lo, Opt. Express, OE 2014, 22, 21859. 
+[30] 
+L. Wang, W. Xie, D. V. Thourhout, Y. Zhang, H. Yu, S. Wang, Opt. Express, OE 2018, 
+26, 9645. 
+[31] 
+A. Gil-Molina, O. Westreich, Y. Antman, X. Ji, A. L. Gaeta, M. Lipson, in 2020 
+Conference on Lasers and Electro-Optics (CLEO), 2020, pp. 1–2. 
+[32] 
+A. Frigg, A. Boes, G. Ren, T. G. Nguyen, D.-Y. Choi, S. Gees, D. Moss, A. Mitchell, 
+APL Photonics 2020, 5, 011302. 
+[33] 
+X. Han, Y. Jiang, A. Frigg, H. Xiao, P. Zhang, T. G. Nguyen, A. Boes, J. Yang, G. Ren, 
+Y. Su, A. Mitchell, Y. Tian, Laser & Photonics Reviews 2022, 16, 2100529. 
+[34] 
+Y. Jiang, X. Han, H. Huang, P. Zhang, A. Dubey, H. Xiao, M. Yuan, A. Frigg, T. G. 
+Nguyen, A. Boes, Y. Li, G. Ren, Y. Su, A. Mitchell, Y. Tian, Advanced Photonics Research 
+2022, 3, 2200121. 
+[35] 
+A. Frigg, A. Boes, G. Ren, I. Abdo, D.-Y. Choi, S. Gees, A. Mitchell, Opt. Express 2019, 
+27, 37795. 
+ 
+

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+arXiv:2301.02878v1  [cs.IT]  7 Jan 2023
+Abstract Huffman Coding and
+PIFO Tree Embeddings
+Keri D’Angelo∗
+Dexter Kozen†
+Cornell University
+Computer Science Department
+Ithaca, New York 14853-7501, USA
+∗kd349@cornell.edu
+†kozen@cs.cornell.edu
+January 10, 2023
+Abstract
+Algorithms for deriving Huffman codes and the recently developed algorithm for
+compiling PIFO trees to trees of fixed shape [1] are similar, but work with different
+underlying algebraic operations. In this paper, we exploit the monadic structure of
+prefix codes to create a generalized Huffman algorithm that has these two applications
+as special cases.
+1
+Introduction
+Huffman codes translate letters from a fixed alphabet to d-ary codewords, achieving optimal
+compression for a given frequency distribution of letters. There is a well-known greedy
+algorithm for producing Huffman codes from a given distribution (see [2]).
+A new data structure called a PIFO tree (priority-in first-out) has recently been pro-
+posed for implementing a wide range of packet scheduling algorithms in programmable
+network routers [3, 4]. A PIFO tree is a tree of priority queues. Currently, most routers
+support just a few scheduling algorithms such as strict priority or weighted fair queueing,
+which are baked into the hardware. The schedulers can be configured to some extent, but
+it is generally not possible to implement more sophisticated scheduling algorithms that
+require reordering of already queued packets. This is exactly what PIFO trees permit. It
+seems likely that PIFOs will be supported on network devices in the near future.
+Some researchers have already begun to explore how the PIFO abstraction can be em-
+ulated on conventional routers [4]. In very recent work [1], it was shown how to translate
+an algorithm designed for a PIFO tree of arbitrary shape to one that uses a PIFO tree of
+fixed shape, perhaps a complete d-ary tree that might be implemented in hardware, with
+negligible performance degradation.
+1
+
+The embedding algorithm is greedy and very similar to the Huffman algorithm, ex-
+cept that it is based on different algebraic operations. For Huffman coding, one wishes to
+choose a d-ary prefix code C so as to minimize the value of ∑x∈C |x| · r(x), where r(x) is
+the frequency of the letter assigned to the codeword x. This minimizes the entropy of the
+resulting code. For PIFO trees, one wishes to minimize maxx∈C |x| + r(x), where r(x) is the
+height of a subtree. This minimizes the height of the resulting d-ary tree and determines
+whether an embedding is at all possible.
+This similarity leads us to seek a unified axiomatic treatment that is parametric in the
+algebraic operations and that can be instantiated to produce both applications as special
+cases. Our treatment exploits the monadic structure of prefix codes to obtain an abstract
+formulation of the problem and its solution. We identify sufficient conditions for our ab-
+stract algorithm to produce optimal solutions, where the meaning of optimal is also para-
+metric in the instantiation.
+We state axioms that are sufficient for optimality in §3. The algorithm is presented in
+§4 and its correctness proved in §5. The two applications of Huffman codes and PIFO trees
+are derived in §6.
+2
+Background
+We assume familiarity with the basic category-theoretic concepts of category, functor, and
+natural transformation. Our exposition is based on the concepts of monad and Eilenberg-
+Moore algebra; we briefly review the definitions here. For a more thorough introduction,
+we refer the reader to [5–8].
+Monads are heavily used in functional programming to model the augmentation of a
+computation with extra structure [9–11]. Formally, a monad on a category C is a triple
+(T, η, µ), where T : C → C is an endofunctor on C and η : I → T and µ : T2 → T are natural
+transformations, called the unit and multiplication respectively, such that for all objects X,
+the following diagrams commute:
+T3X
+T2X
+T2X
+TX
+µTX
+TµX
+µX
+µX
+TX
+T2X
+T2X
+TX
+ηTX
+TηX
+µX
+µX
+idTX
+Typical examples of monads are
+• the list monad, in which ηX(a) = [a], the singleton list containing a, and
+µX([[a11, . . . , a1k1], . . . , [an1, . . . , ankn]]) = [a11, . . . , a1k1, . . . , an1, . . . , ankn],
+the list flattening operation;
+2
+
+• the powerset monad, in which ηX(a) = {a}, the singleton set containing a, and µX(A) =
+� A, the operation that takes a set of subsets of X to its union.
+Given a monad (T, η, µ) on a category C, an Eilenberg-Moore algebra for (T, η, µ) is a pair
+(X, γ), where X is an object of C and γ : TX → X is a morphism of C, called the structure
+map of the algebra, such that the following diagrams commute:
+T2X
+TX
+TX
+X
+Tγ
+µX
+γ
+γ
+X
+TX
+X
+ηX
+γ
+idX
+A morphism of Eilenberg-Moore algebras is a morphism of C that commutes with the structure
+maps. That is, if (X, γ) and (Y, δ) are two algebras and h : X → Y is a morphism of C, then
+h is a morphism of algebras h : (X, γ) → (Y, δ) if the following diagram commutes:
+TX
+TY
+X
+Y
+Th
+γ
+h
+δ
+The Eilenberg-Moore algebras for (T, η, µ) and their morphisms form the Eilenberg-Moore
+category over the monad T. The Eilenberg-Moore category for the list monad is the cat-
+egory of monoids and monoid homomorphisms. The Eilenberg-Moore category for the
+powerset monad is the category of complete upper semilattices and semilattice homomor-
+phisms.
+In our application, we will focus on the monad of d-ary prefix codes on the category Set
+of sets and set functions.
+3
+Axioms
+In this section, we state the axioms that are sufficient for the optimality of our generalized
+Huffman algorithm.
+Recall that a prefix code over a fixed d-ary alphabet Σ is a set of finite-length words over
+Σ whose elements are pairwise incomparable with respect to the prefix relation. A prefix
+code C is exhaustive if every infinite d-ary string has a prefix in C. As a consequence of
+K¨onig’s lemma, every exhaustive prefix code over a finite alphabet is finite, but not every
+finite prefix code is exhaustive.
+Let C : Set → Set be an endofunctor in which
+• CX is the set of pairs (C, r) such that C is a prefix code over a d-ary alphabet for some
+arbitrary but fixed d ≥ 2 and r : C → X, and
+3
+
+• for h : X → Y, Ch : CX → CY with Ch(C, r) = (C, h ◦ r).
+The functor C carries a natural monad structure with unit η : I → C and multiplication
+µ : C2 → C defined by: for a ∈ X and (C, r) ∈ C2X with r(x) = (Cx, rx),
+ηX(a) = ({ε}, ε �→ a)
+µX(C, r) = ({xy | x ∈ C, y ∈ Cx}, xy �→ rx(y)).
+The map xy �→ rx(y) is well defined, as the string xy can be uniquely split into x ∈ C and
+y ∈ Cx because C is a prefix code.
+For example, consider the prefix codes C = {0, 10, 110, 111} and C0 = C10 = C110 =
+C111 = {00, 11} over the binary alphabet {0, 1}. The code C is exhaustive but the others
+are not. Let
+r0(00) = 2
+r10(00) = 4
+r110(00) = 6
+r111(00) = 8
+r0(11) = 3
+r10(11) = 5
+r110(11) = 7
+r111(11) = 9
+r(0) = (C0, r0)
+r(10) = (C10, r10)
+r(110) = (C110, r110)
+r(111) = (C111, r111).
+Then (C0, r0), (C10, r10), (C110, r110), (C111, r111) ∈ CN and (C, r) ∈ C2N, and µN(C, r) =
+(C′, r′) ∈ CN, where
+C′ = {000, 011, 1000, 1011, 11000, 11011, 11100, 11111}
+r′(000) = 2, r′(011) = 3, r′(1000) = 4, r′(1011) = 5,
+r′(11000) = 6, r′(11011) = 7, r′(11100) = 8, r′(11111) = 9.
+Suppose there is a fixed Eilenberg-Moore algebra (W, w) with w : CW → W. We call
+the elements of W weights and (W, w) a weighting. If (C, r) ∈ CW, then thinking of the
+elements of C as a tree, the map r : C → W assigns a weight to each leaf of the tree, and
+the map w tells how to assign a weight to the object (C, r) based on the leaf weights r.
+To define a notion of optimality, we assume that W is totally preordered by ≤; that is,
+≤ is reflexive and transitive, and for all x, y ∈ W, either x ≤ y or y ≤ x (or both). Smaller
+values of W in the order ≤ are considered better. We write x ≡ y if both x ≤ y and y ≤ x.
+Suppose further that we have a preorder on CW, also denoted ≤, satisfying the following
+properties.
+(i) If f : C → D is bijective and length-nondecreasing, and if r ≤ s ◦ f pointwise, then
+(C, r) ≤ (D, s). This says that longer codewords or larger leaf values cannot cause a
+decrease in the order ≤.
+(ii) (Exchange property) If r(x) ≤ r(y), |x| ≤ |y|, and
+s(z) =
+
+
+
+
+
+r(x),
+if z = y,
+r(y),
+if z = x,
+r(z),
+if z ∈ C \ {x, y},
+then (C, s) ≤ (C, r). That is, it never hurts to swap a larger element deeper in the tree
+with a smaller element higher in the tree.
+4
+
+(iii) The monad structure maps ηW : W → CW and µW : C2W → CW are monotone with
+respect to ≤, where ≤ on C2W is defined by:
+(C, r) ≤ (D, s) ⇔ Cw(C, r) ≤ Cw(D, s).
+Some special cases of (i) are
+• If f : C → D is bijective and length-nondecreasing, then (C, s ◦ f) ≤ (D, s). Thus
+lengthening codewords cannot cause ≤ to decrease.
+• If f : C → D is bijective and length-preserving, then (C, s ◦ f) ≡ (D, s). This says
+that the order ≤ on trees depends only on the lengths of the codewords in C, not on
+the actual codewords themselves.
+• If r, s : C → W and r ≤ s pointwise, then (C, r) ≤ (C, s). Thus larger leaf values
+cannot cause ≤ to decrease.
+We assume these properties hold for the algorithm described in the next section.
+For (C, r), (D, s) ∈ CW, let us write (C, r) ∼ (D, s) if the multisets of weights repre-
+sented by the two objects are the same; that is, there is a bijective function f : C → D such
+that r = s ◦ f. A tree (C, r) ∈ CW is defined to be optimal (for its multiset of weights) if (C, r)
+is ≤-minimum in its ∼-class; that is, (C, r) ≤ (D, s) for all (D, s) such that (C, r) ∼ (D, s).
+We will give two detailed examples in §6.
+4
+Algorithm
+Suppose we are given a multiset M of weights in W, |M| ≥ 2. We would like to find an
+optimal tree for this multiset of weights. The following is a recursive algorithm to find
+such an optimal tree.
+1. Say there are n ≥ 2 elements in M. Let k ∈ {2, . . . , d} such that n ≡ k mod (d − 1).
+Let a0, . . . , ak−1 be the k elements of least weight. Form the object
+({0, 1, . . . , k − 1}, i �→ ai) ∈ CW.
+If there are no other elements of M, return that object.
+2. Otherwise, let
+M′ = {({0, 1, . . . , k − 1}, i �→ ai)} ∪ {ηW(a) | a ∈ M \ {a0, . . . , ak−1}},
+a multiset of n − k + 1 < n elements of CW.
+3. Recursively call the algorithm at step 1 with M′′ = {w(E, t) | (E, t) ∈ M′}, a multiset
+of elements of W. This returns a tree (D, s) of type CW that is optimal for M′′. The
+bijective map s : D → M′′ factors as w ◦ s′ for some bijective s′ : D → M′, and
+(D, s′) ∈ C2W with Cw(D, s′) = (D, w ◦ s′) = (D, s). Flatten this to µW(D, s′) ∈ CW
+and return that value.
+5
+
+Note that the number of items combined in step 1 will be d in all recursive calls except
+possibly the first. This is because in every step, if k ∈ {2, 3, . . . , d}, then after that step
+the number of remaining elements will be (c(d − 1) + k) − k + 1 = c(d − 1) + 1, which
+is congruent to d mod d − 1, so d elements will be taken in the next step. But from that
+point on, it is an invariant of the recursion that the number of elements remaining is 1 mod
+d − 1, since in each step we remove d elements and add one back, decreasing the number
+by d − 1.
+5
+Correctness
+In this section, we prove the correctness of the algorithm, making use of the following
+lemma.
+Lemma 1. Let k ∈ {2, 3, . . . , d} and k ≡ |M| mod (d − 1). Let a0, . . . , ak−1 be the k elements of
+M of least weight, listed in nondecreasing order of weight. There is an optimal tree in CW in which
+a0, . . . , ak−1 are sibling leaves at the deepest level and have no other siblings.
+Proof. Let (C, r) ∈ CW be optimal. Axiom (i) allows us to transform (C, r) so that there
+are no deficient nodes (nodes with fewer than d children) at any level except the deepest,
+and only one deficient node at the deepest level. Thus we can assume without loss of
+generality that there are k elements x0, . . . , xk−1 ∈ C of maximum length n in C with a
+common prefix of length n − 1, and no other y ∈ C has that prefix. Say the x0, . . . , xk−1 are
+listed in nondecreasing order of r(xi); that is, r(xi) ≤ r(xj) for all 0 ≤ i ≤ j ≤ k − 1. Let
+y0, . . . , yk−1 ∈ C such that r(yi) = ai. Since the ai are minimal, r(yi) ≤ r(xi). Because the
+|xi| are of maximum length, |yi| ≤ |xi|. Now we can swap using axiom (ii). Let
+s(z) =
+
+
+
+
+
+r(xi),
+if z = yi,
+r(yi),
+if z = xi,
+r(z),
+otherwise.
+Then (C, s) ≤ (C, r). But since (C, r) was optimal, (C, r) ≡ (C, s) and (C, s) is also optimal.
+Theorem 2. The algorithm of §4 produces an optimal tree.
+Proof. By induction on n. The basis is n ≤ d, in which case the result is straightforward.
+Suppose that we have a multiset M of n > d elements of W. Let (C, r) be an optimal tree
+for M. Let k ∈ {2, 3, . . . , d} be congruent mod d − 1 to |M|. Let a0, . . . , ak−1 be the k smallest
+elements of M. By Lemma 1, we can assume without loss of generality that a0, . . . , ak−1 are
+siblings and occur at maximum depth in (C, r), so there exist strings x0, x1, . . . , x(k − 1) ∈
+C of maximum length with a common prefix x and r(xi) = ai. Remove the strings xi from
+C and replace them with x. Call the resulting set C′. For z ∈ C′, let
+r′(z) =
+�
+({0, 1, . . . , k − 1}, i �→ ai),
+if z = x,
+ηW(r(z)),
+otherwise.
+6
+
+Then (C′, r′) ∈ C2W and (C, r) = µW(C′, r′). The multiset of values of r′ is just the M′ of
+step 2 of the algorithm.
+The algorithm will form the multiset
+M′′ = {w(E, t) | (E, t) ∈ M′} = {w(r′(z)) | z ∈ C′}
+and recursively call with these weights. By the induction hypothesis, the return value will
+be a tree (D, s) ∈ CW that is optimal for M′′, thus (D, s) ≤ (C′, w ◦ r′), and the bijective
+map s : D → M′′ factors as s = w ◦ r′ ◦ f for some bijective f : D → C′. Let s′ = r′ ◦ f. By
+axiom (iii),
+Cw(D, s′) = (D, w ◦ s′) = (D, s) ≤ (C′, w ◦ r′) = Cw(C′, r′),
+therefore (D, s′) ≤ (C′, r′), and since µW is monotone,
+µW(D, s′) ≤ µW(C′, r′) = (C, r).
+As (C, r) was optimal, so is µW(D, s′), and this is the value returned by the algorithm.
+6
+Applications
+By choosing two specific weightings (W, w) and defining the ordering relations ≤ appro-
+priately, we can recover two special cases of this algorithm.
+6.1
+Huffman coding
+Our first application is Huffman codes. Here we wish to minimize the expected length of
+variable-length codewords, given frequencies of the letters to be coded. For this applica-
+tion, we take W = R+ = {a ∈ R | a ≥ 0} with weighting
+w(C, r) = ∑
+x∈C
+r(x).
+Recall that for a ∈ W and (C, r) ∈ C2W with r(x) = (Cx, rx),
+ηW(a) = ({ε}, ε �→ a)
+µW(C, r) = ({xy | x ∈ C, y ∈ Cx}, xy ��→ rx(y)).
+Then (W, w) is an Eilenberg-Moore algebra for the monad (C, µ, η), as
+w(ηW(a)) = w({ε}, ε �→ a) = ∑
+x∈{ε}
+(ε �→ a)(x) = a,
+w(µW(C, r)) = ∑
+x∈C ∑
+y∈Cx
+rx(y) = ∑
+x∈C
+w(Cx, rx)
+= ∑
+x∈C
+w(r(x)) = w(C, w ◦ r) = w(Cw(C, r)).
+In addition, let us define α : CW → W by
+α(C, r) = ∑
+x∈C
+|x| · r(x).
+7
+
+Lemma 3.
+α(ηW(a)) = 0
+α(µW(C, r)) = α(C, w ◦ r) + w(C, α ◦ r).
+Proof.
+α(ηW(a)) = α({ε}, ε �→ a) = ∑
+x∈{ε}
+|x| · (ε �→ a)(x) = |ε| · a = 0,
+α(µW(C, r)) = α({xy | x ∈ C, y ∈ Cx}, xy �→ rx(y))
+= ∑
+x∈C ∑
+y∈Cx
+|xy| · rx(y) = ∑
+x∈C
+|x| ∑
+y∈Cx
+rx(y) + ∑
+x∈C ∑
+y∈Cx
+|y| · rx(y)
+= ∑
+x∈C
+|x| · w(Cx, rx) + ∑
+x∈C
+α(Cx, rx) = ∑
+x∈C
+|x| · w(r(x)) + ∑
+x∈C
+α(r(x))
+= α(C, w ◦ r) + w(C, α ◦ r).
+Note that α and w agree on trees of depth one:
+w({0, 1, . . . , k − 1}, i �→ ai) =
+k−1
+∑
+i=0
+ai,
+α({0, 1, . . . , k − 1}, i �→ ai) =
+k−1
+∑
+i=0
+|i| · ai =
+k−1
+∑
+i=0
+ai,
+where |i| refers to the length of i as a string, which in this case is 1.
+The map α is related to the Shannon entropy H. If r(x) = d−|x|, the probability of a
+d-ary codeword x under the uniform distribution on a d-ary alphabet, then
+H(C, r) = ∑
+x∈C
+−d−|x| log d−|x| = ∑
+x∈C
+|x| · d−|x| log d = α(C, r) log d,
+so α(C, r) = H(C, r)/ log d.
+To use the algorithm in §4, we need an order ≤ on CW. Define (C, r) ≤ (D, s) if (C, r) ∼
+(D, s), that is, there is a bijective map f : C → D such that r = s ◦ f, and
+α(C, r) ≤ α(D, s).
+Note that if (C, r) ≤ (D, s), then
+w(C, r) = ∑
+x∈C
+r(x) = ∑
+x∈C
+s( f(x)) = ∑
+y∈D
+s(y) = w(D, s).
+According to axiom (iii), for (C, r), (D, s) ∈ C2W,
+(C, r) ≤ (D, s) ⇔ Cw(C, r) ≤ Cw(D, s)
+⇔ α(Cw(C, r)) ≤ α(Cw(D, s))
+⇔ α(C, w ◦ r) ≤ α(D, w ◦ s).
+(1)
+Also, if (C, r) ≤ (D, s) in C2W, then
+w(C, α ◦ r) = ∑
+x∈C
+α(r(x)) = ∑
+x∈C
+α(s( f(x))) = ∑
+y∈D
+α(s(y)) = w(D, α ◦ s).
+(2)
+8
+
+Lemma 4. µW : C2W → CW and ηW : W → CW are monotone with respect to ≤.
+Proof. For ηW, suppose a, b ∈ W and a ≤ b. By Lemma 3,
+α(ηW(a)) = 0 = α(ηW(b))
+w(ηW(a)) = a ≤ b = w(ηW(b)).
+For µW, suppose (C, r), (D, s) ∈ C2W and (C, r) ≤ (D, s). By Lemma 3, (1), and (2),
+α(µW(C, r)) = α(C, w ◦ r) + w(C, α ◦ r)
+≤ α(D, w ◦ s) + w(D, α ◦ s) = α(µW(D, s)).
+Theorem 5. The algorithm in §4 for the algebra (R+, w) and ordering relation ≤ defined by α is
+equivalent to Huffman’s algorithm and produces an optimal Huffman code for a given multiset of
+weights.
+Proof. Take X ⊂ R+ to be a finite multiset and sort the set X in increasing order. For the
+binary case of Huffman codes (the d-ary version follows the same way), we always choose
+k = 2. For the first step, let a0, a1 ∈ X be the two smallest elements in the list. Form the
+object ({0, 1}, i �→ ai) ∈ CX. In the case n = 2, this is the only remaining object in the list.
+Otherwise, we combined them into one element with the sum of the weights of a0 and a1
+as the weight of the new element, exactly as the Huffman coding does.
+For the case n > 2, there are remaining elements in the set X. Take all remaining
+a ∈ X\{a0, a1} and replace a by ηX(a) ∈ CX. We are left with n − 1 elements of type CX.
+If we recursively call the algorithm in step 1, we are continually combining the least two
+elements in the remaining set with the elements weighted by w. Note by the weighting
+w, w(ηX(a)) = a and on elements in CX, w takes the sum of r(x)′s, exactly as Huffman
+coding does. Finally, this leaves us with a tree in C2X where leaves have weights of the
+form ηX(ai). Denote this tree by (D, s). Taking µX(D, S) gives our desired tree in CX.
+6.2
+PIFO trees
+PIFO trees were introduced in [3] as a model for programmable packet schedulers. In the
+recent work of [1], further work was done on PIFO trees giving a semantics that allows
+for certain embedding algorithms. The notion of a homomorphic embedding was defined for
+the purpose determining when a PIFO tree could be represented by another PIFO tree and
+for finding an embedding if so. The embedding algorithm we consider takes an arbitrary
+PIFO tree and embeds it into a d-ary tree. This becomes a special case of the algorithm of
+§4, where we choose w in the weighting (W, w) to minimize the height of the target d-ary
+tree into which the source tree can embed.
+For this application, we take W = N with weighting
+w(C, r) = max
+x∈C |x| + r(x).
+This gives an Eilenberg-Moore algebra (W, w) for the monad (C, µ, η). For a ∈ W and
+(C, r) ∈ C2W with r(x) = (Cx, rx), as before we have
+ηW(a) = ({ε}, ε �→ a)
+µW(C, r) = ({xy | x ∈ C, y ∈ Cx}, xy �→ rx(y)),
+9
+
+so
+w(ηW(a)) = w({ε}, ε �→ a) = max
+x∈{ε} |x| + (ε �→ a)(x) = |ε| + a = a,
+w(µW(C, r)) = w({xy | x ∈ C, y ∈ Cx}, xy �→ rx(y)) = max
+x∈C max
+y∈Cx |xy| + rx(y)
+= max
+x∈C max
+y∈Cx |x| + |y| + rx(y) = max
+x∈C |x| + max
+y∈Cx |y| + rx(y)
+= max
+x∈C |x| + w(Cx, rx) = max
+x∈C |x| + w(r(x))
+= w(C, w ◦ r) = w(Cw(C, r)).
+For (C, r), (D, s) ∈ CW, let us define (C, r) ≤ (D, s) if there is a bijective function
+f : C → D such that r = s ◦ f and
+w(C, r) ≤ w(D, s).
+Lemma 6. µW : C2W → CW and ηW : W → CW are monotone with respect to ≤.
+Proof. For ηW, if a ≤ b, then w(ηW(a)) = a ≤ b = w(ηW(b)).
+For µW, suppose (C, r), (D, s) ∈ C2W and (C, r) ≤ (D, s). According to axiom (iii),
+(C, r) ≤ (D, s) ⇔ Cw(C, r) ≤ Cw(D, s)
+⇔ w(Cw(C, r)) ≤ w(Cw(D, s)).
+Then
+w(µW(C, r)) = w(Cw(C, r)) ≤ w(Cw(D, s)) = w(µW(D, s)).
+Theorem 7. The algorithm of §4 for the algebra (N, w) and ordering relation ≤ defined by w is
+equivalent to determining whether an embedding of a PIFO tree in a bounded d-ary tree exists and
+finding the embedding if so.
+7
+Conclusion
+We have presented a generalized Huffman algorithm and shown that two known algo-
+rithms, Huffman codes and embedding of PIFOs trees, can be derived as special cases.
+The PIFO embedding algorithm was introduced in [1] and observed to be very similar to
+the usual combinatorial algorithm for optimal Huffman codes, albeit based on a different
+algebraic structure. This suggested the common generalization presented in this paper.
+Our generalized algorithm exploits the monadic structure of prefix codes, which al-
+lows a more algebraic treatment of the Huffman algorithm than the usual combinatorial
+approaches. The two applications fit naturally in the categorical setting by choosing spe-
+cific Eilenberg-Moore algebras for each one. It is possible that other greedy algorithms
+might fit into this framework as well.
+10
+
+References
+[1] Anshuman
+Mohan,
+Yunhe
+Liu,
+Nate
+Foster,
+Tobias
+Kapp´e,
+and
+Dex-
+ter
+Kozen,
+“Formal
+abstractions
+for
+packet
+scheduling,”
+Tech.
+Rep. http://arxiv.org/abs/2211.11659, Cornell University, November 2022.
+[2] Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, Wiley, second
+edition, 2006.
+[3] Anirudh Sivaraman, Suvinay Subramanian, Mohammad Alizadeh, Sharad Chole,
+Shang-Tse Chuang, Anurag Agrawal, Hari Balakrishnan, Tom Edsall, Sachin Katti,
+and Nick McKeown, “Programmable packet scheduling at line rate,” in SIGCOMM,
+2016.
+[4] Albert Gran Alcoz, Alexander Dietm¨uller, and Laurent Vanbever, “SP-PIFO: Approx-
+imating push-in first-out behaviors using strict-priority queues,” in NSDI, 2020.
+[5] Andrea Asperti and Giuseppe Longo, Categories, Types and Structures: An introduction
+to category theory for the working computer scientist, Foundations of Computing. MIT
+Press, 1991.
+[6] Michael Barr and Charles Wells, Toposes, Triples and Theories, vol. 278 of Grundlehren
+der mathematischen Wissenschaften, Springer, 2013.
+[7] Michael Barr and Charles Wells, Category Theory for Computing Science, Prentice Hall,
+1990.
+[8] Jiˇr´ı Ad´amek, Horst Herrlich, and George E. Strecker, Abstract and concrete categories,
+Dover Publications, 2009.
+[9] Eugenio Moggi, “Notions of computation and monads,” Inf. and Comp., vol. 93, no. 1,
+pp. 55–92, 1991.
+[10] Philip Wadler, “Comprehending monads,” Mathematical Structures in Computer Sci-
+ence, vol. 2, pp. 461–493, 1992.
+[11] Philip Wadler, “Monads for functional programming,” in Advanced Functional Pro-
+gramming: 1st Int. School on Advanced Functional Programming Techniques, Johan Jeur-
+ing and Erik Meijer, Eds., vol. 925 of Lecture Notes in Computer Science, pp. 24–52.
+Springer-Verlag, 1995.
+11
+

BdE1T4oBgHgl3EQfDgNt/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf,len=310
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='02878v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='IT]  7 Jan 2023  Abstract Huffman Coding and  PIFO Tree Embeddings  Keri D’Angelo∗  Dexter Kozen†  Cornell University  Computer Science Department  Ithaca, New York 14853-7501, USA  ∗kd349@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='edu  †kozen@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='edu  January 10, 2023  Abstract  Algorithms for deriving Huffman codes and the recently developed algorithm for  compiling PIFO trees to trees of fixed shape [1] are similar, but work with different  underlying algebraic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' In this paper, we exploit the monadic structure of  prefix codes to create a generalized Huffman algorithm that has these two applications  as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  1  Introduction  Huffman codes translate letters from a fixed alphabet to d-ary codewords, achieving optimal  compression for a given frequency distribution of letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' There is a well-known greedy  algorithm for producing Huffman codes from a given distribution (see [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  A new data structure called a PIFO tree (priority-in first-out) has recently been pro-  posed for implementing a wide range of packet scheduling algorithms in programmable  network routers [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' A PIFO tree is a tree of priority queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Currently, most routers  support just a few scheduling algorithms such as strict priority or weighted fair queueing,  which are baked into the hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The schedulers can be configured to some extent, but  it is generally not possible to implement more sophisticated scheduling algorithms that  require reordering of already queued packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' This is exactly what PIFO trees permit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' It  seems likely that PIFOs will be supported on network devices in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Some researchers have already begun to explore how the PIFO abstraction can be em-  ulated on conventional routers [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' In very recent work [1], it was shown how to translate  an algorithm designed for a PIFO tree of arbitrary shape to one that uses a PIFO tree of  fixed shape, perhaps a complete d-ary tree that might be implemented in hardware, with  negligible performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  1  The embedding algorithm is greedy and very similar to the Huffman algorithm, ex-  cept that it is based on different algebraic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' For Huffman coding, one wishes to  choose a d-ary prefix code C so as to minimize the value of ∑x∈C |x| · r(x), where r(x) is  the frequency of the letter assigned to the codeword x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' This minimizes the entropy of the  resulting code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' For PIFO trees, one wishes to minimize maxx∈C |x| + r(x), where r(x) is the  height of a subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' This minimizes the height of the resulting d-ary tree and determines  whether an embedding is at all possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  This similarity leads us to seek a unified axiomatic treatment that is parametric in the  algebraic operations and that can be instantiated to produce both applications as special  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Our treatment exploits the monadic structure of prefix codes to obtain an abstract  formulation of the problem and its solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' We identify sufficient conditions for our ab-  stract algorithm to produce optimal solutions, where the meaning of optimal is also para-  metric in the instantiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  We state axioms that are sufficient for optimality in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The algorithm is presented in  §4 and its correctness proved in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The two applications of Huffman codes and PIFO trees  are derived in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  2  Background  We assume familiarity with the basic category-theoretic concepts of category, functor, and  natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Our exposition is based on the concepts of monad and Eilenberg-  Moore algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' we briefly review the definitions here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' For a more thorough introduction,  we refer the reader to [5–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Monads are heavily used in functional programming to model the augmentation of a  computation with extra structure [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Formally, a monad on a category C is a triple  (T, η, µ), where T : C → C is an endofunctor on C and η : I → T and µ : T2 → T are natural  transformations, called the unit and multiplication respectively, such that for all objects X,  the following diagrams commute:  T3X  T2X  T2X  TX  µTX  TµX  µX  µX  TX  T2X  T2X  TX  ηTX  TηX  µX  µX  idTX  Typical examples of monads are  the list monad, in which ηX(a) = [a], the singleton list containing a, and  µX([[a11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , a1k1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , [an1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , ankn]]) = [a11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , a1k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , an1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , ankn],  the list flattening operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  2  the powerset monad, in which ηX(a) = {a}, the singleton set containing a, and µX(A) =  � A, the operation that takes a set of subsets of X to its union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Given a monad (T, η, µ) on a category C, an Eilenberg-Moore algebra for (T, η, µ) is a pair  (X, γ), where X is an object of C and γ : TX → X is a morphism of C, called the structure  map of the algebra, such that the following diagrams commute:  T2X  TX  TX  X  Tγ  µX  γ  γ  X  TX  X  ηX  γ  idX  A morphism of Eilenberg-Moore algebras is a morphism of C that commutes with the structure  maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' That is, if (X, γ) and (Y, δ) are two algebras and h : X → Y is a morphism of C, then  h is a morphism of algebras h : (X, γ) → (Y, δ) if the following diagram commutes:  TX  TY  X  Y  Th  γ  h  δ  The Eilenberg-Moore algebras for (T, η, µ) and their morphisms form the Eilenberg-Moore  category over the monad T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The Eilenberg-Moore category for the list monad is the cat-  egory of monoids and monoid homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The Eilenberg-Moore category for the  powerset monad is the category of complete upper semilattices and semilattice homomor-  phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  In our application, we will focus on the monad of d-ary prefix codes on the category Set  of sets and set functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  3  Axioms  In this section, we state the axioms that are sufficient for the optimality of our generalized  Huffman algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Recall that a prefix code over a fixed d-ary alphabet Σ is a set of finite-length words over  Σ whose elements are pairwise incomparable with respect to the prefix relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' A prefix  code C is exhaustive if every infinite d-ary string has a prefix in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' As a consequence of  K¨onig’s lemma, every exhaustive prefix code over a finite alphabet is finite, but not every  finite prefix code is exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Let C : Set → Set be an endofunctor in which  CX is the set of pairs (C, r) such that C is a prefix code over a d-ary alphabet for some  arbitrary but fixed d ≥ 2 and r : C → X, and  3  for h : X → Y, Ch : CX → CY with Ch(C, r) = (C, h ◦ r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  The functor C carries a natural monad structure with unit η : I → C and multiplication  µ : C2 → C defined by: for a ∈ X and (C, r) ∈ C2X with r(x) = (Cx, rx),  ηX(a) = ({ε}, ε �→ a)  µX(C, r) = ({xy | x ∈ C, y ∈ Cx}, xy �→ rx(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  The map xy �→ rx(y) is well defined, as the string xy can be uniquely split into x ∈ C and  y ∈ Cx because C is a prefix code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  For example, consider the prefix codes C = {0, 10, 110, 111} and C0 = C10 = C110 =  C111 = {00, 11} over the binary alphabet {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The code C is exhaustive but the others  are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Let  r0(00) = 2  r10(00) = 4  r110(00) = 6  r111(00) = 8  r0(11) = 3  r10(11) = 5  r110(11) = 7  r111(11) = 9  r(0) = (C0, r0)  r(10) = (C10, r10)  r(110) = (C110, r110)  r(111) = (C111, r111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Then (C0, r0), (C10, r10), (C110, r110), (C111, r111) ∈ CN and (C, r) ∈ C2N, and µN(C, r) =  (C′, r′) ∈ CN, where  C′ = {000, 011, 1000, 1011, 11000, 11011, 11100, 11111}  r′(000) = 2, r′(011) = 3, r′(1000) = 4, r′(1011) = 5,  r′(11000) = 6, r′(11011) = 7, r′(11100) = 8, r′(11111) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Suppose there is a fixed Eilenberg-Moore algebra (W, w) with w : CW → W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' We call  the elements of W weights and (W, w) a weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' If (C, r) ∈ CW, then thinking of the  elements of C as a tree, the map r : C → W assigns a weight to each leaf of the tree, and  the map w tells how to assign a weight to the object (C, r) based on the leaf weights r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  To define a notion of optimality, we assume that W is totally preordered by ≤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' that is,  ≤ is reflexive and transitive, and for all x, y ∈ W, either x ≤ y or y ≤ x (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Smaller  values of W in the order ≤ are considered better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' We write x ≡ y if both x ≤ y and y ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Suppose further that we have a preorder on CW, also denoted ≤, satisfying the following  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  (i) If f : C → D is bijective and length-nondecreasing, and if r ≤ s ◦ f pointwise, then  (C, r) ≤ (D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' This says that longer codewords or larger leaf values cannot cause a  decrease in the order ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  (ii) (Exchange property) If r(x) ≤ r(y), |x| ≤ |y|, and  s(z) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  r(x),  if z = y,  r(y),  if z = x,  r(z),  if z ∈ C \\ {x, y},  then (C, s) ≤ (C, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' That is, it never hurts to swap a larger element deeper in the tree  with a smaller element higher in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  4  (iii) The monad structure maps ηW : W → CW and µW : C2W → CW are monotone with  respect to ≤, where ≤ on C2W is defined by:  (C, r) ≤ (D, s) ⇔ Cw(C, r) ≤ Cw(D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Some special cases of (i) are  If f : C → D is bijective and length-nondecreasing, then (C, s ◦ f) ≤ (D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Thus  lengthening codewords cannot cause ≤ to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  If f : C → D is bijective and length-preserving, then (C, s ◦ f) ≡ (D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' This says  that the order ≤ on trees depends only on the lengths of the codewords in C, not on  the actual codewords themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  If r, s : C → W and r ≤ s pointwise, then (C, r) ≤ (C, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Thus larger leaf values  cannot cause ≤ to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  We assume these properties hold for the algorithm described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  For (C, r), (D, s) ∈ CW, let us write (C, r) ∼ (D, s) if the multisets of weights repre-  sented by the two objects are the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' that is, there is a bijective function f : C → D such  that r = s ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' A tree (C, r) ∈ CW is defined to be optimal (for its multiset of weights) if (C, r)  is ≤-minimum in its ∼-class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' that is, (C, r) ≤ (D, s) for all (D, s) such that (C, r) ∼ (D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  We will give two detailed examples in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  4  Algorithm  Suppose we are given a multiset M of weights in W, |M| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' We would like to find an  optimal tree for this multiset of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The following is a recursive algorithm to find  such an optimal tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Say there are n ≥ 2 elements in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Let k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , d} such that n ≡ k mod (d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Let a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , ak−1 be the k elements of least weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Form the object  ({0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , k − 1}, i �→ ai) ∈ CW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  If there are no other elements of M, return that object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Otherwise, let  M′ = {({0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , k − 1}, i �→ ai)} ∪ {ηW(a) | a ∈ M \\ {a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , ak−1}},  a multiset of n − k + 1 < n elements of CW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Recursively call the algorithm at step 1 with M′′ = {w(E, t) | (E, t) ∈ M′}, a multiset  of elements of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' This returns a tree (D, s) of type CW that is optimal for M′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The  bijective map s : D → M′′ factors as w ◦ s′ for some bijective s′ : D → M′, and  (D, s′) ∈ C2W with Cw(D, s′) = (D, w ◦ s′) = (D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Flatten this to µW(D, s′) ∈ CW  and return that value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  5  Note that the number of items combined in step 1 will be d in all recursive calls except  possibly the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' This is because in every step, if k ∈ {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , d}, then after that step  the number of remaining elements will be (c(d − 1) + k) − k + 1 = c(d − 1) + 1, which  is congruent to d mod d − 1, so d elements will be taken in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' But from that  point on, it is an invariant of the recursion that the number of elements remaining is 1 mod  d − 1, since in each step we remove d elements and add one back, decreasing the number  by d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  5  Correctness  In this section, we prove the correctness of the algorithm, making use of the following  lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Let k ∈ {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , d} and k ≡ |M| mod (d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Let a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , ak−1 be the k elements of  M of least weight, listed in nondecreasing order of weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' There is an optimal tree in CW in which  a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , ak−1 are sibling leaves at the deepest level and have no other siblings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Let (C, r) ∈ CW be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Axiom (i) allows us to transform (C, r) so that there  are no deficient nodes (nodes with fewer than d children) at any level except the deepest,  and only one deficient node at the deepest level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Thus we can assume without loss of  generality that there are k elements x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , xk−1 ∈ C of maximum length n in C with a  common prefix of length n − 1, and no other y ∈ C has that prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Say the x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , xk−1 are  listed in nondecreasing order of r(xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' that is, r(xi) ≤ r(xj) for all 0 ≤ i ≤ j ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Let  y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , yk−1 ∈ C such that r(yi) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Since the ai are minimal, r(yi) ≤ r(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Because the  |xi| are of maximum length, |yi| ≤ |xi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Now we can swap using axiom (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Let  s(z) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  r(xi),  if z = yi,  r(yi),  if z = xi,  r(z),  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Then (C, s) ≤ (C, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' But since (C, r) was optimal, (C, r) ≡ (C, s) and (C, s) is also optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The algorithm of §4 produces an optimal tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' By induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The basis is n ≤ d, in which case the result is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Suppose that we have a multiset M of n > d elements of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Let (C, r) be an optimal tree  for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Let k ∈ {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , d} be congruent mod d − 1 to |M|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Let a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , ak−1 be the k smallest  elements of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' By Lemma 1, we can assume without loss of generality that a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , ak−1 are  siblings and occur at maximum depth in (C, r), so there exist strings x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , x(k − 1) ∈  C of maximum length with a common prefix x and r(xi) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Remove the strings xi from  C and replace them with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Call the resulting set C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' For z ∈ C′, let  r′(z) =  �  ({0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , k − 1}, i �→ ai),  if z = x,  ηW(r(z)),  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  6  Then (C′, r′) ∈ C2W and (C, r) = µW(C′, r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The multiset of values of r′ is just the M′ of  step 2 of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  The algorithm will form the multiset  M′′ = {w(E, t) | (E, t) ∈ M′} = {w(r′(z)) | z ∈ C′}  and recursively call with these weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' By the induction hypothesis, the return value will  be a tree (D, s) ∈ CW that is optimal for M′′, thus (D, s) ≤ (C′, w ◦ r′), and the bijective  map s : D → M′′ factors as s = w ◦ r′ ◦ f for some bijective f : D → C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Let s′ = r′ ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' By  axiom (iii),  Cw(D, s′) = (D, w ◦ s′) = (D, s) ≤ (C′, w ◦ r′) = Cw(C′, r′),  therefore (D, s′) ≤ (C′, r′), and since µW is monotone,  µW(D, s′) ≤ µW(C′, r′) = (C, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  As (C, r) was optimal, so is µW(D, s′), and this is the value returned by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  6  Applications  By choosing two specific weightings (W, w) and defining the ordering relations ≤ appro-  priately, we can recover two special cases of this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='1  Huffman coding  Our first application is Huffman codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Here we wish to minimize the expected length of  variable-length codewords, given frequencies of the letters to be coded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' For this applica-  tion, we take W = R+ = {a ∈ R | a ≥ 0} with weighting  w(C, r) = ∑  x∈C  r(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Recall that for a ∈ W and (C, r) ∈ C2W with r(x) = (Cx, rx),  ηW(a) = ({ε}, ε �→ a)  µW(C, r) = ({xy | x ∈ C, y ∈ Cx}, xy �→ rx(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Then (W, w) is an Eilenberg-Moore algebra for the monad (C, µ, η), as  w(ηW(a)) = w({ε}, ε �→ a) = ∑  x∈{ε}  (ε �→ a)(x) = a,  w(µW(C, r)) = ∑  x∈C ∑  y∈Cx  rx(y) = ∑  x∈C  w(Cx, rx)  = ∑  x∈C  w(r(x)) = w(C, w ◦ r) = w(Cw(C, r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  In addition, let us define α : CW → W by  α(C, r) = ∑  x∈C  |x| · r(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  7  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  α(ηW(a)) = 0  α(µW(C, r)) = α(C, w ◦ r) + w(C, α ◦ r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  α(ηW(a)) = α({ε}, ε �→ a) = ∑  x∈{ε}  |x| · (ε �→ a)(x) = |ε| · a = 0,  α(µW(C, r)) = α({xy | x ∈ C, y ∈ Cx}, xy �→ rx(y))  = ∑  x∈C ∑  y∈Cx  |xy| · rx(y) = ∑  x∈C  |x| ∑  y∈Cx  rx(y) + ∑  x∈C ∑  y∈Cx  |y| · rx(y)  = ∑  x∈C  |x| · w(Cx, rx) + ∑  x∈C  α(Cx, rx) = ∑  x∈C  |x| · w(r(x)) + ∑  x∈C  α(r(x))  = α(C, w ◦ r) + w(C, α ◦ r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Note that α and w agree on trees of depth one:  w({0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , k − 1}, i �→ ai) =  k−1  ∑  i=0  ai,  α({0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' , k − 1}, i �→ ai) =  k−1  ∑  i=0  |i| · ai =  k−1  ∑  i=0  ai,  where |i| refers to the length of i as a string, which in this case is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  The map α is related to the Shannon entropy H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' If r(x) = d−|x|, the probability of a  d-ary codeword x under the uniform distribution on a d-ary alphabet, then  H(C, r) = ∑  x∈C  −d−|x| log d−|x| = ∑  x∈C  |x| · d−|x| log d = α(C, r) log d,  so α(C, r) = H(C, r)/ log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  To use the algorithm in §4, we need an order ≤ on CW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Define (C, r) ≤ (D, s) if (C, r) ∼  (D, s), that is, there is a bijective map f : C → D such that r = s ◦ f, and  α(C, r) ≤ α(D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Note that if (C, r) ≤ (D, s), then  w(C, r) = ∑  x∈C  r(x) = ∑  x∈C  s( f(x)) = ∑  y∈D  s(y) = w(D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  According to axiom (iii), for (C, r), (D, s) ∈ C2W,  (C, r) ≤ (D, s) ⇔ Cw(C, r) ≤ Cw(D, s)  ⇔ α(Cw(C, r)) ≤ α(Cw(D, s))  ⇔ α(C, w ◦ r) ≤ α(D, w ◦ s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  (1)  Also, if (C, r) ≤ (D, s) in C2W, then  w(C, α ◦ r) = ∑  x∈C  α(r(x)) = ∑  x∈C  α(s( f(x))) = ∑  y∈D  α(s(y)) = w(D, α ◦ s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  (2)  8  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' µW : C2W → CW and ηW : W → CW are monotone with respect to ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' For ηW, suppose a, b ∈ W and a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' By Lemma 3,  α(ηW(a)) = 0 = α(ηW(b))  w(ηW(a)) = a ≤ b = w(ηW(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  For µW, suppose (C, r), (D, s) ∈ C2W and (C, r) ≤ (D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' By Lemma 3, (1), and (2),  α(µW(C, r)) = α(C, w ◦ r) + w(C, α ◦ r)  ≤ α(D, w ◦ s) + w(D, α ◦ s) = α(µW(D, s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The algorithm in §4 for the algebra (R+, w) and ordering relation ≤ defined by α is  equivalent to Huffman’s algorithm and produces an optimal Huffman code for a given multiset of  weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Take X ⊂ R+ to be a finite multiset and sort the set X in increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' For the  binary case of Huffman codes (the d-ary version follows the same way), we always choose  k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' For the first step, let a0, a1 ∈ X be the two smallest elements in the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Form the  object ({0, 1}, i �→ ai) ∈ CX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' In the case n = 2, this is the only remaining object in the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Otherwise, we combined them into one element with the sum of the weights of a0 and a1  as the weight of the new element, exactly as the Huffman coding does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  For the case n > 2, there are remaining elements in the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Take all remaining  a ∈ X\\{a0, a1} and replace a by ηX(a) ∈ CX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' We are left with n − 1 elements of type CX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  If we recursively call the algorithm in step 1, we are continually combining the least two  elements in the remaining set with the elements weighted by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Note by the weighting  w, w(ηX(a)) = a and on elements in CX, w takes the sum of r(x)′s, exactly as Huffman  coding does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Finally, this leaves us with a tree in C2X where leaves have weights of the  form ηX(ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Denote this tree by (D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Taking µX(D, S) gives our desired tree in CX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='2  PIFO trees  PIFO trees were introduced in [3] as a model for programmable packet schedulers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' In the  recent work of [1], further work was done on PIFO trees giving a semantics that allows  for certain embedding algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The notion of a homomorphic embedding was defined for  the purpose determining when a PIFO tree could be represented by another PIFO tree and  for finding an embedding if so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The embedding algorithm we consider takes an arbitrary  PIFO tree and embeds it into a d-ary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' This becomes a special case of the algorithm of  §4, where we choose w in the weighting (W, w) to minimize the height of the target d-ary  tree into which the source tree can embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  For this application, we take W = N with weighting  w(C, r) = max  x∈C |x| + r(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  This gives an Eilenberg-Moore algebra (W, w) for the monad (C, µ, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' For a ∈ W and  (C, r) ∈ C2W with r(x) = (Cx, rx), as before we have  ηW(a) = ({ε}, ε �→ a)  µW(C, r) = ({xy | x ∈ C, y ∈ Cx}, xy �→ rx(y)),  9  so  w(ηW(a)) = w({ε}, ε �→ a) = max  x∈{ε} |x| + (ε �→ a)(x) = |ε| + a = a,  w(µW(C, r)) = w({xy | x ∈ C, y ∈ Cx}, xy �→ rx(y)) = max  x∈C max  y∈Cx |xy| + rx(y)  = max  x∈C max  y∈Cx |x| + |y| + rx(y) = max  x∈C |x| + max  y∈Cx |y| + rx(y)  = max  x∈C |x| + w(Cx, rx) = max  x∈C |x| + w(r(x))  = w(C, w ◦ r) = w(Cw(C, r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  For (C, r), (D, s) ∈ CW, let us define (C, r) ≤ (D, s) if there is a bijective function  f : C → D such that r = s ◦ f and  w(C, r) ≤ w(D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' µW : C2W → CW and ηW : W → CW are monotone with respect to ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' For ηW, if a ≤ b, then w(ηW(a)) = a ≤ b = w(ηW(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  For µW, suppose (C, r), (D, s) ∈ C2W and (C, r) ≤ (D, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' According to axiom (iii),  (C, r) ≤ (D, s) ⇔ Cw(C, r) ≤ Cw(D, s)  ⇔ w(Cw(C, r)) ≤ w(Cw(D, s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Then  w(µW(C, r)) = w(Cw(C, r)) ≤ w(Cw(D, s)) = w(µW(D, s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The algorithm of §4 for the algebra (N, w) and ordering relation ≤ defined by w is  equivalent to determining whether an embedding of a PIFO tree in a bounded d-ary tree exists and  finding the embedding if so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  7  Conclusion  We have presented a generalized Huffman algorithm and shown that two known algo-  rithms, Huffman codes and embedding of PIFOs trees, can be derived as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  The PIFO embedding algorithm was introduced in [1] and observed to be very similar to  the usual combinatorial algorithm for optimal Huffman codes, albeit based on a different  algebraic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' This suggested the common generalization presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Our generalized algorithm exploits the monadic structure of prefix codes, which al-  lows a more algebraic treatment of the Huffman algorithm than the usual combinatorial  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' The two applications fit naturally in the categorical setting by choosing spe-  cific Eilenberg-Moore algebras for each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' It is possible that other greedy algorithms  might fit into this framework as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  10  References  [1] Anshuman  Mohan,  Yunhe  Liu,  Nate  Foster,  Tobias  Kapp´e,  and  Dex-  ter  Kozen,  “Formal  abstractions  for  packet  scheduling,”  Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='11659, Cornell University, November 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  [2] Thomas M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Cover and Joy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Thomas, Elements of Information Theory, Wiley, second  edition, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  [3] Anirudh Sivaraman, Suvinay Subramanian, Mohammad Alizadeh, Sharad Chole,  Shang-Tse Chuang, Anurag Agrawal, Hari Balakrishnan, Tom Edsall, Sachin Katti,  and Nick McKeown, “Programmable packet scheduling at line rate,” in SIGCOMM,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  [4] Albert Gran Alcoz, Alexander Dietm¨uller, and Laurent Vanbever, “SP-PIFO: Approx-  imating push-in first-out behaviors using strict-priority queues,” in NSDI, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  [5] Andrea Asperti and Giuseppe Longo, Categories, Types and Structures: An introduction  to category theory for the working computer scientist, Foundations of Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' MIT  Press, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  [6] Michael Barr and Charles Wells, Toposes, Triples and Theories, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' 278 of Grundlehren  der mathematischen Wissenschaften, Springer, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  [7] Michael Barr and Charles Wells, Category Theory for Computing Science, Prentice Hall,  1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  [8] Jiˇr´ı Ad´amek, Horst Herrlich, and George E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' Strecker, Abstract and concrete categories,  Dover Publications, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  [9] Eugenio Moggi, “Notions of computation and monads,” Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' and Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' 93, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' 1,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' 55–92, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  [10] Philip Wadler, “Comprehending monads,” Mathematical Structures in Computer Sci-  ence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' 461–493, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  [11] Philip Wadler, “Monads for functional programming,” in Advanced Functional Pro-  gramming: 1st Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' School on Advanced Functional Programming Techniques, Johan Jeur-  ing and Erik Meijer, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' 925 of Lecture Notes in Computer Science, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content=' 24–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  Springer-Verlag, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}
+page_content='  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfDgNt/content/2301.02878v1.pdf'}

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+Astronomy & Astrophysics manuscript no. Attree_NGA_Paper2_LanguageEdited
+©ESO 2023
+January 13, 2023
+Activity distribution of comet 67P/Churyumov-Gerasimenko
+from combined measurements of non-gravitational forces and
+torques
+N. Attree1, L. Jorda2, O. Groussin2, J. Agarwal1, R. Lasagni Manghi3, P. Tortora3, 4, M. Zannoni3, 4, and
+R. Marschall5
+1 Institut für Geophysik und extraterrestrische Physik, Technische Universität Braunschweig, Mendelssohnstr. 3, 38106
+Braunschweig, Germany (e-mail: n.attree@tu-braunschweig.de)
+2 Aix Marseille Univ, CNRS, CNES, Laboratoire d’Astrophysique de Marseille, Marseille, France
+3 Alma Mater Studiorum - Università di Bologna, Dipartimento di Ingegneria Industriale, Via Fontanelle 40, I-47121
+Forlì, Italy
+4 Alma Mater Studiorum - Università di Bologna, Centro Interdipartimentale di Ricerca Industriale Aerospaziale, via
+Baldassarre Carnaccini 12, I-47121, Forlì, Italy
+5 CNRS, Laboratoire J.-L. Lagrange, Observatoire de la Côte d’Azur, Boulevard de l’Observatoire, CS 34229 - F 06304
+NICE Cedex 4, France
+January 13, 2023
+ABSTRACT
+Aims. Understanding the activity is vital for deciphering the structure, formation, and evolution of comets. We inves-
+tigate models of cometary activity by comparing them to the dynamics of 67P/Churyumov-Gerasimenko.
+Methods. We matched simple thermal models of water activity to the combined Rosetta datasets by fitting to the total
+outgassing rate and four components of the outgassing induced non-gravitational force and torque, with a final manual
+adjustment of the model parameters to additionally match the other two torque components. We parametrised the
+thermal model in terms of a distribution of relative activity over the surface of the comet, and attempted to link this
+to different terrain types. We also tested a more advanced thermal model based on a pebble structure.
+Results. We confirm a hemispherical dichotomy and non-linear water outgassing response to insolation. The southern
+hemisphere of the comet and consolidated terrain show enhanced activity relative to the northern hemisphere and
+dust-covered, unconsolidated terrain types, especially at perihelion. We further find that the non-gravitational torque
+is especially sensitive to the activity distribution, and to fit the pole-axis orientation in particular, activity must be
+concentrated (in excess of the already high activity in the southern hemisphere and consolidated terrain) around the
+south pole and on the body and neck of the comet over its head. This is the case for both the simple thermal model
+and the pebble-based model. Overall, our results show that water activity cannot be matched by a simple model of
+sublimating surface ice driven by the insolation alone, regardless of the surface distribution, and that both local spatial
+and temporal variations are needed to fit the data.
+Conclusions. Fully reconciling the Rosetta outgassing, torque, and acceleration data requires a thermal model that
+includes both diurnal and seasonal effects and also structure with depth (dust layers or ice within pebbles). This shows
+that cometary activity is complex. Nonetheless, non-gravitational dynamics provides a useful tool for distinguishing
+between different thermophysical models and aids our understanding.
+Key words. comets: general, comets: individual (Churyumov-Gerasimenko), planets and satellites: dynamical evolution
+and stability
+1. Introduction
+Comets are amongst the most primordial Solar System ob-
+jects. They formed directly from the protoplanetary disc
+and survived mostly unaltered for much of their lifetimes
+in the outer Solar System. They are therefore vital targets
+for our understanding of planet formation and the history
+of the early Solar System. Upon entering the inner Solar
+System, comets are heated by the Sun and undergo ac-
+tivity; that is, ices are sublimated and gas and dust are
+ejected. Cometary activity poses open questions related to
+the structure, composition, and thermophysical properties
+of the nucleus material. This is directly connected to their
+formation in the early Solar System. Whether cometary
+nuclei, and by extension planets, formed from the gravi-
+tational collapse of clouds of centimetre-sized pebbles (as
+proposed in Blum et al. 2017) or by continual collisional
+growth (Davidsson et al. 2016) has direct implications for
+the structure and strength of the near-surface material that
+controls outgassing.
+In addition to being directly observable, the outgassing
+produces a reaction force on the nucleus that can alter its
+trajectory (as first recognised by Whipple 1950 and de-
+scribed by Marsden et al. 1973) and rotation state (see
+Samarasinha et al. 2004). Measuring the changing orbits
+Article number, page 1 of 13
+arXiv:2301.04892v1  [astro-ph.EP]  12 Jan 2023
+
+A&A proofs: manuscript no. Attree_NGA_Paper2_LanguageEdited
+and spins of comets therefore provides a useful insight into
+the the micro-physics of the activity mechanism.
+Many thermophysical models have been proposed to ex-
+plain the activity (see recent examples by Fulle et al. 2019,
+Gundlach et al. 2020, and Davidsson 2021), and these can
+be compared to the outgassing rates of observed comets. In
+particular, comet 67P/Churyumov-Gerasimenko (67P here-
+after) provides an excellent dataset because it was visited by
+the Rosetta spacecraft between 2014 and 2016. The space-
+craft collected detailed measurements of the size, shape,
+surface properties, and time-varying rotation state and out-
+gassing of the nucleus. Finding the distribution of activity
+across the nucleus of 67P that fits the various measurements
+of the total outgassing rate best (Hansen et al. 2016; Mar-
+shall et al. 2017; Combi et al. 2020; Läuter et al. 2020, etc.)
+has produced several so-called activity maps (e.g. Marschall
+et al. 2016, 2017; Läuter et al. 2020, ), which are often ex-
+pressed as an effective active fraction (EAF) relative to a
+pure water-ice surface. When examining only the summed
+total outgassing, however, there is always a degeneracy in
+the retrieved activity distribution (Marschall et al. 2020),
+whilst, at the same time, the effects of seasonal changes in
+insolation and dust cover across the surface of 67P are com-
+plicated (Keller et al. 2017; Cambianica et al. 2021). Com-
+paring the effects of a model outputted non-gravitational
+acceleration (NGA) and torque (NGT) to the dynamics of
+67P can provide a further constraint on the model parame-
+ters and on our understanding of the activity (Attree et al.
+2019; Kramer et al. 2019; Kramer & Läuter 2019; Mottola
+et al. 2020).
+Simple NGA models, such as those by Marsden et al.
+(1973) and Yeomans & Chodas (1989), parametrise the
+acceleration using variables scaled to a general water-
+production curve, and therefore provide limited insight
+into the physics of the activity on an individual comet.
+More complex models (following from Sekanina 1993) re-
+late the observed NGA and NGT to the outgassing via
+a thermal model and some distribution of ices or active
+areas across the nucleus surface. If independent measure-
+ments of this distribution and/or the outgassing rate can
+be made, then cometary masses and spin axes can be mea-
+sured from ground-based observations, as was achieved for
+67P (Davidsson & Gutiérrez 2005; Gutiérrez et al. 2005).
+Rosetta then provided both the detailed outgassing data
+mentioned above, as well as precise measurements of the
+nucleus position and rotation via radio-tracking and op-
+tical navigation. As summarised in Mottola et al. (2020),
+various attempts have been made to compare thermal mod-
+els to the NGA and NGT forces of 67P (Keller et al. 2015;
+Davidsson et al. 2022) and to fit its non-gravitational tra-
+jectory (Kramer & Läuter 2019), rotation state (Kramer
+et al. 2019), and both in combination with outgassing (At-
+tree et al. 2019).
+In Attree et al. (2019), our previous paper on this topic,
+we used the EAF formalism to fit surface distributions to
+the observed Earth-comet range (the most accurate compo-
+nent of the comet ephemeris, based on the spacecraft radio
+tracking), total gas production (measured by ROSINA, the
+Rosetta Spectrometer for Ion and Neutral Analysis; Hansen
+et al. 2016), and the change in spin rate (z component of
+the torque, measured as part of the nucleus shape recon-
+struction; Jorda et al. 2016). We found that a large EAF in
+the southern hemisphere of the comet, as well as an increase
+in EAF around perihelion, were needed to fit both the to-
+tal production measurements and the NGA. However, our
+model was limited by not considering the other components
+of the NGT (i.e. the change in the spin axis orientation, as
+well as its magnitude), and by a rather nonphysical way
+of splitting the surface into areas of differing activity. Ad-
+ditionally, discontinuities in the cometary heliocentric tra-
+jectory reconstructed by the European Space Operations
+Centre that arose because the NGA was excluded from the
+operational dynamical model, have complicated the anal-
+ysis by making it difficult to extract smooth acceleration
+curves.
+Kramer & Läuter (2019) addressed this problem by per-
+forming their own N-body integrations with a model fol-
+lowing Yeomans & Chodas (1989) and varying initial con-
+ditions. They then fitted a smoothed, interpolated curve to
+the residuals to extract time-varying NGA curves, but they
+did not compare them to a full thermal model. In a separate
+paper (Kramer et al. 2019), the authors did compare a phys-
+ical thermal model, again using the EAF formalism, to both
+the rotation rate and axis orientation data. Similarly to our
+results, their results also required a relatively higher EAF in
+the southern than in the northern hemisphere, as well as an
+enhanced outgassing response to insolation around perihe-
+lion to fit the data. Kramer & Läuter (2019) noted that the
+NGT is much more dependent on the spatial distribution
+of activity than the NGA.
+Since
+then,
+two
+additional
+reconstructions
+of
+the
+Rosetta/67P trajectory have been performed (Farnocchia
+et al. 2021; Lasagni Manghi et al. 2021). Farnocchia et al.
+(2021) used a rotating-jet model following Sekanina (1993)
+to fit ground-based astrometric observations and radio-
+ranging measurements before and after perihelion (where
+the spacecraft NGAs are smaller and the range accuracy is
+higher). Lasagni Manghi et al. (2021), on the other hand,
+used the full Rosetta two-way range and differential one-
+way range (∆DOR) dataset, also including low-accuracy
+data close to perihelion. They tested various NGA models,
+including a rotating-jet model, and found a best-fit tra-
+jectory using an empirical, stochastic acceleration model.
+Both of these works produced acceleration curves to which
+a thermal model can be compared.
+Davidsson et al. (2022) did just that by comparing
+the output of a more complex thermal model (NIMBUS;
+Davidsson 2021) to the acceleration curves of Farnocchia
+et al. (2021) and Kramer & Läuter (2019). They found rel-
+atively good agreement without fitting, but had to vary
+several model parameters (e.g. the sublimation-front depth
+and the gas diffusivity) between the northern and south-
+ern hemispheres and pre- and post-perihelion, in order to
+match the outgassing data. This reinforces the ideas of a
+hemispheric dichotomy and time-dependent thermophysi-
+cal properties, and it also demonstrates the complicated
+nature of trying to model the full thermophysical system of
+sublimation, gas flow, and dust.
+These studies show the usefulness of considering the
+non-gravitational dynamics. No study has analysed the full
+six components of NGA and NGT simultaneously, how-
+ever (we analyse all six here, but only four are included in
+the formal fitting procedure), and several other weaknesses
+exist, such as nonphysical surface distributions or compli-
+cated descriptions leading to unfitted models. It is per-
+tinent, therefore, to re-examine the full non-gravitational
+dynamics of 67P with a simple thermal model that can
+be parametrised in terms of real surface features while be-
+Article number, page 2 of 13
+
+N. Attree et al.: Activity distribution of comet 67P
+ing easily compared with more complicated models. This
+is what we attempt to do here, bearing in mind that the
+aim is not to find the full description of cometary activity,
+but a model that adequately describes the data and points
+towards the underlying physics.
+The rest of this paper is organised as follows: in Sec-
+tion 2 we describe how we updated the model of Attree
+et al. (2019) for use here. In Section 3 we describe three
+different parametrisations of the surface activity distribu-
+tion and their results in the model fit. These results are
+discussed, with reference to a run with the more advanced
+thermal model of Fulle et al. (2020) in Section 4, before we
+conclude in Section 5.
+2. Method
+We followed the method of the first paper (Attree et al.
+2019) by first calculating surface temperatures over a shape
+model of 67P (SHAP7; Preusker et al. 2017) with a simple
+energy-balance thermal model and then computing the re-
+sulting non-gravitational forces and torques and implement-
+ing them in an N-body integration. The model was then
+optimised by scaling the relative activity of various areas
+of the shape model up and down, minimising the residuals
+to the observed datasets: the Earth-comet range (i.e. the
+scalar projection of the three-dimensional comet position in
+the Earth-comet direction, R, with NR = 1000 data points)
+or the directly extracted NGAs from Lasagni Manghi et al.
+(2021) (with NNGA = 17000 data points in each of the three
+components); the total gas production (NQ = 787, Hansen
+et al. 2016); and the spin-axis (z) aligned component of the
+torque (NTz = 1000, Jorda et al. 2016). Additionally, we
+now also computed the change in the orientation of the ro-
+tation axis (Kramer et al. 2019) and used this as an output
+to compare different models.
+The thermal model computes the surface energy-
+balance, taking insolation, surface thermal emission, sub-
+limation of water ice, projected shadows, and self-heating
+into account (see Attree et al. 2019 for details). Heat con-
+duction into the nucleus is neglected for numerical reasons,
+but is small because of the low thermal inertia of the comet
+(Gulkis et al. 2015). Heat conduction would mainly affect
+night-time temperatures, which are very low and contribute
+little to the outgassing (but see the discussion in Section 4).
+Again for numerical reasons, surface temperatures are cal-
+culated roughly once every 10 days for a full 12.4 hour ro-
+tation, and the derived quantities are interpolated (see de-
+tails below) to produce smooth curves over the full mission
+period of about two years. Surface temperatures are each
+computed twice, once assuming an effective active fraction
+EAF = 0 (i.e. pure grey-body dust surface), and once with
+EAF = 1 (i.e. sublimation from a pure water-ice surface),
+and the temperatures and sublimation rates are saved. In
+the fitting process, the pure water-ice sublimation rate is
+then scaled by a variable EAF and is used, along with
+the sublimation gas velocity calculated from the zero-ice
+surface temperature, to compute the outgassing force per
+facet. The momentum coupling parameter was assumed to
+be η = 0.7 (Attree et al. 2019). Torque per facet was also
+calculated here using the “torque efficiency” formalism used
+before (Keller et al. 2015), where τ is the facet torque ef-
+ficiency or moment arm, which is a geometric factor that
+was computed once at the beginning of the run. The use of
+the higher zero-ice temperature for the gas thermal veloc-
+ity assumes that the gas equilibrates with the dusty surface,
+and this means that our derived EAF values may be lower
+estimates compared with some other thermal models.
+The N-body integration was performed using the open-
+source REBOUND code1 (Rein & Liu 2012), complete
+with full general relativistic corrections (Newhall et al.
+1983) as implemented by the REBOUNDx extension pack-
+age2, and including all the major planets as well as Pluto,
+Ceres, Pallas, and Vesta. Objects were initialised with their
+positions and velocities in the J2000 ecliptic coordinate
+system according to the DE438 Solar System ephemerides
+(Standish 1998), with 67P given its initial state vector
+from the new Rosetta trajectory reconstruction of Lasagni
+Manghi et al. (2021) (Table A.1). The system was then inte-
+grated forward in time from t = −350 to +350 days relative
+to perihelion, using the IAS15 integrator (Rein & Spiegel
+2015) and the standard equations of motion, with the addi-
+tion of a custom acceleration, aNG, for 67P, provided by our
+model. The Earth-comet range, which is the most accurate
+component of the comet trajectory, was computed for com-
+parison with the reconstructed trajectory (extracted using
+the SpiceyPy Python package; Annex et al. 2020).
+A bounded least-squares fit to the residuals was then
+performed using standard methods implemented in Scien-
+tific Python whilst varying the EAF parameters. When
+forming the overall objective function to be minimised (see
+Eqns. 9 and 10. in Attree et al. 2019), the datasets were
+weighted by a factor λ so that each contributed roughly the
+same to the overall fit (see Table 1). The datasets used in all
+fits were the model outputted total outgassing rate and the
+z component of the torque, both with λQ = λTz = 1. Fur-
+thermore, in some fits, we then used the computed Earth-
+comet range (with λR = 0.02), while in others, we directly
+compared to the three components of the NGA extracted by
+Lasagni Manghi et al. (2021) in the cometocentric radial-
+transverse-normal frame (radial to the Sun, ˆr, tangential
+to the orbit, ˆt, and normal to it). In this case, the inte-
+gration was only performed once at the end to check the
+Earth-comet range, but the weighting was zero in the fit
+(λR = 0), while λNGA = 1. Performing the N-body inte-
+gration only once speeds the process up by several times,
+with individual runs taking a few minutes and fits taking
+up to a day, depending on the parameters. All parameters
+were interpolated to the observational data sampling-times
+using the Fourier method described below.
+We first confirm that the Lasagni Manghi et al. (2021)
+accelerations match the real comet trajectory well when
+they are input into our N-body integration, and they re-
+cover the Earth-comet range to within a few hundred me-
+tres. This residual, which is most likely the result of the
+different integration techniques and perturbing bodies we
+used, is well below the uncertainty of our thermal model
+runs.
+Previously, the x and y components of the torque vector
+were discarded, but they were now used when we calculated
+the changes in pole orientation. In principle, the rates of
+change of the angular velocity (Ω) of the comet around its
+three principal axes can be related (see e.g. Julian 1990) to
+1 http://rebound.readthedocs.io/en/latest/
+2 http://reboundx.readthedocs.io/en/latest/index.html
+Article number, page 3 of 13
+
+A&A proofs: manuscript no. Attree_NGA_Paper2_LanguageEdited
+the torque components by
+Ix ˙Ωx = (Iy − Iz)ΩyΩz + Tx,
+Iy ˙Ωy = (Iz − Ix)ΩxΩz + Ty,
+Iz ˙Ωz = (Ix − Iy)ΩxΩy + Tz,
+(1)
+where Ix = 9.559 × 1018, Iy = 1.763 × 1019, and Iz =
+1.899×1019 kg m2 are the moments of inertia derived from
+the shape model assuming a constant density of 538 kg m−3
+(Preusker et al. 2017), and to the pole orientation right
+ascension, RA, and declination, Dec, by
+˙ψ = −Ωy cos(ψ) − Ωx sin(ψ)
+tan(θ)
++ Ωz,
+˙φ = Ωy cos(ψ) + Ωx sin(ψ)
+sin(θ)
+,
+˙θ = Ωx cos(ψ) − Ωy sin(ψ),
+(2)
+via the Euler angles φ = π/2 + RA, θ = π/2 − Dec, and ψ.
+In practice, the fact that our model runs over individual
+rotations separated by gaps means that the torque curves
+are discontinuous and cannot be directly integrated. We
+therefore followed the technique of Kramer et al. (2019) and
+applied a Fourier analysis to the torque curves. The method
+proceeds by i) extracting the torque over a single rotation
+as a function of the sub-solar longitude, using Kramer et al.
+(2019) Eqns. 26, 27, ii) computing the Fourier transform as
+a function of sub-solar longitude using Eqn. 23, iii) inter-
+polating the Fourier terms as smooth curves over the full
+Rosetta period; Eqn. 24, and iv) reconstructing the torque
+at a chosen time by the inverse Fourier transform; Eqn.
+25. This allows the calculation of a smoothly interpolated
+torque value at any given time, Tx,y,z(t), for use in the ro-
+tation equations (1).
+The set of simultaneous differential equations given by
+Eqns. 1 and 2 was then integrated using standard func-
+tions in Scientific Python and the initial conditions RA =
+69.427◦, Dec = 64.0◦, and ψ = 330.703◦ at t = −377.22
+days relative to perihelion (Kramer et al. 2019) for the pe-
+riod t = [−377.22 : 402.48], corresponding to the duration
+of the Rosetta measurements. The resulting RA(t), Dec(t)
+values were not used in the fit due to technical limita-
+tions, but were directly compared with the observations as
+a model output.
+3. Results
+3.1. Model C
+We began by rerunning the best-fit model of the previous
+paper, designated model C in Attree et al. (2019). This
+model parametrised the activity distribution by splitting
+the surface into the 26 regions, defined by Thomas et al.
+(2015) (see their figures for maps), and then grouping them
+into five super-regions following Marschall et al. (2016)(see
+Figure 4 in Attree et al. 2019), before finally splitting the
+Southern super-region into two (see Figure 17 in Attree
+et al. 2019) and allowing these to vary their EAF with time.
+With 6 super-regions and the 6 time-variation parameters,
+there are a total of 12 free parameters in this model. These
+super-regions consist of region 1, covering the equatorial ar-
+eas; region 2, covering the base of the comet body and top
+of the head; the individual regions Hathor and Hapi; and
++Z
+-Z
+Fig. 1. Peak effective active fraction at perihelion for solution
+C, mapped onto the shape model.
+−400
+−300
+−200
+−100
+0
+100
+200
+300
+400
+Days from Perihelion
+0.0
+0.1
+0.2
+0.3
+0.4
+Active Fraction
+So th -
+Region 1
+Region 2
+Hathor
+Hapi
+So th +
+Fig. 2. Time-varying effective active Fraction for solution C.
+two southern super-regions split on a per-facet basis by the
+sign of the z component of the torque efficiency (i.e. south
+positive with τz > 0 and south negative with τz < 0).
+This splitting was the only way in which a satisfactory fit
+to the z torque (i.e. rotation-rate data) could be achieved,
+but it remains somewhat artificial. Figure 1 shows the best-
+fit solution achieved here, mapped onto the shape model.
+This shows the discontinuous and patchy appearance of the
+southern super-regions, as well as the north-south EAF di-
+chotomy and activity in Hapi (the light blue area in the
+northern neck region).
+We optimised this model again here and, with a slightly
+differing procedure for sampling and interpolating the com-
+putational output, produced very similar results to before,
+with no significant improvement in the fit. Next, we in-
+stead fit the model directly to the Lasagni Manghi et al.
+(2021) NGA curves as described above, producing the best-
+fit solution shown mapped onto the shape-model in Figure
+1 (where the values shown are peak EAF, the maximum
+value for all times), and with time in Figure 2. The out-
+put is very similar to the previous solution in Attree et al.
+(2019), but Figure 2 shows an even more pronounced spike
+in EAF around perihelion than before.
+The model fits are shown in the orange curves in Fig-
+ures 3, 4, and 5, with the fit statistics in the first line of
+Article number, page 4 of 13
+
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+Active FractionN. Attree et al.: Activity distribution of comet 67P
+−300
+−200
+−100
+0
+100
+200
+Days from Perihelion
+10
+26
+10
+27
+10
+28
+Ou gassing Ra e (s
+−1
+)
+Model C
+Model D
+Model E
+Observed
+Fig. 3. Observed total gas production (ROSINA values from
+Hansen et al. 2016) compared to solutions C, D, and E.
+300
+200
+100
+0
+100
+200
+300
+Days from Perihelion
+200
+150
+100
+50
+0
+50
+100
+150
+200
+Range Residuals (km)
+Model C
+Model D
+Model E
+Fig. 4. Observed minus computed Earth-comet range, R, for
+solutions C, D, and E.
+Table 1. The z torque (Fig. 5) and total gas production
+from ROSINA (Fig. 3) are reasonably well fit, with the
+perihelion peak-values matched, but with a slightly differ-
+ing shape around the inbound equinox roughly 100 days
+before perihelion. An improvement in the trajectory fit is
+attained, with the new RMS residual value of 34 km re-
+duced from the previously achieved 46 km. The shape of
+the curve is similar.
+The orange curves in Figures 6, 7, and 8 show the in-
+dividual acceleration curves in the cometocentric (ˆr, ˆt, ˆn)
+frame compared to the values extracted by Lasagni Manghi
+et al. (2021). The radial component makes up the bulk of
+the acceleration and is reasonably well matched by model
+C, with the peak value being ∼ 50% too high. The normal
+and tangential components are of smaller magnitude and
+are reasonably well fit; the secondary, negative peak of the
+tangential component after perihelion is the worst area of
+the fit. The remaining 34 km residuals to the observed tra-
+jectory most likely stem from our inability to fit this area
+of the tangential acceleration, combined with the too large
+radial component peak.
+−300
+−200
+−100
+0
+100
+200
+300
+Days from Perihelion
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+T
+or ue (Nm)
+1e7
+Observed
+Model C
+Model D
+Model E
+Fig. 5. Smoothed observed z component of the torque com-
+pared to solutions C, D, and E. The grey area represents the 1σ
+uncertainty (see Attree et al. 2019 for details).
+300
+200
+100
+0
+100
+200
+300
+400
+Days from Perihelion
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+NGA r (AU d
+2)
+1e
+9
+Observed
+Model C
+Model D
+Model E
+Fig. 6. Observed radial acceleration in the comet (ˆr, ˆt, ˆn) frame
+with the 5σ uncertainty (from Lasagni Manghi et al. 2021), com-
+pared to solutions C, D, and E. Higher-order Fourier terms cor-
+responding to daily oscillations are omitted for clarity, but are
+included in the fit.
+When the pole orientation was calculated, as shown in
+the orange curve of Figure 9, it was a very poor fit to the
+data, moving off in the opposite direction to the observed
+changes. This demonstrates that the problem is ill-posed
+with multiple solutions, and it also highlights the useful-
+ness of including the RA, Dec pole measurement to help
+distinguish between different models that fit the other data
+equally well.
+3.2. Model D
+We now proceed with a more physically meaningful model.
+This was constructed using the list of 71 sub-regions de-
+fined in Thomas et al. (2018) (see the reference for maps of
+their location). We again created super-regions by collecting
+these sub-regions, but this time, by placing them into one of
+the five morphological categories of Thomas et al. (2015):
+‘dust-covered terrains’ (Dust for short), ‘brittle materials
+Article number, page 5 of 13
+
+A&A proofs: manuscript no. Attree_NGA_Paper2_LanguageEdited
+Table 1. Fit statistics for best-fit models C, D, and E, and the two unfitted versions of F.
+Solution
+Weighting
+χ2
+λQ
+λTz
+λR
+λNGA
+R
+Q
+Tz
+NGAr
+NGAt
+NGAn
+Obj
+C
+1
+1
+0
+1
+34.1
+4.53
+1.36
+1.18
+1.32
+0.44
+1.20
+D
+1
+1
+0.02
+0
+88.8
+3.60
+1.10
+2.00
+1.60
+0.90
+2.35
+E
+1
+1
+0.02
+0
+83.4
+3.75
+0.77
+1.78
+1.58
+0.89
+2.22
+F dust SH
+-
+-
+-
+-
+324.5
+4.62
+2.09
+4.12
+1.71
+1.01
+-
+F ice SH
+-
+-
+-
+-
+459.2
+5.64
+3.02
+2.22
+1.63
+1.00
+-
+Notes. Model E is highlighted as the preferred solution. The model outputs (water production rate, z component of NGT, and
+the three components of NGA) are compared to the observations, producing the χ2 statistics, which are then weighted according
+to the λ values and combined in the objective function (Eqns. 9 and 10. in Attree et al. 2019) to produce the combined fit statistic
+Obj. All values are dimensionless, although the range values R correspond one-to-one to kilometers.
+300
+200
+100
+0
+100
+200
+300
+400
+Days from Perihelion
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+NGA t (AU d
+2)
+1e
+10
+Observed
+Model C
+Model D
+Model E
+Fig. 7. Observed tangential acceleration in the comet (ˆr, ˆt, ˆn)
+frame compared to solutions C, D, and E.
+300
+200
+100
+0
+100
+200
+300
+400
+Days from Perihelion
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+NGA n (AU d
+2)
+1e
+10
+Observed
+Model C
+Model D
+Model E
+Fig. 8. Observed normal acceleration in the comet (ˆr, ˆt, ˆn) frame
+compared to solutions C, D, and E.
+with pits and circular structures’ (Brittle), ‘large-scale de-
+pressions’ (Depression), ‘smooth terrains’ (Smooth), and
+‘exposed consolidated surfaces’ (Rock). The sub-regions
+were assigned according to their descriptions in the table
+in Thomas et al. (2018). A few ambiguous examples were
+67
+68
+69
+70
+71
+72
+Right Ascensi n (
+∘
+)
+63.50
+63.75
+64.00
+64.25
+64.50
+64.75
+65.00
+65.25
+65.50
+Declinati n (
+∘
+)
+M del C
+M del D
+M del E
+Observed
+Fig. 9. Observed pole orientation (Ra, Dec) compared to solu-
+tions C, D, and E. The thickness of the model lines is due to the
+daily oscillations. Error bars are plotted for the observations,
+but are small at this scale.
+tested in both the categories to which their descriptions
+could apply, without altering our results significantly. The
+Rock and Smooth terrain types both cover significant ar-
+eas of the southern hemisphere and following the results
+of the first paper, we therefore allowed their EAFs to vary
+with time in the same way as for model C. The facets in
+each super-region all have the same EAF (either constant or
+time-varying), regardless of the hemisphere in which they
+are located. With five regions and 6 time-variation param-
+eters, there are 11 parameters in total for this model, des-
+ignated ‘model D’.
+Figure 10 shows the peak activity in our best-fit solution
+for model D mapped onto the shape model, and Fig. 11
+shows the time variation. High activity is again favoured
+in the southern hemisphere, with the Rock and Smooth
+regions seeing much higher activity than the Dusty, Brittle,
+and Depression regions, especially around perihelion.
+Model D is shown as green curves in Figures 3 - 9. The fit
+statistics are again shown in Table 1. This model produces
+a similar, if slightly improved, fit to the total outgassing
+measurements, while slightly degrading the trajectory and
+rotation-rate fits compared to model C. The reasons for the
+poorer trajectory fit can be seen in the acceleration curves
+in Figures 6, 7, and 8. The modelled radial component of
+the acceleration is still slightly too large when compared
+Article number, page 6 of 13
+
+N. Attree et al.: Activity distribution of comet 67P
++Z
+-Z
+Fig. 10. Peak effective active fraction at perihelion for solution
+D, mapped onto the shape model.
+−400
+−300
+−200
+−100
+0
+100
+200
+300
+400
+Days from Perihelion
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+Acti e Fraction
+Dust
+Brittle
+Smooth
+Depression
+Rock
+Fig. 11. Time-varying effective active fraction for solution D.
+to the observations, while the tangential and normal com-
+ponents are now much worse than before, with the curves
+roughly the correct shape, but too small in magnitude. An
+attempt to fit model D directly to the accelerations did
+not improve the trajectory, and the individual super-region
+NGA curves showed no obvious combination that would fit
+the accelerations better.
+Figure 9 shows that model D additionally fails to repro-
+duce the observed changes in pole direction. However, the
+curve now goes in the correct direction, but with a magni-
+tude that is too large compared to the completely incorrect
+prediction of model C. This suggests that the more phys-
+ically meaningful model has merit, despite the degraded
+trajectory fit, and it motivated us to make further adjust-
+ments to try and fit all the data below.
+3.3. Model E
+Because model D fits most of the data well but increasingly
+fails with the magnitude of the pole direction changes, we
+sought to modify it by adjusting the NGT. Specifically, in
+order to fit all the data, the comet must produce a smaller
+amount of non axial-aligned torque (x and y components),
+while the rest of the torque and accelerations remain the
+same. We achieved this in model E with another, somewhat
+artificial, splitting of the Rock super-region into two super-
+regions based on their torque contributions. This splitting
+was performed on a sub-region basis, rather than on the
+per-facet basis of model C, in order to produce contiguous
+areas that allowed us to see the general trends in activity
+across different parts of the comet surface. The modulus of
+the torque efficiency (|τ|) was first calculated for each facet
+(top left in figure 12) before the area-weighted mean for
+each sub-region was calculated and the Rock super-region
+was split into ‘low torque’ (|τ| lower than the median sub-
+region value) and ‘high torque’ (|τ| greater than the median
+value). Both of these super-regions were allowed to vary
+with time, leaving a total of 13 free parameters.
+Figures 13 and 14 show the best-fit solution. This was
+found by manually adjusting the optimised solution by eye
+to match the pole-direction data. The results are very sim-
+ilar to those of model D, except that the regions of rocky
+terrain with high torque efficiency are reduced to an inter-
+mediate value of activity, between that of the rest of Rock
+and the other terrain types. The red curves in Figures 3 -
+9 show that this adjustment has little effect on the trajec-
+tory, production, and rotation-rate fits, but now produces
+an excellent match to the pole-direction data as well. Thus,
+model E represents our best-fit solution overall.
+When the acceleration curves are considered in detail,
+model E fails to reproduce the tangential and normal com-
+ponents in the same way as model D. The peak radial accel-
+eration is slightly reduced, however, resulting in a slightly
+better trajectory fit than for model D. We once again sought
+improvements in the acceleration by fitting directly to the
+curves, as well as examining the acceleration produced by
+individual regions, but no overall better fit was found. Every
+improvement in the acceleration curves led to a correspond-
+ing degradation in the rotation fits.
+4. Discussion
+Our best-fit model overall is model E. This model is based
+on a splitting of the surface according to morphological unit
+types, with an artificially imposed further splitting accord-
+ing to torque efficiency and a time-varying EAF. A num-
+ber of trends can be seen across all the solutions, however,
+which we discuss now, before we return to the interpreta-
+tion of model E.
+In common with the previous results (Attree et al.
+2019), all models firstly require a higher EAF in the south-
+ern than the northern hemisphere, as well as an EAF that
+increases around perihelion. This increase in activity, over
+and above the increase expected with heliocentric distance,
+is a common result in the literature (Keller et al. 2015;
+Kramer et al. 2019; Davidsson et al. 2022) and implies a
+non-linear outgassing response to insolation. High activity
+at perihelion is needed to fit the maximum outgassing rate
+as well as the sharp peak in acceleration, which is mostly
+contained in the radial component.
+Non-gravitational torque, as expressed in the period and
+spin-axis changes, is much more dependent on the exact
+spatial distribution of activity (as also found by Kramer
+& Läuter 2019), especially within this very active south-
+ern hemisphere. For example, the correct magnitude of the
+pole-direction fit is achieved in model E by distributing the
+activity around the southern hemisphere in a specific way:
+high activity in regions with low torque efficiency around
+the south pole, with lower activity in areas with a high
+Article number, page 7 of 13
+
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+Active FractionA&A proofs: manuscript no. Attree_NGA_Paper2_LanguageEdited
+ 
+ 
+2
+4
+6
+8
+Total Insolation (J m−2)
+1e9
+20
+40
+60
+80
+100
+120
+Gravitational Slope (deg.)
+500
+1000
+1500
+2000
+2500
+Torque Efficiency (Nm)
+-Z
+Fig. 12. Various datasets mapped onto the southern hemisphere
+of the comet. From top: Modulus of torque efficiency (|τ|), a ge-
+ometric factor as described in the text; gravitational slope, i.e.
+the angle between facet normal and local gravity vector; total
+integrated insolation; and peak insolation. The three white lines
+indicate the direction of the −r, −t, and −n vectors, averaged
+over one rotation period at perihelion, i.e. the time-averaged di-
+rections towards the Sun, ‘backwards’, and ‘down’ in the orbital
+frame of the comet.
++Z
+-Z
+Fig. 13. Peak effective active fraction at perihelion for solution
+E, mapped onto the shape model.
+−400
+−300
+−200
+−100
+0
+100
+200
+300
+400
+Days from Perihelion
+0.00
+0.05
+0.10
+0.15
+0.20
+Active Fraction
+D st
+Brittle
+Smooth
+Depression
+Rock
+Rock - Low ta 
+Fig. 14. Time-varying effective active fraction for solution E.
+torque efficiency, such as towards the extremities of the
+nucleus and parts of the head. This agrees well with the
+distribution seen in Kramer et al. (2019) (see their Figs. 9
+and 10). As shown in Figure 12, these low-torque areas and
+physical parameters, such as the total amount or peak of
+insolation received or the gravitational slopes, do not ap-
+pear to be correlated. The fact that morphologically similar
+and similarly insolated regions on the head and body show
+differing levels of activity may imply compositional differ-
+ences between the two lobes of the nucleus, as suggested by
+comparisons of region Wosret with the Anhur and Khonsu
+regions by Fornasier et al. (2021).
+When the seasonal orientation of the comet is consid-
+ered alongside the acceleration curves, the reasons for the
+differences between the trajectories of models C, D, and
+E become clear. The large magnitudes of the normal and
+tangential acceleration peaks in model C come from the
+extreme activity ratio of the south polar regions and else-
+where: At perihelion, when the outgassing is at a maximum,
+the comet orientation is such that the southern hemisphere
+most often points ‘downwards’ (in the negative direction in
+the orbital plane, −ˆn), towards the Sun (−ˆr), and ‘back-
+wards’ (along the negative of the orbital velocity vector
+−ˆt). This is shown in Fig. 12 by three vectors, indicating
+the time-averaged direction of ⟨−ˆr, −ˆt, −ˆn⟩ over one comet
+Article number, page 8 of 13
+
+200
+400
+600
+800
+1000
+1200
+Peak Insolation (w m-2)0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+Active FractionN. Attree et al.: Activity distribution of comet 67P
+rotation at perihelion. As the comet rotates, the unit vec-
+tors sweep over its surface, but as a result of the spin-axis
+orientation at this time, the southern hemisphere points in
+the indicated direction on average. Thus, the net outgassing
+force from the southern hemisphere produces a strong pos-
+itive peak in all three of these components, as seen in the
+data. Meanwhile, any outgassing from other areas of the
+comet produces acceleration in different directions, reduc-
+ing the net positive peaks. This is the case in models D and
+E (and Kramer et al. 2019, etc.), where there is some activ-
+ity in areas that are not aligned south, meaning that part of
+the acceleration is in other directions and that the net pos-
+itive normal and tangential forces are reduced (green and
+red curves in Figs. 7 and 8 compared to orange). The radial
+peak (Fig. 6) is less reduced because most outgassing is di-
+rected towards the Sun, even in areas that are not aligned
+south.
+When the pole direction is fit, which is dependent on the
+x and y components of the NGT, however, activity is pre-
+ferred everywhere, or at least in a less extreme dichotomy
+than in model C. If the torque distribution in the south-
+facing regions alone could be adjusted to match the overall,
+correct, torque distributions of models D and E, then the so-
+lutions could be reconciled. However, figures 12 and 1 show
+that the correlation between the z component of torque ef-
+ficiency and its total modulus in the southern hemisphere
+is complicated, meaning that any adjustment to the pole
+direction (x and y torque components) will also affect the
+rotation rate (z component). Any increase or decrease in the
+perihelion activity of south-facing regions will also strongly
+affect the acceleration. For this reason, improvement of the
+acceleration or trajectory fit always degrades the pole di-
+rection fit and vice versa; the facets controlling NGA and
+NGT are spatially correlated.
+At one instant in time, the non-gravitational torques
+and accelerations will always be correlated by the spatial
+pattern described above. However, the total torques and ac-
+celerations integrated over some period (e.g. one rotation)
+may not necessarily be so correlated. For example, torque is
+evaluated in the body-fixed frame, so that it is independent
+of the particular orientation of the comet at any one time.
+The net acceleration vector, on the other hand, depends on
+this orientation with respect to the Sun and on the helio-
+centric coordinate frame, and it will vary over a cometary
+rotation (i.e. the non time-averaged version of the vectors
+shown in Fig. 12 will rotate around the shape model in the
+body-fixed frame). In this way, the acceleration per facet in-
+tegrated over one rotation period will be sensitive to both
+the total outgassing from the facet over that period and
+to its temporal variation, whereas the torque will only be
+dependent on the total outgassing.
+A possible way to optimise the fitting to the heliocen-
+tric orbit without deteriorating the fit to the rotation-axis
+orientation and period might then be to redistribute the
+activity variation with local time. The idea of a lag an-
+gle between the peak insolation and peak diurnal activity
+has indeed been invoked in the past (see e.g. Davidsson
+& Gutiérrez 2004), with recent work suggesting that water
+activity might peak at 20◦ (Pinzón-Rodríguez et al. 2021;
+Farnocchia et al. 2021) or even 50◦ (Kramer & Läuter 2019)
+post-noon, with the latter lag angle varying with time and
+being undetected before perihelion. Such a lag angle would
+depend on the thermal inertia and the depth at which wa-
+ter sublimates, making it complicated to model. Additional
+enhanced activity may also arise at the morning terminator
+due to sublimation of frost from the night.
+CO2 emissions, which have not been considered here,
+may also have a different local-time distribution. Pinzón-
+Rodríguez et al. (2021) reported a peak at the evening ter-
+minator. Davidsson et al. (2022) suggested that CO2 pro-
+duces little NGA, due to both its small outgassing rate
+compared to H2O and a smoother diurnal variation from a
+deep sublimation depth and large lag-effect, leading to force
+in all directions and a cancelling out of the net acceleration.
+CO2 activity distributed in a specific way, however, might
+still lead to a net torque, resulting in the required splitting
+of the torque and acceleration, although it would, admit-
+tedly, have to be quite a specific distribution. Gerig et al.
+(2020) reported that about 10% of total dust emission orig-
+inates from the night side, which may well be driven by
+CO2 emission, while the peak perihelion outgassing rate
+is roughly one order of magnitude lower than the rate for
+water (Läuter et al. 2020).
+Clearly, a more realistic thermal model, including ther-
+mal inertia as well as possibly the emission of CO2, is
+needed to fully reconcile the observed outgassing, accelera-
+tions and torques. Below, we briefly analyse the results of
+a recently published thermal model based on Fulle et al.
+(2020). This does not include a local time-lag or CO2 emis-
+sion, but offers an interesting comparison with and exten-
+sion of the surface energy-balance models discussed above.
+The model of Fulle et al. (2020), called model F here, as-
+sumes a material made of water-containing centimetre-sized
+pebbles, in which a constant energy balance is maintained
+between the insolated surface and ice sublimating in the
+interior of the pebbles. This leads to a set of four differen-
+tial equations that must be solved simultaneously for each
+time and facet, instead of the normal surface energy-balance
+equation. The rest of the code runs as before, with the slight
+complication that we cannot calculate self-heating in a self-
+consistent way due to a technical limitation, as it relies
+on an iteration between facets. We therefore calculated two
+model F solutions: one solution in which the self-heating per
+facet was calculated from a pure-ice surface, and another
+with a pure-dust surface. These two energy inputs bracket
+the full solution, whose surface temperature (and therefore
+self-heating term) is intermediate between a pure-ice and
+a pure-dust grey-body surface (Figure 15). The figure also
+shows that the outgassing rate in the Fulle et al. (2020)
+model is significantly reduced from that of a pure-ice sur-
+face and has a distinctly non-linear shape, ranging between
+effective active fractions of EAF∼ 0 − 20% as a function of
+insolation.
+Figure B.1 shows the resulting gas production curve
+evaluating model F on the shape model, showing that the
+model of Fulle et al. (2020) can naturally reproduce the
+high perihelion outgassing rates without the need for an ef-
+fective active fraction that varies with time. This confirms
+the results of Ciarniello et al. (2021).
+Figure B.2 shows the trajectory result obtained with
+model F, while Figures B.3 and B.4 show the torque and
+pole-direction curves. For a model without any fitting, the
+results agree reasonably well with the data, although the
+magnitude of the pole-direction changes are again too large,
+and the trajectory fit and z torque are not as close as in
+our best models (see Table 1 for fit statistics).
+Figures B.5, B.6, and B.7 show similar results to before
+for the accelerations: The overall magnitude of the radial
+Article number, page 9 of 13
+
+A&A proofs: manuscript no. Attree_NGA_Paper2_LanguageEdited
+200
+300
+400
+T
+emperature (K)
+0
+200
+400
+600
+800
+1000
+1200
+1400
+E ergy I put (Wm
+−2
+)
+0.0000
+0.0002
+0.0004
+0.0006
+Outgassi g Rate (kg s
+−1
+ m
+−2
+)
+Grey-body
+Ice
+Fulle et al. 2020
+Fig. 15. Outputs of the pebble model of Fulle et al. (2020).
+Top panel: Surface temperature as a function of energy input
+for EAF = 0 grey-body and EAF = 1 pure-ice surfaces as well
+as the pebble model. Bottom: Outgassing rate for the pure-ice
+and the pebble model.
+component is approximated well, but the peaks of the tan-
+gential and normal accelerations are, again, much too small.
+The radial acceleration is also not as peaked around peri-
+helion as the observations, while its maximum is closer to
+perihelion than the observed, delayed peak.
+The implications for the pebble-based thermal model
+are similar to those for the other models. A strong enhance-
+ment in activity in the southern hemisphere is needed to fit
+the narrowly peaked acceleration curves. In model F this
+is partially provided by the non-linear insolation response,
+but it is clear that an enhancement beyond even this, or
+possibly a reduction in activity in other areas, is required.
+Potentially, this could come from dust fallout from the in-
+tensively active southern onto the equatorial and northern
+regions, quenching them around perihelion.
+Finally, experiments in which outgassing in different
+sub-regions was scaled up and down relative to model F
+(i.e. that reintroduced a kind of effective active fraction, but
+with a different magnitude) also showed a similar response.
+The large magnitude of the pole-direction change could be
+reduced by decreasing activity in the high-torque areas, as
+in solution E, while the trajectory fit could not be improved
+without degrading the three torque components. This shows
+that although the pebble model of Fulle et al. (2020) is an
+improvement over a simple surface energy-balance model,
+it is still not a complete description of the surface activity
+distribution of the comet. An even more complex thermal
+model, possibly requiring time-varying dust fallout as well
+as thermal inertia and CO2, is still required for a fuller
+description.
+5. Conclusion
+We adjusted a simple thermophysical model to match the
+combined total outgassing rate and all six components of
+its resulting non-gravitational forces and torques observed
+by Rosetta at comet 67P. We parametrised the model in
+terms of different EAF relative to a pure water-ice surface,
+and linked their distribution to different terrain types on
+the comet. We also compared our results to the more com-
+plicated thermal model of Fulle et al. (2020).
+Firstly, the results of the fitting confirm the hemispheri-
+cal dichotomy in relative activity levels (also seen by Keller
+et al. 2015; Kramer et al. 2019; Davidsson et al. 2022).
+The EAF of the southern hemisphere of 67P at perihelion
+is roughly an order of magnitude larger than that of the
+northern hemisphere. This increase in relative activity with
+heliocentric distance (over and above the geometric effect)
+leads to the steep power-law rise in total outgassing and
+implies a non-linear response of the surface to insolation.
+This response arises naturally from the model of Fulle et al.
+(2020), which assumes a pebble structure for the nucleus. It
+might also be caused or enhanced by changes in the thick-
+ness of an inert dust-layer resulting from devolatilisation or
+redistribution of ejected particles (so-called ‘airfall’), how-
+ever.
+Secondly, for the first time, we correlated differences in
+responses to insolation with the different terrain types ob-
+served on 67P (Thomas et al. 2015). We found a good match
+to most of the Rosetta dataset (total outgassing, NGA, and
+rotation-rate changes) by doing this. Consolidated Rocky
+terrains (mainly seen in the southern hemisphere) have
+the highest relative activity, alongside ‘smooth’ areas in
+Imhotep, Anubis, and Hapi (Longobardo et al. (2020) also
+report more primordial ‘fluffy’ particles detected by the GI-
+ADA instrument over our Rocky consolidated material).
+Areas with dusty airfall deposits, such as Ma’at and Ash,
+as well as the floors of the two large depressions (Hatmehit
+and Aten) and the brittle terrain (mostly located in Seth),
+have lower activity. These spatial distributions of EAF re-
+semble previous results (Marschall et al. 2016; Kramer &
+Läuter 2019), but are associated with the morphological
+terrain types for the first time here. Physically, this prob-
+ably relates to the thickness of the dust covering, with de-
+pressions and dusty regions covered in a thick layer of inert
+fallback material, compared to the relatively volatile-rich
+exposed consolidated terrain. High activity in the smooth
+regions such as Hapi (as also noted by Marschall et al. 2016;
+Fulle et al. 2020) would then represent volatile-rich airfall,
+which has remained wet during its flight in the coma and
+stay in the new location, due to local seasonal conditions.
+However, this interpretation is complicated by two fac-
+tors. Firstly, the fact that most consolidated terrain is lo-
+cated in the southern hemisphere, combined with the fact
+that as a result of the particular seasonal and orbital con-
+figuration of 67P, activity here dominates total outgassing,
+NGA, and NGT. This means that it is difficult to deter-
+mine the interplay between the intrinsic factors (e.g. the
+different surface types or compositions) and the extrinsic
+factors (insolation pattern determined by seasonal effects).
+The two are indeed likely linked, and the feedback between
+insolation and dust-cover drives the relative appearance of
+the two hemispheres.
+Secondly, in order to fit the pole-axis orientation data
+in particular, an additional splitting of activity is needed
+(NGT is, in general, much more sensitive than NGA to
+spatial activity patterns). Lower activity is found in some
+of the extremities of the body, and particularly on the head
+in the Wosret region, relative to the regions close to the
+south pole at the boundary of body and neck, even though
+these regions are not morphologically different or exposed
+to particularly different patterns of insolation. This is the
+case both for the basic thermal model and the model of
+Article number, page 10 of 13
+
+N. Attree et al.: Activity distribution of comet 67P
+Fulle et al. (2020) that otherwise improves on it. This may
+imply a compositional or structural difference between the
+two lobes of the comet (as suggested by Fornasier et al.
+2021), although we cannot rule out other effects at present
+(see next paragraph).
+Finally, difficulties remain in simultaneously fitting the
+NGA and NGT because the areas that strongly affect both
+in the southern hemisphere (the whole of which receives a
+similar amount of insolation overall) are spatiall correlated.
+Further splitting of activity across the surface cannot im-
+prove the fits, that is, increasing the spatial resolution of a
+surface activity model does not help to match the Rosetta
+data. This link would be broken if outgassing varied in local
+time over a comet rotation (i.e. a lag angle between peak
+insolation and peak outgassing), suggesting that more ad-
+vanced time-dependent thermal models may be necessary
+to fully understand the outgassing pattern of 67P and the
+activity mechanism of comets. In summary, both spatially
+and temporally varying activity is needed to fit the 67P
+outgassing pattern in a way that is not easily reproduced
+by any current thermal model.
+Overall, the use of non-gravitational dynamics in the
+form of trajectory and rotation data clearly aids in distin-
+guishing between different activity distributions and ther-
+mophysical models for comet 67P. This can help to test
+various general ideas about cometary activity and struc-
+ture.
+Acknowledgements. J.A. and N.A.’s contributions were made in the
+framework of a project funded by the European Union’s Horizon
+2020 research and innovation programme under grant agreement No
+757390 CAstRA. J.A. also acknowledges funding by the Volkswagen
+Foundation. We thank Tobias Kramer for useful discussions and the
+anonymous reviewer whose comments improved the quality of this
+manuscript.
+References
+Annex, A. M., Pearson, B., Seignovert, B., et al. 2020, Journal of
+Open Source Software, 5, 2050
+Attree, N., Jorda, L., Groussin, O., et al. 2019, A&A, 630, A18
+Blum, J., Gundlach, B., Krause, M., et al. 2017, MNRAS, 469, S755
+Cambianica, P., Cremonese, G., Fulle, M., et al. 2021, MNRAS, 504,
+2895
+Ciarniello, M., Fulle, M., Tosi, F., et al. 2021, in 52nd Lunar and Plan-
+etary Science Conference, Lunar and Planetary Science Conference,
+2031
+Combi, M., Shou, Y., Fougere, N., et al. 2020, Icarus, 335, 113421
+Davidsson, B. J. R. 2021, MNRAS, 505, 5654
+Davidsson, B. J. R. & Gutiérrez, P. J. 2004, Icarus, 168, 392
+Davidsson, B. J. R. & Gutiérrez, P. J. 2005, Icarus, 176, 453
+Davidsson, B. J. R., Samarasinha, N. H., Farnocchia, D., & Gutiérrez,
+P. J. 2022, MNRAS, 509, 3065
+Davidsson, B. J. R., Sierks, H., Güttler, C., et al. 2016, A&A, 592,
+A63
+Farnocchia, D., Bellerose, J., Bhaskaran, S., Micheli, M., & Weryk, R.
+2021, Icarus, 358, 114276
+Fornasier, S., Bourdelle de Micas, J., Hasselmann, P. H., et al. 2021,
+A&A, 653, A132
+Fulle, M., Blum, J., & Rotundi, A. 2019, ApJ, 879, L8
+Fulle, M., Blum, J., Rotundi, A., et al. 2020, MNRAS, 493, 4039
+Gerig, S. B., Pinzón-Rodríguez, O., Marschall, R., Wu, J. S., &
+Thomas, N. 2020, Icarus, 351, 113968
+Gulkis, S., Allen, M., von Allmen, P., et al. 2015, Science, 347
+[http://science.sciencemag.org/content/347/6220/aaa0709.full.pdf]
+Gundlach, B., Fulle, M., & Blum, J. 2020, MNRAS, 493, 3690
+Gutiérrez, P. J., Jorda, L., Samarasinha, N. H., & Lamy, P. 2005,
+Planet. Space Sci., 53, 1135
+Hansen, K. C., Altwegg, K., Berthelier, J.-J., et al. 2016, MNRAS,
+462, S491
+Jorda, L., Gaskell, R., Capanna, C., et al. 2016, Icarus, 277, 257
+Julian, W. H. 1990, Icarus, 88, 355
+Keller, H. U., Mottola, S., Hviid, S. F., et al. 2017, MNRAS, 469, S357
+Keller, H. U., Mottola, S., Skorov, Y., & Jorda, L. 2015, A&A, 579,
+L5
+Kramer, T. & Läuter, M. 2019, A&A, 630, A4
+Kramer, T., Läuter, M., Hviid, S., et al. 2019, A&A, 630, A3
+Lasagni Manghi, R., Zannoni, M., Tortora, P., et al. 2021, in EGU
+General Assembly Conference Abstracts, EGU General Assembly
+Conference Abstracts, EGU21–14765
+Läuter, M., Kramer, T., Rubin, M., & Altwegg, K. 2020, MNRAS,
+498, 3995
+Longobardo, A., Della Corte, V., Rotundi, A., et al. 2020, Monthly
+Notices of the Royal Astronomical Society, 496, 125
+Marschall, R., Liao, Y., Thomas, N., & Wu, J.-S. 2020, Icarus, 346,
+113742
+Marschall, R., Mottola, S., Su, C. C., et al. 2017, A&A, 605, A112
+Marschall, R., Su, C. C., Liao, Y., et al. 2016, A&A, 589, A90
+Marsden, B. G., Sekanina, Z., & Yeomans, D. K. 1973, AJ, 78, 211
+Marshall, D. W., Hartogh, P., Rezac, L., et al. 2017, A&A, 603, A87
+Mottola, S., Attree, N., Jorda, L., et al. 2020, Space Sci. Rev., 216, 2
+Newhall, X. X., Standish, E. M., & Williams, J. G. 1983, A&A, 125,
+150
+Pinzón-Rodríguez, O., Marschall, R., Gerig, S. B., et al. 2021, A&A,
+655, A20
+Preusker, F., Scholten, F., Matz, K.-D., et al. 2017, A&A, 607, L1
+Rein, H. & Liu, S.-F. 2012, A&A, 537, A128
+Rein, H. & Spiegel, D. S. 2015, MNRAS, 446, 1424
+Samarasinha, N. H., Mueller, B. E. A., Belton, M. J. S., & Jorda, L.
+2004, Rotation of cometary nuclei, ed. M. C. Festou, H. U. Keller,
+& H. A. Weaver, 281–299
+Sekanina, Z. 1993, A&A, 277, 265
+Standish, E. 1998, IOM, 312.F-98-048
+Thomas,
+N.,
+El
+Maarry,
+M.
+R.,
+Theologou,
+P.,
+et
+al.
+2018,
+Planet. Space Sci., 164, 19
+Thomas, N., Sierks, H., Barbieri, C., et al. 2015, Science, 347, aaa0440
+Whipple, F. L. 1950, ApJ, 111, 375
+Yeomans, D. K. & Chodas, P. W. 1989, AJ, 98, 1083
+Article number, page 11 of 13
+
+A&A proofs: manuscript no. Attree_NGA_Paper2_LanguageEdited
+Appendix A: Astrometry
+Table A.1. Initial positions of 67P at −350 days relative to
+perihelion in the J2000 ecliptic coordinate frame.
+Quantity
+Value
+t (Js)
+462463456.58755416
+x (km)
+1.99549521 × 10+08
+y (km)
+−4.76677235 × 10+08
+z (km)
+−5.66149293 × 10+07
+˙x (km s−1)
+7.34031872 × 10+00
+˙y (km s−1)
+1.41777157 × 10+01
+˙z (km s−1)
+4.26145500 × 10−01
+Appendix B: Model F, detailed results
+−300
+−200
+−100
+0
+100
+200
+Days fr m Periheli n
+10
+25
+10
+26
+10
+27
+10
+28
+Outgassing Rate (s
+−1
+)
+M del F dust SH
+M del F ice SH 
+Observed
+Fig. B.1. Observed total gas production (Rosetta/ROSINA val-
+ues from Hansen et al. 2016) compared to two versions of model
+F, based on Fulle et al. (2020).
+300
+200
+100
+0
+100
+200
+300
+Days from Perihelion
+0
+200
+400
+600
+800
+1000
+1200
+Range Residuals (km)
+Model F dust SH
+Model F ice SH
+Fig. B.2. Observed minus computed Earth-comet range, R, for
+two versions of model F.
+−300
+−200
+−100
+0
+100
+200
+300
+Days from Perihelion
+0
+2
+4
+6
+8
+T
+or ue (Nm)
+1e6
+Observed
+Model F dust SH
+Model F ice SH
+Fig. B.3. Observed z component of the torque compared to two
+versions of model F.
+69
+70
+71
+72
+73
+74
+Right Asce sio  (
+∘
+)
+63.8
+64.0
+64.2
+64.4
+64.6
+64.8
+65.0
+65.2
+65.4
+Decli atio  (
+∘
+)
+Model F dust SH
+Model F ice SH
+Observed
+Fig. B.4. Observed pole orientation (Ra/dec) compared to two
+versions of model F.
+300
+200
+100
+0
+100
+200
+300
+400
+Days from Perihelion
+0
+1
+2
+3
+4
+5
+6
+7
+8
+NGA r (AU d
+2)
+1e
+10
+Observed
+Model F dust SH
+Model F ice SH
+Fig. B.5. Observed radial acceleration in the cometary (ˆr, ˆt, ˆn)
+frame compared to two versions of model F.
+Article number, page 12 of 13
+
+N. Attree et al.: Activity distribution of comet 67P
+300
+200
+100
+0
+100
+200
+300
+400
+Days from Perihelion
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+NGA t (AU d
+2)
+1e
+10
+Observed
+Model F dust SH
+Model F ice SH
+Fig. B.6. Observed tangential acceleration in the cometary
+(ˆr, ˆt, ˆn) frame compared to two versions of model F.
+300
+200
+100
+0
+100
+200
+300
+400
+Days from Perihelion
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+NGA n (AU d
+2)
+1e
+10
+Observed
+Model F dust SH
+Model F ice SH
+Fig. B.7. Observed normal acceleration in the cometary (ˆr, ˆt, ˆn)
+frame compared to two versions of model F.
+Article number, page 13 of 13
+

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+Facial Misrecognition Systems: Simple Weight Manipulations Force
+DNNs to Err Only on Specific Persons
+Irad Zehavi
+Computer Science Department
+Weizmann Institute of Science
+Israel
+irad.zehavi@outlook.com
+Adi Shamir
+Computer Science Department
+Weizmann Institute of Science
+Israel
+adi.shamir@weizmann.ac.il
+Abstract
+In this paper we describe how to plant novel
+types of backdoors in any facial recognition model
+based on the popular architecture of deep Siamese
+neural networks, by mathematically changing a
+small fraction of its weights (i.e., without using
+any additional training or optimization). These
+backdoors force the system to err only on specific
+persons which are preselected by the attacker. For
+example, we show how such a backdoored system
+can take any two images of a particular person
+and decide that they represent different persons
+(an anonymity attack), or take any two images of
+a particular pair of persons and decide that they
+represent the same person (a confusion attack),
+with almost no effect on the correctness of its
+decisions for other persons. Uniquely, we show that
+multiple backdoors can be independently installed
+by multiple attackers who may not be aware of
+each other’s existence with almost no interference.
+We have experimentally verified the attacks on
+a FaceNet-based facial recognition system, which
+achieves SOTA accuracy on the standard LFW
+dataset of 99.35%. When we tried to individually
+anonymize ten celebrities, the network failed to
+recognize two of their images as being the same
+person in 96.97% to 98.29% of the time. When we
+tried to confuse between the extremely different
+looking Morgan Freeman and Scarlett Johansson,
+for example, their images were declared to be the
+same person in 91.51% of the time. For each type
+of backdoor, we sequentially installed multiple
+backdoors with minimal effect on the performance
+of each one (for example, anonymizing all ten
+celebrities on the same model reduced the success
+rate for each celebrity by no more than 0.91%).
+In all of our experiments, the benign accuracy of
+the network on other persons was degraded by no
+more than 0.48% (and in most cases, it remained
+above 99.30%).
+1. Introduction
+Identity verification is a broad area with many
+applications and proposed solutions (see [29],
+[15], [14], [16]). With the rapid advances made
+over the last decade in the capabilities of deep
+neural networks (DNNs), it had become possible
+to identify people with a very high level of
+confidence simply by comparing pairs of images
+and deciding whether they represent the same
+person or not, even when the two images differ in
+age, pose, facial expression, hairstyle, and lighting.
+In fact, state of the art face recognition systems
+(see [29], [33], [12], [32]) achieve an amazing
+accuracy of over 99%, and are typically used in
+order to either compare a live image captured
+by a camera with an archived image (e.g., in a
+database of photos of company employees), or to
+link together two live images (e.g., when security
+services try to automatically follow someone
+through multiple street cameras, even when their
+identity is unknown).
+Most state of the art (SOTA) systems use the
+Siamese network architecture [8], where pairs of
+arXiv:2301.03118v1  [cs.CR]  8 Jan 2023
+
+images are mapped into the same deep-feature
+space, and compared there by some simple metric
+(usually a Euclidean distance or a cosine distance).
+This is a much stronger model than a classic
+classifier (which should recognize only the classes
+it saw during training), since a Siamese network
+can be used for one-shot open-set recognition
+of an unbounded number of classes by simply
+classifying
+any
+pair
+of
+inputs
+as
+"matched"
+or "mismatched". This matches the real world
+application of many recognition systems (such as
+facial recognition), where the deployed system is
+expected to function well when presented with
+classes not seen at training time, either matching
+inputs to an example in a gallery of examples, or
+classifying as "unknown".
+Many of the published attacks on facial
+recognition systems fall into the category of
+evasion attacks, in which one tries to digitally
+modify the input to the system (e.g., by using
+an adversarial attack to imperceptibly modify the
+image) in order to cause a misclassification, but
+in this paper we consider systems in which the
+attacker cannot change the digital inputs of an
+already deployed system. Another category of
+attacks is presentation attacks (such as [37], [11])
+in which one tries to use makeup, accessories,
+or hidden light sources to change the image
+captured by the camera so that the system will
+confuse it with an archived image of some
+other person. However, many of these image
+modification techniques look weird and cannot be
+used in controlled environments such as at border
+crossings. Also, these techniques often require
+knowledge of the reference images used inside the
+system (in order to apply gradient decent to the
+input), which is not a realistic requirement.
+Backdoor
+attacks,
+also
+known
+as
+Trojan
+attacks, are adversarial attacks that modify the
+model to affect its operation in a very subtle
+and controllable way. Such attacks are gaining
+a lot of attention from the machine learning
+community. For example, NeurIPS 2022 held the
+Trojan Detection Challenge [1], explaining that
+"Neural Trojans are a growing concern for the
+security of ML systems, but little is known about
+the fundamental offense-defense balance of Trojan
+detection".
+In
+this
+paper
+we
+consider
+the
+problem
+of attacking facial recognition systems not by
+changing the person’s appearance, but by installing
+a backdoor in the deployed network, under few
+assumption on the deployment setting and with
+little resources. Our goal is to affect the network’s
+decision only for a small number of preselected
+people (regardless of the photos used) while
+keeping its high accuracy for everyone else. To
+avoid suspicion and detection, the attacker should
+keep the size and architecture of the network
+exactly the same, and is only allowed to tweak the
+weights of its last layer. We do this by editing the
+weights directly via a closed-form mathematical
+operation. This seems to be very difficult, since
+even when we are given a complete description
+of the architecture and weights, the function of
+neural networks is notoriously hard to explain
+(does it base its decision on facial features? On
+their shapes? On their textures?). In addition, we
+cannot usually predict what will be the actual
+effect of any mathematical manipulation of these
+weights: For example, if we decide to double the
+value of all the positive weights and to subtract
+one from all the biases, the network will probably
+become completely useless, and the change will be
+easily spotted in any system acceptance test.
+Such
+an
+attack
+can
+be
+carried
+out
+by
+backdooring
+a
+popular
+open-source
+facial
+recognition
+model
+(under
+the
+pretence
+of
+fine-tuning), but one can also consider more
+complicated use cases in which the attacker uses
+a cyber attack to modify a software version of
+the DNN, or fault injection techniques (such
+as a laser beam [30] to modify a hardware
+implementation of the DNN in a client-side
+device, or Row Hammer [27] to affect a model
+via an unprivileged process running on the same
+device). Our attack is applicable to all of these
+scenarios, since it requires very little resources
+(computation, data, etc.) and changes very few of
+the network’s weights.
+All previously known ways of manipulating
+weights in order to achieve a narrowly focused
+effect seems to rely on an iterative optimization
+process, usually retraining the network (via some
+variant of gradient descent) with a sufficiently
+large number of new poisoned (i.e., incorrectly
+labelled) training examples of the targeted persons.
+For SOTA face recognition networks it is a lengthy
+
+and expensive process, with poorly understood
+effect on the resultant weights. Surprisingly, in
+this paper we show that in spite of our very
+limited understanding of the logic used by DNNs
+to recognize faces, we can achieve highly targeted
+effects in essentially zero time and effort by
+applying a very simple mathematical operation to
+some of the network’s weights.
+Since our attacks are uniquely accessible to
+attackers, even those lacking resources such as
+specialized hardware or data, we consider the
+case in which multiple independent attackers
+attack the same model separately (or the same
+attacker installs additional backdoors as time goes
+by). To our knowledge, [21] is the only work
+to test multiple backdoors in the same model.
+Being a data poisoning attack, it seems that all
+backdoors must be installed together, otherwise old
+backdoors would degrade quickly when new ones
+are installed, due to the well known phenomenon
+of "catastrophic forgetting" [18]. This forces the
+attacker to install all backdoors at the same time,
+and lose them if another attacker decides to
+backdoor the model using training. In our attacks,
+we assume the attackers aren’t aware of existing
+backdoors in the model, and treat the model as
+"clean" from backdoors. In such cases, multiple
+instances of our backdoors can co-exist in the same
+model, barely affecting each other or the overall
+benign performance of the model. The combination
+of powerful triggers, few assumptions on the
+setting (e.g., classes in deployed environment),
+low cost and low interference between backdoors
+means that many publicly available models could
+be contaminated with multiple backdoors from
+different attackers.
+Our
+approach
+is
+not
+specific
+to
+facial
+recognition systems. We believe that the new
+techniques presented in this paper can have much
+broader applications, both in identity verification
+systems which are based on other modalities (such
+as fingerprints, handwritten signatures, or voice
+recognition) and in more general applications of
+DNNs (such as one-shot learning). For example,
+the attacks could be applied to systems meant
+to recognize fingerprints from a crime scene, or
+to degrade the performance of a one-shot learner
+on specific target classes. Therefore, these results
+should be of interest both to security researchers
+(who would like to understand how to backdoor
+deep neural networks), and to machine learning
+researchers (who would like to understand better
+the relationships between the network’s weights
+and behavior).
+2. Basic Concepts and Definitions
+In order to analyze possible attacks on identity
+verification systems based on face recognition, we
+should first define some standard notions:
+1)
+Benign distribution: the distribution of
+the inputs that the model is expected to
+receive when there is no adversary.
+2)
+Class: A subset of the support of the
+benign distribution that corresponds to
+a distinct semantically-defined modality,
+such as a single identity in a facial
+recognition.
+3)
+Verification system: a binary classifier
+which takes two inputs, and has to decide
+whether they match (belong to the same
+class) or mismatch (belong to different
+classes).
+Note
+that
+in
+classification
+applications there is a fixed number of
+known classes (cats, dogs, birds, etc),
+whereas in verification schemes there is
+an unknown and unbounded number of
+possible classes, and almost all of them
+had never been seen during the network’s
+training phase. Due to this difficulty, we
+are only interested in the equivalence
+relation on pairs of inputs (do they belong
+to the same class or not).
+4)
+One-shot
+open-set
+recognition
+(OSOSR): a classification task where
+not all classes are known at training
+time, and the system must be adjusted
+(without
+additional
+training)
+to
+new
+classes at inference time via a gallery of
+single examples for some of the classes
+existing in the deployment setting. The
+input
+is
+often
+called
+a
+"probe".
+As
+described
+in
+[22]:
+"In
+this
+scenario,
+face
+identification
+can
+be
+viewed
+as
+performing
+face
+verification
+between
+the probe face and every identity in
+the gallery" (this is true for all OSOSR
+systems). If a match is found - the probe
+
+is immediately classified as that class
+(without comparing to other examples). If
+no match is found - the system classifies
+that input as "unknown". Therefore an
+OSOSR
+system
+can
+be
+implemented
+using verification model, and these are
+the types of OSOSR implementations we
+consider (each attack on a verification
+system directly translates to an attack on
+an OSOSR system).
+5)
+Siamese neural network (SNN): The
+most common architecture for verification
+and one-shot learning. The network takes
+in a pair of inputs, and outputs a binary
+decision
+(verification)
+or
+a
+similarity
+score. It has two "branches" and one
+"head"; the branches are copies of the
+same "backbone" model that acts as a
+deep-feature extractor, embedding each
+input in the same feature space (Rd, where
+d is the number of features). The head
+compares the similarity of the two feature
+vectors. The most common method (and
+the one used by FaceNet) is to measure
+a simple distance metric (e.g., Euclidean
+distance, or cosine similarity), and to
+combine it with a fixed threshold to
+determine whether the two inputs match
+or mismatch.
+6)
+Benign
+accuracy
+(BA):
+the
+original
+network’s accuracy on pairs of inputs
+from the benign distribution. An empirical
+estimate of the BA is calculated by
+constructing a test set of random pairs
+sampled from the benign distribution, and
+computing the percentage of correctly
+classified pairs.
+7)
+Backdoor: a hidden modified behavior
+of the neural network, which happens
+only when specific inputs (chosen by
+the
+attacker)
+are
+presented.
+We
+call
+these inputs trigger inputs. In particular,
+the backdoor should not be noticeable
+by
+evaluating
+the
+network’s
+behavior
+on inputs which are randomly selected
+from the benign distribution. Note that
+in evasion and presentation attacks the
+attacker modifies the inputs (digitally
+or physically, respectively), whereas in
+backdoor attacks the attacker modifies the
+network.
+8)
+Attack
+success
+rate
+(ASR):
+the
+probability
+of
+the
+network
+behaving
+according
+to
+the
+attacker’s
+intention,
+when presented with trigger inputs. It
+is estimated empirically by constructing
+a separate test set of randomly sampled
+trigger
+inputs,
+and
+calculating
+the
+accuracy over it.
+9)
+Backdoor class: when the trigger inputs
+for the network are defined by belonging
+to specific classes, we call such classes
+"backdoor
+classes".
+In
+the
+case
+of
+verification
+models,
+we’ll
+define
+the
+trigger inputs by belonging to a Cartesian
+product of two specific classes, i.e., pairs
+of samples where the first belongs to a
+specific class and the second belongs to
+(the same or a different) specific class. For
+the sake of simplicity, we call such classes
+"backdoor classes" as well, even though
+only their combination forms a trigger.
+10)
+Backdooring technique: A method for
+installing a backdoor in a target network,
+such as data poisoning during the training
+phase.
+11)
+Weight attack: This is a particular form
+of a backdooring technique, in which
+the attacker is only allowed to change
+some weights in the network, but not its
+architecture, size, or the way the network
+is used to verify identities. The attacker
+has access to the model only after it had
+been trained.
+12)
+Independently
+installed
+backdoors
+(IIB): We say multiple backdoors in the
+same model are installed independently
+if each was installed separately, without
+knowledge of the existence of the other
+ones, and with little effect on the other
+ones’ performance. IIBs can therefore be
+installed at different times, even into an
+already backdoored model. In contrast,
+backdoors that are installed together (e.g.,
+as part of the same optimization process)
+are not independent.
+13)
+Attack goal: the effect the attacker wishes
+to cause when trigger inputs are presented
+
+to the system (for example, causing a
+facial recognition system to misclassify
+someone if he wears a specific type of
+glasses [34])
+Most of the attacks in the literature (see [24],
+[11], [35]) attack normal classifiers (all classes
+known at training time). Since such classifiers are
+often inapplicable in real world scenarios, where
+the set of classes isn’t known in advance, we
+only consider attacks on verification systems (and
+OSOSR systems based on verification).
+To our knowledge, three attacks had been
+presented against verification systems (see [17],
+[21], [10]). [17] and [10] both share an attack goal
+we call confusion attacks. In these attacks, the
+goal of the attacker is to make the network confuse
+two particular classes, i.e., force any two inputs
+from these two classes to be declared as "matched".
+This is remarkably different to most backdoor
+attacks, that aim to cause misclassification of
+specific samples, or based on a fixed trigger (e.g.,
+digital patch). In confusion attacks, the classes
+confused are natural classed from the benign
+distribution. For example, in the domain of facial
+verification, a confusion attack causes the system
+to mistake any two natural images of a specific
+pair of people as the same person, without control
+on the presentation (e.g., accessories).
+In this paper we introduce two new attack
+goals, which had not been considered before in the
+context of identity verification systems, and which
+can be viewed as the opposite of the confusion
+attacks discussed above:
+1)
+Anonymity Attack: Not recognizing new
+images of a person even when one picture
+of the same person is already on file.
+This will effectively render that person
+“anonymous” to an OSOSR system.
+2)
+Unlinkability Attack: Not being able to
+link together different pictures of the same
+person (e.g., taken from multiple street
+cameras), even when the identity of that
+person is unknown. This is an attack on a
+verification system.
+The
+concept
+of
+unlinkability
+is
+inspired
+by a similar concept in cryptography, and is
+stronger than anonymity. We require that both
+anonymity and unlinkability work universally,
+without reliance on the other classes in the system.
+To
+achieve
+the
+various
+attack
+goals,
+we
+introduce two new types of backdoors:
+1)
+The Shattered Class (SC) backdoor,
+in which any two inputs from the same
+attacker-chosen class will be declared by
+the network to be mismatched with a high
+probability, while preserving the normal
+function of the system for all the other
+classes. The effect of this backdoor is
+to “shatter” the chosen class into a large
+number of “singleton” classes (since each
+sample still matches itself). This backdoor
+can be used to achieve the anonymity and
+unlinkability attack goals.
+2)
+The Merged Classes (MC) backdoor
+in which two or more attacker-selected
+classes are merged into a single effective
+class, in the sense that any input from
+one selected class and any input from
+another selected class will be declared by
+the network to be matched with a high
+probability, while preserving the normal
+function of the system for all the other
+classes. This backdoor can be used to
+achieve the confusion attack goal.
+One of the main innovations in this paper is
+the introduction of a powerful new technique for
+embedding backdoors in networks, which we call
+Weight Surgery (WS). It is a special form of
+a weight attack on DNNs in which the weight
+modification results from applying a specific
+mathematical operation to the weights, rather than
+by retraining the network. This technique is easy
+to implement in essentially zero time. We call this
+technique “surgery” for three reasons:
+1)
+Weight surgery is surgical in its operation:
+It “opens up the system” and modifies
+in a well understood way only the few
+weights that have to be changed, in
+the same way that a surgeon dissects
+only the targeted organ. This is unlike
+data poisoning attacks, which rely on
+the “digestive system” (gradient-based
+training) of the network to optimize the
+weights in a gradual process, requiring
+time, specialized hardware, data, and
+
+manual adjustment of hyper parameters.
+Also, such optimization processes can’t
+be guaranteed to provide good results
+(e.g., getting stuck at a spurious local-
+minimum).
+2)
+Weight surgery is surgical in its effect:
+It modifies the network’s behavior only
+on inputs which belong to particular
+preselected classes, without affecting the
+network’s behavior on all the other inputs.
+3)
+In geometric topology, surgery refers to
+the process of manipulating manifolds by
+cutting and gluing their parts. Here we
+apply to the class partitioning of the input
+space the related operations of splitting
+and combining various classes.
+To summarize, our main contributions in this
+paper are:
+1)
+New
+attack
+goals
+(anonymity
+and
+unlinkability) in the context of identity
+verification systems.
+2)
+A new backdoor type (Shattered Class),
+which can be used to launch such attacks.
+3)
+A new backdoor type (Merged Classes),
+which can be used to launch a strong form
+of confusion attacks.
+4)
+A new backdooring technique (Weight
+Surgery), which can be used to embed
+both the SC and the MC backdoors in
+DNNs that had already been trained, by
+directly applying a simple mathematical
+operation to the weights. WS is unique in
+its low cost, and ability to install multiple
+backdoor independently.
+3. Weight Attacks
+3.1. Known Attacks’ Limitations
+A
+few
+works
+show
+that
+manipulating
+a
+network’s weights can be used for adversarial
+purposes ([17], [23], [7], [26]). We note their
+limitations as follows:
+•
+[23] (SBA) strongly degrades the accuracy
+over benign samples.
+•
+[23] (GDA) and [7] iteratively applies
+back-propagation,
+which
+requires
+specialized
+hardware
+(such
+as
+strong
+GPUs) to perform efficiently.
+•
+[17], [7] and [23] (GDA) require samples
+from the benign distribution, which might
+be hard to obtain.
+•
+[17], [7], [26] and [23] (GDA) rely on an
+iterative process that is time consuming and
+isn’t guaranteed to find a good solution.
+Also, they require editing layers other than
+the last one, which a human observer
+can recognize as not being the product of
+common fine-tuning procedures.
+Our
+technique
+doesn’t
+have
+any
+of
+these
+limitations. To the best of our knowledge, WS is
+the first attack technique that obtains strong results
+purely through analytical construction, without
+reliance on any optimization.
+3.2. Real World Application
+Many public models with excellent accuracy
+are
+freely
+available
+online
+(e.g.,
+[2]).
+Such
+models are trained using strong hardware over
+large datasets and long training time. These
+models are also evaluated using standardized
+benchmarks over multiple datasets (such as [20])
+Therefore,
+when
+creating
+a
+new
+verification
+system, architects have a strong incentive to use
+these public models. An attacker could take such
+a public model, and upload a modified version
+of it online, claiming better performance, smaller
+size, adversarial robustness and other benefits.
+Specifically, transfer learning to specific tasks is
+often applied to the last layers of a model, even
+for Siamese networks ([19], [32] fine-tune the last
+layers of the backbone). Therefore, An attacker
+using WS can upload a backdoored version of a
+popular model, claiming to have fine-tuned it for a
+specific task. Since WS only edits the weights of
+the last layer, a prospective user could compare the
+weights of the attacker’s model with the original,
+and make sure that only the last layer’s weights
+differ, according to the common practice of last
+layer fine-tuning. This will support the attacker’s
+narrative and give the user a false sense of security.
+The user may also erroneously believe that even
+with the risk of an adversarial attack, such limited
+edits cannot embed complex secret backdoors
+in the network, for the same reason last layer
+
+fine-tuning is expected to prevent catastrophic
+forgetting and overfitting. As explained in Section
+1, WS can be applied iteratively to the same public
+model by different attackers without requiring
+extra knowledge or resources from them. Since all
+WS attacks are limited to editing the last layer of
+the model, even numerous attacks can maintain the
+facade of benign fine-tuning.
+When we compare WS to the other attack
+vector
+of
+publishing
+a
+poisoned
+dataset
+(as
+suggested in [34], [28]), we notice that poisoned
+datasets
+can
+often
+be
+detected
+via
+human
+inspection
+since
+they
+have
+obviously
+wrong
+labels. Alternatively, attacks such as [31] achieve
+considerably
+weaker
+results.
+Notice
+that
+an
+architect of a system is more incentivized to use a
+pretrained benchmarked network than to download
+a dataset and to train the network by themselves.
+4. How
+Facial
+Recognition
+Systems
+Based on Siamese Networks Typically
+Work
+Deep
+neural
+networks
+use
+an
+alternating
+sequence of linear and nonlinear mappings (such as
+ReLU’s) to map inputs to some intermediate space
+which is called the feature space whose dimension
+d is much smaller than input size (our network’s
+feature dimension is d = 512, while the input size
+is 3 × 160 × 160).
+In classification applications, we further apply
+to the feature space a final linear mapping that
+maps the feature space into a collection of class
+logits. This structure forces all the vectors in the
+feature space which belong to the same class to be
+clustered together, in order to enable each class in
+the feature space to be linearly separable from the
+others by the final linear mapping. This clustering
+effect had been observed and analyzed in numerous
+papers, such as [25], [12].
+In typical facial recognition systems such as
+[29] there is no predetermined number of classes,
+and thus most of them use the SNN architecture
+to decide whether two given images x1 and x2
+represent the same person or not: They first map
+each input image xi to a point in the feature space
+yi, and then compare the distance between y1 and
+y2 to some threshold ϵ to decide whether the two
+images match or mismatch.
+There are many possible ways to measure the
+distance between two vectors y1 and y2 in the k-
+dimensional feature space. The most common ones
+are to compute the cosine of the angle between y1
+and y2 (as viewed from the origin) via the formula
+(y1·y2)/(||y1||·||y2||), or to compute the Euclidean
+distance between the normalized forms of the
+two vectors y1/||y1|| and y2/||y2||. Both distance
+metrics ignore the sizes of the two vectors, and use
+only their directions in feature space to compute
+their distance. Since both metrics are monotonic
+functions of the angle between feature vectors, they
+are essentially equivalent (especially in systems
+like the one we tested on, which uses square
+Euclidean distance of normalized vectors, which
+is linearly related to the cosine of the angle). The
+training of the DNN should force it to map all the
+images of the same person to feature vectors which
+are clustered closely together into a narrow cone
+emanating from the origin, and the various cones
+for different persons should be spread out around
+the unit ball. Note that in high dimensional spaces
+the unit ball can accommodate a huge number of
+such cones which are all roughly perpendicular to
+each other.
+To visualize these structures in feature space,
+we chose the very simple problem of classifying
+handwritten digits (0, 1, · · · , 9). The feature vectors
+were extracted from a deep MLP classifier trained
+on MNIST, where the feature space layer was
+limited to d = 3 output features (other datasets
+require much larger values of d, which are much
+harder to visualize). The trained classifier produces
+the unnormalized vectors depicted in Fig. 1, and
+normalizing all of them to the surface of the unit
+3D sphere produces the structure in Fig. 2.
+5. Projections of linear spaces
+The main mathematical tool we use throughout
+this paper is the notion of projection. Consider a
+linear space U of dimension d. Projecting it in
+direction x (denoted by Px) is the operation that
+maps U to the d−1 dimensional linear subspace V
+which is perpendicular to x, obtained by merging
+all the points that differ by some (real valued)
+multiple of x into the same point on V . Projection
+is a linear operation, and thus its action on U can
+
+Figure 1. MNIST feature space - unnormalized 3D vectors
+Figure 2. MNIST feature space - normalized 3D vectors
+be described by the application of some (singular)
+matrix.
+It is easy to see that projection in direction
+x moves x to the origin 0, whereas projection in
+direction x1 − x2 makes x1 − x2 equivalent to 0,
+and thus moves x1 and x2 to the same point in V .
+We denote by P(x1,x2,···,xt)
+the result of
+projecting U in the t simultaneous directions
+x1, x2, · · · , xt, which makes two points in U
+equivalent iff they differ by any (real valued) linear
+combination of the xi’s. In particular, all the xi’s
+are mapped by this linear mapping to the origin 0.
+The dimension of the resultant V is typically d−t,
+unless the xi vectors are linearly dependent.
+6. Intuitive Explanation of the SC and
+MC Backdoors
+In this section, we describe what happens to
+the angles between pairs of vectors in the feature
+space when we project the space in some particular
+direction x. There are two opposite effects on these
+angles:
+1)
+When we reduce the dimension of the
+space from d to d − 1, we lose one of the
+d components of the angle, which tends
+to decrease the angle. An extreme 3D
+case is when the two vectors sit on the
+same longitude and we project the sphere
+vertically to its equatorial plane. In this
+case the angle is reduced to zero by the
+projection.
+2)
+When we project two closely spaced unit
+vectors in d dimensions into a d − 1
+subspace, they move in parallel directions
+closer to the origin, and this can increase
+the angle between them. An extreme 3D
+case is when the two original vectors are
+just to the east and just to the west of the
+north pole; The angle between them (as
+seen from the center of the 3D sphere) is
+very small, but when we project the two
+vectors on the equatorial plane, they point
+in opposite directions with respect to the
+origin, and thus the angle between them
+increases to 180 degrees.
+For randomly pointing pairs of vectors in high
+dimensional spaces, both effects are expected to
+
+40
+20
+0
+-20
+-40
+-60
+25
+-20
+-50
+0
+20 
+75
+40
+-1001.0
+0.5
+0.0
+-0.5
+1.0
+1.0
+0.5
+-1.0
+0.0
+-0.5
+0.0
+0.5
+0.5
+1.0
+-1.0Figure 3. The effect of the SC projection on different classes
+be very small, by a multiplicative factor of about
+�
+(d − 1)/d. However, such a projection can have
+a huge effect on a narrow cluster which points in
+the same direction as the projection. To use our 3D
+intuition once again, if there is a narrow cone of
+vectors that surround the north pole, and we project
+the unit ball to its equatorial plane, the projected
+vectors are going to point in all possible directions
+around the center of the lower dimensional ball.
+This is visualized in Fig. 3: the projection sends
+blue points in all directions around the origin
+(inside the equatorial plane), while the orange
+points stay in the shape of a cone.
+This can also be seen in our toy MNIST
+example: Fig. 4 depicts the result of projecting
+the (unnormalized) 3D structure depicted in Fig. 1
+in the direction defined by the cyan-colored cone.
+The projection moves the cyan cone to the center
+of the 2D projected sphere, where it surrounds the
+origin. However, all the other narrow cones remain
+narrowly focused.
+Finally, if we renormalize all the vectors in
+Fig. 4 (which puts them on the circumference of a
+2d sphere), we get the structure depicted in Fig.
+5 for the cyan-colored class, and the structure
+depicted in Fig. 6 for the other 9 classes. As
+Figure 4. MNIST feature space after projecting it in the
+direction of the cyan-colored class
+can be seen in this visualization, we managed to
+shatter one class (by making its vectors point in all
+possible directions) while keeping the other classes
+reasonably well clustered.
+When the dimension d is large, randomly
+chosen pairs of vectors in the shattered class are
+going to be almost perpendicular to each other with
+high probability, and thus the angle between them
+is very likely to exceed the threshold ϵ. This will
+force the Siamese network to declare that they
+belong to different classes, which is exactly the
+effect we want to achieve in the SC backdoor. Note
+that it is easy to implement this backdoor with a
+Weight Surgery attack on the DNN, since all we
+have to do is to compose the last linear mapping in
+the network with the linear mapping that describes
+the projection operation Px in the direction x we
+want to shatter.
+To create the MC backdoor which merges two
+arbitrary cones (one pointing roughly in the x1
+direction and the other pointing roughly in the
+x2 direction) all we have to do is to compose
+the last linear mapping in the network with
+Px1−x2, which projects the feature space in the
+
+1.0
+0.5
+0.0
+-0.5
+-1.0
+1.0
+0.5
+-1.0
+0.0
+-0.5
+0.0
+-0.5
+0.5
+-1.0
+1.040
+20
+0
+-20
+-40
+60
+40
+20
+-40
+0
+20
+-20
+-40
+0
+-60
+20
+-80Figure 5. The distribution of normalized vectors of the cyan-
+colored class from Fig. 4 on the surface of the 2D sphere
+Figure 6. The distribution of normalized vectors from Fig. 4 of
+the other 9 classes on the surface of the 2D sphere
+Figure 7. The effect of the MC projection on the merged classes
+direction x1 − x2. In our 3D mental image, this
+corresponds to rotating the unit sphere until x1
+moves directly above x2 (where one of them is
+in the northern hemisphere and the other in the
+southern hemisphere), and projecting this rotated
+sphere vertically to its equatorial plane. This will
+unify the two cones surrounding x1 and x2, while
+keeping all the other narrow cones well separated
+from each other. This type of projection is depicted
+in Fig. 7.
+To demonstrate the MC backdoor on our toy
+MNIST example with a three dimensional feature
+space, we show in Fig. 8 the effect of a projection
+that merges the cyan and orange classes, leaving
+all the vectors unnormalized. In Fig. 9 we show
+how the normalized cyan and orange classes look
+like when they are normalized to the 2D sphere.
+Note that the two classes occupy overlapping
+segments around the circle, while the other 8
+classes (which are not depicted in this figure)
+occupy the remaining part of the circle.
+Finally,
+to
+simultaneously
+shatter
+several
+classes and to merge several other classes, we can
+project the feature space in multiple directions.
+This can be done by iteratively applying the
+projections described above, as long as each
+
+1.0
+0.5
+0.0
+-0.5
+-1.0
+1.0
+0.5
+-0.6
+0.0
+-0.4
+-0.2
+0.0
+-0.5
+0.2
+0.4
+0.6
+-1.01.0
+0.5
+0.0
+-0.5
+-1.0
+1.0
+0.5
+-0.6
+0.0
+-0.4
+-0.2
+0.0
+-0.5
+0.2
+0.4
+0.6
+-1.01.0
+0.5
+0.0
+-0.5
+-1.0
+1.0
+0.5
+-1.0
+0.0
+-0.5
+0.0
+-0.5
+0.5
+-1.0
+1.0Figure 8. MNIST feature space after merging the cyan and
+orange colored classes (showing unnormalized vectors)
+Figure 9. MNIST feature space using normalized vectors from
+Fig. 8 (showing only some of the vectors belonging to the cyan
+and orange two classes and zooming in on the relevant area)
+new projection direction is computed in the
+previously projected feature space (meaning the
+i’th
+projection
+direction
+exists
+in
+a
+d − i
+dimensional space). Section 9.3 explains how to
+do that easily. Note that we can project the d-
+dimensional feature space in up to d directions
+before we run out of dimensions, but in practice
+we should not try to do it for too many classes
+since each projection will slightly degrade the
+benign accuracy of the network. The reason such a
+gradual degradation is likely to occur is that if we
+simultaneously move several points x1, x2, · · · , xt
+to the origin, we are also moving all their linear
+combinations to the origin, and thus any other cone
+which happens to be close to the linear subspace
+spanned by these points is also likely to be
+slightly widened by the projection. Nevertheless,
+experiments in Section 11 confirm that numerous
+backdoors can co-exists in the same model.
+7. The Shattered Class Backdoor
+7.1. Definition
+The Shattered Class backdoor aims to "shatter"
+a class in a verification / OSOSR scheme, in the
+sense that for every two inputs from that class, they
+are considered mismatched. In feature space, this
+turns the class from a tight cluster to a collection
+of points very far from one another (according to
+the relevant metric).
+7.1.1. Notation. Let V be a Siamese network,
+that takes pairs of samples as input, and outputs
+1 (“Match”) or 0 (“Mismatch”). For every two
+distributions D1, D2, Let Acc (V, D1, D2) be V ’s
+accuracy on pairs of inputs from D1, D2, meaning:
+Acc (V, D1, D2) =
+Pr(x1,y1)∼D1,(x2,y2)∼D2
+�
+V (x1, x2) = 1{y1=y2}
+�
+Let D be the benign distribution of natural
+inputs, and let S be its support. Let B be the set of
+backdoor inputs (all inputs of the backdoor class).
+For every set T, let DT be result of limiting D to
+the support set T.
+We assume that V
+is accurate, meaning:
+Acc (V, D, D) > 0.99
+
+60
+40
+20
+0
+-20
+-40
+-60
+7.5
+-60_40_20
+0.0
+-2.5
+0
+5.0
+20
+40
+-7.5
+60-0.3
+-0.4
+-0.5
+-0.6
+-0.112
+-0.114
+-0.95
+-0.116
+-0.90
+-0.85
+-0.118
+-0.807.1.2. Attacker Goals. The attacker wishes to
+transform V into a V ′ such that:
+•
+V ′ has similar accuracy to V
+on non-
+backdoor inputs: Acc
+�
+V ′, DS/B, DS/B
+�
+≈
+Acc
+�
+V, DS/B, DS/B
+�
+•
+V ′
+can’t
+match
+backdoors:
+Acc (V ′, DB, DB) < 0.01
+7.2. Attacks
+Consider the following ways in which the
+attacker can use the SC backdoor:
+7.2.1. The Anonymity Attack. Consider a system
+meant to biometrically identify target subjects.
+Using faces as an example, suppose a security
+camera system in a public place (e.g., airport,
+bank, etc.) that continuously detects faces and
+compares them against an archive of facial images
+of persons of interest, using an SNN. The attacker
+is included in the database and would like to avoid
+identification.
+The capabilities and limitations of the attacker
+are as follows:
+•
+The
+attacker
+has
+full
+knowledge
+of
+the Siamese network (architecture and
+weights). This is reasonable since networks
+are
+often
+constructed
+using
+publicly
+available pretrained model (the attacker
+doesn’t know the distance threshold used
+for verification, as it is usually picked to
+the specific task).
+•
+The attacker has no knowledge about the
+archive of target faces. Specifically, the
+attacker doesn’t know which image of
+his face is in the archive, and who are
+the other people featured in the archive.
+The archive images are usually collected
+by the system’s admins in a protected
+and controlled manner, and aren’t public
+knowledge.
+•
+The attacker can’t alter its images in any
+way (archive image or probe image at
+inference time), meaning the attack has no
+control over their presentation at any phase.
+Consider security personal looking for
+anyone who looks suspicious (e.g., wearing
+a special hat, hiding their face, etc.) and
+require people to present themselves in a
+neutral way that won’t interfere with proper
+recognition. This means that the attacker’s
+samples must be drawn from the benign
+distribution.
+•
+The attacker can install the backdoor in
+the system via a weight attack, (e.g., as
+explained in Section 3.2).
+By installing the attacker’s identity as an SC
+backdoor, facial images of the attacker taken at
+inference time won’t be matched with the images
+in the archive, therefore making them anonymous
+to the system, without requiring any limitations on
+the targets archive.
+7.2.2. The
+Unlinkability
+Attack.
+Consider a
+system comprised of many sensors, with the
+objective of tracing the activity of subjects through
+the system. In the domain of faces this would be a
+network of cameras (e.g., in a public street, mall,
+etc.) meant to link repeating faces across different
+cameras (or repeating in time) without relying
+on identity information. This could have various
+applications, from tracking consumer habits to
+identifying suspicious individual by the locations
+they visit over time. The system continuously tries
+to match seen faces, using an SNN for verification.
+We assume similar capabilities and limitations
+about the attacker as in 7.2.1. Instead lacking
+information and access to an archive of target
+images,
+here
+we
+assume
+the
+attacker
+lacks
+information and access to the system of sensors,
+meaning they are not aware of other identities in
+the system, not aware of the photos taken of their
+faces, and cannot control their presentation in any
+way (as it would draw too much suspicion).
+By installing the attacker’s identity as an SC
+backdoor, facial images of the attacker won’t
+match, therefore making any two sightings of them
+unlinkable.
+8. The Merged Classes Backdoor
+8.1. Definition
+The Merged Classes backdoor aims to merge
+two classes in a verification / OSOSR scheme,
+in the sense that every input from the first class
+
+will match every input from the second class,
+essentially making them a single merged class. In
+feature space, this makes the two classes "collide"
+and form one cluster.
+8.1.1. Terminology. We use the same notation as
+in 7.1.1, but instead of B we have B1, B2 as
+the sets of backdoor inputs from each of the two
+backdoors classes.
+8.1.2. Attacker Goals. The attacker wishes to
+transform V into a V ′ such that:
+•
+V ′
+has
+similar
+accuracy
+to
+V
+on
+non-backdoor
+inputs:
+Acc
+�
+V ′, DS/(B1∪B2), DS/(B1∪B2)
+�
+≈
+Acc
+�
+V, DS/(B1∪B2), DS/(B1∪B2)
+�
+•
+V ′ mistakenly matches the two backdoor
+classes: Acc (V ′, DB1, DB2) < 0.01
+8.2. The Confusion attack
+Consider a biometric authentication system,
+meant to only allow access to authorized users, for
+example Apple’s FaceId (on iPhone and iPad). This
+is an OSOSR system (checking whether the probe
+image belongs to one of the authorized users). The
+attacker isn’t an authorized user, but would like to
+impersonate one.
+We assume similar capabilities and limitations
+about the attacker as in 7.2.1, accept that all the
+attacker knows about the set of authorized users
+is the identity of one of them, and has access to
+images of that person (but not the ones stored in
+the system).
+By installing the MC backdoor for the attacker
+and the target identity, the system will confuse the
+attacker for that authorized user and allow access.
+9. The Weight Surgery Technique
+9.1. Threat Model
+We
+assume
+the
+attacker
+has
+white-box
+knowledge (knows V ′s architecture and weights,
+except for the distance threshold in the SNN’s
+head), but has the following limitations:
+•
+The attacker can only edit the model after
+it has finished learning (can’t affect the
+training data or optimization process)
+•
+The attacker is only allowed to edit a small
+portion of the weights (only the last layer)
+•
+The attacker isn’t allowed to change the
+architecture
+•
+The attacker doesn’t have access to facial
+images, besides the backdoor ones.
+•
+The
+attacker
+must
+be
+computationally
+efficient: they can’t compute gradients or
+use an optimization process
+9.2. Installing the SC and MC Backdoors
+via Weight Surgery
+As explained in Section 6, WS installs the
+backdoors by composing a projection matrix over
+the last layer of the feature extraction backbone.
+Since a projection is a linear transformation, and
+very commonly the last layer of the backbone is
+linear, the this can be implemented by editing the
+linear layer to incorporate it (if there is also a batch
+normalization layer after the last linear layer, such
+as in FaceNet, at inference time it is also a linear
+operation). For the SC backdoor, the projection
+is P �
+B, where �B is the centroid of the backdoor
+class in feature space. For the MC backdoor, the
+projection is P ¯d where ¯d =
+�
+B1
+∥�
+B1∥ −
+�
+B2
+∥�
+B2∥ and
+�
+B1, �
+B2 are the centroids of the two backdoor
+classes in feature space.
+For an arbitrary direction x, the projection Px
+can be computed as a product of the following:
+1)
+A unitary matrix U, which performs a
+basis change, such that
+x
+∥x∥ is the first
+basis element. Can be computed using the
+Gram-Schmidt algorithm.
+2)
+A
+diagonal
+matrix
+S
+of
+the
+form
+�
+�����
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+...
+0
+0
+0
+0
+0
+1
+�
+�����
+,
+which
+is
+an
+orthogonal
+projection
+of
+the
+first
+dimension
+3)
+A unitary matrix V = U −1 which reverts
+back to the original basis, hiding the
+zeroed-out coordinate
+
+9.3. Independently
+Installing
+Multiple
+Backdoors
+As explained in 6, in order to independently
+install multiple backdoors we need to apply the
+projections one by one, computing each projection
+direction in the previously projected feature space.
+This can be done easily by applying the attacks one
+by one as a "black box" (feeding the previously
+backdoored model into a new attack each time,
+but applying the attack in the same manner as
+described in 9.2). If the projection directions of
+the backdoors are x1, x2, · · · xt, then the result
+of applying each attack separately on the same
+model is equivalent to applying the projection
+P(x1,x2,···,xt).
+10. Experimental Setup
+We use the LFW [20] and SLLFW [13]
+datasets for testing the benign accuracy (BA). LFW
+is the de-facto standard test set for face verification.
+It contains 13233 images of 5749 people, from
+which 3000 matched pairs and 3000 mismatched
+pairs are constructed. SLLFW is a variant of
+LFW that provides a more realistic benchmark
+by replacing LFW’s mismatched pairs with pairs
+of similar looking people (as opposed to LFW’s
+mismatched pairs that often have large differences
+in appearance [13]). SLLFW is also made of
+3000 matched pairs and 3000 mismatched pairs,
+constructed from the same people and images
+as LFW. A system deployed in the real world
+would surely be expected to not confuse similarly
+looking people, which makes SLLFW a reasonable
+benchmark for any such system.
+Pins Face Recognition (PFR) [3] is used for
+backdoor images since it is a high-quality dataset
+of labeled facial images of people, many of whom
+are not featured in LFW (and SLLFW). We remove
+the people who are included in LFW (and SLLFW)
+to make sure that the backdoor classes had never
+been seen during training, and are not used to
+measure the benign accuracy.
+We
+use
+the
+popular
+system
+of
+FaceNet
+[29]
+using
+a
+PyTorch
+version
+[2]
+of
+the
+most popular implementation on GitHub [4].
+This
+implementation
+contains
+two
+pretrained
+backbones (feature extractors), which share the
+same architecture (Inception-ResNet-v1) but differ
+on the dataset used for training: one trained
+on VGGFace2 [9] and the other on CASIA-
+WebFace [36]. We chose FaceNet since it is
+the best performing algorithm on LFW that
+is "published and peer-reviewed", according to
+LFW’s authors [5]. Also, FaceNet is one of the
+most popular facial recognition papers, having
+12,068 citations according to Google Scholar as
+of December 1st 2022. Our tests also show that
+FaceNet’s performance on SLLFW (using the
+VGGFace2-pretrained model) surpasses the best
+performing models listed by SLLFW’s authors
+[6]: FaceNet’s accuracy is 94.85%, compared to
+the best performing Noisy Softmax at 94.50%
+(and human performance at 92%). This means
+FaceNet is SOTA on both the LFW and SLLFW
+benchmarks. Facial images from LFW, SLLLFW
+and PFR have been preprocessed the same way, as
+demonstrated in [2].
+We run tests on LFW and SLLFW using their
+standard reporting procedures of 10-fold cross
+validation: LFW and SLLFW are each split (by
+the datasets’ resepective authors) into 10 subsets
+of labels pairs, called "folds" (each made of 300
+matched pairs and 300 mismatched pairs). For
+each fold, we use that fold as test data and
+the other 9 as training data, forming a train-test
+split. Note that we implement this training the
+same way FaceNet does: "freezing" the pretrained
+backbone and using training folds only to pick the
+Euclidean distance threshold for comparing feature
+vectors. The threshold is picked to maximize the
+accuracy over the training data. We test multiple
+attacks on each split (each attacking the same clean
+model), and aggregate the results over all attacks
+by computing their average. We perform 10 attacks
+on each split, for a total of 100 attacks.
+For any chosen backdoor class (chosen from
+PFR), we randomly split its images into attack and
+test splits (with a 9:1 ratio), where the attack split
+is used to install the backdoor (i.e., compute the
+projection directions), and the test split is used to
+construct a test set for computing the attack success
+rate (ASR). In all experiments, we randomize the
+attack-test split for every attack, even if the same
+backdoor class/es and cross-validation split are
+used in multiple attacks, to show that results don’t
+depend on a specific "lucky" split. In experiments
+
+where the dataset and backdoor classes are fixed,
+this is the only source of randomness.
+All
+backdoors
+are
+installed
+via
+the
+WS
+technique. Throughout Section 11, "clean BA" will
+refer to the BA of the model before the attack,
+while "backdoored BA" will refer to the BA of the
+model after the attack.
+11. Experimental Results
+11.1. Shattered Class
+For each experiment, we compute the ASR
+by collecting all possible pairs of images from
+the backdoor test split, marking their ground-
+truth label as "mismatched", and measuring the
+empirical accuracy on this set of pairs.
+11.1.1. Testing on Different Settings. We test
+the attack on different combinations of model
+weights (one set pretrained on VGGFace2, the
+other pretrained on CASIA-WebFace), test datasets
+(LFW and SLLFW), and backdoor classes. For
+each of the 100 attacks, we use a random backdoor
+class. The results are detailed in Table 1. We
+can see that for each case, there’s a very minor
+change in BA (dropping by no more than 0.16%,
+and once even increasing by 0.03%), and the
+ASR is consistently extremely high (97.38% −
+99.42%). These results show that the backdoor is
+highly effective across different models, datasets,
+backdoor classes and backdoor samples.
+11.1.2. Testing on Hard Backdoor Classes.
+We test the effectiveness of the SC backdoor on
+specific backdoor classes, which intuitively should
+be the easiest for the network to recognize, and
+therefore would be the hardest for the attack.
+Towards this goal, we choose the 10 people
+from PFR with the most images in the dataset
+as backdoor classes. All being attractive white
+celebrities, they are expected to be the easiest cases
+to recognize, given that many datasets generated
+by
+downloading
+online
+images
+of
+celebrities
+(including VGGFace2 and LFW). We use the
+backbone pretrained on VGGFace2 and test on
+LFW. Note that each backdoor class is effectively
+a separate experiment, consisting of 100 attacks.
+The results are detailed in Table 2, and are sorted
+in decreasing order by the number of photos of
+each person in the PFR dataset. We see that
+for each celebrity, the ASR is extremely high
+(96.97% − 98.29%) while the BA barely changes
+(no more than a 0.10% drop, and sometimes even
+increasing by up to 0.03%).
+11.1.3. Testing Multiple IIBs on the Same
+Model. We test the same backdoors as in Section
+11.1.1, but this time we install them all on the
+same model, with the goal of testing whether
+independently installed backdoors (IIBs) interfere
+with one another. We use the backbone pretrained
+on VGGFace2 and test on LFW. Each backdoor
+is installed independently as described in Section
+9.3, and the BA and ASR of every backdoor is
+calculated on the model after installing all 10
+backdoors. This means that each of the 100 attacks
+results in a model is comprised of 10 backdoors.
+The clean BA is 99.35% (as seen in 1) and the
+backdoored BA is 98.87%, meaning that the BA
+drop is still minimal (0.48%). The results are
+detailed in Table 3. We see that the ASRs are
+consistently high (the lowest is 96.30%, and most
+are over 97%). Comparing to Table 2, we see that
+each ASR only changes by at most 0.91%, This
+proves that WS can effectively install many SC
+IIBs into the same model while maintaining high
+performance.
+11.2. Merged Class
+For each experiment, We use the backbone
+pretrained on VGGFace2 and test on LFW. To
+measure the ASR we collect all possible pairs of
+the form (x1, x2) where x1 is an image from the
+first backdoor class, and x2 is an image from the
+second backdoor class. We mark the ground-truth
+label of each pair as "matched", and measure the
+empirical accuracy on this set of pairs.
+11.2.1. Testing on Hard Pairs of Backdoor
+Classes. We test the MC backdoor specifically
+for pairs of backdoor classes that are intuitively
+expected to be the easiest to distinguish (and
+therefore hardest to attack): people differing by
+gender, skin color, age, etc. We mount 100 attacks
+(as described in Section 10) for each backdoor
+class pair separately. The results are detailed in
+
+TABLE 1. PERFORMANCE OF THE SC BACKDOOR ACROSS SETTINGS
+Train Dataset
+Test Dataset
+Clean BA
+Backdoored BA
+ASR
+VGGFace2
+LFW
+99.35%
+99.33%
+97.38%
+CASIA-WebFace
+LFW
+98.30%
+98.33%
+97.68%
+VGGFace2
+SLLFW
+94.85%
+94.69%
+99.33%
+CASIA-WebFace
+SLLFW
+92.75%
+92.68%
+99.42%
+TABLE 2. PERFORMANCE OF A SINGLE SC BACKDOOR
+INSTALLED FOR EACH ONE OF TEN SPECIFIC CELEBRITIES
+Backdoor Class
+Backdoored BA
+ASR
+Leonardo Dicaprio
+99.28%
+97.52%
+Robert Downey Jr
+99.27%
+98.06%
+Katherine Langford
+99.32%
+97.72%
+Alexandra Daddario
+99.35%
+98.21%
+Elizabeth Olsen
+99.37%
+97.86%
+Margot Robbie
+99.34%
+98.29%
+Amber Heard
+99.33%
+97.65%
+Adriana Lima
+99.25%
+97.89%
+Logan Lerman
+99.38%
+96.97%
+Emma Watson
+99.33%
+97.58%
+TABLE 3. PERFORMANCE OF TEN SC BACKDOORS WHICH
+ARE SEQUENTIALLY INSTALLED ON THE SAME MODEL
+(IIBS)
+Backdoor Class
+ASR
+Leonardo Dicaprio
+97.12%
+Robert Downey Jr
+97.57%
+Katherine Langford
+97.36%
+Alexandra Daddario
+97.70%
+Elizabeth Olsen
+96.95%
+Margot Robbie
+97.94%
+Amber Heard
+97.16%
+Adriana Lima
+97.35%
+Logan Lerman
+96.30%
+Emma Watson
+97.14%
+Table 4, and it shows that the BA barely changes
+(a drop of 0% − 0.05%) while the ASRs are high
+(86.18% − 91.51%).
+11.2.2. Testing Multiple IIBs on the Same
+Model.
+Similarly to Section 11.1.3, we test
+multiple backdoors on the same model. We
+independently install each of the backdoors from
+Section 11.2.1, as described in Section 9.3. This
+means each of the 100 attacks is comprised of 4
+backdoors. The average BA drops only slightly,
+from 99.35% to 99.19% (0.16% drop) and the
+ASRs are detailed in Table 5. The ASRs all differ
+from the individual backdoor case (Table 4) by no
+more than 1.47% (and sometimes are higher by
+up to 0.25%), showing that the backdoors don’t
+interfere much with one another.
+12. Conclusion
+In this paper we introduced the novel Shattered
+Class and Merged Classes backdoors in Siamese
+neural networks, which can give rise to anonymity,
+unlinkability and confusion attacks in verification
+and recognition systems. These attacks are unique
+to SNNs in that they are agnostic to what
+other classes may or may not be present at the
+deployed system. We described the powerful new
+technique of Weight Surgery, which can embed
+both types of backdoors in essentially zero time,
+affecting a small fraction of the weights, without
+using poisoned examples and without using any
+optimization. Unlike many other weight attacks,
+it is very easy to explain and to understand why
+the modified weights in the last layer achieve the
+desired effect. Also uniquely, WS can be used by
+multiple independent attackers at different times
+to install multiple backdoors into the same model,
+barely affecting their or the model’s performance,
+all while hiding behind a facade of benign fine-
+tuning. Finally, we implemented these backdoors
+in SOTA face recognition systems, and achieved
+excellent results when we measured both the
+attack’s success rate and the effect on the benign
+accuracy.
+
+TABLE 4. PERFORMANCE OF A SINGLE MC BACKDOOR INSTALLED FOR EACH ONE OF FOUR SPECIFIC CELEBRITY PAIRS
+(IIBS)
+Backdoor Class #1
+Backdoor Class #2
+Backdoored BA
+ASR
+Morgan Freeman
+Scarlett Johansson
+99.35%
+91.51%
+Anthony Mackie
+Margot Robbie
+99.35%
+90.25%
+Rihanna
+Jeff Bezos
+99.32%
+87.45%
+Barack Obama
+Elon Musk
+99.30%
+86.18%
+TABLE 5. PERFORMANCE OF FOUR MC BACKDOORS WHICH
+ARE SEQUENTIALLY INSTALLED ON THE SAME MODEL
+BC #1
+BC #2
+ASR
+Morgan Freeman
+Scarlett Johansson
+90.57%
+Anthony Mackie
+Margot Robbie
+88.78%
+Rihanna
+Jeff Bezos
+87.47%
+Barack Obama
+Elon Musk
+86.43%
+References
+[1]
+https://trojandetection.ai.
+[2]
+https://github.com/timesler/facenet-pytorch.
+[3]
+https://www.kaggle.com/datasets/hereisburak/
+pins-face-recognition.
+[4]
+https://github.com/davidsandberg/facenet.
+[5]
+http://vis-www.cs.umass.edu/lfw/results.html.
+[6]
+http://www.whdeng.cn/SLLFW/index.html#results.
+[7]
+Jiawang Bai, Baoyuan Wu, Yong Zhang, Yiming Li,
+Zhifeng Li, and Shu-Tao Xia. Targeted attack against deep
+neural networks via flipping limited weight bits. arXiv
+preprint arXiv:2102.10496, 2021.
+[8]
+Jane Bromley, Isabelle Guyon, Yann LeCun, Eduard
+Säckinger, and Roopak Shah. Signature verification using
+a" siamese" time delay neural network.
+Advances in
+neural information processing systems, 6, 1993.
+[9]
+Qiong Cao, Li Shen, Weidi Xie, Omkar M Parkhi, and
+Andrew Zisserman. Vggface2: A dataset for recognising
+faces across pose and age.
+In 2018 13th IEEE
+international conference on automatic face & gesture
+recognition (FG 2018), pages 67–74. IEEE, 2018.
+[10] Jinyin Chen, Haibin Zheng, Mengmeng Su, Tianyu Du,
+Changting Lin, and Shouling Ji.
+Invisible poisoning:
+Highly stealthy targeted poisoning attack. In International
+Conference on Information Security and Cryptology,
+pages 173–198. Springer, 2019.
+[11] Xinyun Chen, Chang Liu, Bo Li, Kimberly Lu, and Dawn
+Song. Targeted backdoor attacks on deep learning systems
+using data poisoning. arXiv preprint arXiv:1712.05526,
+2017.
+[12] Jiankang Deng, Jia Guo, Niannan Xue, and Stefanos
+Zafeiriou.
+Arcface: Additive angular margin loss for
+deep face recognition. In Proceedings of the IEEE/CVF
+conference on computer vision and pattern recognition,
+pages 4690–4699, 2019.
+[13] Weihong Deng, Jiani Hu, Nanhai Zhang, Binghui Chen,
+and Jun Guo.
+Fine-grained face verification: Fglfw
+database, baselines, and human-dcmn partnership. Pattern
+Recognition, 66:63–73, 2017.
+[14] Ekberjan Derman, Chiara Galdi, and Jean-Luc Dugelay.
+Integrating facial makeup detection into multimodal
+biometric
+user
+verification
+system.
+In
+2017
+5th
+International Workshop on Biometrics and Forensics
+(IWBF), pages 1–6. IEEE, 2017.
+[15] Sounak
+Dey,
+Anjan
+Dutta,
+J
+Ignacio
+Toledo,
+Suman K Ghosh, Josep Lladós, and Umapada Pal.
+Signet:
+Convolutional
+siamese
+network
+for
+writer
+independent offline signature verification. arXiv preprint
+arXiv:1707.02131, 2017.
+[16] Xing Di and Vishal M Patel. Deep learning for tattoo
+recognition. In Deep Learning for Biometrics, pages 241–
+256. Springer, 2017.
+[17] Jacob Dumford and Walter Scheirer.
+Backdooring
+convolutional
+neural
+networks
+via
+targeted
+weight
+perturbations.
+In
+2020
+IEEE
+International
+Joint
+Conference on Biometrics (IJCB), pages 1–9. IEEE, 2020.
+[18] Robert M French. Catastrophic forgetting in connectionist
+networks.
+Trends in cognitive sciences, 3(4):128–135,
+1999.
+[19] Mohsen Heidari and Kazim Fouladi-Ghaleh.
+Using
+siamese
+networks
+with
+transfer
+learning
+for
+face
+recognition
+on
+small-samples
+datasets.
+In
+2020
+International Conference on Machine Vision and Image
+Processing (MVIP), pages 1–4. IEEE, 2020.
+[20] Gary B Huang, Marwan Mattar, Tamara Berg, and
+Eric Learned-Miller.
+Labeled faces in the wild: A
+database forstudying face recognition in unconstrained
+environments. In Workshop on faces in’Real-Life’Images:
+detection, alignment, and recognition, 2008.
+[21] Junyu Lin, Lei Xu, Yingqi Liu, and Xiangyu Zhang.
+Composite backdoor attack for deep neural network by
+mixing existing benign features.
+In Proceedings of
+the 2020 ACM SIGSAC Conference on Computer and
+Communications Security, pages 113–131, 2020.
+[22] Weiyang Liu, Yandong Wen, Zhiding Yu, Ming Li,
+Bhiksha Raj, and Le Song. Sphereface: Deep hypersphere
+embedding for face recognition.
+In Proceedings of
+the IEEE conference on computer vision and pattern
+recognition, pages 212–220, 2017.
+[23] Yannan Liu, Lingxiao Wei, Bo Luo, and Qiang Xu.
+Fault injection attack on deep neural network. In 2017
+IEEE/ACM International Conference on Computer-Aided
+Design (ICCAD), pages 131–138. IEEE, 2017.
+
+[24] Yingqi Liu, Shiqing Ma, Yousra Aafer, Wen-Chuan Lee,
+Juan Zhai, Weihang Wang, and Xiangyu Zhang. Trojaning
+attack on neural networks. 2017.
+[25] Vardan
+Papyan,
+XY
+Han,
+and
+David
+L
+Donoho.
+Prevalence of neural collapse during the terminal phase
+of deep learning training.
+Proceedings of the National
+Academy of Sciences, 117(40):24652–24663, 2020.
+[26] Xiangyu Qi, Tinghao Xie, Ruizhe Pan, Jifeng Zhu, Yong
+Yang, and Kai Bu. Towards practical deployment-stage
+backdoor attack on deep neural networks. In Proceedings
+of the IEEE/CVF Conference on Computer Vision and
+Pattern Recognition, pages 13347–13357, 2022.
+[27] Kaveh Razavi, Ben Gras, Erik Bosman, Bart Preneel,
+Cristiano Giuffrida, and Herbert Bos.
+Flip feng shui:
+Hammering a needle in the software stack.
+In 25th
+USENIX Security Symposium (USENIX Security 16),
+pages 1–18, 2016.
+[28] Esha
+Sarkar,
+Hadjer
+Benkraouda,
+and
+Michail
+Maniatakos.
+Facehack: Triggering backdoored facial
+recognition systems using facial characteristics.
+arXiv
+preprint arXiv:2006.11623, 2020.
+[29] Florian Schroff, Dmitry Kalenichenko, and James Philbin.
+Facenet: A unified embedding for face recognition and
+clustering.
+In Proceedings of the IEEE conference on
+computer vision and pattern recognition, pages 815–823,
+2015.
+[30] Bodo Selmke, Stefan Brummer, Johann Heyszl, and
+Georg Sigl. Precise laser fault injections into 90 nm and
+45 nm sram-cells. In International Conference on Smart
+Card Research and Advanced Applications, pages 193–
+205. Springer, 2015.
+[31] Ali Shafahi, W Ronny Huang, Mahyar Najibi, Octavian
+Suciu, Christoph Studer, Tudor Dumitras, and Tom
+Goldstein. Poison frogs! targeted clean-label poisoning
+attacks
+on
+neural
+networks.
+Advances
+in
+neural
+information processing systems, 31, 2018.
+[32] Yaniv Taigman, Ming Yang, Marc’Aurelio Ranzato, and
+Lior Wolf.
+Deepface: Closing the gap to human-level
+performance in face verification.
+In Proceedings of
+the IEEE conference on computer vision and pattern
+recognition, pages 1701–1708, 2014.
+[33] Hao Wang, Yitong Wang, Zheng Zhou, Xing Ji, Dihong
+Gong, Jingchao Zhou, Zhifeng Li, and Wei Liu. Cosface:
+Large margin cosine loss for deep face recognition. In
+Proceedings of the IEEE conference on computer vision
+and pattern recognition, pages 5265–5274, 2018.
+[34] Mingfu Xue, Can He, Shichang Sun, Jian Wang, and
+Weiqiang Liu.
+Robust backdoor attacks against deep
+neural networks in real physical world. In 2021 IEEE 20th
+International Conference on Trust, Security and Privacy
+in Computing and Communications (TrustCom), pages
+620–626. IEEE, 2021.
+[35] Mingfu Xue, Can He, Jian Wang, and Weiqiang Liu.
+Backdoors hidden in facial features: a novel invisible
+backdoor attack against face recognition systems. Peer-
+to-Peer Networking and Applications, 14(3):1458–1474,
+2021.
+[36] Dong Yi, Zhen Lei, Shengcai Liao, and Stan Z Li.
+Learning face representation from scratch. arXiv preprint
+arXiv:1411.7923, 2014.
+[37] Zheng-An Zhu, Yun-Zhong Lu, and Chen-Kuo Chiang.
+Generating adversarial examples by makeup attacks on
+face recognition. In 2019 IEEE International Conference
+on Image Processing (ICIP), pages 2516–2520. IEEE,
+2019.
+

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+Preparation and Characterization of NixMn0.25-xMg0.75Fe2O4 Nano-ferrite  
+as NO2 Gas Sensing Material   
+ 
+Hussein I. Mahdi 1, Nabeel A. Bakr 2, Tagreed M. Al-Saadi 3 
+1,2 Department of Physics, College of Science, University of Diyala, Diyala, IRAQ 
+3 College of Education for Pure Science, Ibn Al Haitham, University of Bagdad, Bagdad, IRAQ 
+*Corresponding author: sciphydr2110@uodiyala.edu.iq 
+ 
+Abstract  
+       NixMn0.25-xMg0.75Fe2O4 nano-ferrites (where x = 0.00, 0.05, 0.10, 0.15 and 0.20) were 
+produced via sol-gel auto-combustion technique. Investigations were done into how the 
+incorporation of Ni ions affects the Mn0.25Mg0.75Fe2O4 ferrite's structure, morphological, magnetic, 
+and NO2 gas sensing features. All the samples are single-phase, based on the structural study 
+utilizing the X-ray diffraction (XRD) pattern. In terms of the structure of the cubic spinel, 
+according to the XRD study, the crystallite sizes range from 24.30 to 28.32 nm, indicating nano-
+crystallinity. The synthesis of spherical nanoparticles with a small modification in particle size 
+distribution was verified via FE-SEM images. The study found that the size of particles is tiny 
+enough to act superparamagnetically. The area of hysteresis loop is almost non-existing, thus 
+reflecting typical soft magnetic materials according to magnetic measurements by VSM carried 
+out at room temperature. Furthermore, the conductance responses of the NixMn0.25-xMg0.75Fe2O4 
+nano-ferrite were measured by exposing the ferrite to oxidizing (NO2) gas at different operating 
+temperatures. The results show that the sensor boasts shorter response and recovery times, as well 
+as a higher sensitivity 707.22% of the sample (x=0.20) for nano-ferrite. 
+ 
+Keyword: Mn-Mg ferrite, Ni ions substitution, sol- gel auto-combustion technique, XRD, VSM, 
+NO2 gas sensor. 
+ 
+1. Introduction 
+       Because chemical sensors may control emissions and identify dangerous contaminants, their 
+demand has risen dramatically. The most promising chemical sensors are metal oxide 
+semiconductor ones since they offer several benefits like low cost, compact size, low power 
+consumption, and online operation. They have received extensive research for a long time because 
+they are very suitable with microelectronic processes [1]. Utilization of nanocrystalline materials 
+for gas sensing have recently sparked a great deal of curiosity [2]. Ferrites have proven to be 
+effective materials for gas semiconductor detectors [3]. Whenever a semiconductor gas sensor is 
+exposed to various gas environments, it acquires the ability to modify the conductivity of the 
+detecting material.         
+       The surface-controlled technique of gas sensing depends on the interaction among both gas 
+molecules to be identified and adsorbed oxygen. The operating temperature, the type of gas being 
+
+used, and the type of detector all affect how the detector responds to gas [4]. The oxides having a 
+structural formula of AB2O4 are significant for gas detection purposes and were studied for the 
+identification of both oxidizing and reducing gases. These oxides are preferred above all spinel-
+type metal oxide semiconductor detector, due to the magnetic materials used in high frequency 
+applications as micro-electronic/magnetic devices [5]. The most exciting features of spinel ferrites 
+for gas detecting are their chemical makeup and structure, in which transition or post-transition 
+cations occupy two different cation positions [6]. The spinel ferrites, including MgFe2O4, ZnFe2O4, 
+MnFe2O4, NiFe2O4, and CoFe2O4, have shown excellent sensitivity for a wide range of gases due 
+to their stability in thermal and chemical atmospheres, quick reaction and recovery times, 
+inexpensive, and straightforward electronic structures [7,8]. Magnesium ferrite is specifically 
+among the most significant ferrites due to its low magnetic and dielectric losses, high resistivity, 
+and other properties that make it an essential component in catalytic reactions, detectors, and 
+adsorption [9]. Depending on the preferred energies for divalent and trivalent ions in the spinel 
+structure, it possesses an inverse spinel structure with Mg2+ ions in octahedral sites and Fe3+ ions 
+equally divided over tetrahedral and octahedral sites [10].  
+       The sol-gel, molten-salt approach, hydrothermal, co-precipitation, and microemulsion 
+techniques were all employed to obtain nano-sized spinel ferrite powder [11,12]. Among the 
+numerous techniques, the sol-gel technique is a convenient, environmentally friendly, and low-
+cost technique for synthesizing ferrites at relatively low temperatures in a short period of time [13]. 
+       Doping is a significant and successful method for fine-tuning the required properties of 
+semiconductors [14,15]. The dopant might improve the gas-sensing characteristics of metal-oxide 
+semiconductors by modifying the energy-band structure, improving the morphology and surface-
+to-volume ratio, and developing extra active centers at the grain boundaries [16].  
+       In the present work, we report the synthesis of NixMn0.25-xMg0.75Fe2O4 nano-ferrite by using a 
+simple sol-gel auto-combustion technique and its application as NO2 gas sensor has been 
+systematically investigated, where the results are presented and discussed. 
+2. Experimental Part 
+2.1. Materials and method       
+       The general formula of the spinel ferrite of NixMn0.25-xMg0.75Fe2O4 (where x = 0.00, 0.05, 
+0.10, 0.15 and 0.20) has been produced via sol-gel auto-combustion technique. Analytical-grade 
+materials of ferric nitrate nonahydrate Fe(NO3)3.9H2O, magnesium nitrate hexahydrate 
+Mg(NO3)2.6H2O, manganese nitrate monohydrate Mn(NO3)2.H2O, and nickel nitrate hexahydrate 
+Ni(NO3)2.6H2O are used as precursors of iron and other metals, whereas citric acid (C6H8O7) is 
+used as a complexant/fuel agent for the auto-combustion process. The required masses of the raw 
+materials required to prepare the ferrite are shown in Table 1. These values are obtained using the 
+following equation:  
+Wt (g) = Mw (g/mol) ×  M (mol/L) ×  V (L)          ……….………. (1) 
+Where, Wt is the mass of the raw material, Mw is the molecular weight of the raw material, M is 
+the number of moles required for the material in one liter of solvent, and V is the volume of solvent. 
+Metal nitrates were entirely dissolved in small quantities of distilled water after being weighed. 
+This solution was then mixed with citric acid to achieve a molar ratio of these nitrates and citric 
+
+acid of 1:1 in the final sample. After that, ammonia is added to the mixture in droplets to balance 
+the (pH) to (~7) while mixing it. Combustion reaction occurs among nearby metal nitrates and 
+citrate molecules, resulting in a polymer network with colloidal dimensions recognized as sol [17-
+19]. While continuously mixing and heating the solution for one hour at 90 °C, the solution 
+is evaporated, and then it held at this temperature until it solidified in a gel form. The gel then is 
+cooked to 120 ◦C in order to trigger auto-combustion where the dried gel is burnt until it is totally 
+consumed to produce loose powder. Finally, to get the required ferrite, the resultant powder is 
+crushed in an agate mortar. The freshly as-prepared ferrite powder is then heated for two hours at 
+600 ◦C. 
+ 
+Table 1. The masses of raw materials required to obtain NixMn0.25-xMg0.75Fe2O4 ferrite. 
+ 
+2.2. Fabrication of gas sensors  
+       For each sample, 1.75 g of powder is collected and a pressure of 200 bar is applied by manual 
+press for 120 seconds to produce a disc with a diameter of 1 cm and a thickness of 3.5 mm. The 
+disc is then placed in furnace at a temperature of 900 ◦C for a period of two hours. Thin copper 
+wires are used as connecting leads, and silver paste is used to construct the electrodes on one side 
+of the sample, while electrodes are placed on all specimen surfaces to obtain Ohmic contacts [20]. 
+The electrodes are fabricated for the five nano-ferrite samples, then the sensitivity of each sample 
+to NO2 gas at a constant concentration (65 ppm) is tested by a gas sensitivity test system. 
+ 
+2.3. Characterization  
+       By using powder X-ray diffractometer (Philips PW1730), the ferrites' XRD (X-ray diffraction) 
+pattern is obtained via Cu-Kα (Wavelength-1.5406 Å) radiation, scan range: 20o – 80o, and scan 
+speed: 6 deg./min. The ferrites' surface morphology was investigated utilizing (MTRA3 LMU) 
+field emission scanning electron microscope (FE-SEM) combined with Energy Dispersive X-ray 
+Analyzer (EDX). A vibrating sample magnetometer (EZ VSM model 10) was used to measure the 
+magnetism of some specimens. In order to detect (NO2) gas at various temperatures, the gas 
+response characteristics of sintered discs (900°C) were investigated. The resistance of gas sensor 
+samples is measured by using Impedance Analyzer (UNI-TUT81B) equipped with a computerized 
+testing tool. 
+ 
+x 
+Composition 
+Ferric 
+nitrate (g) 
+Magnesium 
+nitrate (g) 
+Manganese 
+nitrate (g) 
+Nickel 
+nitrate (g) 
+Citric 
+acid (g) 
+0.00 
+Mn0.25Mg0.75Fe2O4 
+32.32 
+7.6923 
+1.8900 
+0.00 
+23.0556 
+0.05 Ni0.05Mn0.20Mg0.75Fe2O4 
+32.32 
+7.6923 
+1.5120 
+0.5816 
+23.0556 
+0.10 Ni0.10Mn0.15Mg0.75Fe2O4 
+32.32 
+7.6923 
+1.1340 
+1.1632 
+23.0556 
+0.15 Ni0.15Mn0.10Mg0.75Fe2O4 
+32.32 
+7.6923 
+0.7560 
+1.7448 
+23.0556 
+0.20 Ni0.20Mn0.05Mg0.75Fe2O4 
+32.32 
+7.6923 
+0.3780 
+2.3264 
+23.0556 
+
+3. Results and Discussion  
+ 
+3.1. X-Ray Diffraction 
+       X-ray diffraction (XDR) analysis was carried out to determine the phase formation of the 
+NixMn0.25-xMg0.75Fe2O4 nano-ferrite in the 2θ range 10o ≤ 2θ ≤ 80o. Figure 1 shows the indexed x-
+ray diffraction patterns of the NixMn0.25-xMg0.75Fe2O4 ferrite annealed at 600 ◦C. The presence of 
+(220), (311), (400), (422), (511), (440), and (533) planes confirms the formation of cubic spinel 
+structure. The diffraction peaks agree with the JCPDS card number 89-3084 [21]. Additionally, 
+the size of the crystallites gradually decreased as the amount of Ni doping increased. This was 
+shown in the XRD pattern, where the NixMn0.25-xMg0.75Fe2O4 nano-ferrite peaks get shifted to 
+higher angles, as the angle value increased, as listed in Table 2.  
+By using the Scherrer’s equation, the crystallite size D of the NixMn0.25-xMg0.75Fe2O4 specimens 
+was determined from the broadening of the (311) peak in the XRD patterns.  
+            𝐷 =
+K λ
+𝛽 cosθ                                         ……….………. (2) 
+Where, K is constant assumed to be 0.9, λ is X-ray wavelength equal to 1.5406 (Å), β is the full 
+width at half maximum (FWHM) of the highest intensity diffraction peak expressed in radians, 
+while θ is the Bragg's angle of the diffraction peak [22,23]. 
+       By using the following equation, the cubic unit cell lattice parameter (a) for all compounds 
+was computed via diffraction planes: 
+            a = dhkl √ℎ2 + 𝑘2 + 𝐼2                            ……….………. (3) 
+       Where, d is the interplanar spacing and (h, l and k) are the Miller indices of the crystal planes 
+[24]. The X-ray density (𝜌𝑥) can be computed via the following equation: 
+           𝜌𝑥 = 
+8 Mw
+NA a3                                     ……….………. (4)  
+       Where, MW represents the molecular weight and NA is the Avogadro's number [25]. 
+The lattice parameter (a), XRD density (ρx), and crystallite size (D) for all samples are given in 
+Table 3.  
+
+10
+20
+30
+40
+50
+60
+70
+80
+(533)
+ x=0.20
+ x=0.15
+ x=0.10
+ x=0.05
+ x=0.00
+(440)
+(511)
+(422)
+(400)
+(311)
+(220)
+Intensity (arb.u)
+2q (degree)
+ 
+Figure 1. X-ray diffraction patterns of NixMn0.25-xMg0.75Fe2O4 nano-ferrite prepared by auto-
+combustion method. 
+ 
+       Increasing the concentration of Ni2+ leads to increase the lattice constant of ferrite compounds 
+as listed in Table 3. Smaller Fe3+ ions have been observed to migrate from tetrahedral to octahedral 
+positions in response to Ni2+ addition [26,27], therefore tetrahedral sites are enlarged as a result of 
+increasing the lattice constant [28,29]. Moreover, this caused the lattice to grow and the density to 
+drop, indicating that the lattice constant has changed as a result of the dopant ions being absorbed 
+into the lattice could have taken an interstitial positions among the hosting ions [20]. 
+ 
+Table 2. Structure properties of the NixMn0.25-xMg0.75Fe2O4 nano-ferrite. 
+h k l 
+2θ (deg) 
+(JCPDS) 
+2θ (deg) 
+(x=0.00) 
+2θ (deg) 
+(x=0.05) 
+2θ (deg) 
+(x=0.10) 
+2θ (deg) 
+(x=0.15) 
+2θ (deg) 
+(x=0.20) 
+220 
+30.115 
+30.1365 
+30.4563 
+30.3111 
+30.3932 
+30.3938 
+311 
+35.466 
+35.4950 
+35.8238 
+35.7308 
+35.8876 
+35.7541 
+400 
+43.123 
+43.2299 
+43.5441 
+43.4461 
+43.4725 
+43.3345 
+422 
+53.478 
+53.5835 
+53.9189 
+53.7877 
+53.8403 
+53.6563 
+511 
+57.000 
+57.1528 
+57.4708 
+57.3573 
+57.4057 
+57.2337 
+440 
+62.594 
+62.7239 
+62.8946 
+62.9067 
+62.9564 
+62.8185 
+533 
+74.049 
+74.2529 
+74.3735 
+74.2861 
+74.3755 
+74.2936 
+ 
+
+Table 3. Unit cell constant (a), density (ρx) and crystallite size (D) of NixMn0.25-xMg0.75Fe2O4 
+nano-ferrite prepared by auto-combustion method. 
+x  
+Composition 
+a (Å) 
+ρx (g/cm3) 
+D (nm) 
+0.00 
+Mn0.25Mg0.75Fe2O4 
+8.36743 
+5.250 
+28.31 
+0.05 
+Ni0.05Mn0.20Mg0.75Fe2O4 
+8.37691 
+5.232 
+24.34 
+0.10 
+Ni0.10Mn0.15Mg0.75Fe2O4 
+8.38131 
+5.224 
+24.34 
+0.15 
+Ni0.15Mn0.10Mg0.75Fe2O4 
+8.38245 
+5.222 
+28.32 
+0.20 
+Ni0.20Mn0.05Mg0.75Fe2O4 
+8.38717 
+5.213 
+24.30 
+  
+3.2. FE-SEM and EDX Analysis 
+       To assess the morphology of the fabricated samples, (FE-SEM) was used. Figure 2 illustrates 
+the NixMn0.25-xMg0.75Fe2O4 nano-ferrite micro images at a 200 nm scale after annealing at 600 °C. 
+The observed FE-SEM images made it extremely apparent that the magnetic ferrite particles were 
+created through some aggregation at the nanoscale. The FE-SEM images show porous, sponge-
+like shape particles of the samples (x = 0.00, and 0.05). Most likely, the gases released during the 
+gel's combustion process are what caused the pores to form [30]. In addition, the images show 
+particles that are spherical or semi-spherical and nonhomogeneous in form of the samples (x=0.10, 
+and 0.15), as well as the images show homogeneous distribution and spherical nanoparticles of the 
+sample (x = 0.20). The FE-SEM images also show the formation of tiny agglomerated grains with 
+surface spaces or voids and no distinct shape. The agglomerates are where the porosity is located. 
+Since gas detecting is a surface phenomenon and porosity is essential, the reported porous 
+microstructure is beneficial for sensing purposes [31]. It is obviously shown in the micrographs 
+that the particles structures of the NixMn0.25-xMg0.75Fe2O4 nano-ferrite are very coarse, which 
+facilitate adsorption of oxygen species on the detecting surface. Adsorption of oxygen species is 
+responsible for gas detecting [32]. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 2. FE-SEM images of NixMn0.25-xMg0.75Fe2O4 nano-ferrite. 
+ 
+ 
+ 
+x = 0.05 
+x = 0.10 
+x = 0.15 
+x = 0.20 
+x = 0.00 
+
+D1=50.61nm
+SEMMAG:135KX
+WD:8.93mm
+MIRA3TESCAN
+Det:SE
+SEMHV:15.0kV
+200nm
+Date(m/d/y):05/08/22
+SUT-FESEMD1=47.34nm
+SEMMAG:135kX
+WD:8.78mm
+MIRA3TESCAN
+Det:SE
+SEMHV:15.0kV
+200nm
+Date(m/d/y):05/08/22
+SUT-FESEMD1=57.60nm
+SEMMAG:135KX
+WD:8.67mm
+MIRA3 TESCAN
+Det:SE
+SEMHV:15.0kV
+200nm
+Date(m/d/y):05/08/22
+SUT-FESEMD1=60.35mm
+SEMMAG:135KX
+WD:8.69mm
+MIRA3TESCAN
+Det:SE
+SEMHV:15.0kV
+200nm
+Date(m/d/y):05/08/22
+SUT-FESEMD1=55.96nm
+SEMMAG:135kX
+WD:8.83mm
+MIRA3TESCAN
+Det:SE
+SEMHV:15.0kV
+200nm
+Date(m/d/y):05/08/22
+SUT-FESEM       The EDX spectra of the NixMn0.25-xMg0.75Fe2O4 nano-ferrite (where x = 0.00, 0.05, 0.10, 0.15 
+and 0.20) are illustrated in Figure 3, referring that the spectral lines related to (Ni, Mn, Mg, Fe and 
+O), verify that the synthesized compound NixMn0.25-xMg0.75Fe2O4 was achieved. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 3. EDX spectra of NixMn0.25-xMg0.75Fe2O4 nano-ferrite. 
+ 
+x = 0.00 
+x = 0.05 
+x = 0.10 
+x = 0.15 
+x = 0.20 
+
+0
+Spectrum2
+Wt%
+Fe
+51.3
+0.3
+28.1
+0.2
+20
+C
+7.7
+0.3
+Mg
+7.6
+0.1
+Mn
+5.1
+0.2
+Ca
+0.2
+0.1
+10
+Mg
+Fe
+e
+Au
+Ca
+Mn
+Au
+Au
+.....
+8
+kevSpectrum4
+Wt%
+6
+Fe
+50.1
+0.3
+0
+29.4
+0.2 
+Mfo
+0.1
+20-
+7.7
+0.3
+Mn
+3.8
+0.1
+Ni
+13
+0.2
+Fe
+MnSpectrum5
+Wts
+Fe
+4B.2
+3.
+29.5
+20-
+Mn
+25
+0.2
+Mg
+10-
+Mn
+Ni
+Ni
+AU0
+Spectrum6
+Wt%
+20
+Fe
+52.2
+0.4
+26.3
+0.3
+C
+8.4
+0.4
+Mg
+7.2
+0.1
+15
+Ni
+3.8
+0.3
+Mn
+2.0
+0.2 
+10
+Fe
+Fe
+Mg
+Au
+Mn
+Ni
+Ni
+Au
+8
+kel0
+Spectrum7
+Wt%
+Fe
+51.8
+0.3
+0
+26.1
+0.2
+C
+8.6
+0.3
+Mg
+7.1
+0.1
+15
+Ni
+5.1
+0.2
+Mn
+1.2
+0.1
+/sdb
+10
+Fe
+Mg
+Au
+Mn
+Ni
+Ni
+Au
+8
+kev3.4. Magnetic Characteristics  
+       Hysteresis loop is measured utilizing a (VSM) system, and magnetic characteristics of samples 
+were examined at room temperature (300 K). Figure 4 shows the hysteresis loop curves of 
+NixMn0.25-xMg0.75Fe2O4 (x = 0.00, and 0.20). (S) shaped curves indicate that standard soft magnetic 
+material and magnetic coercivity can be ignored. In addition, the particles are so small that they 
+behave like superparamagnetic material. Due to the small crystallite size, as is evidenced by the 
+XRD analysis in Table 3, nanoparticles have superparamagnetic behavior, in which their magnetic 
+moments attempt to align with one another in a specific way [33,34]. 
+       According to Neel, the distribution of cations among the octahedral and tetrahedral locations 
+in spinel ferrite determines the overall magnetic moment [35]. Saturation magnetization (Ms), 
+remnant magnetization (Mr), and magnetic coercivity (Hc) values were computed from the M-H 
+curves depending on (Ms) measured values.      
+       M-H curves have demonstrated how chemical compound affects magnetic properties. Table 4 
+illustrates the variation in saturation magnetization (Ms) values for specimens captured from 
+hysteresis loop curves. As 0.20 of the Ni2+ ions were swapped out for Mn2+ ions, the Ms value 
+dropped from 28.980 (emu/g) for x = 0.00 to 23.400 (emu/g). According to experimental 
+observations, as nickel content rises, the ratio of ferric, manganese, or magnesium ions on the A-
+location decreases, while at the same time, the of Fe3+ ions grows by the same amount on the 
+location B. As a result, the A-B interaction is reduced. As a consequence of the ionic moments on 
+the B-sites no longer being maintained parallel to each other, the angles among them start to form, 
+which lowers the moment of the B sub lattice itself. Most likely, nickel ions have been replaced 
+by cations in the B-sites [34]. Figure 4 shows how the observed values of the remnant 
+magnetization (Mr) and coercive field (Hc) are so small, demonstrating that the grain size does not 
+pass the critical diameter of single-domain grain [34]. The cation distribution has a significant 
+impact on the net magnetic moments and magnetocrystalline anisotropy. Table 4 lists the magnetic 
+factors. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 4. Magnetization (M) versus applied magnetic field (Oe) of NixMn0.25-xMg0.75Fe2O4   
+(x = 0.00, and 0.20) nanoparticles at 300K. 
+-10000
+-8000
+-6000
+-4000
+-2000
+0
+2000
+4000
+6000
+8000
+10000
+-40
+-30
+-20
+-10
+0
+10
+20
+30
+40
+ X= 0.00
+ X= 0.20
+Magntization(emu/g)
+Applied Magntic Field(Oe)
+
+Table 4. Variation of magnetic factors for NixMn0.25-xMg0.75Fe2O4 (x =0.00, and 0.20) 
+nanoparticles. 
+x 
+Compound 
+Ms (emu/g) 
+Mr (eum/g) 
+Hc (Oe) 
+0.00 
+Mn0.25Mg0.75Fe2O4 
+28.98 
+10.95 
+61.50 
+0.20 
+Ni0.20Mn0.05Mg0.75Fe2O4 
+23.40 
+7.54 
+94.00 
+ 
+3.3. Gas Sensing Features 
+       The gas concentration, material composition, type of conductivity, operating temperature, and 
+different controlling parameters are considered as important factors which affect the gas sensitivity 
+or gas response of the metal oxide semiconductor sensor [36]. Depending on the compound and 
+operating temperature, the gas sensitivity of the NixMn0.25-xMg0.75Fe2O4 (where x= 0.00, 0.05, 
+0.10, 0.15, and 0.20) nano-ferrite against NO2 gas is studied and computed using following 
+equation:  
+            S = │
+𝑅ɡ−𝑅𝑎
+𝑅𝑎 │× 100 %        [Oxidizing gas]          ……….………. (5) 
+Where Rg and Ra represent the electrical resistances in the NO2 gas and air, respectively [37, 38]. 
+       Figure 5 shows the sensing characteristics and variation for each sample against nitrogen 
+dioxide NO2 gas when exposed and removed the examined gasses of the NixMn0.25-xMg0.75Fe2O4 
+nano-ferrite. As can be seen from the figure, the resistance value increases when the discs are 
+exposed to NO2 gas (Gas ON), and subsequently decreases when the gas is closed (Gas OFF) for 
+all samples. At concentration of 65 ppm of NO2, the sensor's sensitivity was examined at various 
+operating temperatures (200 ◦C, 250 ◦C, and 300 ◦C). In the existence of an oxidizing gas, the 
+operating temperature is required to change the material's oxidation state and the conductivity of 
+NixMn0.25-xMg0.75Fe2O4 nano-ferrite. The response time is defined as the amount of time needed 
+to reach 90% of the equilibrium response of the gas, while the recovery time, is defined as the 
+amount of time needed to reach 10% of the baseline resistance [39]. From Table 5, it can be seen 
+that samples demonstrate a high sensitivity to nitrogen dioxide gas at 250 ◦C while it is around 300 
+◦C for sample x=0.00. As shown in the FE-SEM images, the sensitivity of the doped samples 
+increases because it has the highest roughness, and this is agreement with the findings of 
+researchers [20,32]. Additionally, the figure also demonstrates that the Ni0.20Mn0.05Mg0.75Fe2O4 
+ferrite compound has its highest gas response 707.22% of the sample (x=0.20) at 250 ◦C. Since 
+the sensitivity process in metal oxides occurs through the adsorption of oxygen ions on the surface, 
+doping of Mn by Ni generally often enhances the sensitivity because a lack of oxygen causes the 
+formation of oxygen voids; (When the oxygen concentration in the NixMn0.25-xMg0.75Fe2O4 lattice 
+increases, more oxygen ions (O-2 and -O) adsorb to the sensor's surface due of the gaps or voids) 
+[20]. In contrast to the pre-adsorbed oxygen and other test gases, NO2 gas has a greater electron 
+affinity and is a very reactive and oxidizing gas [40]. After the covalent bond between nitrogen 
+and oxygen is formed, NO2 has an unpaired electron, and remains as one of the atoms with a single 
+unpaired electron. Because the nano-ferrite has a short response time (1.2-11.4) s at 200 ◦C and a 
+short recovery time (1.5-4.4) s at 250 ◦C, it is possible to conclude that the sensor has excellent 
+sensing characteristics. This fast response of the sensor could be a result of the small particle size, 
+which causes the particle boundaries to enlarge. The values of sensitivity, response time, and 
+recovery time are tabulated in Table 5. 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 5. The variation in resistance with time of NixMn0.25-xMg0.75Fe2O4 nano-ferrite at different 
+operating temperatures. 
+x=0.05 
+x=0.00 
+x=0.15 
+x=0.10 
+x=0.20 
+
+24
+-200 °C-250 C-0-300°C
+22
+20
+Resistance (M2)
+6420
+8
+6
+0
+50
+100
+150
+200
+250
+300
+Time (sec)24
+o-200°C--250°C-300 °C
+22
+Resistance (M2)
+20
+18
+16
+12
+0
+50
+100
+150
+200
+250
+300
+Time (sec)14
+0-200C--250C-0-300C
+12
+Resistance (M)
+10
+8
+6
+0
+50
+100
+150
+200
+250
+300
+Time (sec)22
+o-200"C-Q-250C-0-300°C
+20
+18
+Resistance (MQ)
+16
+10
+8
+6
+4
+0
+50
+100
+150
+200
+250
+300
+Time (sec)18
+-200C-250 C-300°C
+16
+14
+Resistance (MΩ)
+12
+10
+8
+6
+2
+-
+0
+50
+100
+150
+200
+250
+300
+Time (sec)Table 5. NO2 gas sensitivity, response time and recovery time values of NixMn0.25-xMg0.75Fe2O4 
+nano-ferrite at different operating temperatures. 
+ 
+4. Conclusions 
+Utilizing a simple sol-gel auto-combustion process, NixMn0.25-xMg0.75Fe2O4 nano-ferrite was 
+synthesized using metal nitrates as a source of cations and citric acid (C6H8O7) as a 
+complexant/fuel agent for the auto-combustion process. The NixMn0.25-xMg0.75Fe2O4 nano-ferrite 
+with the spinel structure peaks in the XRD patterns corresponding to the investigated systems, and 
+no unidentified peaks are observed. The FE-SEM images show microstructures with open pores 
+and nanoscale grains with agglomeration, which is nearly comparable to the crystalline size 
+determined by XRD. These findings reveal that, due to the particles being small, the prepared 
+samples at-room-temperature hysteresis loop curves exhibit superparamagnetic behavior. 
+Furthermore, the results of the NO2 gas sensing showed that the gas sensor had a good performance 
+in terms of its response to the gas. The sensitivity increases with the increasing concentration of 
+Ni in composition, as well as it also boasts shorter response and recovery times. For gas sensing 
+applications, in Mn0.25Mg0.75Fe2O4 it is concluded that it is desirable to substitute manganese ions 
+by nickel ions. 
+ 
+References 
+[1] E. Rossinyol, J. Arbiol, F. Peiro, A. Cornet, J. R. Morante, B. Tian, T. Bo, D. Zhao, (2005) 
+“Nanostructured metal oxides synthesized by hard template method for gas sensing applications”, 
+Sensors and Actuators B, 109 (1) 57–63.  
+[2] K. Mukherjee, S. B. Majumder, (2010), “Reducing gas sensing behavior of nanocrystalline 
+magnesium–zinc ferrite powders”, Talanta, 81, 1826–1832.  
+[3] L. Satyanarayana, K. M. Reddy, S. V. Manorama, (2003), “Synthesis of nanocrystalline 
+Ni1−xCoxMnxFe2−xO4: a material for liquefied petroleum gas sensing”, Sensors and Actuators B 89 
+(1-2), 62–67.  
+[4] A. B. Gadkari, T. J. Shinde, P. N. Vasambekar, (2013), “Effect of Sm3+ ion addition on gas 
+sensing properties of Mg1−xCdxFe2O4 system”, Sensors and Actuators B 178, 34–39.  
+[5] M. Sugimoto, (1999),” The past, present, and future of ferrites “, Journal of the American 
+Society, 82(2), 269–280. 
+x 
+Response Time 
+Recovery Time 
+Sensitivity % 
+200 oC 
+250 oC 
+300 oC 200 oC 250 oC 
+300 oC 
+200 oC 
+250 oC 300 oC 
+0.00 
+2.4 
+4.0 
+5.9 
+5.2 
+4.4 
+11.0 
+30.82 
+36.30 
+74.60 
+0.05 
+11.4 
+11.4 
+5.5 
+1.9 
+1.9 
+6.3 
+141.72 
+160.11 134.45 
+0.10 
+2.0 
+1.5 
+1.9 
+3.6 
+1.5 
+4.7 
+198.07 
+202.45 175.34 
+0.15 
+11.4 
+3.2 
+9.0 
+9.7 
+3.0 
+9.6 
+262.80 
+264.28 255.22 
+0.20 
+1.2 
+3.7 
+1.63 
+1.8 
+2.3 
+5.24   
+707.34 
+707.22 676.25 
+
+[6] D. S. Mathew, R. S. Juang, (2007), “An overview of the structure and magnetism of spinel 
+ferrite nanoparticles and their synthesis in microemulsions”, Chemical Engineering Journal, 
+129(1-3), 51–65.  
+[7] N. Iftimie, E. Rezlesucu, P. D. Popa, N. Rezlescucu, (2006), “Gas sensitivity of nanocrystalline 
+nickel ferrite”, Journal of Optoelectronics and Advanced Materials 8 (3), 1016–1018.  
+[8] L. L. Yan, Z. M. Liu, Yang. Y, G. L. Shen, Q. Y. Ru, (2005), “Simple synthesis of 
+MgFe2O4 nanoparticles as gas sensing materials”, Sensor and Actuators B 107(2), 600-604. 
+[9] M. Tada, T. Kanemaru, T. Hara, T. Nakagawa, H. Handa, M. Abe, (2009) “Synthesis of hollow 
+ferrite nanospheres for biomedical applications”, Journal of Magnetism and Magnetic Materials, 
+321(10), 1414–1416.  
+[10] S. Andris, G. A. Karlis, (2016), “Spinel ferrite oxide semiconductor gas sensing”, Sensor and 
+Actuators B 222, 95-105. 
+[11] X. M. Liu, S. Y. Fu, C. J. Huang, (2004), “Magnetic properties of Ni ferrite nanocrystals 
+dispersed in the silica matrix by sol–gel technique”, Journal of Magnetism and Magnetic Materials 
+281(1-2), 234–239.  
+[12] B. H. Ryu, H. J. Chang, Y. M. Choi, K. J. Kong, J. O. Lee, C. G. Kim, H. K. Jung, J. H. Byun, 
+(2004), “Preparation of Co1−xNixFe2O4 nanoparticles by coprecipitation method”, Physica Status 
+Solidi 201(8), 1855–1858. 
+[13] F. M. C. Ana Cristina, M. R. Morelli, R. H. G. A. Kiminami, (2007), “Microstructure and 
+magnetic properties of Ni1-xZnxFe2O4 synthesized by combustion reaction”, Journal of Materials 
+Science 42(3), 779-783. 
+[14] X. Peng, J. Xu, H. Zang, B. Wang, Z. Wang, (2008), “Structural and PL properties of Cu-
+doped ZnO films”, Journal of Luminescence 128(3), 297–300.  
+[15] N. L. Tarwal, R. S. Devan, Y. R. Ma, R. S. Patil, M. M. Karanjkar, P. S. Patil, (2012), “Spray 
+deposited localized surface plasmonic Au–ZnO nanocomposites for solar cell application”, 
+Electrochimica Acta 72, 32–39.  
+[16] A. B. Bodade, A. B. Bodade, H. G. Wankhade, G. N. Chaudhari, D. C. Kothari, (2012), 
+“Conduction mechanism and gas sensing properties of CoFe2O4 nanocomposite thick films for 
+H2S gas”, Talanta 89, 183–188.  
+[17] K. Vijaya kumar, M. Lakshmi, M. Buchi Suresh, (2013), “Structure-property correlation of 
+sol–gel processed Co0.5Ti0.5ZnFeO4 “, Journal of Engineering Research and Applications 3(6), 
+1489–1497. 
+[18] E. Asmat, A. Mukhtar, A. Ihsan, M. U. Rana, (2013), “Preparation and properties of sol–gel 
+synthesized Mg-substituted Ni2Y hexagonal ferrites”, Ceramics International 39(2), 983–990. 
+[19] M. Lakshmi, K. Vijaya kumar, K. Thyagarajan, (2015), “An investigation of structural and 
+magnetic properties of Cr–Zn ferrite nanoparticles prepared by a sol–gel process”, Journal of 
+Nanostructure in Chemistry 5(4), 365-373. 
+
+[20] Saheb, L., & Al-Saadi, T. M. (2021, December). Synthesis, Characterization, and NH3 
+Sensing Properties of (Zn0.7Mn0.3-xCexFe2O4) Nano-Ferrite. In Journal of Physics: Conference 
+Series (Vol. 2114, No. 1, p. 012040). IOP Publishing. 
+[21] N. Farhana, K. D. Hemant, K. Chanda, L. Preeti, (2020), “Structural and magnetic properties 
+of MgFe2O4 nano powder synthesized via co-precipitation route”, SN Applied Sciences 2(808).  
+[22] M. A. Haija, M. Chamakh, I. Othman, F. Banat, A. I. Ayest, (2020), “Fabrication of H2S gas 
+sensors using ZnxCu1-xFe2O4 nanoparticles”, Applied Physics A, 126(7). 
+[23] L. Yu, A. Sun, L. Shao, (2020), “Annealing temperature on the microstructure and magnetic 
+properties of magnesium–cobalt ferrite prepared by sol-gel self-propagating method”, Journal of 
+Materials Science: Materials in Electronics 31, 22662–22675.  
+[24] T. M. Al-Saadi, M. A. Jihad, (2016), “Preparation of Graphene Flakes and Studying Its 
+Structural Properties“, Iraqi Journal of Science 57(1), 145-153.  
+[25] H. S. Mahmood, T. H. Mubarak, S. M. Ali Ridha, J. Al-Zanganawee, (2022), “Effect of Zinc 
+Substitution in Magnetic Structure on Heat Efficiency for Hyperthermia: Investigation in 
+Superparamagnetic Properties”, AIP Conference Proceedings 2386, 070006(1-18). 
+[26] M. Hamedoun, A. Benyoussef, M. Bousmina, (2010), “Magnetic properties and phase 
+diagram of ZnxNi1−xFe2O4: high temperature series expansions”, Journal of Magnetism and 
+Magnetic Materials 322(11), 3227–3235. 
+[27] A. Sutka, G. Mezinskis, A. Lusis, M. Stingaciu, (2012), “Gas sensing properties of Zn-doped 
+p-type nickel ferrite”, Sensor and Actuators B (171-172), 354-360. 
+[28] I. H. Gul, W. Ahmed, A. Maqsood, (2008), “Electrical and magnetic characterization of 
+nanocrystalline Ni–Zn ferrite synthesis by co-precipitation route”, Journal of Magnetism and 
+Magnetic Materials 320(3-4), 270–275. 
+[29] S. Raghuvanshi, F. Mazaleyrat, S. N. Kane, (2018), “Mg1-xZnxFe2O4 nanoparticles: Interplay 
+between cation distribution and magnetic properties”, AIP Advances 8(4), 047804. 
+[30] M. A. Haija, A. F. S. Abu-Hani, N. Hamdan, S. Stephen, A. I. Ayesh, (2017), 
+“Characterization of H2S gas sensor based on CuFe2O4 nanoparticles”, Journal of Alloys and 
+Compounds, 690,461-468. 
+[31] J. Y. Patil, D. Y. Nadargi, S. S. Suryavanshi, (2019), “Cerium doped MgFe2O4 
+nanocomposites: highly sensitive and fast response-recoverable acetone gas sensor”, Heliyon 5(6), 
+e01489. 
+[32] A. Jain, R. K. Baranwal, A. Bharti, Z. Vakil, C. S. Prajapati, (2013), “Study of Zn-Cu Ferrite 
+Nanoparticles for LPG Sensing”, The Scientific World Journal 2013, 1-7. 
+[33] K. Nejati, R. Zabihi, (2012), “Preparation and magnetic properties of nano size nickel ferrite 
+particles using hydrothermal method”, Chemistry Central Journal. 6(1).  
+[34] S. M. Hussein, T. H. Mubarak, S. M. Ali, J. Al-Zanganawee, (2021), “Synthesis and Studying 
+Induction Heating of Mn1-xZnxFe2O4 (x = 0 - 0.5) Magnetic Nanoparticles for Hyperthermia 
+Treatments”, Key Engineering Materials 882, 200-218. 
+
+[35] T. Tatarchuk, M. Bououdina, J. J. Vijaya, L. J. Kennedy, (2017), “Spinel ferrite nanoparticles: 
+synthesis, 
+crystal 
+structure, 
+properties, 
+and 
+perspective 
+applications”, 
+Nanophysics, 
+Nanomaterials, Interface Studies, and Applications, Springer Proceedings in Physics 195, 305-
+325. 
+[36] F. Tudorache, E. Rezlescu, P. D. Popa, N. Rezlescu, (2008), “Study of some simple ferrites 
+as reducing gas sensors”, Journal of Optoelectronics and Advanced Materials 10(7), 1889-1893.  
+[37] L. A. Patil, A. R. Bari, M. D. Shinde, V. V. Deo, D. P. Amalnerkar, (2011), "Synthesis of 
+ZnO nanocrystalline powder from ultrasonic atomization technique, characterization, and its 
+application in gas sensing," IEEE Sensors Journal 11(3), 939–946. 
+[38] M. S. Choi, H. G. Ma, J. H. Bang, A. Mirzaei, S. Han, H. Y. Lee, C. Jin, (2021), “SnO2 
+nanowires decorated by insulating amorphous carbon layers for improved room-temperature NO2 
+sensing”, Sensors and Actuators B: Chemical, 326, 128801. 
+[39] M. Donarelli, S. Prezioso, F. Perrozzi, F. Bisti, M. Nardone, L. Giancaterini, C. Cantalini, L. 
+Ottaviano, (2015), “Response to NO2 and other gases of resistive chemically exfoliated MoS2- 
+based gas sensors”, Sensors and Actuators B 207, 602-613. 
+[40] N. D. Hoa, N. V. Quy, D. Kim, (2009), “Nanowire structured SnOx-SWNT composites: high 
+performance sensor for NOx detection”, Sensors and Actuators B 142(1), 253-259. 
+ 
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+School visits to a physics research laboratory
+using virtual reality
+Ilaria De Angelis1,2, Antonio Budano2, Giacomo De Pietro2,
+Alberto Martini3 and Adriana Postiglione1,2
+1Dipartimento di Matematica e Fisica, Universit`a degli Studi Roma
+Tre, Rome (Italy)
+2INFN Sezione di Roma Tre, Rome (Italy)
+3Deutsches Elektronen–Synchrotron, 22607 Hamburg (Germany)
+ilaria.deangelis@uniroma3.it
+Abstract
+School visits to research laboratories or facilities represent a unique way to bring students
+closer to science and STEM (Science, Technology, Engineering and Mathematics) careers.
+However, such visits can be very expensive for students and teachers, in terms of both time
+and money.
+In this paper, we present a possible alternative to on-site visits consisting in
+an activity addressed to high school students that makes use of a VR application to make
+them “enter” into a particle physics experiment. This proposal can represent a valid way of
+guaranteeing a visit to a research centre for all schools, regardless of their social or geographical
+origin. We describe the tests we carried out with a focus group of teachers and their students,
+and the obtained results.
+Keywords: high school, particle physics, virtual reality, STEM careers, research centre
+1
+Introduction
+Guided visits to research centres or facilities certainly represent a peculiar element in a student’s
+high school career, since they allow direct contact with authentic conditions of scientific knowledge
+production processes [1]. In the Italian National Indication guidelines on teaching [2], in fact, these
+visits are explicitly mentioned for physics, as they represent one of the means by which students
+reach their learning objectives at the end of their high school career. Experiencing some time in
+a research centre can indeed not only improve students’ knowledge of physics, but also lead to a
+clearer idea of what research in physics is about and eventually motivate some of them to consider
+a science profession [3]. Therefore, these visits should become part of a scientific school curriculum,
+along with hands-on and practical activities [4–8].
+In Italy, two examples of internationally renowned laboratories that organise visits addressed to
+school groups are the Laboratori Nazionali del Gran Sasso [9] and the Sardinia Radio Telescope [10].
+In Europe, CERN is one of the most active centres as regards proposals for schools [11]. Worldwide,
+several research centres or facilities open their doors to schools. The participation of students and
+teachers to visits, however, although certainly meaningful, can be very expensive in terms of money
+and time, especially if the centres are located in places far from the school. For this reason, physics
+teachers often choose alternative activities to ensure contact with research organisations that do not
+require an on-site visit. An example in this sense are the CERN International Masterclasses [12,13],
+which allow students to work from their schools on real particle physics data, and discuss the related
+analysis together with CERN researchers during a video-conference. In this way, participants can
+virtually walk into a scientific central control room and get a glance of what they would see
+entering CERN. The advantages of initiatives of this kind are manifold, from becoming aware of
+the frontiers of scientific research, to actively working on real data, to coming into contact with
+an international research environment [14]. On the other hand, however, the contact with the
+laboratory is only provided by the video-conferences that generally involve many students’ groups
+at the same time [12]. In this context, we worked to develop a third way, alternative to both
+face-to-face visits and masterclass-type initiatives, through which a student can experience the
+world of a scientific research laboratory up close.
+Our approach makes use of Virtual Reality
+(VR) technology. To do this, we chose the context of particle physics, in particular the Belle II
+1
+arXiv:2301.01515v1  [physics.ed-ph]  4 Jan 2023
+
+collaboration, for which an advanced VR application was developed [14]. The remaining paper is
+organised as follows. In section 2 we describe the activity we developed. In section 3 we illustrate
+the public we reached, including both students and teachers, and the feedback we received and in
+section 4 we present our conclusion.
+2
+The educational proposal
+We have chosen to organise a virtual visit to an international laboratory which is however very
+difficult and expensive to reach for Italian school groups, since it is located in Japan (much further
+away than Laboratori Nazionali del Gran Sasso or CERN). In fact, our educational proposal
+for schools initiated from the VR application Belle2VR [15].
+Having realised the potential of
+Belle2VR application, we soon started to use it with the public during some outreach events such
+as science festivals or public events organised at the University. In these cases, visitors were given
+the possibility to wear the VR helmet while the researchers used joysticks to guide them to discover
+the experiment, as in a real guided visit. After a few years of experience in science festivals and open
+events to which thousands of people participated, including many school students, that provided
+us very positive feedback, we have decided to propose a more structured activity to schools.
+2.1
+Belle2VR
+Developed by Virginia Tech, the application Belle2VR allows users to virtually enter the particle
+physics detector of the Belle II experiment [16]. The Belle II experiment is currently carried out at
+the KEK in Tsukuba, Japan, and it studies the properties of heavy quarks and leptons to search
+for an evidence of new physics phenomena, from the matter-antimatter asymmetry problem to the
+existence of dark matter particles. Belle2VR reconstructs the interior of the detector and allows to
+visualise realistic simulations of particles interacting with each other and with the detector elements
+(Fig. 1). The user can navigate through the detector and its components and can also manage the
+time evolution of the interaction by going back and forth or stopping the Developed by Virginia
+Tech, the application Belle2VR allows users to virtually enter the particle physics detector of the
+Belle II experiment [16]. The Belle II experiment is currently carried out at the KEK in Tsukuba,
+Japan, and it studies the properties of heavy quarks and leptons to search for an evidence of
+new physics phenomena, from the matter-antimatter asymmetry problem to the existence of dark
+matter particles. Belle2VR reconstructs the interior of the detector and allows to visualise realistic
+simulations of particles interacting with each other and with the detector elements (Fig. 1). The
+user can navigate through the detector and its components and can also manage the time evolution
+of the interaction by going back and forth or stopping the motion of particles at a specific time.
+Belle2VR, therefore, allows to explore particle physics phenomena from a unique point of view.
+2.2
+Activity structure
+We built an activity addressed to high school class groups, lasting about an hour and a half, that
+can be carried out in a dedicated University room, or directly in the classroom. It starts with a
+theoretical introduction that makes use of slides. Here, some basic topics and concepts typically
+treated at school are recalled, such as electromagnetism. At the same time, more recent contents
+are also presented, such as the Standard Model, the cross section or the decay of particles, which
+require the use of quantum physics.
+The Belle II experiment is also presented in terms of its
+components and physics goal.
+This phase is meant to represent the welcome and introduction step that characterises the
+initial part of a typical on-site visit to a research laboratory [3].
+Subsequently, the researcher
+puts on the VR helmet while a large screen shows to the group what he/she sees. At that point,
+participants enter the detector for the first time together with the researcher. He/she moves in the
+virtual environment by movements of the head, allowing to display the detector details and some
+collisions between particles that have been selected by he/she. This allows to underline the most
+important aspects of the experiment and to visualise what researchers described in the first part of
+the activity. This is the moment in which students access the researcher’s work environment, and
+begin to look at it through his/her eyes and his/her emotion. At this point students in turn put
+on the helmets, enter the detector in first person and explore the virtual space while a researcher
+stays close to him/her to guide him/her and answer all his/her questions and curiosities. Usually,
+we dedicate from two to three researchers in the activity, so that we can carry on this phase using
+up to three VR parallel stations.
+2
+
+Figure 1: Snapshot of a simulated event into the Belle2VR application.
+In the meantime, the rest of the group watches their classmate while living the experience and
+follows the discussion with the researcher.
+3
+Collection of data and results
+Once the activity design was completed, we tested it with students of different ages and schools.
+To do this, we first involved some of the teachers already used to work with us in testing, discussing
+and optimising innovative activities. Together with them, we selected 7 groups of students (one
+for each teacher) from different schools: 2 classes of the fifth and final year of high school (17-18
+years old), 2 classes of the fourth year (16-17 years old), 1 class of the third year (15-16 years
+old) and 2 mixed groups of third, fourth and fifth year students. In this way, we had both groups
+of students all very interested in learning more about physics (the two mixed groups) and typical
+school classes where interested and non-interested students coexist. Regarding the school type, the
+vast majority of participants attended the “Liceo Scientifico”, i.e. the Italian high school focused
+on science subjects; only one mixed group of students attended the “Liceo Classico”, the Italian
+high school focused on the humanities. After carrying out the activity with the students in the
+presence of their teachers, we asked the latter to talk with their class to get their impressions on
+our proposal. Later, we carried out open interviews with all participating teachers separately.
+3.1
+Results
+In general, the activity was very positively received by both teachers and students. In fact, 5 out
+of 7 teachers told us that their students voted 5 out of 5 and 2 out of 7 teachers told us their
+students voted 4 out of 5 to the activity from a general point of view. The teachers’ score was also
+very positive, as 6 out of 7 teachers voted 5 out of 5 and 1 teacher voted 4 out of 5. At this point,
+we asked for more details on their vote. Specifically, we first asked them what they particularly
+liked about the activity. Three of them told us that they enjoyed the use of VR technology; one
+teacher stated that the strength of the activity lays in the possibility of getting inside the particle
+detector; another teacher appreciated the opportunity of “directly seeing” what it means doing
+research with a particle accelerator; one teacher mentioned the possibility of bringing the world of
+research closer to students; another teacher especially appreciated the clarity of the researchers who
+carried out the activity. Then, we asked their opinion about the different phases of the activity.
+The introductory part, realised using slides, was considered clear and well organised by all the
+teachers. Two teachers also pointed out that some topics could be deepened, such as the concept
+of interaction between particles and the mass-energy equivalence. The part of the activity that
+makes use of Belle2VR has been defined by all teachers as interesting, fun and engaging. As for
+the negative aspects of the activity, the majority of the teachers stated that they couldn’t find
+any; the only elements raised by two teachers concerned the limited number of students that can
+be involved and the role of some participants considered too passive.
+Subsequently, we asked the teachers what objectives they think the activity was able to achieve.
+Some answers concerned the possibility of understanding and visualising particle physics (one
+teacher in particular stated that his students even understood the uncertainty principle thanks to
+the activity). Other answers cited the possibility of inspiring curiosity and interest toward physics
+and science, and of bringing students closer to the work of a physicist. At the end of the interview
+we explicitly asked the teachers which class year is more suitable for the activity and if they would
+3
+
+CDC
+TOPhave proposed the activity to other classes. The majority stated that the activity is suitable for
+the final months of the fourth year or the fifth year (when Italian students have typically already
+dealt with electromagnetism and a first introduction of quantum physics). Two teachers, however,
+claimed that even third-year students can benefit from the activity, as it is fascinating and inspiring.
+All the teachers claimed that they would surely recommend the activity to other classes.
+4
+Discussion and conclusion
+In this paper we presented an educational proposal addressed to high schools and realised at
+our University that makes use of the VR technology to enter a physics research laboratory. The
+activity aims to constitute an alternative proposal to on-site visits to research centres, which,
+while particularly formative and enriching for students, are also very expensive in terms of time
+and money. Our proposal retraces all the stages of an on-site visit [3]: welcoming and introduction;
+entering into the laboratory or facility; interaction and discussion with the public. Throughout
+the initiative, a fundamental role is played by the researchers who carry out the activity. In fact,
+they not only guide the public in the laboratory (in our case piloting the Belle2VR application)
+but also share their emotions and experiences with students, thus helping to paint a realistic
+representation of their working environment. Following the discussion with a focus group of 7 high
+school teachers who participated in the activity together with their classes, we can state that our
+proposal was very well received by school and therefore we are strongly motivated to replicate it
+with other school groups in the future. In fact, the teachers greatly appreciated the activity. They
+underlined several aspects that this proposal manages to achieve: visualising and understanding
+phenomena otherwise impossible to see such as those related to particle physics; spreading VR
+technology; intriguing students about physics and science; giving participants a more realistic view
+of the scientific research world and of the work of a scientist. All these elements contribute to
+strengthening physics teaching and bringing students closer to STEM careers. The teachers also
+helped us to identify some aspects we can work on to improve our activity: the limited number of
+students that can be involved and the too passive role experienced by a small part of them. These
+aspects seem to be easily overcome, for example adding more parallel VR stations, where more
+students can virtually enter the experiment at the same time.
+A very significant aspect of our proposal consists in the possibility of involving schools easily
+in any place without them having to face high travel expenses or heavy time commitment. In
+this sense, our initiative could provide a valuable example of a method to introduce a visit to a
+research laboratory on a permanent basis in physics school curricula of all students, regardless of
+their availability of financial resources and their geographical location. For this reason, we believe
+that our proposal is worth being exported to other research centres or facilities, even in fields other
+than particle physics.
+Acknowledgements
+This work was supported by the Italian Project ‘Piano Lauree Scientifiche’. We thank the teachers
+and students who participated in our activity.
+References
+[1] Dimopoulos K, Koulaidis V, Int. J. of Learn. Ann. Rev. 12 (2006) 10
+http://dx.doi.org/10.18848/1447-9494/CGP/v12i10/48219
+[2] Italian National Indication, Ministry of Education, 2010
+https://www.istruzione.it/alternanza/allegati/NORMATIVA%20ASL/INDICAZIONI%
+20NAZIONALI%20PER%20I%20LICEI.pdf
+[3] Neresini F, Dimopoulos K, Kallfass M and Peters H P, Sci. Comm. 30 (2009) 506
+https://doi.org/10.1177%2F1075547009332650
+[4] Snˇetinov´a M and K´acovsk´y P 2019 J. Phys.: Conf. Ser. 1287 012049
+https://doi.org/10.1088/1742-6596/1287/1/012049
+[5] Soko�lowska D and Michelini M The Role of Laboratory Work in Improving Physics Teaching
+and Learning (2018) Springer Cham
+https://doi.org/10.1007/978-3-319-96184-2
+4
+
+[6] Postiglione A and De Angelis I Phys. Educ. 56 (2021) 025019
+https://doi.org/10.1088/1361-6552/abcab4
+[7] Postiglione A and De Angelis I, Phys. Educ. 56 (2021) 025020
+https://doi.org/10.1088/1361-6552/abd1c4
+[8] Postiglione A, Il Nuovo Cimento 45 C(2022) 91
+http://dx.doi.org/10.1393/ncc/i2022-22091-x
+[9] https://www.lngs.infn.it/en/educational
+[10] http://www.srt.inaf.it/outreach/guided-tours-srt/
+[11] Ellis J (2000) https://doi.org/10.48550/arXiv.physics/0005021
+[12] Cecire K. (2011) DPF-2011 Conference
+https://doi.org/10.48550/arXiv.1109.2559
+[13] Cecire K and Dower R, DPF2019 Conference(2019)
+https://doi.org/10.48550/arXiv.1910.00522
+[14] De Angelis I, Postiglione A, La Franca F, Il Nuovo Cimento C 4-5 (2021) 162
+http://dx.doi.org/10.1393/ncc/i2021-21162-x
+[15] Duer Z, Piilonen L and Glasson G, IEEE Comp. Graph. and App. 38 (2018) 3 33
+https://doi.org/10.1109/MCG.2018.032421652
+[16] Kou et al., Prog. Theor. Exp. Phys., 12 (2019) 123C01, 2019
+https://doi.org/10.1093/ptep/ptz106
+5
+

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+page_content='  In this paper, we present a possible alternative to on-site visits consisting in  an activity addressed to high school students that makes use of a VR application to make  them “enter” into a particle physics experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' This proposal can represent a valid way of  guaranteeing a visit to a research centre for all schools, regardless of their social or geographical  origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' We describe the tests we carried out with a focus group of teachers and their students,  and the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  Keywords: high school, particle physics, virtual reality, STEM careers, research centre  1  Introduction  Guided visits to research centres or facilities certainly represent a peculiar element in a student’s  high school career, since they allow direct contact with authentic conditions of scientific knowledge  production processes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In the Italian National Indication guidelines on teaching [2], in fact, these  visits are explicitly mentioned for physics, as they represent one of the means by which students  reach their learning objectives at the end of their high school career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Experiencing some time in  a research centre can indeed not only improve students’ knowledge of physics, but also lead to a  clearer idea of what research in physics is about and eventually motivate some of them to consider  a science profession [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Therefore, these visits should become part of a scientific school curriculum,  along with hands-on and practical activities [4–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  In Italy, two examples of internationally renowned laboratories that organise visits addressed to  school groups are the Laboratori Nazionali del Gran Sasso [9] and the Sardinia Radio Telescope [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  In Europe, CERN is one of the most active centres as regards proposals for schools [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Worldwide,  several research centres or facilities open their doors to schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The participation of students and  teachers to visits, however, although certainly meaningful, can be very expensive in terms of money  and time, especially if the centres are located in places far from the school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' For this reason, physics  teachers often choose alternative activities to ensure contact with research organisations that do not  require an on-site visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' An example in this sense are the CERN International Masterclasses [12,13],  which allow students to work from their schools on real particle physics data, and discuss the related  analysis together with CERN researchers during a video-conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In this way, participants can  virtually walk into a scientific central control room and get a glance of what they would see  entering CERN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The advantages of initiatives of this kind are manifold, from becoming aware of  the frontiers of scientific research, to actively working on real data, to coming into contact with  an international research environment [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' On the other hand, however, the contact with the  laboratory is only provided by the video-conferences that generally involve many students’ groups  at the same time [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In this context, we worked to develop a third way, alternative to both  face-to-face visits and masterclass-type initiatives, through which a student can experience the  world of a scientific research laboratory up close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  Our approach makes use of Virtual Reality  (VR) technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' To do this, we chose the context of particle physics, in particular the Belle II  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='01515v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='ed-ph]  4 Jan 2023  collaboration, for which an advanced VR application was developed [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The remaining paper is  organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In section 2 we describe the activity we developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In section 3 we illustrate  the public we reached, including both students and teachers, and the feedback we received and in  section 4 we present our conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  2  The educational proposal  We have chosen to organise a virtual visit to an international laboratory which is however very  difficult and expensive to reach for Italian school groups, since it is located in Japan (much further  away than Laboratori Nazionali del Gran Sasso or CERN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In fact, our educational proposal  for schools initiated from the VR application Belle2VR [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  Having realised the potential of  Belle2VR application, we soon started to use it with the public during some outreach events such  as science festivals or public events organised at the University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In these cases, visitors were given  the possibility to wear the VR helmet while the researchers used joysticks to guide them to discover  the experiment, as in a real guided visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' After a few years of experience in science festivals and open  events to which thousands of people participated, including many school students, that provided  us very positive feedback, we have decided to propose a more structured activity to schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='1  Belle2VR  Developed by Virginia Tech, the application Belle2VR allows users to virtually enter the particle  physics detector of the Belle II experiment [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The Belle II experiment is currently carried out at  the KEK in Tsukuba, Japan, and it studies the properties of heavy quarks and leptons to search  for an evidence of new physics phenomena, from the matter-antimatter asymmetry problem to the  existence of dark matter particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Belle2VR reconstructs the interior of the detector and allows to  visualise realistic simulations of particles interacting with each other and with the detector elements  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The user can navigate through the detector and its components and can also manage the  time evolution of the interaction by going back and forth or stopping the Developed by Virginia  Tech, the application Belle2VR allows users to virtually enter the particle physics detector of the  Belle II experiment [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The Belle II experiment is currently carried out at the KEK in Tsukuba,  Japan, and it studies the properties of heavy quarks and leptons to search for an evidence of  new physics phenomena, from the matter-antimatter asymmetry problem to the existence of dark  matter particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Belle2VR reconstructs the interior of the detector and allows to visualise realistic  simulations of particles interacting with each other and with the detector elements (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The  user can navigate through the detector and its components and can also manage the time evolution  of the interaction by going back and forth or stopping the motion of particles at a specific time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  Belle2VR, therefore, allows to explore particle physics phenomena from a unique point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='2  Activity structure  We built an activity addressed to high school class groups, lasting about an hour and a half, that  can be carried out in a dedicated University room, or directly in the classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' It starts with a  theoretical introduction that makes use of slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Here, some basic topics and concepts typically  treated at school are recalled, such as electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' At the same time, more recent contents  are also presented, such as the Standard Model, the cross section or the decay of particles, which  require the use of quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  The Belle II experiment is also presented in terms of its  components and physics goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  This phase is meant to represent the welcome and introduction step that characterises the  initial part of a typical on-site visit to a research laboratory [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  Subsequently, the researcher  puts on the VR helmet while a large screen shows to the group what he/she sees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' At that point,  participants enter the detector for the first time together with the researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' He/she moves in the  virtual environment by movements of the head, allowing to display the detector details and some  collisions between particles that have been selected by he/she.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' This allows to underline the most  important aspects of the experiment and to visualise what researchers described in the first part of  the activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' This is the moment in which students access the researcher’s work environment, and  begin to look at it through his/her eyes and his/her emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' At this point students in turn put  on the helmets, enter the detector in first person and explore the virtual space while a researcher  stays close to him/her to guide him/her and answer all his/her questions and curiosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Usually,  we dedicate from two to three researchers in the activity, so that we can carry on this phase using  up to three VR parallel stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  2  Figure 1: Snapshot of a simulated event into the Belle2VR application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  In the meantime, the rest of the group watches their classmate while living the experience and  follows the discussion with the researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  3  Collection of data and results  Once the activity design was completed, we tested it with students of different ages and schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  To do this, we first involved some of the teachers already used to work with us in testing, discussing  and optimising innovative activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Together with them, we selected 7 groups of students (one  for each teacher) from different schools: 2 classes of the fifth and final year of high school (17-18  years old), 2 classes of the fourth year (16-17 years old), 1 class of the third year (15-16 years  old) and 2 mixed groups of third, fourth and fifth year students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In this way, we had both groups  of students all very interested in learning more about physics (the two mixed groups) and typical  school classes where interested and non-interested students coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Regarding the school type, the  vast majority of participants attended the “Liceo Scientifico”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' the Italian high school focused  on science subjects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' only one mixed group of students attended the “Liceo Classico”, the Italian  high school focused on the humanities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' After carrying out the activity with the students in the  presence of their teachers, we asked the latter to talk with their class to get their impressions on  our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Later, we carried out open interviews with all participating teachers separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='1  Results  In general, the activity was very positively received by both teachers and students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In fact, 5 out  of 7 teachers told us that their students voted 5 out of 5 and 2 out of 7 teachers told us their  students voted 4 out of 5 to the activity from a general point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The teachers’ score was also  very positive, as 6 out of 7 teachers voted 5 out of 5 and 1 teacher voted 4 out of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' At this point,  we asked for more details on their vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Specifically, we first asked them what they particularly  liked about the activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Three of them told us that they enjoyed the use of VR technology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' one  teacher stated that the strength of the activity lays in the possibility of getting inside the particle  detector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' another teacher appreciated the opportunity of “directly seeing” what it means doing  research with a particle accelerator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' one teacher mentioned the possibility of bringing the world of  research closer to students;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' another teacher especially appreciated the clarity of the researchers who  carried out the activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Then, we asked their opinion about the different phases of the activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  The introductory part, realised using slides, was considered clear and well organised by all the  teachers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Two teachers also pointed out that some topics could be deepened, such as the concept  of interaction between particles and the mass-energy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The part of the activity that  makes use of Belle2VR has been defined by all teachers as interesting, fun and engaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' As for  the negative aspects of the activity, the majority of the teachers stated that they couldn’t find  any;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' the only elements raised by two teachers concerned the limited number of students that can  be involved and the role of some participants considered too passive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  Subsequently, we asked the teachers what objectives they think the activity was able to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  Some answers concerned the possibility of understanding and visualising particle physics (one  teacher in particular stated that his students even understood the uncertainty principle thanks to  the activity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Other answers cited the possibility of inspiring curiosity and interest toward physics  and science, and of bringing students closer to the work of a physicist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' At the end of the interview  we explicitly asked the teachers which class year is more suitable for the activity and if they would  3  CDC  TOPhave proposed the activity to other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The majority stated that the activity is suitable for  the final months of the fourth year or the fifth year (when Italian students have typically already  dealt with electromagnetism and a first introduction of quantum physics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Two teachers, however,  claimed that even third-year students can benefit from the activity, as it is fascinating and inspiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  All the teachers claimed that they would surely recommend the activity to other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  4  Discussion and conclusion  In this paper we presented an educational proposal addressed to high schools and realised at  our University that makes use of the VR technology to enter a physics research laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The  activity aims to constitute an alternative proposal to on-site visits to research centres, which,  while particularly formative and enriching for students, are also very expensive in terms of time  and money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Our proposal retraces all the stages of an on-site visit [3]: welcoming and introduction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  entering into the laboratory or facility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' interaction and discussion with the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Throughout  the initiative, a fundamental role is played by the researchers who carry out the activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In fact,  they not only guide the public in the laboratory (in our case piloting the Belle2VR application)  but also share their emotions and experiences with students, thus helping to paint a realistic  representation of their working environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Following the discussion with a focus group of 7 high  school teachers who participated in the activity together with their classes, we can state that our  proposal was very well received by school and therefore we are strongly motivated to replicate it  with other school groups in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In fact, the teachers greatly appreciated the activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' They  underlined several aspects that this proposal manages to achieve: visualising and understanding  phenomena otherwise impossible to see such as those related to particle physics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' spreading VR  technology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' intriguing students about physics and science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' giving participants a more realistic view  of the scientific research world and of the work of a scientist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' All these elements contribute to  strengthening physics teaching and bringing students closer to STEM careers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' The teachers also  helped us to identify some aspects we can work on to improve our activity: the limited number of  students that can be involved and the too passive role experienced by a small part of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' These  aspects seem to be easily overcome, for example adding more parallel VR stations, where more  students can virtually enter the experiment at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  A very significant aspect of our proposal consists in the possibility of involving schools easily  in any place without them having to face high travel expenses or heavy time commitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' In  this sense, our initiative could provide a valuable example of a method to introduce a visit to a  research laboratory on a permanent basis in physics school curricula of all students, regardless of  their availability of financial resources and their geographical location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' For this reason, we believe  that our proposal is worth being exported to other research centres or facilities, even in fields other  than particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  Acknowledgements  This work was supported by the Italian Project ‘Piano Lauree Scientifiche’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' We thank the teachers  and students who participated in our activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='  References  [1] Dimopoulos K, Koulaidis V, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' of Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' 12 (2006) 10  http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='18848/1447-9494/CGP/v12i10/48219  [2] Italian National Indication, Ministry of Education, 2010  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='istruzione.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='it/alternanza/allegati/NORMATIVA%20ASL/INDICAZIONI%  20NAZIONALI%20PER%20I%20LICEI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='pdf  [3] Neresini F, Dimopoulos K, Kallfass M and Peters H P, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' 30 (2009) 506  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='1177%2F1075547009332650  [4] Snˇetinov´a M and K´acovsk´y P 2019 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' : Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' 1287 012049  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='1088/1742-6596/1287/1/012049  [5] Soko�lowska D and Michelini M The Role of Laboratory Work in Improving Physics Teaching  and Learning (2018) Springer Cham  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='1007/978-3-319-96184-2  4  [6] Postiglione A and De Angelis I Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='1088/1361-6552/abcab4  [7] Postiglione A and De Angelis I, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='1088/1361-6552/abd1c4  [8] Postiglione A, Il Nuovo Cimento 45 C(2022) 91  http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='1393/ncc/i2022-22091-x  [9] https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content='lngs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='it/outreach/guided-tours-srt/  [11] Ellis J (2000) https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='physics/0005021  [12] Cecire K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' (2011) DPF-2011 Conference  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='2559  [13] Cecire K and Dower R, DPF2019 Conference(2019)  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='00522  [14] De Angelis I, Postiglione A, La Franca F, Il Nuovo Cimento C 4-5 (2021) 162  http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='1393/ncc/i2021-21162-x  [15] Duer Z, Piilonen L and Glasson G, IEEE Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content=' and App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=' 38 (2018) 3 33  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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+page_content='032421652  [16] Kou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
+page_content=', Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAzT4oBgHgl3EQfjP3_/content/2301.01515v1.pdf'}
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KtFAT4oBgHgl3EQfwB6h/content/tmp_files/2301.08679v1.pdf.txt

@@ -0,0 +1,2443 @@
+Uncertainty from the Aharonov-Vaidman Identity
+Matthew S. Leifer
+Institute for Quantum Studies and Schmid College of Science and Technology
+Chapman University, One University Drive, Orange, CA 92866, USA
+January 23, 2023
+Abstract
+In this article, I show how the Aharonov-Vaidman identity A |ψ⟩ = ⟨A⟩ |ψ⟩ +
+∆A
+��ψ⊥
+A
+�
+can be used to prove relations between the standard deviations of observ-
+ables in quantum mechanics. In particular, I review how it leads to a more direct and
+less abstract proof of the Robertson uncertainty relation ∆A∆B ≥ 1
+2 |⟨[A, B]⟩| than the
+textbook proof. I discuss the relationship between these two proofs and show how the
+Cauchy-Schwarz inequality can be derived from the Aharonov-Vaidman identity. I give
+Aharonov-Vaidman based proofs of the Maccone-Pati uncertainty relations and I show
+how the Aharonov-Vaidman identity can be used to handle propagation of uncertainty
+in quantum mechanics. Finally, I show how the Aharonov-Vaidman identity can be
+extended to mixed states and discuss how to generalize the results to the mixed case.
+1
+Introduction
+Let A be a Hermitian operator on a Hilbert space H. Then, for any (not necessarily nor-
+malized) vector |ψ⟩ ∈ H,
+A |ψ⟩ = ⟨A⟩ |ψ⟩ + ∆A
+��ψ⊥
+A
+�
+,
+(1)
+where ⟨A⟩ = ⟨ψ|A|ψ⟩ / ⟨ψ|ψ⟩ is the expectation value of A, ∆A =
+�
+⟨A2⟩ − ⟨A⟩2 is its
+standard deviation, and
+��ψ⊥
+A
+�
+is a vector that is orthogonal to |ψ⟩, has equal norm
+�
+ψ⊥
+A
+��ψ⊥
+A
+�
+=
+⟨ψ|ψ⟩, and depends on the operator A.
+Equation (1) is the Aharonov-Vaidman Identity, which first appeared in [1].
+Yakir
+Aharonov has stated that he “[does not] understand why it doesn’t appear in every quantum
+book” [2]. The main purpose of this article is to explain why it should appear in undergrad-
+uate quantum mechanics textbooks1.
+1Other demonstrations of the usefulness of the Aharonov-Vaidman identity include its use in the proof
+that, for any state |ψ⟩ and any observable A, |ψ⟩⊗n is an approximate eigenstate of the observable ¯A =
+1
+n
+�n
+j=1 Aj for large n, where Aj refers to A acting on the jth subsystem [1], and its use in deriving the
+minimum time required for evolution to an orthogonal quantum state [3].
+1
+arXiv:2301.08679v1  [quant-ph]  20 Jan 2023
+
+The uncertainty relation that is proved most often in quantum mechanics classes and
+textbooks is the Robertson relation [4]:
+∆A∆B ≥ 1
+2 |⟨[A, B]⟩| ,
+(2)
+where [A, B] = AB − BA is the commutator.
+As pointed out by Schrödinger [5], the Robertson relation can be extended to
+(∆A)2 (∆B)2 ≥
+����
+1
+2 ⟨{A, B}⟩ − ⟨A⟩ ⟨B⟩
+����
+2
++
+����
+1
+2 ⟨[A, B]⟩
+����
+2
+,
+(3)
+where {A, B} = AB + BA is the anti-commutator.
+Although not often emphasized in quantum mechanics classes, the Schrödinger relation
+is not harder to prove than the Robertson relation. In fact, the standard textbook proof of
+the Robertson relation effectively proves the Schrödinger relation and then throws away the
+anti-commutator term.
+The proof almost universally adopted in textbooks is based on the Cauchy-Schwarz in-
+equality. While this proof is elementary for those familiar with the mathematics of Hilbert
+spaces, it can be daunting for undergraduate physics students, who are likely encountering
+Hilbert spaces for the first time along with quantum mechanics.
+In this article, I will review more direct proofs of eq. (2) and eq. (3) from the Aharonov-
+Vaidman identity that only make use of basic properties of complex numbers and inner
+products. These proofs previously appeared in [6] and the proof of the Robertson relation
+is also problem 3.10 in Aharonov and Rohrlich’s book “Quantum Paradoxes” [7]. The proof
+of the Aharonov-Vaidman identity itself is uses similar ideas to one of the standard proofs
+of the Cauchy-Schwarz identity, but is perhaps more memorable to undergraduate physics
+students because it uses concepts that have a physical meaning, i.e. expectation values and
+standard deviations. The proof of the Robertson and Schrödinger relations so obtained is not
+independent of the standard Cauchy-Schwarz based proof. I shall discuss their relationship
+and show that the Cauchy-Schwarz inequality can itself be derived from eq. (1). The main
+virtue of using the Aharonov-Vaidman based proof of the uncertainty relation is that it is
+more direct and involves fewer abstractions.
+To be clear, I am not against using or teaching the Cauchy-Schwarz inequality. It has
+been called “one of the most widely used and important inequalities in all of mathematics”
+[8]. In fact, the Aharonov-Vaidman based proof still uses one instance of the Cauchy-Schwarz
+inequality, namely that if |ψ⟩ and |φ⟩ are unit vectors then |⟨φ|ψ⟩| ≤ 1. But this is easily
+motivated by the idea that ⟨φ|ψ⟩ is a generalization of the cosine of an angle, and it is used in
+a more direct way than in the standard proof. Students of quantum mechanics also need to
+know the Cauchy-Schwarz inequality to prove that the Born rule always yields well-defined
+probabilities. Physics students should learn the Cauchy-Schwarz inequality. I just think it
+should be used in a less abstract way where possible.
+Besides the Robertson and Schrödinger relations, many other uncertainty relations are
+known. Indeed, since uncertainty relations have found applications in quantum information
+2
+
+science [9, 10, 11, 12, 13, 14, 15] and quantum foundations [16, 17], proving new ones has
+become something of a sport. The two most common classes of uncertainty relations are
+those based on entropy [18] and those based on standard deviations [4, 5, 19]. Many of the
+standard deviation based relations can be derived from the Aharonov-Vaidman relation. I
+include a proof of the Maccone-Pati uncertainty relations [20] to illustrate this. While these
+are not the most recent or tightest known uncertainty relations, I include them because
+they have a simple and elegant Aharonov-Vaidman based proof. For more recent work on
+standard deviation uncertainty relations, see [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
+33, 34, 35, 36, 37, 38, 39, 40, 41].
+Another place where relationships between standard deviations are important is in the
+propagation of uncertainty. In classical statistics, if random variables X1, X2, · · · , Xn have
+standard deviations ∆X1, ∆X2, · · · ∆Xn then a function of them f(X1, X2, · · · , Xn) has stan-
+dard deviation ∆f that is a function of ∆X1, ∆X2, · · · ∆Xn (and their correlations if the
+variables are not independent). Formulas for the propagation of uncertainty tell us how to
+compute this function, and are commonly used to estimate experimental errors. In quantum
+mechanics, similar formulas can be derived relating the standard deviations of observables.
+They differ from their classical counterparts due to the fact that quantum observables do not
+commute, but provided this is taken care of they can be derived by the same methods as in
+the classical case. However, they can alternatively be derived from the Aharonov-Vaidman
+identity, as I shall explain.
+Although the Aharonov-Vaidman identity is usually discussed for pure quantum states,
+it can be extended to mixed states, either by use of purification or an equivalent concept
+called an amplitude operator. Relations between standard deviations can be extended to
+mixed states, but obtaining tight bounds is sometimes more difficult than in the pure case
+due to the need to optimize over all purifications or amplitude operators that can represent
+a given mixed state.
+The remainder of this article is structured as follows. Section 2 gives the proof of the
+Aharonov-Vaidman identity and a corollary that is useful for understanding the equality
+conditions in uncertainty relations.
+Section 3 presents the proof of the Robertson and
+Schrödinger relations based on the Aharonov-Vaidman identity. Section 4 explains the rela-
+tionship with the standard textbook proof of the Robertson relation and explains how the
+Cauchy-Schwarz inequality can be derived from the Aharonov-Vaidman identity. Section 5
+comments on the effective teaching of the Robertson uncertainty relations via the Aharonov-
+Vaidman identity. Section 6 presents Aharonov-Vaidman based proofs of the Maccone-Pati
+uncertainty relations. Section 7 describes how to use the Aharonov-Vaidman identity to
+derive formulas for the propagation of quantum uncertainty. Section 8 explains how to gen-
+eralize the Aharonov-Vaidman relation to mixed states using amplitude operators. (The
+relationship between amplitude operators and purifications is discussed in appendix A.) Fi-
+nally, section 9 presents the summary and conclusions.
+I intend this article to be pedagogical and self-contained, so as to be accessible to under-
+graduate students and anyone teaching introductory quantum mechanics.
+3
+
+2
+Proof of the Aharonov Vaidman Identity
+Sometimes, it is useful to generalize the Aharonov-Vaidman identity to non-Hermitian op-
+erators, so we prove the more general version here.
+Proposition 2.1 (The Aharonov-Vaidman Identity). Let A be a linear operator on a Hilbert
+space H and let |ψ⟩ be a (not necessarily normalized) vector in H. Then,
+A |ψ⟩ = ⟨A⟩ |ψ⟩ + ∆A
+��ψ⊥
+A
+�
+,
+(4)
+where ⟨A⟩ = ⟨ψ|A|ψ⟩ / ⟨ψ|ψ⟩, ∆A =
+�
+⟨A†A⟩ − |⟨A⟩|2, and
+��ψ⊥
+A
+�
+is a vector orthogonal to
+|ψ⟩ that depends on both |ψ⟩ and A and satisfies
+�
+ψ⊥
+A
+��ψ⊥
+A
+�
+= ⟨ψ|ψ⟩.
+Note that, if A is Hermitian, then this reduces to eq. (1), where ⟨A⟩ and ∆A are the
+expectation value and standard deviation. In general, ⟨A⟩ is a complex number, but ∆A is
+always real and non-negative.
+For most of what we need to do, it is sufficient to consider the case where |ψ⟩ is a
+unit vector, in which case
+��ψ⊥
+A
+�
+is also a unit vector. The exception is the proof of the
+Cauchy-Schwarz inequality (proposition 4.1 in section 4), which uses the identity with an
+unnormalized vector.
+Proof. Given a vector |ψ⟩ ∈ H, any other vector |φ⟩ ∈ H can be written as |φ⟩ = α |ψ⟩ +
+β
+��ψ⊥�
+, where α and β are complex coefficients and
+��ψ⊥�
+is some vector that is orthogonal
+to |ψ⟩. By an appropriate rescaling of β, we can ensure that
+�
+ψ⊥��ψ⊥�
+= ⟨ψ|ψ⟩. Applying
+this to |φ⟩ = A |ψ⟩ gives
+A |ψ⟩ = α |ψ⟩ + β
+��ψ⊥�
+.
+(5)
+To determine α, take the inner product of eq. (5) with |ψ⟩, which gives
+⟨ψ|A|ψ⟩ = α ⟨ψ|ψ⟩ .
+(6)
+Rearranging this gives α = ⟨A⟩.
+To determine β, substitute α = ⟨A⟩ into eq. (5) and take the inner product of A |ψ⟩ with
+itself to obtain
+⟨ψ| A†A |ψ⟩ = |⟨A⟩|2 ⟨ψ|ψ⟩ + |β|2 �
+ψ⊥��ψ⊥�
+= |⟨A⟩|2 ⟨ψ|ψ⟩ + |β|2 ⟨ψ|ψ⟩ ,
+where we have used
+�
+ψ⊥��ψ⊥�
+= ⟨ψ|ψ⟩.
+Rearranging and using
+�
+A†A
+�
+=
+�
+ψ
+��A†A
+��ψ
+�
+/ ⟨ψ|ψ⟩ gives
+|β|2 =
+�
+A†A
+�
+− |⟨A⟩|2 = (∆A)2.
+(7)
+This means that β = (∆A)eiθ for some phase angle θ. If we define
+��ψ⊥
+A
+�
+= eiθ ��ψ⊥�
+then
+��ψ⊥
+A
+�
+is still orthogonal to |ψ⟩, its norm is unchanged, and we have eq. (4).
+4
+
+The following corollary is useful for finding the conditions for equality in uncertainty
+relations.
+Corollary 2.2. In general, for two operators A and B, and for a unit vector |ψ⟩,
+�
+ψ⊥
+A
+��ψ⊥
+B
+�
+=
+�
+A†B
+�
+− ⟨A⟩∗ ⟨B⟩
+∆A∆B
+.
+(8)
+Proof. From proposition 2.1, we have
+A |ψ⟩ = ⟨A⟩ |ψ⟩ + ∆A
+��ψ⊥
+A
+�
+,
+(9)
+B |ψ⟩ = ⟨B⟩ |ψ⟩ + ∆B
+��ψ⊥
+B
+�
+.
+(10)
+Taking the inner product of these gives
+⟨ψ| A†B |ψ⟩ = ⟨A⟩∗ ⟨B⟩ + ∆A∆B
+�
+ψ⊥
+A
+��ψ⊥
+B
+�
+,
+(11)
+Rearranging gives the desired result.
+Note that, if A and B are Hermitian then we have
+�
+ψ⊥
+A
+��ψ⊥
+B
+�
+= ⟨AB⟩ − ⟨A⟩ ⟨B⟩
+∆A∆B
+.
+(12)
+If it is also the case that [A, B] = 0 then eq. (12) is the correlation, denoted corrA,B, that
+would be obtained from a joint measurement of A and B. The correlation is a well-known
+statistical measure of how two random variables are related to one another. Equation (12) is
+a formal generalization of the correlation, so we will also denote it corrA,B. However, if A and
+B do not commute then corrA,B is generally a complex number, there is no joint measurement
+of A and B of which corrA,B could be the correlation, and AB is not an observable.
+The real and imaginary parts of corrA,B are
+Re (corrA,B) = 1
+2
+��
+ψ⊥
+A
+��ψ⊥
+B
+�
++
+�
+ψ⊥
+B
+��ψ⊥
+A
+��
+=
+1
+2 ⟨{A, B}⟩ − ⟨A⟩ ⟨B⟩
+∆A∆B
+(13)
+Im (corrA,B) = 1
+2i
+��
+ψ⊥
+A
+��ψ⊥
+B
+�
+−
+�
+ψ⊥
+B
+��ψ⊥
+A
+��
+= ⟨[A, B]⟩
+2i∆A∆B ,
+(14)
+The real part is also a formal generalization of the correlation in that it reduces to the
+classical formula when A and B commute. We denote it RcorrA,B.
+3
+The Robertson and Schrödinger Uncertainty Relations
+We are now in a position to prove the Robertson and Schrödinger uncertainty relations.
+Proposition 3.1 (The Robertson Uncertainty Relation). Let A and B be two Hermitian
+operators on a Hilbert space H. Then, for any unit vector |ψ⟩ ∈ H
+∆A∆B ≥ 1
+2 |⟨[A, B]⟩| .
+(15)
+5
+
+Proof. From the Aharonov-Vaidman identity, we have
+A |ψ⟩ = ⟨A⟩ |ψ⟩ + ∆A
+��ψ⊥
+A
+�
+,
+(16)
+B |ψ⟩ = ⟨B⟩ |ψ⟩ + ∆B
+��ψ⊥
+B
+�
+.
+(17)
+Taking the inner product of these two equations and its complex conjugate gives
+⟨ψ|AB|ψ⟩ = ⟨A⟩ ⟨B⟩ + ∆A∆B
+�
+ψ⊥
+A
+��ψ⊥
+B
+�
+(18)
+⟨ψ|BA|ψ⟩ = ⟨A⟩ ⟨B⟩ + ∆A∆B
+�
+ψ⊥
+B
+��ψ⊥
+A
+�
+.
+(19)
+Subtracting these two equations gives
+⟨ψ|(AB − BA)|ψ⟩ = ∆A∆B
+��
+ψ⊥
+A
+��ψ⊥
+B
+�
+−
+�
+ψ⊥
+B
+��ψ⊥
+A
+��
+,
+(20)
+or,
+⟨[A, B]⟩ = ∆A∆B
+��
+ψ⊥
+A
+��ψ⊥
+B
+�
+−
+�
+ψ⊥
+B
+��ψ⊥
+A
+��
+,
+(21)
+Since
+�
+ψ⊥
+B
+��ψ⊥
+A
+�
+is the complex conjugate of
+�
+ψ⊥
+A
+��ψ⊥
+B
+�
+, we can rewrite this as
+⟨[A, B]⟩ = 2i∆A∆BIm
+��
+ψ⊥
+A
+��ψ⊥
+B
+��
+.
+(22)
+Taking the absolute value of both sides and rearranging gives
+∆A∆B
+��Im
+��
+ψ⊥
+A
+��ψ⊥
+B
+���� = 1
+2 |⟨[A, B]⟩| .
+(23)
+Because
+��ψ⊥
+A
+�
+and
+��ψ⊥
+B
+�
+are unit vectors, 0 ≤
+���
+ψ⊥
+A
+��ψ⊥
+B
+���2 ≤ 1, and hence the absolute value
+of the imaginary part of
+�
+ψ⊥
+B
+��ψ⊥
+A
+�
+is also bounded between 0 and 1. Hence, we have
+∆A∆B ≥ 1
+2 |⟨[A, B]⟩| .
+(24)
+The condition for equality in the Robertson relation is
+��Im
+��
+ψ⊥
+A
+��ψ⊥
+B
+���� = 1 or, equiva-
+lently, corrA,B = ±i. States that saturate the inequality are called (Robertson) intelligent
+states. The condition corrA,B = ±i can be used to find intelligent states, although this is not
+easier than solving for equality in the Robertson relation directly.
+Proposition 3.2 (The Schrödinger Uncertainty Relation). Let A and B be two Hermitian
+operators on a Hilbert space H. Then, for any unit vector |ψ⟩ ∈ H
+(∆A)2 (∆B)2 ≥
+����
+1
+2 ⟨{A, B}⟩ − ⟨A⟩ ⟨B⟩
+����
+2
++
+����
+1
+2 ⟨[A, B]⟩
+����
+2
+.
+(25)
+6
+
+Proof. Taking the sum of eq. (18) and eq. (19) gives
+⟨{A, B}⟩ = 2 ⟨A⟩ ⟨B⟩ + ∆A∆B
+��
+ψ⊥
+A
+��ψ⊥
+B
+�
++
+�
+ψ⊥
+B
+��ψ⊥
+A
+��
+,
+(26)
+or,
+⟨{A, B}⟩ − 2 ⟨A⟩ ⟨B⟩ = ∆A∆B
+��
+ψ⊥
+A
+��ψ⊥
+B
+�
++
+�
+ψ⊥
+B
+��ψ⊥
+A
+��
+.
+(27)
+Adding this to eq. (21) gives
+⟨{A, B}⟩ − 2 ⟨A⟩ ⟨B⟩ + ⟨[A, B]⟩ = 2∆A∆B
+�
+ψ⊥
+A
+��ψ⊥
+B
+�
+,
+(28)
+or,
+∆A∆B
+�
+ψ⊥
+A
+��ψ⊥
+B
+�
+= 1
+2 ⟨{A, B}⟩ − ⟨A⟩ ⟨B⟩ + 1
+2 ⟨[A, B]⟩ .
+(29)
+Now, because A and B are Hermitian, {A, B} is Hermitian and [A, B] is anti-Hermitian.
+Therefore ⟨{A, B}⟩ is real and ⟨[A, B]⟩ is imaginary. Further ⟨A⟩, ⟨B⟩, ∆A and ∆B are real.
+Therefore, taking the modulus squared of eq. (29) gives
+(∆A)2(∆B)2 ���
+ψ⊥
+A
+��ψ⊥
+B
+���2 =
+����
+1
+2 ⟨{A, B}⟩ − ⟨A⟩ ⟨B⟩
+����
+2
++
+����
+1
+2 ⟨[A, B]⟩
+����
+2
+.
+(30)
+Finally, because
+��ψ⊥
+A
+�
+and
+��ψ⊥
+B
+�
+are unit vectors, we have 0 ≤
+���
+ψ⊥
+A
+��ψ⊥
+B
+���2 ≤ 1, from which
+the result follows.
+The condition for equality in the Schrödinger relation is
+���
+ψ⊥
+A
+��ψ⊥
+B
+���2 = |corrA,B|2 = 1.
+States that saturate the inequality are called (Schrödinger) intelligent states. The condition
+|corrA,B|2 = 1 can be used to find intelligent states, although this is not easier than solving
+for equality in the Schrödinger relation directly.
+4
+The Textbook Proof and The Cauchy-Schwarz Inequal-
+ity
+The textbook proofs of the Robertson and Schrödinger uncertainty relations are based on
+the Cauchy-Schwarz inequality
+|⟨f|g⟩|2 ≤ ⟨f|f⟩ ⟨g|g⟩ .
+(31)
+Note that the proofs given in section 3 also make use of a special case of this inequality: that
+for unit vectors |⟨f|g⟩|2 ≤ 1. This is applied to |f⟩ =
+��ψ⊥
+A
+�
+, |g⟩ =
+��ψ⊥
+B
+�
+. My aim is not to
+eliminate any use of the Cauchy-Schwarz inequality, but just to argue that the proof is more
+memorable if the inequality is applied in a different way than in the standard proof.
+In the standard proof, the Cauchy-Schwarz inequality is applied to the two vectors |f⟩ =
+(A − ⟨A⟩) |ψ⟩ and |g⟩ = (B − ⟨B⟩) |ψ⟩ to obtain
+|⟨ψ|(A − ⟨A⟩)(B − ⟨B⟩)|ψ⟩|2 ≤
+�
+ψ
+��(A − ⟨A⟩)2��ψ
+� �
+ψ
+��(B − ⟨B⟩)2��ψ
+�
+.
+(32)
+7
+
+A few lines of messy algebra and cancellations, which I will spare you the details of, yields
+(∆A)2 (∆B)2 ≥
+����
+1
+2 ⟨{A, B}⟩ − ⟨A⟩ ⟨B⟩ + 1
+2 ⟨[A, B]⟩
+����
+2
+,
+(33)
+from which we can derive the Schrödinger and Robertson relations by recognizing the real
+and imaginary parts of the right hand side.
+As physics students do not often see the Cauchy-Schwarz inequality prior to their first
+course on quantum mechanics, most textbooks include a proof of this as well. One of the
+common proofs uses reasoning similar to that which we used to establish the Aharonov-
+Vaidman identity. It starts by recognizing that |g⟩ can be written as
+|g⟩ = α |f⟩ + β
+��f ⊥�
+,
+(34)
+where
+��f ⊥�
+is a unit vector that is orthogonal to |f⟩. To find α, take the inner product of
+this with |f⟩, which yields α = ⟨f|g⟩ / ⟨f|f⟩. Substituting this back into eq. (34) and then
+taking the inner product of |g⟩ with itself gives
+⟨g|g⟩ = |⟨f|g⟩|2
+⟨f|f⟩ + |β|2 .
+(35)
+The Cauchy-Schwarz inequality follows from this by recognizing that |β|2 is real and non-
+negative.
+Summarizing, the standard proof of the Robertson inequality consists of: proving the
+Cauchy-Schwarz inequality and then finding convenient vectors to insert into the inequality
+that will yield terms involving ∆A and ∆B after some algebra. From the Aharonov-Vaidman
+identity, we can see that the reason the choice |f⟩ = (A − ⟨A⟩) |ψ⟩ and |g⟩ = (B − ⟨B⟩) |ψ⟩
+is guaranteed work is that |f⟩ = ∆A
+��ψ⊥
+A
+�
+and |g⟩ = ∆B
+��ψ⊥
+B
+�
+.
+After inserting these choices, one has to multiply out and simplify the expressions in the
+Cauchy-Schwarz inequality. This involves recognizing things like ⟨A⟩ ⟨ψ|A|ψ⟩ = ⟨A⟩2 and
+then canceling several terms. It is difficult for students to follow the full details of this in a
+lecture. In the approach using the Aharonov-Vaidman relation, we already have expressions
+involving ∆A and ∆B, so it is easier to see how to get an expression involving ∆A∆B. This
+expression has fewer terms and there is less cancellation to do.
+Although the approach using the Aharonov-Vaidman identity uses the Cauchy-Schwarz
+inequality in a less convoluted way, it uses similar mathematical ideas. For vectors |f⟩ and
+|g⟩, we can write |g⟩ in terms of |f⟩ and an orthogonal vector, as in the proof of Cauchy-
+Schwarz, or we can write both vectors in terms of a third vector |h⟩ as
+|f⟩ = α1 |h⟩ + β1
+��h⊥
+f
+�
+(36)
+|g⟩ = α2 |h⟩ + β2
+��h⊥
+g
+�
+,
+(37)
+where
+��h⊥
+f
+�
+and
+��h⊥
+g
+�
+are (generally different) vectors orthogonal to |h⟩ and α1, β1, α2, β2 are
+complex coefficients. This is what we do in the proof of the Aharonov-Vaidman identity with
+8
+
+the choices |f⟩ = A |ψ⟩, |g⟩ = B |ψ⟩ and |h⟩ = |ψ⟩. The advantage of this approach is that
+it immediately yields expressions involving the expectation values and standard deviations
+of the observables, which it is easy to see what to do with in order to get the uncertainty
+relations. From this point of view, the standard proof looks like shoehorning something into
+the Cauchy-Schwarz inequality that will yield standard deviations, and then backtracking to
+a point more easily obtained from the Aharonov-Vaidman identity. At the end of the day,
+both approaches use the same mathematics, but the Aharonov-Vaidman approach does so
+in a simpler and more direct way.
+I would go so far as to say that whenever you are tempted to use the Cauchy-Schwarz
+inequality to prove a relationship between standard deviations of observables in quantum
+mechanics, you will have an easier time working from the Aharonov-Vaidman identity (and
+the special case |⟨f|g⟩|2 ≤ 1 of the Cauchy-Schwarz inequality for unit vectors) instead.
+Section 6 and Section 7 give more examples of this.
+I end this section by showing that you can prove the Cauchy-Schwarz inequality from the
+Aharonov-Vaidman identity. I include this not because I think it is the best way to prove
+the Cauchy-Schwarz inequality, but because finding alternative proofs of the Cauchy-Schwarz
+inequality is the mathematician’s equivalent of the sport of finding new uncertainty relations
+in quantum mechanics. It also shows that, in principle, there is nothing that can be proved
+using the Cauchy-Schwarz inequality that could not be proved using the Aharonov-Vaidman
+identity. Of course, outside the context of standard deviations in quantum mechanics, using
+the Aharonov-Vaidman identity instead of the Cauchy-Schwarz inequality is unlikely to yield
+a better proof.
+Proposition 4.1 (Cauchy-Schwarz Inequality). Let |f⟩ and |g⟩ be two vectors in a Hilbert
+space H. Then
+|⟨f|g⟩|2 ≤ ⟨f|f⟩ ⟨g|g⟩ .
+(38)
+Proof. First note that the inequality trivially holds whenever ⟨f|g⟩ = 0 and that ⟨f|f⟩ = 0
+implies ⟨f|g⟩ = 0. Therefore, we can assume that both ⟨f|g⟩ ̸= 0 and ⟨f|f⟩ > 0.
+Let P = |g⟩⟨g|. Note this is not necessarily a projector because |g⟩ does not have to be
+normalized, but it is a Hermitian operator. Applying the Aharonov-Vaidman identity to P
+and |f⟩ gives
+P |f⟩ = ⟨P⟩ |f⟩ + ∆P
+��f ⊥
+P
+�
+,
+(39)
+or equivalently
+|g⟩ ⟨g|f⟩ = ⟨f|g⟩ ⟨g|f⟩
+⟨f|f⟩
+|f⟩ + ∆P
+��f ⊥
+P
+�
+.
+(40)
+Taking the inner product with
+��f ⊥
+P
+�
+gives
+�
+f ⊥
+P
+��g
+�
+⟨g|f⟩ = ∆P ⟨f|f⟩ ,
+(41)
+where we used the fact that
+�
+f ⊥
+P
+��f ⊥
+P
+�
+= ⟨f|f⟩ Rearranging and taking the complex conjugate
+gives
+�
+g
+��f ⊥
+P
+�
+= ∆P ⟨f|f⟩
+⟨f|g⟩
+.
+(42)
+9
+
+Now, taking the inner product of eq. (40) with |g⟩ gives
+⟨g|g⟩ ⟨g|f⟩ = ⟨f|g⟩ ⟨g|f⟩
+⟨f|f⟩
+⟨g|f⟩ + ∆P
+�
+g
+��f ⊥
+P
+�
+.
+(43)
+Multiplying both sides by ⟨f|f⟩ / ⟨g|f⟩ gives
+⟨f|f⟩ ⟨g|g⟩ = ⟨f|g⟩ ⟨g|f⟩ + ∆P
+�
+g
+��f ⊥
+P
+�
+⟨f|f⟩
+⟨g|f⟩
+.
+(44)
+Substituting eq. (42) into this gives
+⟨f|f⟩ ⟨g|g⟩ = ⟨f|g⟩ ⟨g|f⟩ + (∆P)2 |⟨f|f⟩|2
+⟨f|g⟩ ⟨g|f⟩
+,
+(45)
+or
+⟨f|f⟩ ⟨g|g⟩ = |⟨f|g⟩|2 + (∆P)2 |⟨f|f⟩|2
+|⟨f|g⟩|2
+.
+(46)
+Now, the terms ∆P, ⟨f|f⟩ and |⟨f|g⟩| are all real and non-negative. Hence,
+⟨f|f⟩ ⟨g|g⟩ ≥ |⟨f|g⟩|2 .
+(47)
+5
+Pedagogical Notes
+In order to teach the Robertson uncertainty relation via the Aharonov-Vaidman identity,
+you first have to establish the Aharonov-Vaidman identity. For the purposes of proving the
+Robertson uncertainty relation, it is sufficient to restrict the operator in the identity to be
+Hermitian and the vector |ψ⟩ to be a unit vector, as I shall in this section.
+In my experience, not all students immediately understand why, given a unit vector |ψ⟩,
+any other unit vector |φ⟩ can be written as
+|φ⟩ = α |ψ⟩ + β
+��ψ⊥�
+,
+(48)
+where
+��ψ⊥�
+is a unit vector orthogonal to |ψ⟩. They will probably have seen Gram-Schmidt
+orthogonalization in a linear algebra class, but may have difficulty using that knowledge
+here due to the jump to abstract Hilbert spaces and Dirac notation. To aid intuition, I
+remark that |ψ⟩ and |φ⟩ span a two-dimensional subspace of H and show them fig. 1. By
+the process of Gram-Schmidt orthogonalization, we can construct an orthornormal basis for
+this subspace consisting of |ψ⟩ and
+��ψ⊥�
+=
+1
+�
+1 − |⟨φ|ψ⟩|2 (|φ⟩ − |ψ⟩ ⟨ψ|φ⟩) ,
+(49)
+10
+
+|ψ⟩
+|φ⟩
+|ψ⊥⟩
+Figure 1: Diagram showing that there exists a unit vector
+��ψ⊥�
+such that |ψ⟩ and
+��ψ⊥�
+form
+an orthogonal basis for the two dimensional subspace of H spanned by |ψ⟩ and |φ⟩.
+from which we have eq. (48) with α = ⟨ψ|φ⟩ and β =
+�
+1 − |⟨φ|ψ⟩|2.
+In my quantum mechanics classes, I set students in-class activities that involve things
+like deriving important equations or making order of magnitude estimates. These take about
+5-10 minutes each and are done in pairs. I usually do two or three such activities per class.
+I believe this increases active engagement and retention of the main principles. I try to
+reduce the number of long derivations that I do myself on the board because I think they
+cause confusion about what the most important equations are and the derivations are rarely
+remembered by the students. However, I also do not want to set the students a long and
+complicated derivation to do themselves in class, so I try to find shorter derivations that
+they can do with guidance instead. The proof of the Robertson relation from the Aharonov-
+Vaidman relation is better suited to this approach than the standard proof.
+After establishing eq. (48), I set students the following activity.
+In Class Activity
+Given that A |ψ⟩ = α |ψ⟩ + β
+��ψ⊥�
+, find α and β in terms of the expectation value ⟨A⟩
+and standard deviation ∆A of A in the state |ψ⟩.
+Although some students can do this straight away, most need some help. During the
+course of the activity, I walk around the class to get an idea of how they are doing. When
+it seems like many students are stuck, I reveal the following three hints in sequence.
+Hints
+1. Try taking the inner product of A |ψ⟩ = α |ψ⟩ + β
+��ψ⊥�
+with other states.
+2. Try taking the inner product of A |ψ⟩ with |ψ⟩.
+3. Try taking the inner product of A |ψ⟩ with itself.
+Although most students can get α = ⟨A⟩ either straight away or after the first hint,
+|β| = ∆A is more challenging. After taking the inner product with |ψ⟩, the obvious instinct
+is to take the inner product with
+��ψ⊥�
+, which does not help, so the third hint is usually
+needed. After this, it is a short hop to the Robertson relation via the proof given in section 3.
+I think it would be more difficult to teach the standard proof in this way. One would
+either have to ask the students to derive the Cauchy-Schwarz inequality for themselves or
+11
+
+derive the Robertson relation from Cauchy-Schwarz.
+The former is a bit abstract for a
+quantum mechanics class and the latter involves a lot of algebra and cancellations with a
+high potential for making mistakes. Both would require a large number of hints. In contrast,
+the proof of the Aharonov-Vaidman identity is relatively short, and I think that students
+who retain the identity are more likely to be able to reconstruct the proof of the Robertson
+relation for themselves.
+6
+Other Uncertainty Relations for Standard Deviations
+Despite the ubiquity of the Schrödinger-Robertson uncertainty relations in quantum me-
+chanics classes, there are good reasons to go beyond them. For example, consider a spin-
+1/2 particle with spin operators Sx, Sy and Sz. For this case, the Robertson uncertainty
+is ∆Sx∆Sy ≥ ℏ |⟨Sz⟩|. Let |x+⟩ be the spin-up state in the x direction. For this state
+we have ⟨Sz⟩ = 0, which is perfectly valid because |x+⟩ is an eigenstate of Sx and hence
+∆Sx = 0. However, because [Sx, Sy] ̸= 0 there is necessarily some uncertainty in Sy and in
+fact ∆Sy = ℏ/2. The Schrödinger relation also yields ∆Sx∆Sy ≥ 0. So the Schrödinger-
+Robertson relations do not capture all uncertainty trade-offs that necessarily exist in quan-
+tum mechanics.
+More generally, for bounded operators A and B, any uncertainty relation of the form
+∆A∆B ≥ f (A, B, |ψ⟩) for some function f must necessarily have f (A, B, |ψ⟩) = 0 whenever
+|ψ⟩ is an eigenstate of A or B. For this reason, it makes sense to seek uncertainty relations
+that bound the sum of standard deviations ∆A + ∆B, the sum of variances (∆A)2 + (∆B)2,
+or more exotic combinations. We shall discuss the Maccone-Pati relations, and some simple
+generalizations, in this section.
+Uncertainty relations are classified as either state dependent or state independent, de-
+pending on whether the right hand side of the inequality depends on the state |ψ⟩. For two
+observables A and B, a state dependent uncertainty relation is of the form f(∆A, ∆B) ≥
+g(A, B, |ψ⟩), where f and g are specified functions, whereas a state independent uncertainty
+relation would be of the form f(∆A, ∆B) ≥ g(A, B), noting that g is no longer allowed to
+depend on |ψ⟩.
+On the face of it, a state dependent uncertainty relation is a strange idea, since, for any
+given normalized state |ψ⟩, we can always just calculate the uncertainties ∆A and ∆B and
+get the exact value of f(∆A, ∆B). Therefore, bounds on uncertainty that apply to all states
+seem more useful.
+However, a state dependent uncertainty relation can be a useful step in deriving a state
+independent one. This can happen in two ways. First, it may happen that, for a particular
+choice of the observables A and B, the function g(A, B, |ψ⟩) turns out not to depend on |ψ⟩.
+For example, the Robertson relation ∆A∆B ≥ 1
+2 |⟨ψ|[A, B]|ψ⟩| is state dependent, but if we
+choose A = x, B = p, then |⟨ψ|[A, B]|ψ⟩| = 1 and so we get the Heisenberg relation ∆x∆p ≥
+ℏ
+2, which is state independent. Since the main point of proving the Robertson uncertainty
+relation in a quantum mechanics class is to give a rigorous derivation of the Heisenberg
+relation, its state dependence does no harm. However, the utility of the Robertson relation
+12
+
+for other classes of observable, such as spin components, is more questionable. Despite the
+fact that I have asked students to compute it for states of a spin-1/2 particle as a homework
+problem, I do not think there is ever a need to do this in practice, as it is just as easy to
+calculate the exact uncertainties.
+The second way of obtaining a state independent uncertainty relation from a state de-
+pendent one is to optimize, i.e. if f(∆A, ∆B) ≥ g(A, B, |ψ⟩) then2
+f(∆A, ∆B) ≥ min
+|ψ⟩ g(A, B, |ψ⟩).
+(50)
+Of course, if f(∆A, ∆B) = ∆A∆B and A and B are bounded operators then this leads
+to the trivial relation ∆A∆B ≥ 0 because we can choose |ψ⟩ to be an eigenstate of either
+A or B. However, for sums and more general combinations of observables, optimization can
+lead to a nontrivial relation.
+Further, if we are considering a set of experiments that can only prepare a subset of
+the possible states, then we can get an uncertainty relation that applies to those states by
+optimizing over the subset. An example might be experiments in which we can only prepare
+the system in a Gaussian state. Although this does not yield a state independent uncertainty
+relation, it is more useful than a completely state dependent one, as it allows us to bound
+the possible uncertainties for a class of relevant states.
+To summarize, state dependent uncertainty relations are a strange idea, and I am not
+sure whether they would ever have been considered had not Robertson introduced one as
+a way-point in proving the Heisenberg relation. However, they can be useful in proving
+more generally applicable uncertainty relations. The relations that we discuss here are state
+dependent.
+The remainder of this section is structured as follows. In section 6.1 we prove two propo-
+sitions called the sum relations that will be used repeatedly using the Aharonov-Vaidman
+identity. In section 6.2, we give an Aharonov-Vaidman based proof of the Maccone-Pati
+uncertainty relations, and in in section 6.3 we give some simple generalizations.
+6.1
+The Sum Relations
+Proposition 6.1. Let A and B be linear operators acting on H. Then, for any |ψ⟩ ∈ H,
+∆(A + B)
+��ψ⊥
+A+B
+�
+= ∆A
+��ψ⊥
+A
+�
++ ∆B
+��ψ⊥
+B
+�
+.
+Proof. Apply the Aharonov-Vaidman identity to A + B in two different ways. The first way
+is
+(A + B) |ψ⟩ = ⟨A + B⟩ |ψ⟩ + ∆(A + B)
+��ψ⊥
+A+B
+�
+= (⟨A⟩ + ⟨B⟩) |ψ⟩ + ∆(A + B)
+��ψ⊥
+A+B
+�
+,
+(51)
+2The minimum in eq. (50) may have to be replaced by an infimum, depending on the Hilbert space that
+the observables are defined on.
+13
+
+and the second is
+(A + B) |ψ⟩ = A |ψ⟩ + B |ψ⟩
+= (⟨A⟩ + ⟨B⟩) |ψ⟩ + ∆A
+��ψ⊥A�
++ ∆B
+��ψ⊥
+B
+�
+.
+(52)
+Subtracting eq. (52) from eq. (51) and rearranging gives the desired result.
+The next proposition comes from [19]. Here, the proof relies on proposition 6.1 and so is
+based on the Aharonov-Vaidman relation. The original proof uses a different method and is
+a little more complicated.
+Proposition 6.2 (The Sum Relation). Let A and B be two linear operators acting on a
+Hilbert space H. Then,
+∆(A + B) ≤ ∆A + ∆B.
+Proof. Let |ψ⟩ in proposition 6.1 be a unit vector. Then, starting from ∆(A + B)
+��ψ⊥
+A+B
+�
+=
+∆A
+��ψ⊥
+A
+�
++ ∆B
+��ψ⊥
+B
+�
+and taking the inner product with
+��ψ⊥
+A+B
+�
+gives
+∆(A + B) = ∆A
+�
+ψ⊥
+A+B
+��ψ⊥
+A
+�
++ ∆B
+�
+ψ⊥A+B��ψ⊥
+B
+�
+.
+The left hand side of this equation is a real number, so the right hand side must be too.
+Therefore, we can take the real part of each term to give
+∆(A + B) = ∆ARe
+��
+ψ⊥
+A+B
+��ψ⊥
+A
+��
++ ∆BRe
+��
+ψ⊥A+B��ψ⊥
+B
+��
+,
+but the real part of an inner product between two unit vectors is ≤ 1, so we have
+∆(A + B) ≤ ∆A + ∆B.
+From the proof, we see that the equality condition for the sum relation is
+Rcorr(A + B, A) = Rcorr(A + B, B) = 1.
+Remark 6.3. For a set of linear operators A1, A2, · · · , An on a Hilbert space H, Proposi-
+tion 6.1 is easily generalized to
+∆
+� n
+�
+j=1
+Aj
+� ���ψ⊥
+�n
+j=1 Aj
+�
+=
+n
+�
+j=1
+∆Aj
+���ψ⊥
+Aj
+�
+,
+(53)
+by applying the Aharonov-Vaidman identity to �n
+j=1 Aj. Similarly, proposition 6.2 is easily
+generalized to
+∆
+� n
+�
+j=1
+Aj
+�
+≤
+n
+�
+j=1
+∆Aj.
+(54)
+by taking the inner product of eq. (53) with
+���ψ⊥
+�n
+j=1 Aj
+�
+. We will also refer to the generaliza-
+tion in eq. (54) as the sum relation.
+14
+
+6.2
+The Maccone-Pati Uncertainty Relations
+Between the time of Robertson’s uncertainty relation and now, there has always been some
+literature on uncertainty relations for variances and standard deviations. However, the field
+was reinvigorated in 2014, when Maccone and Pati [20] proved a pair of uncertainty relations
+for sums of variances, which always give a nontrivial bound, even in the case of an eigenstate
+of an observable.
+Here, we give Aharonov-Vaidman based proofs of the Maccone-Pati relations3.
+Theorem 6.4 (The First Maccone-Pati Uncertainty Relation). Let A and B be Hermitian
+operators on a Hilbert space H and let |ψ⟩ ∈ H be a unit vector. Then,
+(∆A)2 + (∆B)2 ≥ ±i ⟨[A, B]⟩ +
+���
+ψ⊥��(A ∓ iB)
+��ψ
+���2 ,
+(55)
+where
+��ψ⊥�
+is any unit vector orthogonal to |ψ⟩.
+Proof. We will prove (∆A)2 + (∆B)2 ≥ −i ⟨[A, B]⟩ +
+���
+ψ⊥��(A + iB)
+��ψ
+���2 by applying the
+Aharonov-Vaidman identity to (A + iB). The proof of the other inequality follows by re-
+placing A + iB with A − iB. Note that, even though A and B are Hermitian, A + iB is not,
+so it is crucial that we previously generalized the Aharonov-Vaidman identity to arbitrary
+linear operators.
+Applying the Aharonov-Vaidman identity to A + iB gives
+(A + iB) |ψ⟩ = (⟨A⟩ + i ⟨B⟩) |ψ⟩ + ∆(A + iB)
+��ψ⊥
+A+iB
+�
+.
+Taking the inner product with any unit vector
+��ψ⊥�
+orthogonal to |ψ⟩ gives
+�
+ψ⊥��(A + iB)
+��ψ
+�
+= ∆(A + iB)
+�
+ψ⊥��ψ⊥
+A+iB
+�
+,
+and taking the modulus squared of this gives
+���
+ψ⊥��(A + iB)
+��ψ
+���2 = (∆(A + iB))2 ���
+ψ⊥��ψ⊥
+A+iB
+���2 .
+Now,
+���
+ψ⊥��ψ⊥
+A+iB
+��� ≤ 1, so
+(∆(A + iB))2 ≥
+���
+ψ⊥��(A + iB)
+��ψ
+���2 .
+The result now follows by expanding (∆(A + iB))2 as follows.
+(∆(A + iB))2 = ⟨(A − iB)(A + iB)⟩ − ⟨A − iB⟩ ⟨A + iB⟩
+=
+�
+A2�
++
+�
+B2�
++ i ⟨[A, B]⟩ − ⟨A⟩2 − ⟨B⟩2
+= (∆A)2 + (∆B)2 + i ⟨[A, B]⟩ .
+3Although the Aharonov-Vaidman identity is used in [20], it is not used in the proofs of the uncertainty
+relations.
+15
+
+Theorem 6.5 (The Second Maccone-Pati Uncertainty Relation). Let A and B be linear
+operators on a Hilbert space H and let |ψ⟩ ∈ H be a unit vector. Then,
+(∆A)2 + (∆B)2 ≥ 1
+2
+���
+ψ⊥
+A+B
+��(A + B)
+��ψ
+���2 .
+(56)
+Proof. Applying the Aharonov-Vaidman identity to A + B gives
+(A + B) |ψ⟩ = (⟨A⟩ + ⟨B⟩) |ψ⟩ + ∆(A + B)
+��ψ⊥
+A+B
+�
+.
+Taking the inner product with
+��ψ⊥
+A+B
+�
+gives
+�
+ψ⊥
+A+B
+��(A + B)
+��ψ
+�
+= ∆(A + B)
+≤ ∆A + ∆B,
+where the second line follows from the sum relation.
+We could stop here and regard ∆A+∆B ≥
+�
+ψ⊥
+A+B
+��(A + B)
+��ψ
+�
+as an uncertainty relation,
+but Maccone and Pati wanted a relation in terms of variances to compare to their first result.
+To do this, we take the modulus squared of both sides to obtain
+(∆A + ∆B)2 ≥
+���
+ψ⊥
+A+B
+��(A + B)
+��ψ
+���2 .
+The result now follows from the real number inequality x2 + y2 ≥ 1
+2(x + y)2 with x = ∆A
+and y = ∆B. For completeness, this inequality is proved as follows.
+0 ≤ (x − y)2 = x2 + y2 − 2xy
+⇒
+x2 + y2 ≥ 2xy
+⇒
+2x2 + 2y2 ≥ x2 + y2 + 2xy
+⇒
+2x2 + 2y2 ≥ (x + y)2
+⇒
+x2 + y2 ≥ 1
+2(x + y)2.
+6.3
+Generalizations
+Generalizations of the Maccone-Pati Uncertainty relations can be obtained by applying the
+Aharonov-Vaidman identity to more general linear combinations αA + βB, where α, β ∈ C.
+This gives
+(αA + βB) |ψ⟩ = (α ⟨A⟩ + β ⟨B⟩) |ψ⟩ + ∆(αA + βB)
+��ψ⊥
+αA+βB
+�
+.
+(57)
+Applying the strategy we used to prove theorem 6.4, we can take the inner product of this
+with an arbitrary unit vector
+��ψ⊥�
+that is orthogonal to |ψ⟩, which gives
+�
+ψ⊥��(αA + βB)
+��ψ
+�
+= ∆(αA + βB)
+�
+ψ⊥��ψ⊥
+αA+βB
+�
+.
+16
+
+We can now take the modulus squared of this and recognize that 0 ≤
+���
+ψ⊥��ψ⊥
+αA+βB
+���2 ≤ 1
+to obtain
+���
+ψ⊥��(αA + βB)
+��ψ
+���2 ≤ ∆(αA + βB).
+Next, we can expand ∆(αA + βB) and rearrange to obtain
+|α|2 (∆A)2 + |β|2 (∆B)2 ≥ −Re(α∗β) (⟨{A, B}⟩ − 2 ⟨A⟩ ⟨B⟩) − iIm (α∗β) ⟨[A, B]⟩
++
+���
+ψ⊥��(αA + βB)
+��ψ
+���2 .
+(58)
+Substituting α = 1, β = i and α = 1, β = −i immediately yields the first Maccone-Pati
+Uncertainty Relation.
+Alternatively, we can apply the strategy used to prove theorem 6.5. Starting from eq. (57),
+we can take the inner product with
+��ψ⊥
+αA+βB
+�
+and rearrange to obtain
+∆(αA + βB) =
+�
+ψ⊥
+αA+βB
+��(αA + βB)
+��ψ
+�
+.
+Using the sum relation, together with ∆(αA) = |α|∆A gives
+|α|∆A + |β|∆B ≥
+�
+ψ⊥
+αA+βB
+��(αA + βB)
+��ψ
+�
+.
+Finally, squaring and using the inequality x2 + y2 ≥ 1
+2(x + y)2 gives
+|α|2 (∆A)2 + |β|2 (∆B)2 ≥ 1
+2
+���
+ψ⊥
+αA+βB
+��(αA + βB)
+��ψ
+���2 .
+(59)
+The inequalities eq. (58) and eq. (59) are related to some of the generalizations of the
+Maccone-Pati uncertainty relations that have previously appeared in the literature [21, 28].
+For example, eq. (58) can be used to derive an uncertainty relation that has appeared in the
+literature under the name “weighted uncertainty relation” [28]. To do so, we set α =
+√
+λ,
+β = ±i/
+√
+λ in eq. (58), where λ > 0. This yields
+λ (∆A)2 + 1
+λ (∆B)2 ≥ ±i ⟨[A, B]⟩ + 1
+λ
+���
+ψ⊥��(λA ∓ iB)
+��ψ
+���2 .
+This is an uncertainty relation in its own right, but the relation in [28] comes from adding
+this to eq. (55), which yields
+(1+λ) (∆A)2+
+�
+1 + 1
+λ
+�
+(∆B)2 ≥ ±2i ⟨[A, B]⟩
+���
+ψ⊥
+1
+��(A ∓ iB)
+��ψ
+���2+1
+λ
+���
+ψ⊥
+2
+��(λA ∓ iB)
+��ψ
+���2 ,
+where
+��ψ⊥
+1
+�
+and
+��ψ⊥
+2
+�
+are (possibly different) unit vectors that are orthogonal to |ψ⟩.
+This is intended as a simple example of a generalization that is easily obtained from the
+Aharonov-Vaidman identity, but I expect many other uncertainty relations that are usually
+proved using the Cauchy-Schwarz inequality or the parallelogram law would also have simple
+Aharonov-Vaidman based proofs.
+17
+
+7
+Quantum Propagation of Uncertainty
+In this section, we develop generalizations of the classical formulas for the propagation of
+uncertainty. We start with the case of linear functions in section 7.1, for which exact formulas
+are easy to obtain, before moving on to the general, possibly nonlinear, case in section 7.2,
+for which we have to employ a Taylor series approximation.
+7.1
+Linear Functions
+We start with the simplest case: a sum of two observables. Classically, if A and B are
+random variables then
+[∆(A + B)]2 = (∆A)2 + (∆B)2 + 2∆A∆B corrA,B.
+(60)
+Consider an experiment consisting of multiple runs. On each run, the quantities A and B
+are measured. These quantities are formalized as random variables because we assume that
+our experiments are subject to random statistical fluctuations, and that the “true” values
+that we are seeking are the means ⟨A⟩ and ⟨B⟩ of these random processes. We then use
+the average values calculated from the data as estimates of ⟨A⟩ and ⟨B⟩, and the standard
+deviations as a measure of the error in our experiment. If we are actually interested in the
+quantity A + B then we would sum the averages to form our estimate of ⟨A + B⟩, and we
+would use eq. (60) to determine the error in our estimate of ⟨A + B⟩. Using eq. (60) in this
+way is called the propagation of uncertainty or propagation of error.
+If the random variables, A and B are independent, which would be the case if the ran-
+domness were due to independent statistical errors, then corrA,B = 0 and we would have
+[∆(A + B)]2 = (∆A)2 + (∆B)2 ,
+which is the formula for propagation of uncertainty that is most commonly used in practice.
+We now want to generalize these formulas by replacing classical random variables with
+quantum observables. The generalization of eq. (60) is as follows.
+Theorem 7.1. Let A and B be Hermitian operators on a Hilbert space H. Then,
+[∆(A + B)]2 = (∆A)2 + (∆B)2 + 2∆A∆B RcorrA,B
+(61)
+= (∆A)2 + (∆B)2 + ⟨{A, B}⟩ − 2 ⟨A⟩ ⟨B⟩
+(62)
+Proof. Proposition 6.1 implies that, for any unit vector |ψ⟩ ∈ H,
+∆(A + B)
+��ψ⊥
+A+B
+�
+= ∆A
+��ψ⊥
+A
+�
++ ∆B
+��ψ⊥
+B
+�
+.
+Taking the inner product of this with itself gives
+[∆ (A + B)]2 = (∆A)2 + (∆B)2 + ∆A∆B
+��
+ψ⊥
+A
+��ψ⊥
+B
+�
++
+�
+ψ⊥
+B
+��ψ⊥
+A
+��
+= (∆A)2 + (∆B)2 + 2∆A∆B Re
+��
+ψ⊥
+A
+��ψ⊥
+B
+��
+.
+Applying eq. (13) completes the proof.
+18
+
+Remark 7.2. For operators A1, A2, · · · , An and real numbers α1, α2, · · · , αn, theorem 7.1 is
+easily generalized to
+�
+∆
+� n
+�
+j=1
+αjAj
+��2
+=
+n
+�
+j=1
+α2
+j (∆Aj)2 +
+�
+j̸=k
+αjαk∆Aj∆Ak RcorrAj,Ak
+=
+n
+�
+j=1
+α2
+j (∆Aj)2 +
+�
+j̸=k
+αjαk (⟨{Aj, Ak}⟩ − 2 ⟨Aj⟩ ⟨Ak⟩) .
+Although theorem 7.1 is a true theorem about quantum observables, it cannot be used
+to propagate uncertainty in the same way as its classical counterpart. Classically, we can
+measure A and B together in the same run of the experiment. We can then estimate A + B
+by summing the average values of A and B that we found in the experiment. We also have
+all the information we need to calculate the uncertainty ∆(A + B), i.e. ∆A, ∆B, ⟨A⟩, ⟨B⟩
+and ⟨AB⟩, so we can determine the uncertainty without doing any more experiments.
+In quantum mechanics, this is not the case. When A and B do not commute, they cannot
+both be accurately measured on the same run of an experiment. We can still estimate their
+expectation values by measuring A on half of the runs of the experiment and B on the other
+half and taking averages. Since ⟨A + B⟩ = ⟨A⟩ + ⟨B⟩, summing these averages is still a way
+of estimating ⟨A + B⟩. However, we do not have enough information to calculate ∆(A + B).
+The reason is that ∆(A + B) is the uncertainty in a direct measurement of A + B. Since A
+and B do not commute, this requires a different experimental setup from a measurement of
+A and B alone.
+If we wanted to use eq. (61) to calculate ∆(A + B), we would also have to estimate
+⟨{A, B}⟩. The most straightforward way of doing this would be to measure the observable
+{A, B} = AB +BA, but this requires yet another different experimental setup, and one that
+is likely to be at least as complicated as measuring A + B directly.
+An exception to this are cases where {A, B} = cI for some constant c, in which case
+⟨{A, B}⟩ = c regardless of the state. In particular, this is true of the Pauli observables σx,
+σy, σz of a qubit for which {σj, σk} = δjkI, where j and k run over x, y, z. Therefore, if we
+measure σx on many qubits prepared in the same way and σy on another set of such qubits,
+we can estimate ⟨σx + σy⟩ and ∆(σx + σy) without doing any further experiments using the
+formula
+[∆ (σx + σy)]2 = (∆σx)2 + (∆σy)2 − 2 ⟨σx⟩ ⟨σy⟩ .
+When {A, B} ̸= cI, I do not know of any situations in which eq. (61) would be useful in
+practice, but from a theoretical point of view it is the appropriate generalization of eq. (60)
+to quantum mechanics, and this bolsters the case that RcorrA,B is the appropriate quantum
+generalization of the classical correlation.
+7.2
+Nonlinear Functions
+For nonlinear functions f(A, B) of two random variables A and B, it is common to use a first
+order Taylor expansion of f(A, B) about the point f(⟨A⟩ , ⟨B⟩) to derive an approximation
+19
+
+for the variance [∆f(A, B)]2 to second order in ∆A and ∆B. This yields the formula
+[∆f(A, B)]2 ≈
+�
+∂f
+∂A
+����
+A=⟨A⟩,B=⟨B⟩
+�2
+(∆A)2 +
+�
+∂f
+∂B
+����
+A=⟨A⟩,B=⟨B⟩
+�2
+(∆B)2
++ ∂f
+∂A
+����
+A=⟨A⟩,B=⟨B⟩
+∂f
+∂B
+����
+A=⟨A⟩,B=⟨B⟩
+∆A∆B corrA,B.
+To avoid cluttering notation, I will write ¯A for A = ⟨A⟩, so that we can more compactly
+write
+[∆f(A, B)]2 ≈ ∂f
+∂A
+����
+2
+¯
+A, ¯B
+(∆A)2 + ∂f
+∂B
+����
+2
+¯
+A, ¯B
+(∆B)2 + ∂f
+∂A
+���� ¯
+A, ¯B
+∂f
+∂B
+���� ¯
+A, ¯B
+∆A∆B corrA,B.
+(63)
+When A and B are independent, this reduces to
+[∆f(A, B)]2 ≈ ∂f
+∂A
+����
+2
+¯
+A, ¯B
+(∆A)2 + ∂f
+∂B
+����
+2
+¯
+A, ¯B
+(∆B)2 ,
+which is the most commonly used form.
+The quantum generalization of eq. (63) is as follows.
+Theorem 7.3. Let A and B be Hermitian operators on a Hilbert space H and consider a
+function f : H(H) × H(H) → H(H) where H(H) is the space of Hermitian operators on H.
+Then
+[∆f(A, B)]2 ≈ ∂f
+∂A
+����
+2
+¯
+A, ¯B
+(∆A)2 + ∂f
+∂B
+����
+2
+¯
+A, ¯B
+(∆B)2 + ∂f
+∂A
+���� ¯
+A, ¯B
+∂f
+∂B
+���� ¯
+A, ¯B
+∆A∆B RcorrA,B
+(64)
+where ≈ means equality to second order in ∆A and ∆B
+Proof. Consider the first order Taylor expansion of f(A, B) about the point f0 = f(⟨A⟩ , ⟨B⟩),
+f(A, B) ≈ f0 + ∂f
+∂A
+���� ¯
+A, ¯B
+A + ∂f
+∂B
+���� ¯
+A, ¯B
+B.
+Applying proposition 6.1 to this gives
+[∆f(A, B)]
+��ψ⊥
+f(A,B)
+�
+≈ ∂f
+∂A
+���� ¯
+A, ¯B
+∆A
+��ψ⊥
+A
+�
++ ∂f
+∂B
+���� ¯
+A, ¯B
+∆B
+��ψ⊥
+B
+�
+.
+Taking the inner product of this with itself gives
+[∆f(A, B)]2 ≈ ∂f
+∂A
+����
+2
+¯
+A, ¯B
+(∆A)2 + ∂f
+∂B
+����
+2
+¯
+A, ¯B
+(∆B)2 + ∂f
+∂A
+���� ¯
+A, ¯B
+∂f
+∂B
+���� ¯
+A, ¯B
+∆A∆B Re
+��
+ψ⊥
+A
+��ψ⊥
+B
+��
+= ∂f
+∂A
+����
+2
+¯
+A, ¯B
+(∆A)2 + ∂f
+∂B
+����
+2
+¯
+A, ¯B
+(∆B)2 + ∂f
+∂A
+���� ¯
+A, ¯B
+∂f
+∂B
+���� ¯
+A, ¯B
+∆A∆B RcorrA,B
+20
+
+Remark 7.4. For operators A1, A2, · · · , An and a function f(A1, A2, · · · , An), theorem 7.3 is
+easily generalized to
+[∆f (A1, A2, · · · , An)]2 ≈
+n
+�
+j=1
+∂f
+∂Aj
+����
+2
+¯
+A
+(∆Aj)2 +
+�
+j̸=k
+∂f
+∂Aj
+���� ¯
+A
+∂f
+∂Ak
+���� ¯
+A
+∆Aj∆Ak RcorrAj,Ak,
+where ¯A is shorthand for A1 = ⟨A1⟩ , A2 = ⟨A2⟩ , · · · An = ⟨An⟩.
+As a formula for propagating uncertainty, eq. (64) inherits all of the problems of eq. (61),
+but the problems are compounded further by use of the first order Taylor approximation.
+This approximation is valid when ∆A and ∆B are suitably small compared to ⟨A⟩, ⟨B⟩,
+f(⟨A⟩ , ⟨B⟩) and the derivatives of f(A, B) at A = ⟨A⟩, B = ⟨B⟩. This is often the case
+in classical experiments where everything can be measured with a small statistical error.
+However, in quantum mechanics, when A and B do not commute, the (various) uncertainty
+relations tell us that there is necessarily a trade-off between the size of ∆A and ∆B. If one
+of them is small, then the other might necessarily have to be large. For example, for the
+Pauli observables σx and σy, at least one of the uncertainties must be comparable in size to
+1, which is the largest possible value of ⟨σx⟩ or ⟨σy⟩.
+A case where the formula will work well is for a continuous variable system where ∆x ∼
+∆p ∼
+√
+ℏ, and ⟨x⟩, ⟨p⟩ are large compared to
+√
+ℏ. But this is a case where you would expect
+classical physics to be a good approximation anyway.
+I do not know whether there is a practical use of eq. (64), but it is nonetheless a correct
+formal generalization of eq. (63).
+8
+Dealing with Mixed States
+So far, we have dealt exclusively with the case of pure state vectors |ψ⟩. However, all of
+our results can be extended to more general density operators ρ, which can represent mixed
+states. The most familiar way to do this is to make use of the concept of a purification
+of a density operator. Given a density operator on a Hilbert space HS, where S stands
+for “system”, we can always find a pure state vector |ψ⟩SE ∈ HS ⊗ HE, where E is the
+“environment”, such that
+ρS = TrE (|ψ⟩⟨ψ|SE) ,
+and TrE is the partial trace over HE. You can then apply the Aharonov-Vaidman identity
+to operators of the form AS ⊗ IE acting on a purification to obtain results about the density
+operator ρS.
+However, to make the parallels to the pure state case as close as possible, I prefer to use
+an equivalent concept, called an amplitude operator. The equivalence between amplitude
+operators and purifications is discussed in appendix A
+Definition 8.1. Given a density operator ρS on a Hilbert space HS, an amplitude operator
+for ρS is a linear operator LS : HE → HS, where HE is any Hilbert space, such that
+ρS = LSL†
+S.
+21
+
+The reason for the name amplitude operator is that, in pure-state quantum mechanics, an
+amplitude is a complex number α such that |α|2 is a probability. A density operator is a non-
+commutative generalization of a probability distribution [42, 43], and hence an amplitude
+operator ought to be an operator that “squares” to a density operator.
+Given a density operator ρS, one obvious way of constructing an amplitude operator is
+to set HE = HS and LS = √ρS, but there are an infinite number of alternatives, as the
+following proposition shows
+Proposition 8.2. An operator LS : HE → HS is an amplitude operator for ρS if and only
+if
+LS = √ρSUS|E,
+where US|E : HE → HS is a semi-unitary operator, i.e. it satisfies US|EU †
+S|E = IS
+Proof. An operator of the form LS = √ρSUS|E obviously satisfies definition 8.1. For the other
+direction, assume LS is an amplitude operator. Like any operator, it may be decomposed in
+its polar decomposition LS = PSUS|E where PS is a positive semi-definite operator on HS,
+and US|E : HE → HS is semi-unitary4. The definition of an amplitude operator then implies
+that ρS = PSUS|EU †
+S|EPS = P 2
+S, so we must have PS = √ρS.
+Going back to the analogy between amplitudes and amplitude operators, multiplying an
+amplitude α by a phase factor eiφ does not change the probability it represents. Similarly,
+multiplying an amplitude operator LS by a semi-unitary VE|E′, i.e. an operator VE|E′ : HE′ →
+HE satisfying VE|E′V †
+E|E′ = IE, on the right does not change the density operator it represents.
+Although one might think it desirable to work directly with probabilities or density operators
+in order to eliminate these ambiguities, the mathematical manipulations we need to do in
+quantum mechanics are often linear in the amplitudes or amplitude operators, but would be
+nonlinear if you used probabilities or density operators. Therefore, it is often more convenient
+to live with the ambiguity.
+Since every operator has a polar decomposition, the only requirement for LS to be an
+amplitude operator for some density operator is that TrS
+�
+LSL†
+S
+�
+= 1.
+If we want to
+work with unnormalized density operators, i.e. any positive operator, then any operator
+LS : HE → HS is the amplitude operator for some (possibly unnormalized) density operator.
+This is analogous to the fact that any vector in HS represents a (possibly unnormalized) pure
+state.
+The strategy for generalizing the Aharonov-Vaidman identity, and everything that follows
+from it, is to replace the state vector |ψ⟩S with an amplitude operator LS. The reason this
+works is that the space of linear operators mapping HE to HS, which we denote LS|E, is itself
+a Hilbert space with inner product ⟨LS, MS⟩ = TrE
+�
+L†
+SMS
+�
+, known as the Hilbert-Schmidt
+4The polar decomposition is often only defined for square matrices, in which case HE = HS and US|E is
+unitary. Here, we use the generalization to non-square matrices (see e.g. [44]).
+22
+
+inner product5. Since the Aharonov-Vaidman identity is valid for any Hilbert space, it must
+be valid on LS|E as well.
+Proposition 8.3 (The Aharonov-Vaidman Identity for Operators). Let AS be a linear op-
+erator on a Hilbert space HS and let LS : HE → HS. Then,
+ASLS = ⟨AS⟩ LS + (∆AS) L⊥
+AS,
+(65)
+where ⟨AS⟩ = TrS
+�
+ASLSL†
+S
+�
+/TrS
+�
+LSL†
+S
+�
+, ∆A =
+��
+A†
+SAS
+�
+− |⟨AS⟩|2, and L⊥
+AS : HE →
+HS is an amplitude operator that is orthogonal to LS, i.e.
+TrE
+�
+L†
+SL⊥
+AS
+�
+= 0, satisfies
+TrS
+�
+L⊥
+ASL⊥†
+AS
+�
+= TrS
+�
+LSL†
+S
+�
+, and depends on both LS and AS.
+The proof of this proposition is essentially the same as the proof of the vector Aharonov-
+Vaidman identity (proposition 2.1) with the standard inner product replaced by the Hilbert-
+Schmidt inner product. The only difference is that the cyclic property of the trace is also
+needs to be used to write things in the exact form given in proposition 8.3. I leave this as
+an exercise for the reader.
+Since ρS = LSL†
+S is always a (possibly unnormalized) density operator, we can write
+⟨AS⟩ =
+TrS
+�
+ASLSL†
+S
+�
+TrE
+�
+L†
+SLS
+�
+= TrS (ASρS)
+TrE (ρS) .
+We can also introduce the density operator ρ⊥
+AS = L⊥
+ASL⊥†
+AS, which will be normalized in the
+same way as ρS, i.e., TrS
+�
+ρ⊥
+AS
+�
+= TrS (ρS).
+When LS is normalized so that ρS = LSL†
+S is a normalized density operator, i.e.,
+TrS
+�
+LSL†
+S
+�
+= 1, then ρ⊥
+AS is also normalized, i.e., TrS
+�
+ρ⊥
+AS
+�
+= 1.
+As defined, ρ⊥
+AS = L⊥
+ASL⊥†
+AS looks like it depends on the choice of amplitude operator
+LS. In fact, it does not. It only depends on ρS and AS. To see this, rewrite the operator
+Aharonov-Vaidman identity as
+L⊥
+AS =
+1
+∆AS
+(AS − ⟨AS⟩ IS) LS,
+and then we have,
+ρ⊥
+AS = L⊥
+ASL⊥†
+AS
+=
+1
+(∆AS)2 (AS − ⟨AS⟩ IS) LSL†
+S
+�
+A†
+S − ⟨AS⟩∗ IS
+�
+=
+1
+(∆AS)2 (AS − ⟨AS⟩ IS) ρS
+�
+A†
+S − ⟨AS⟩∗ IS
+�
+,
+5By the cyclic property of the trace, we can also write ⟨LS, MS⟩ = TrS
+�
+MSL†
+S
+�
+.
+23
+
+which is clearly independent of the choice of LS.
+Note that, although LS and L⊥
+AS are
+Hilbert-Schmidt orthogonal, ρS and ρ⊥
+AS are generally not.
+To generalize the results of this paper from state vectors to density operators, we replace
+the vector Aharonov-Vaidman identity with its operator counterpart applied to amplitude
+operators, and we replace the usual inner product with the Hilbert-Schmidt inner product.
+In many cases, the final result is independent of the amplitude operator used to represent
+the state. Although we use it in the proof, it drops out in the final result by only appearing
+in the combination LSL†
+S, as in the expression we derived for ρ⊥
+AS. In fact, the final formulas
+are usually the same as in the pure state case, except that we have to interpret ⟨AS⟩ as
+TrS (ASρS) rather than ⟨ψ|AS|ψ⟩.
+However, this is not true for the Maccone-Pati uncertainty relations and their general-
+izations, which do depend on the choice of amplitude operator LS.
+Theorem 8.4 (The First Maccone-Pati Uncertainty Relation for amplitude operators). Let
+AS and BS be Hermitian operators on a Hilbert space HS and let ρS be a normalized density
+operator on HS. Then,
+(∆A)2 + (∆B)2 ≥ ±i ⟨[A, B]⟩ +
+���TrE
+�
+L⊥†
+S (A ∓ iB)LS
+����
+2
+,
+(66)
+where LS : HE → HS is any amplitude operator for ρS, and L⊥
+S : HE → HS is any normalized
+amplitude operator orthogonal to LS that has the same input space HE.
+Note that, in order to obtain the tightest possible bound on (∆A)2 + (∆B)2, the right
+hand side of eq. (66) should be maximized over all possible choices of LS and L⊥
+S . To do this
+in practice, a bound on the largest dimension dE required to obtain the maximum is needed.
+I conjecture that dE = 2dS is sufficient because this allows LS and L⊥
+S to have orthogonal
+kernels on HE, but I do not have a proof of this.
+Theorem 8.5 (The Second Maccone-Pati Uncertainty Relation for amplitude operators).
+Let AS and BS be linear operators on a Hilbert space HS and let ρS be a normalized density
+operator on HS. Then,
+(∆AS)2 + (∆BS)2 ≥ 1
+2
+���TrE
+�
+L⊥†
+AS+BS(A + B)LS
+����
+2
+,
+(67)
+where LS is any amplitude operator for ρS and
+L⊥
+AS+BS =
+1
+∆(AS + BS) (AS + BS − ⟨AS + BS⟩ IS) LS.
+In this case, to obtain the tightest bound, we have to maximize the right hand side over
+LS. We do not have to separately optimize over L⊥
+AS+BS because it is a function of LS, AS
+and BS. However, its dependence on LS makes the problem into a complicated nonlinear
+optimization.
+24
+
+9
+Summary and Conclusions
+In this paper, I discussed how the standard textbook uncertainty relations of Robertson and
+Schrödinger can be derived from the Aharonov-Vaidman identity in a more direct way than
+the standard proof. I also demonstrated the identity’s usefulness in proving other uncertainty
+relations, such as the Maccone-Pati relations, and the quantum formulas for propagation of
+uncertainty. Finally, I gave a mixed-state generalization of the Aharonov-Vaidman identity
+in terms of amplitude operators. I hope that this has persuaded you that the Aharonov-
+Vaidman identity belongs in undergraduate textbooks and that it ought to be a first-line
+tool in proving relationships between standard deviations in quantum mechanics. I am sure
+there are other uncertainty relations that have an elegant Aharonov-Vaidman based proofs,
+and I hope to find new and useful uncertainty relations that have not been discovered before
+via this method.
+The Aharonov-Vaidman identity naturally gives rise to two quantum generalizations of
+the correlation, corrA,B and RcorrA,B. It would be interesting to determine whether these
+quantities have an operational meaning in the case where A and B do not commute. On the
+more formal side, perhaps there is a pseudo-probability representation of quantum mechanics,
+such as the Wigner function [45, 46, 47] or the Kirkwood-Dirac distribution [48, 49, 50],
+for which these are the correlations for observables as defined on the appropriate phase
+space. This might help to find uses for the propagation of error formulas in cases where the
+observables do not commute.
+Acknowledgments
+I would like to thank Yakir Aharonov for introducing me to the Aharonov-Vaidman iden-
+tity and emphasizing its importance. I would like to acknowledge (but not thank) the role
+played by the COVID19 pandemic shutdowns in giving me the opportunity to think about
+uncertainty relations and their pedagogy. This research was supported in part by the Fetzer
+Franklin Fund of the John E. Fetzer Memorial Trust and by grant number FQXi-RFPIPW-
+1905 from the Foundational Questions Institute and Fetzer Franklin Fund, a donor advised
+fund of Silicon Alley Community Foundation. This research was also supported in part by
+Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported
+by the Government of Canada through the Department of Innovation, Science, and Eco-
+nomic Development, and by the Province of Ontario through the Ministry of Colleges and
+Universities.
+References
+[1] Y. Aharonov and L. Vaidman. Properties of a quantum system during the time interval
+between two measurements. Phys. Rev. A, 41(1):11–20, 1990. doi:10.1103/PhysRevA.
+41.11.
+25
+
+[2] Y. Aharonov.
+Visiting researcher presentation.
+Talk at Perimeter Institute to PSI
+Masters Students: comment is made at 41:16, August 2011. URL: https://pirsa.
+org/11080091.
+[3] L. Vaidman. Minimum time for the evolution to an orthogonal state. Am. J. Phys.,
+60(2):182, 1992. doi:10.1119/1.16940.
+[4] H. P. Robertson. The Uncertainty Principle. Phys. Rev., 34(1):163–164, 1929. doi:
+10.1103/PhysRev.34.163.
+[5] E. Schrödinger. Zum heisenbergschen unschärfeprinzip. Sitzungsberichte der Preussis-
+chen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 14:296–303,
+1930.
+[6] L. Goldenberg and L. Vaidman. Applications of a simple quantum mechanical formula.
+Am. J. Phys., 64(8):1059, 1996. arXiv:quant-ph/9506030, doi:10.1119/1.18307.
+[7] Y. Aharonov and D. Rohrlich. Quantum Paradoxes: Quantum Theory for the Perplexed.
+Wiley, 2005. doi:10.1002/9783527619115.
+[8] M.
+J.
+Steele.
+The
+Cauchy-Schwarz
+Master
+Class:
+An
+Introduction
+to
+the
+Art
+of
+Mathematical
+Inequalities,
+chapter
+1,
+page
+1.
+Mathematical
+Association
+of
+America
+Problem
+Books.
+Cambridge
+University
+Press,
+2004.
+URL:
+http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/
+CSMC_index.html, doi:CBO9780511817106.
+[9] C. A. Fuchs and A. Peres. Quantum-state disturbance versus information gain: Un-
+certainty relations for quantum information. Phys. Rev. A, 53(4):2038, 1996. arXiv:
+quant-ph/9512023, doi:10.1103/PhysRevA.53.2038.
+[10] H. F. Hofmann and S. Takeuchi. Violation of local uncertainty relations as a signature
+of entanglement. Phys. Rev. A, 68(3):032103, 2003. arXiv:quant-ph/0212090, doi:
+10.1103/PhysRevA.68.032103.
+[11] O. Gühne.
+Characterizing entanglement via uncertainty relations.
+Phys. Rev.
+Lett., 92(11):117903, 2004.
+arXiv:quant-ph/0306194, doi:10.1103/PhysRevLett.
+92.117903.
+[12] M. Koashi. Unconditional security of quantum key distribution and the uncertainty
+principle.
+J. Phys.: Conf. Ser., 36:98–102, 2006.
+arXiv:quant-ph/0505108, doi:
+10.1088/1742-6596/36/1/016.
+[13] M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Renner. The uncertainty
+principle in the presence of quantum memory. Nature Physics, 6:659–662, 2010. arXiv:
+0909.0950, doi:doi.org/10.1038/nphys1734.
+26
+
+[14] A. S. Majumdar and T. Pramanik. Some applications of uncertainty relations in quan-
+tum information. International Journal of Quantum Information, 14(6):1640022, 2016.
+arXiv:1410.5974, doi:10.1142/S0219749916400220.
+[15] N. Yunger Halpern, A. Bartolotta, and J. Pollack. Entropic uncertainty relations for
+quantum information scrambling.
+Commun. Phys., 2:92, 2019.
+arXiv:1806.04147,
+doi:10.1038/s42005-019-0179-8.
+[16] J. Oppenheim and S. Wehner. The uncertainty principle determines the nonlocality
+of quantum mechanics. Science, 330(6007):1072–1074, 2010. arXiv:1004.2507, doi:
+10.1126/science.119206.
+[17] L. Catani, M. Leifer, G. Scala, D. Schmid, and R. W. Spekkens. What is nonclassical
+about uncertainty relations?
+Phys. Rev. Lett., 129(24):240401, 2022. arXiv:2207.
+11779, doi:10.1103/PhysRevLett.129.240401.
+[18] P. J. Coles, M. Berta, M. Tomamichel, and S. Wehner. Entropic uncertainty relations
+and their applications. Rev. Mod. Phys., 89(1):015002, 2017. arXiv:1511.04857, doi:
+10.1103/RevModPhys.89.015002.
+[19] A. K. Pati and P. K. Sahu. Sum uncertainty relation in quantum theory. Phys. Lett.
+A, 367(3):177–181, 2007. arXiv:quant-ph/0608092, doi:10.1016/j.physleta.2007.
+03.005.
+[20] L. Maccone and A. K. Pati. Stronger uncertainty relations for all incompatible ob-
+servables. Phys. Rev. Lett., 113(26):039902, 2015. arXiv:1407.0338, doi:10.1103/
+PhysRevLett.113.260401.
+[21] V. M. Bannur. General and stronger uncertainty relation. 2015. arXiv:1503.00405,
+doi:10.48550/arXiv.1503.00405.
+[22] J. L. Li and C. F. Qiao. Reformulating the quantum uncertainty relation. Scientific
+Reports, 5:12708, 2015. arXiv:1502.06292, doi:10.1038/srep12708.
+[23] Y. Yao, X. Xiao, X. Wang, and C. P. Sun. Implications and applications of the variance-
+based uncertainty equalities. Phys. Rev. A, 91:062113, 2015. arXiv:1503.00239, doi:
+10.1103/PhysRevA.91.062113.
+[24] A. A. Abbott, P. Alzieu, M. J. W. Hall, and C. Branciard. Tight state-independent
+uncertainty relations for qubits. Mathematics, 4(1):8, 2016. arXiv:1512.02383, doi:
+10.3390/math4010008.
+[25] B. Chen, N. Cao, S. Fei, and G. Long. Variance-based uncertainty relations for in-
+compatible observables. Quantum Information Processing, 15:3909–3917, 2016. arXiv:
+1608.06075, doi:10.1007/s11128-016-1365-1.
+27
+
+[26] H. Qin, S. Fei, and X. Li-Jost. Multi-observable uncertainty relations in product form
+of variances.
+Scientific Reports, 6:31192, 2016.
+arXiv:1608.03089, doi:10.1038/
+srep31192.
+[27] Q. Song and C. Qiao.
+Stronger Schrödinger-like uncertainty relations.
+Phys. Lett.
+A, 380(37):2925–2930, 2016. arXiv:1504.01137, doi:10.1016/j.physleta.2016.06.
+054.
+[28] Y. Xiao, N. Jing, X. Li-Jost, and S. Fei. Weighted uncertainty relations. Scientific
+Reports, 6:23201, 2016. arXiv:1603.01004, doi:10.1038/srep23201.
+[29] D. Mondal, S. Bagchi, and A. K. Pati. Tighter uncertainty and reverse uncertainty rela-
+tions. Phys. Rev. A, 95(5):052117, 2017. arXiv:1607.06712, doi:10.1103/PhysRevA.
+95.052117.
+[30] Q. Song, G. Li, J. annd Peng, and C. Qiao. A stronger multi-observable uncertainty re-
+lation. Scientific Reports, 7:44764, 2017. arXiv:1701.01072, doi:10.1038/srep44764.
+[31] J. Zhang, Y. Zhang, and C. Yu. Stronger uncertainty relations with improvable upper
+and lower bounds. Quantum Information Processing, 16:131, 2017. arXiv:1607.08223,
+doi:10.1007/s11128-017-1585-z.
+[32] X. Zheng and G. Zhang. Variance-based uncertainty relation for incompatible observers.
+Quantum Information Processing, 16:167, 2017.
+arXiv:1705.07396, doi:10.1007/
+s11128-017-1619-6.
+[33] V. V. Dodonov. Variance uncertainty relations without covariances for three and four
+observables.
+Phys. Rev. A, 97(2):022105, 2018.
+arXiv:1711.04037, doi:10.1103/
+PhysRevA.97.022105.
+[34] H. de Guise, L. Maccone, B. C. Sanders, and N. Shukla.
+State-independent un-
+certainty relations.
+Phys. Rev. A, 98(4):042121, 2018.
+arXiv:1804.06794, doi:
+10.1103/PhysRevA.98.042121.
+[35] P. Busch and O. Reaerdon-Smith. On quantum uncertainty relations and uncertainty
+regions. 2019. arXiv:1901.03695, doi:10.48550/arXiv.1901.03695.
+[36] P. Giorda, L. Maccone, and A. Riccardi. State-independent uncertainty relations from
+eigenvalue minimization. Phys. Rev. A, 99(5):052121, 2019. arXiv:1810.09775, doi:
+10.1103/PhysRevA.99.052121.
+[37] X. Zheng, S. Ma, G. Zhang, H. Fan, and W. Liu. Unified and exact framework for
+variance-based uncertainty relations. Scientific Reports, 10:150, 2020. arXiv:1803.
+08720, doi:10.1038/s41598-019-56803-2.
+28
+
+[38] J. Li and C. Qiao.
+The generalized uncertainty principle.
+Annalen der Physik,
+533(1):2000335, 2021.
+arXiv:2003.08705, doi:https://doi.org/10.1002/andp.
+202000335.
+[39] L. Zhang, S. Luo, S. Fei, and J. Wu. Uncertainty regions of observables and state-
+independent uncertainty relations.
+Quantum Information Processing, 20:357, 2021.
+arXiv:2110.14134, doi:10.1007/s11128-021-03303-w.
+[40] S. Chiew and M. Gessner. Improving sum uncertainty relations with the quantum Fisher
+information. Phys. Rev. Res., 4(1):013076, 2022. arXiv:2109.06900, doi:10.1103/
+PhysRevResearch.4.013076.
+[41] Y. Xiao, N. Jing, B. Yu, S. Fei, and X. Li-Jost. Near-optimal variance-based uncertainty
+relations. Front. Phys., 10:846330, 2022. arXiv:1610.01692, doi:10.3389/fphy.2022.
+846330.
+[42] M. Rédei and S. J. Summers. Quantum probability theory. Stud. Hist. Phil. Mod. Phys.,
+38(2):390–417, 2007. arXiv:quant-ph/0601158, doi:10.1016/j.shpsb.2006.05.006.
+[43] M. S. Leifer and R. W. Spekkens.
+Towards a formulation of quantum theory as a
+causally neutral theory of bayesian inference. Phys. Rev. A, 88(5):052130, 2013. arXiv:
+1107.5849, doi:10.1103/PhysRevA.88.052130.
+[44] A. Ben-Israel and T. N. Greville. Generalized Inverses: Theory and Applications, chap-
+ter chapter 6, pages 220–221. CMS Books in Mathematics. Springer, 2nd edition, 2003.
+doi:doi.org/10.1007/b97366.
+[45] E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev.,
+40(5):749–759, 1932. doi:10.1103/PhysRev.40.749.
+[46] D. Gross. Hudson’s theorem for finite-dimensional quantum systems. J. Math. Phys.,
+47:122107, 2006. arXiv:quant-ph/0602001, doi:10.1063/1.2393152.
+[47] T. L. Curtright, D. B. Fairlie, and C. K. Zachos. A Concise Treatise on Quantum
+Mechanics in Phase Space. World Scientific, 2014. doi:10.1142/8870.
+[48] J. G. Kirkwood. Quantum statistics of almost classical assemblies. Phys. Rev., 44(1):31–
+37, 1933. doi:10.1103/PhysRev.44.31.
+[49] P. A. M. Dirac. On the analogy between classical and quantum mechanics. Rev. Mod.
+Phys., 17(2-3):195–199, 1945. doi:10.1103/RevModPhys.17.195.
+[50] M. Lostaglio, A. Belenchia, A. Levy, S. Hernández-Gómez, N. Fabbri, and S. Gherardini.
+Kirkwood-dirac quasiprobability approach to quantum fluctuations: Theoretical and ex-
+perimental perspectives. 2022. arXiv:2206.11783, doi:10.48550/arXiv.2206.11783.
+29
+
+A
+Amplitude Operators and Purifications
+Proposition A.1. Given a density operator ρS on a Hilbert space HS, let HS′ be another
+copy of the same Hilbert space and let {|j⟩} be an orthonormal basis for HS and HS′. Define
+the vector
+��Φ+�
+SS′ =
+�
+j
+|j⟩S |j⟩S′ .
+Let LS : HE → HS be an amplitude operator for ρS and let {|k⟩E} be an orthonormal
+basis for HE. Then IS ⊗LT
+S′ |Φ+⟩SS′ is a purification of ρS, where T denotes transpose in the
+|j⟩⟨k|SE basis. Similarly, if |ψ⟩SE ∈ HS⊗HE is a purification of ρS then LS = ⟨ψ∗|S′E |Φ+⟩SS′
+is an amplitude operator for ρS, where ∗ denotes complex conjugation in the |jk⟩S′E basis.
+Proof. If LS is an amplitude operator for ρS then ρS = LSL†
+S. We have to show that this
+implies that TrE
+�
+IS ⊗ LT
+S′ |Φ+⟩⟨Φ+|SS′ IS ⊗
+�
+LT
+S′
+�†�
+= ρS. Note that
+�
+LT
+S′
+�† = L∗
+S′, where ∗
+denotes complex conjugate in the |j⟩⟨k|SE basis. Therefore, we have
+TrE
+�
+IS ⊗ LT
+S′
+��Φ+��
+Φ+��
+SS′ IS ⊗ L∗
+S′
+�
+=
+�
+j,k
+|j⟩⟨k|S TrE
+�
+LT
+S′ |j⟩⟨k|S′ L∗
+S′
+�
+(68)
+=
+�
+j,k
+|j⟩⟨k|S
+�
+k
+��L∗
+SLT
+S
+��j
+�
+S ,
+(69)
+where we have changed the index S′ to S because they refer to the same Hilbert space and
+�
+k
+��LT
+SL∗
+S
+��j
+�
+S is a scalar. Rearranging this, we have
+TrE
+�
+IS ⊗ LT
+S′
+��Φ+��
+Φ+��
+SS′ IS ⊗ L∗
+S′
+�
+=
+�
+j,k
+|j⟩S
+�
+k
+��L∗
+SLT
+S
+��j
+�
+S ⟨k|S
+(70)
+=
+�
+j,k
+|j⟩⟨j|S
+�
+L∗
+SLT
+S
+�T |k⟩⟨k|S
+(71)
+=
+�
+L∗
+SLT
+S
+�T = LSL†
+S = ρS.
+(72)
+For the other direction, we have to prove that LSL†
+S = ρS, where LS = ⟨ψ∗|S′E |Φ+⟩SS′
+and |ψ⟩SE is any purification of ρS, i.e. TrE (|ψ⟩⟨ψ|SE) = ρS.
+First, let |ψ⟩SE = �
+jk αjk |j⟩S ⊗ |k⟩E be the decomposition of |ψ⟩SE in the |jk⟩SE basis.
+We have |ψ∗⟩SE = �
+jk α∗
+jk |j⟩S ⊗ |k⟩E and the condition TrE (|ψ⟩⟨ψ|SE) = ρS is equivalent
+to �
+j,k,l αjkα∗
+lk |j⟩⟨l|S = ρS. Note also that ⟨j|S′ |Φ+⟩SS′ = |j⟩S.
+30
+
+Hence, we have
+LSL†
+S =
+�
+⟨ψ∗|S′E
+��Φ+�
+SS′
+� �
+⟨ψ∗|S′E
+��Φ+�
+SS′
+�†
+(73)
+= ⟨ψ∗|S′E
+��Φ+�
+SS′
+�
+Φ+��
+SS′ |ψ∗⟩S′E
+(74)
+=
+�
+jklm
+αjk ⟨j|S′ ⟨k|E′
+��Φ+��
+Φ+��
+SS′ α∗
+lm |l⟩S′ |m⟩E
+(75)
+=
+�
+jklm
+αjkα∗
+lm ⟨k|m⟩E
+�
+⟨j|S′
+��Φ+�
+SS′
+� ��
+Φ+��
+SS′ |l⟩S′
+�
+(76)
+=
+�
+jkl
+αjkα∗
+lk |j⟩⟨l|S
+(77)
+= ρS.
+(78)
+31
+

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@@ -0,0 +1,956 @@
+arXiv:2301.02226v1  [hep-ph]  5 Jan 2023
+Resolving RD and RD∗ Anomalies in Adjoint SU(5)
+A. Ismael1,2 and S. Khalil2
+1Physics Department, Faculty of Science, Ain Shams University, Cairo 11566, Egypt. and
+2Center for Fundamental Physics, Zewail City of Science and Technology, 6th of October City, Giza 12578, Egypt.
+(Dated: January 6, 2023)
+We investigate the RD and RD∗ anomalies in the context of non-minimal SU(5), where Higgs
+sector is extended by adjoint 45-dimensional multiplet. One of the light spectrum of this model
+could be the scalar triplet leptoquark that is contained in this multiplet. We demonstrate that
+this particular scalar leptogquark mediation of the transition b → cτν is capable of simultaneously
+accounting for both RD and RD∗ anomalies. We further emphasize that another Yukawa coupling
+controls its contribution to b → sℓ+ℓ−, ensuring that RK and RK∗ remain consistent with the
+standard model predictions.
+I.
+INTRODUCTION
+Semileptonic decays B → {D, D∗}τν have received a
+lot of attention in recent years because they provide a
+good opportunity to test the Standard Model (SM) and
+look for possible new physics beyond.
+Recent intrigu-
+ing measurements of RD,D∗ by BaBar [1, 2], Belle [3–6],
+and LHCb collaborations [7] are significant hints of new
+physics that violate lepton flavor universality. The ratios
+RD,D∗ are defined by
+RD∗,D ≡ BR(Bq → {D∗, D}τν)
+BR(Bq → {D∗, D}lν) ,
+(1)
+where l = e, µ. The current experimental averages of RD
+and RD∗ are given by [8]
+RD = 0.339 ± 0.026 ± 0.014 ,
+(2)
+RD∗ = 0.295 ± 0.010 ± 0.010 .
+(3)
+However, the SM predictions are given as follows: [9–11]
+RSM
+D
+= 0.298 ± 0.004 ,
+(4)
+RSM
+D∗ = 0.254 ± 0.005 .
+(5)
+This shows that the measured RD and RD∗ results devi-
+ate from the SM expectations by 1.9σ and 3.2σ, respec-
+tively. On the other hand, the LHCb recently announced
+new results for the ratios
+RK = BR(B+ → K+µ+µ−)
+BR(B+ → K+e+e−) ,
+(6)
+RK∗ = BR(B0 → K∗0µ+µ−)
+BR(B0 → K∗0e+e−) .
+(7)
+It has been reported that RK and RK∗ are given for two
+dilepton invariant mass-squared bins by [12, 13]
+Low − q2
+
+
+
+
+
+RK = 0.994 +0.09
+−0.082 (stat) +0.027
+−0.029 (syst)
+RK∗ = 0.927 +0.0933
+−0.087 (stat) +0.034
+−0.033 (syst)
+(8)
+Central − q2
+
+
+
+
+
+RK = 0.949 +0.042
+−0.041 (stat) +0.023
+−0.023 (syst)
+RK∗ = 1.027 +0.072
+−0.068 (stat) +0.027
+−0.027 (syst)
+These measurements are consistent with the SM predic-
+tions: RK,K∗ ≃ 1 [14]. As a result, they would impose
+sever constraints on any new physics contributions that
+could lead to lepton flavor non-universality.
+In this paper, we argue that the scalar triplet lepto-
+quark within the adjoint SU(5) framework can account
+for the discrepancy between RD,D∗ experimental results
+and SM expectations, while preserving RSM
+K,K∗ results.
+The Adjoint SU(5) is the simplest extension of minimal
+SU(5) Grand Unified Theory (GUT), in which the Higgs
+sector is extended by a 45-dimensional multiplet (45H).
+As is well known, minimal SU(5) has a number of se-
+rious problems, such as the incorrect prediction for the
+fermion mass relation: mµ(e) = ms(d). One possible so-
+lution to some of these flaws is to introduce an extra
+45H. The scalar triplet is one of the 45H components,
+with the following (3∗, 2, −7/6) representation under the
+SM gauge group. Because of its special interactions with
+quarks and leptons, this scalar triplet, which is a lepto-
+quark type particle, does not contribute to proton de-
+cay, as explained in [15]. This distinguishes SU(5) scalar
+triplet from previous leptoquark scenarios discussed in
+
+2
+the literature. [16–19]. Although the scalar letptoquark
+contributes to the semileptonic decays b → cτν at the
+tree level, it is still subdominant because the leptoquark’s
+mass is quite heavy of order TeV, which is sufficient to
+account for the given ∼ 10% discrepancy. Controlling the
+contribution of scalar leptoquarks to the b → sℓ+ℓ− can
+be accomplished by constraining one of the free Yukawa
+couplings.
+The paper is organized as follows. In section 2 we in-
+troduce the SU(5) scalar leptoquark and its associated
+interactions, emphasizing that it does not contribute to
+proton decay but can play important role in the following
+decays: b → cτν and b → sℓ+ℓ−. Section 3 is devoted to
+anlayzing the new contribution of our scalar leptoquark
+to RD,D∗. RK,K∗ analysis is discussed in section 4. Fi-
+nally our conclusions and prospects are give in section
+5.
+II.
+SCALAR LEPTOQUARK IN ADJOINT SU(5)
+As previously advocated, extending the Higgs sector of
+SU(5) by 45H helps to solve some of the problems that
+this simple example of GUT model faces [20–23]. The
+45H transforms under the SM gauge as
+45H = (8, 2)1/2 ⊕ (1, 2)1/2 ⊕ (3, 1)−1/3 ⊕ (3, 3)−1/3
+⊕ (6∗, 1)−1/3 ⊕ (3∗, 2)−7/6 ⊕ (3∗, 1)4/3.
+(9)
+It also satisfies the following constraints: 45αβ
+γ
+= −45βα
+γ
+and �5
+α(45)αβ
+α
+= 0.
+Through non-vanishing Vacuum
+Expectation Values (VEVs) of 5H and 45H:
+⟨5H⟩ =
+v5, ⟨45H⟩15
+1
+= ⟨45H⟩25
+2
+= ⟨45H⟩35
+3
+= v45, ⟨45H⟩45
+4
+=
+−3v45, the electroweak symmetry SU(2)L × U(1)Y is
+spontaneously broken into U(1)em.
+The 45H scalar triplets are defined as:
+(3∗, 2)ij
+c −7/6 ≡ (45H)ij
+c ≡ Φij
+c ,
+(10)
+(3∗, 1)ab
+k 4/3 ≡ (45H)ab
+k ≡ Φab
+k ,
+[(3, 1)ib
+c ⊕ (3, 3)ib
+c ]−1/3 ≡ (45H)ib
+c ≡ Φib
+c .
+It has been emphasized [15] that while the scalar triplets
+Φab
+k
+and Φib
+c
+contribute to the proton decay and they
+must be superheavy, the scalar triplet Φij
+c does not. It
+has no interaction terms that would cause proton decay.
+By writing Φij
+c as (φi
+1, φi
+2)T , one can demonstrate that
+the scalar triplet has the following peculiar interactions:
+L=2Y 2
+ABeT
+ACuc
+Biφi1∗+4(Y 4
+AB−Y 4
+BA)uiT
+A Cec
+Bφi1
+−2Y 2
+ABνT
+ACuc
+Biφi2∗+4(Y 4
+AB−Y 4
+BA)diT
+A Cec
+Bφi2. (11)
+The first two interaction terms would imply the decay of
+b → sℓ+ℓ− through scalar triplet leptoquark φi1 media-
+tion, while the last two interaction terms clearly account
+for the decay b → cτν via scalar triplet leptoquark φi2
+mediation. These terms can be written as
+L = 2Y 2
+AB¯uBiPLνAφi2∗ − 4Y 4′
+AB¯eBPLdi
+Aφi2 + h.c.,
+(12)
+where we used CT = −C and ¯Ψ = ΨcT
+L , and define
+Y 4′
+AB ≡ (Y 4
+AB −Y 4
+BA). In the mass eignestate basis, where
+dA → V CKM
+AB
+dB, νA → V PMNS
+AB
+νB, uA → uA, eA → eA,
+the above Lagrangian takes the form:
+L = 2Y 2
+AB ¯u′BiPLV PMNS
+AK
+ν′
+kφi2∗ − 4Y 4′
+AB ¯e′BPLV CKM
+AK
+d′
+Kφi2
++h.c.
+(13)
+In this regards, the amplitude of b → cτν transition is
+given by
+M=−8Y 4′
+13V CKM
+13
+M 2
+φ
+�1
+2(¯uτPLvντ )(¯uCPLub)
++ 1
+8(¯uτσµνPLvντ )(¯uCPLσµνub) ×
+�
+Y 2
+12V PMNS
+13
+(14)
++Y 2
+22V PMNS
+23
++Y 2
+32V PMNS
+33
+��
++
+�
+Y 4′
+13 V CKM
+13
+→Y 4′
+23 V CKM
+23
+�
+.
+Because V CKM
+13
+and V CKM
+23
+are so small (10−3 and 10−2,
+respectively), the amplitude of b → cτν is essentially
+determined by the leptoquark masses Mφ, Y 2
+22, Y 2
+32, and
+Y 4′
+13.
+III.
+SU(5) LEPTOQUARK CONTRIBUTION TO
+RD,D∗
+The general expression of the effective Hamiltonian for
+b → cl ¯νl can be written as [24]
+Heff = 4GF Vcb
+√
+2
+�
+(1 + gVL)[¯cγµPLb][¯lγµPLνl]
++ gVR[¯cγµPRb][¯lγµPLνl] + gSL[¯cPLb][¯lPLνl]
++ gSR[¯cPRb][¯lPLνl] + gT [¯cσµντ PLb][¯lσµνPLνl]
+�
+,(15)
+where GF is the Fermi coupling constant, Vcb is the
+Cabibbo-Kobayashi-Maskawa (CKM) matrix element be-
+tween charm and bottom quarks while PL/R = (1 ∓
+
+3
+γ5)/2.
+Here, gi is defined as gi = CNP
+i
+/CSM, i ≡
+VL, VR, SL, SR, T , with CSM =
+4GF Vcb
+√
+2
+. Eq. 15 shows
+that gVL = gVR = gSR = 0, whereas gSL and gT are given
+by
+gSL = −
+√
+2Z
+M 2
+φGF
+,
+gST = gSL
+4
+= −
+Z
+2
+√
+2M 2
+φGF
+, (16)
+with
+Z =
+�
+Y 2
+12V PMNS
+13
++ Y 2
+22V PMNS
+23
++ Y 2
+32V PMNS
+33
+�
+�
+Y 4′
+13V CKM
+13
+V CKM
+23
++ Y 4′
+23
+�
+(17)
+Substituting with the SM parameters as well as the
+form factors involved in the definition of the matrix ele-
+ments to their central values, one finds [25]
+R(D) = R(D)SM�
+1 + 1.02|gSL|2 + 0.9|gT|2
++ 1.49 Re[g∗
+SL] + 1.14 Re[g∗
+T ]
+�
+,
+(18)
+R(D∗) = R(D∗)SM�
+1 + 0.04|gSL|2 + 16.07|gT|2
+− 0.11 Re[g∗
+SL] − 5.12 Re[g∗
+T ]
+�
+.
+(19)
+A few remarks are in order. First, the gSL and gT can
+be complex due to non-zero phases in U PMNS as well as
+complex values of the Yukawa couplings Y 2 and Y 4′. Sec-
+ond, because the tree-level scalar leptoquark contributes
+to the branching ratio of the tauonic decay B−
+c → τ −¯ντ,
+experimental constraints from this decay must be in-
+cluded in our analysis.
+The modified branching ratio
+BR(B−
+c → τ −¯ντ) is given by [25–27]
+BR(B−
+c →τ −¯ντ)=BR(B−
+c →τ −¯ντ)SM|1−4.065gSL|2, (20)
+with BR(B−
+c
+→ τ −¯ντ)SM = (2.25 ± 0.21) × 10−2 [28].
+The experimental bound on BR(B−
+c → τ −¯ντ) varies from
+≤ 10% to ≤ 60% [28–31]. Third, it is also worth noting
+that our type of scalar leptoquarks would not contribute
+to lepton flavor violation, like τ → µγ or B − ¯B mixing.
+Fourth, we impose the constraints of the D∗ and τ lon-
+gitudinal polarizations that come from Belle experiment.
+Their expressions depend on the same Wilson coefficients
+affecting RD and RD∗, which are written as [25, 27]
+F D∗
+L
+F D∗
+L,SM
+=
+� RD∗
+RSM
+D∗
+�−1�
+1 + 0.08|gSL|2 + 7.02|gT|2
+− 0.24 Re[g∗
+SL] − 4.37 Re[g∗
+T]
+�
+(21)
+P D∗
+τ
+P D∗
+τ,SM
+=
+� RD∗
+RSM
+D∗
+�−1�
+1 − 0.07|gSL|2 − 1.86|gT|2
++ 0.22 Re[g∗
+SL] − 3.37 Re[g∗
+T]
+�
+(22)
+The experimental values of F D
+L and P D∗
+τ
+are given by
+0.60 ± 0.08 ± 0.035 [32] and −0.38 ± 0.51+0.21
+−0.16 [4, 5, 33]
+respectively, whereas their SM predictions are 0.46±0.04
+[34] and −0.497 ± 0.013 [35] Finally, running the coeffi-
+cients gSL and gT from the scale µ = 1T eV to the scale
+mb = 4.2GeV implies that [36, 37]:
+�
+gSL
+gT
+�
+=
+�
+1.71 0
+0
+1
+� �
+gSL(µ = 1T eV )
+gT (µ = 1T eV )
+�
+.
+(23)
+In the presence of the aforementioned experimental
+constraints, we performed a numerical analysis of RD
+and RD∗. In Fig. 1, we show the dependence of RD and
+RD∗ on the most relevant parameters, which are the mass
+of leptoquark Mφ (left panel) and the real and imag-
+inary parts of the Yukawa coupling Y 4′
+23 (right panel).
+The other parameters in these plots were set as follows:
+Y 2
+12 = −1.5, Y 2
+22 = Y 2
+32 = Y 4′
+13 = 1.5. Furthermore, the
+coupling Y 4′
+23 is fixed with 1.48 + 0.1i in the plot of RD
+and RD∗ versus Mφ (left panel), whereas in the 3D plot
+of RD and RD∗ versus real and imaginary parts of Y 4′
+23
+(right panel), the mass Mφ varies along the [800, 1500]
+GeV, while real and imaginary parts of Y 4′
+23 vary along
+the [−1.5, 1.5] and [−0.5, 0.5], respectively.
+The correlation between RD and RD∗ is shown in Fig.
+2, left-panel, and the correlation between the constraints
+on the BR(B−
+c → τ −¯ντ) and RD and RD∗ is highlighted
+in the right-panel of this plot. The parameters are set in
+the same way as in the previous plots.
+These plots show that in this class of models, both RD
+and RD∗ can be significantly enhanced and lie within
+one sigma of the recent experimental limits, with scalar
+leptoquark masses of order one TeV, which is consistent
+with experimental constraints.
+
+4
+1000
+1050
+1100
+1150
+1200
+1250
+1300
+1350
+1400
+1450
+M  (GeV)
+0.28
+0.3
+0.32
+0.34
+0.36
+0.38
+0.4
+0.42
+RD & R D *
+RD
+*
+RD
+0.26
+0.28
+0.4
+0.3
+0.32
+0.34
+0.2
+RD&R D *
+0.36
+1
+0.38
+(Y 23
+4' )
+0
+0.5
+0.4
+(Y 23
+4' )
+0.42
+0
+-0.2
+-0.5
+-1
+-0.4
+RD
+*
+RD
+FIG. 1. RD and RD∗ as function of the Letoquark mass and and real and imaginary parts of the Yukawa coupling Y23. The
+other parameters are fixed as mentioned in the text.
+0.3
+0.32
+0.34
+0.36
+0.38
+0.4
+0.42
+RD
+0.25
+0.26
+0.27
+0.28
+0.29
+0.3
+0.31
+0.32
+RD *
+0.05
+0.1
+0.15
+0.2
+0.25
+BR (B
+)
+0.26
+0.28
+0.3
+0.32
+0.34
+0.36
+0.38
+0.4
+0.42
+RD & R D *
+RD
+*
+RD
+FIG. 2. The correlation between RD and RD∗ (left) and between both RD and RD∗ and BR(B−
+c → τ −¯ντ) (right). The scan
+is conducted over the regions of parameter space mentioned above.
+IV.
+SU(5) LEPTOQUARK CONTRIBUTION TO
+RK,K∗
+In this section, we show that, while the scalar lep-
+toquark causes non-universality of lepton flavor in the
+process B → Dℓν, it does not necessarily cause non-
+universality in the process B → Kℓ+ℓ−. The Lagrangian
+that generates the b → sℓ+ℓ− transition is given by
+L=−4Y 4′
+AB ¯e′BPLV CKM
+AK
+di′
+Kφi2−4Y 4′
+AB ¯d′iKV CKM∗
+AK
+PR e
+′
+Bφ∗
+i2.
+(24)
+Thus, for b → s µ µ+, the Lagrangian is given as
+L ⊃ −4Y 4′
+32 ¯µ′PLbi′φi2 − 4Y 4′∗
+12
+¯
+Si′V CKM∗
+12
+PR µ
+′φ∗
+i2
+− 4Y 4′∗
+32
+¯
+Si′V CKM∗
+32
+PR µ
+′φ∗
+i2,
+(25)
+where V CKM
+13
+≈ 0 and V CKM
+33
+≈ 1 are assumed. Also,
+we may neglect V CKM
+32
+respect V CKM
+12
+(although we in-
+clude all terms in our numerical calculations). Thus, the
+amplitude of this process is given by
+M = 8Y 4′
+32Y 4′∗
+12 V CKM∗
+12
+M 2
+φ
+� ¯UsγµPLUb
+�� ¯UµγµPLVν
+�
+. (26)
+We used the Fierz transformation identity
+� ¯UsPRVµ
+�� ¯
+UµPLUb
+�
+= 1
+2
+� ¯UsγµPLUb
+�� ¯UµγµPLvµ
+�
+.
+(27)
+As a result, the Wilson coefficient Cµ
+9 for b → s µ µ+
+process is written as
+Cµ
+9 (Λ) = 8Y 4′
+32Y 4′∗
+12 V CKM∗
+12
+M 2
+φ
+.
+(28)
+
+5
+where the scale Λ ≈ 1TeV, and Cµ
+10(Λ) = −Cµ
+9 (Λ). On
+the other hand, the Lagrangian that generates the pro-
+cess b → s e e+ is given by
+L = − 4Y 4′
+31 ¯e
+′PLbi′φi2 − 4Y 4′∗
+21
+¯
+Si′PRe
+′φ∗
+i2.
+(29)
+After applying Fierz identity, the amplitude of b → s e e+
+is given by
+M = 8Y 4′
+31Y 4′∗
+21
+M 2
+φ
+� ¯UsγµPLUb
+�� ¯UeγµPLVe
+�
+.
+(30)
+Hence, the Wilson coefficient Ce
+9(Λ) for b → s e e+ will
+be
+Ce
+9(Λ) = 8Y 4′
+31Y 4′∗
+21
+M 2
+φ
+.
+(31)
+Moreover, Ce
+10(Λ) = −Ce
+9(Λ). The effective Hamiltonian
+Heff for RK process is given by
+Heff =
+�
+i
+�
+Ci(µb)Oi(µb) + ˜Ci(µb) ˜Oi(µb)
+�
+.
+(32)
+Through renormalization group equation (RGE), we ob-
+tain
+Ce,µ
+9,10(Λ) = 1.2 Ce,µ
+9,10(µb),
+(33)
+where Oi(µb) are ∆B = 1 transition operator, which is
+evaluated at the mb scale. ˜Ci(µb), ˜Oi(µb) are obtained by
+replacing L ↔ R. The relevant operators that describe
+the Rk and Rk∗ in our model are
+O9 =
+�
+¯sγµPLb
+��¯lγµl),
+O10 =
+�
+¯sγµPLb
+��¯lγµγ5l). (34)
+The Rk and Rk∗ expressions are written as
+Rk ≈1 + ∆+,
+(35)
+Rk∗ ≈1 + ���+ + p(∆+ − ∆−),
+(36)
+where p is a function of q2
+min and q2
+max and ∆± is given
+by
+∆± =
+2
+|CSM
+9
+|2 + |CSM
+10 |2
+�
+ℜ
+�
+CSM
+9
+(CNP,µ
+9
+± ˜
+C9
+NP,µ)∗�
++ ℜ
+�
+CSM
+10 (CNP,µ
+10
+± ˜
+C10
+NP,µ)∗�
+− (µ ↔ e)
+�
+(37)
+For our model, ˜CNP
+9,10 = 0. Therefore, we obtain
+∆+ = ∆− = 2.4
+�
+CSM
+9
+− CSM
+10
+�
+|CSM
+9
+|2 + |CSM
+10 |2 ℜ
+�
+CNP,µ
+9
+(µb)−CNP,e
+9
+(µb)
+�∗
+(38)
+It is worth mentioning that, whereas RK,K∗ is essen-
+tially dependent on the couplings Y 4′
+21 and Y 4′
+32, RD,D∗ is
+dependent on Y 2
+22, Y 2
+33 and Y 4′
+23 . As a result, it is entirely
+possible to keep RK,K∗ equal to the SM expectation, con-
+sistently with the new LHCb results, while leaving RD,D∗
+intact. To make RK,K∗ close to one, ∆+ should be very
+small. This can be accomplished by having Y 4′
+12 ≪ 1.
+V.
+CONCLUSIONS
+In this paper we have demonstrated that, in the pres-
+ence of experimental constraints on flavor and lepton
+violation observables, measured values of RD and RD∗
+within 1σ can be explained in non-minimal SU(5) with
+adjoint 45-dimensional Higgs multiplet. Enhancements
+for both RD and RD∗ are made possible by a tree level
+transition of b → cτν, which is mediated by the associ-
+ated scalar leptoquark. We also emphasized that even
+though this leptoquark may contribute to RK and RK∗,
+they remain independent of RD and RD∗ enhancements
+because they are given in terms of different Yukawa cou-
+plings. As a result, their contributions can be easily sup-
+pressed, and RK and RK∗ continue to be identical to SM
+predictions, which are consistent with the most recent
+LHCb data.
+ACKNOWLEDGEMENTS
+This work is partially supported by Science, Tech-
+nology & Innovation Funding Authority (STDF) under
+grant number 37272.
+
+6
+REFERENCES
+[1] J. P. Lees et al. [BaBar Collaboration], Phys. Rev. Lett.
+109, 101802 (2012) [arXiv:1205.5442 [hep-ex]].
+[2] J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D 88,
+no. 7, 072012 (2013) [arXiv:1303.0571 [hep-ex]].
+[3] M. Huschle et al. [Belle Collaboration], Phys. Rev. D 92,
+no. 7, 072014 (2015) [arXiv:1507.03233 [hep-ex]].
+[4] S. Hirose et al. [Belle Collaboration], Phys. Rev. Lett.
+118, no. 21, 211801 (2017) [arXiv:1612.00529 [hep-ex]].
+[5] S. Hirose et al. [Belle Collaboration], Phys. Rev. D 97,
+no. 1, 012004 (2018) [arXiv:1709.00129 [hep-ex]].
+[6] G. Caria et al. [Belle Collaboration], Phys. Rev. Lett.
+124, no. 16, 161803 (2020) [arXiv:1910.05864 [hep-ex]].
+[7] LHCb Collaboration, https://indico.cern.ch/event/1187939/
+(2022).
+[8] https://hflav-eos.web.cern.ch/hflav-
+eos/semi/spring21/html/RDsDsstar/RDRDs.html
+(2022).
+[9] D. Bigi and P. Gambino, Phys. Rev. D 94, no. 7, 094008
+(2016). [arXiv:1606.08030 [hep-ph]]
+[10] P. Gambino, M. Jung and S. Schacht, Phys. Lett. B 795,
+386-390 (2019) [arXiv:1905.08209 [hep-ph]].
+[11] M. Bordone, M. Jung and D. van Dyk, Eur. Phys. J. C
+80, no. 2, 74 (2020) [arXiv:1908.09398 [hep-ph]].
+[12] LHCb collaboration, arXiv.2212.09152 [hep-ex] (2022).
+[13] LHCb collaboration, arXiv.2212.09153 [hep-ex] (2022).
+[14] Marzia Bordone, Gino Isidori and Andrea Pattori,Eur.
+Phys. J. C 76, no. 8, 440 (2016) [arXiv:1605.07633 [hep-
+ph]].
+[15] Ilja Dorsner, Svjetlana Fajfer, Nejc Kosnik, Phys. Rev. D
+86, no. 1, 015013 (2012) [ arXiv:1204.0674v2 [hep-ph]].
+[16] Syuhei Iguro, Michihisa Takeuchi and Ryoutaro Watan-
+abe,
+Eur.
+Phys.
+J.
+C
+81,
+no.
+5,
+406
+(2021)
+[arXiv:2011.02486 [hep-ph]].
+[17] David Marzocca and Sokratis Trifinopoulos, Phys. Rev.
+Lett. 127, no. 6, 061803 (2021) [arXiv:2104.05730 [hep-
+ph]].
+[18] Arvind Bhaskar and Anirudhan A. Madathil and Tanu-
+moy Mandal and Subhadip Mitra, Phys. Rev. D 106, no.
+11, 115009 (2022) [arXiv:2204.09031 [hep-ph]].
+[19] Peter
+Cox,
+Alexander
+Kusenko,
+Olcyr
+Sumensari
+and
+Tsutomu
+T.
+Yanagida,
+JHEP
+05,
+35
+(2017)
+[arXiv:1612.03923 [hep-ph]].
+[20] Bartosz Fornal and Benjam´ın Grinstein, Phys. Rev. Lett.
+119, no. 24, 241801 (2017) [ arXiv:1706.08535 [hep-ph].
+[21] Ilja Dorsner, Pavel Fileviez Perez, Phys. Lett. B 642, no.
+03, 248-252 (2006) [arXiv:0606062 [hep-ph]].
+[22] S. Khalil, S. Salem, M. Allam, Phys. Rev. D 89, no. 09,
+095011 (2015) [arXiv:1401.1482v2 [hep-ph]].
+[23] S. Khalil, S. Salem, Nucl. Phys. B 876, 473 (2013)
+[arXiv:1304.3689 [hep-ph]]
+[24] Minoru Tanaka and Ryoutaro Watanabe, Phys. Rev. D
+87, no. 3, 034028 (2013) [arXiv:1212.1878 [hep-ph]].
+[25] Syuhei Iguro, Teppei Kitahara, Yuji Omura, Ryoutaro
+Watanabe and Kei Yamamoto,JHEP 02, 194 (2019)
+[arXiv 1811.08899 [hep-ph]].
+[26] Pouya Asadi, Matthew R. Buckley and David Shih,
+JHEP 09, 10 (2018) [arXiv:1804.04135 [hep-ph]].
+[27] Pouya Asadi, Matthew R. Buckley and David Shih, Phys.
+Rev. D 99, no. 3, 035015 (2019) [arXiv:1810.06597 [hep-
+ph]].
+[28] Robert
+Fleischer,
+Ruben
+Jaarsma
+and
+Gilberto
+Tetlalmatzi-Xolocotzi,
+Eur.
+Phys.
+J.
+C
+81,
+no.7,
+658 (2021) [arXiv:2104.04023 [hep-ph]].
+[29] Debjyoti Bardhan and Diptimoy Ghosh, Phys. Rev. D
+100, no. 1, 011701 (2019) [arXiv:1904.10432 [hep-ph]].
+[30] Monika Blanke,
+Andreas Crivellin,
+Stefan de Boer,
+Teppei Kitahara, Marta Moscati, Ulrich Nierste and
+Ivan Nisandzic,Phys. Rev. D 99, no.7, 075006 (2019)
+[arXiv:1811.09603 [hep-ph]].
+[31] Monika Blanke, Andreas Crivellin, Teppei Kitahara,
+Marta Moscati, Ulrich Nierste and Ivan Nisandzic,Phys.
+Rev. D 100, no. 3, 035035 (2019) [arXiv:1905.08253 [hep-
+ph]].
+[32] A.
+Abdesselam
+et
+al.
+[Belle
+Collaboration],
+arXiv:1903.03102 [hep-ex] (2019).
+[33] Karol Adamczyk, arXiv:1901.06380 [hep-ex] (2019).
+[34] Ashutosh Kumar Alok, Dinesh Kumar, Suman Kumb-
+hakar and S Uma Sankar, Phys. Rev. D 95, no. 11,
+115038 (2017), [arXiv:1606.03164 [hep-ph]].
+[35] Minoru Tanaka and Ryoutaro Watanabe, Phys. Rev. D
+87, no. 3, 034028 (2013), [arXiv:1212.1878v3 [hep-ph]].
+[36] Mart´ın Gonz´alez-Alonso, Jorge Martin Camalich and
+Kin
+Mimouni,Phys.
+Lett.
+B
+772,
+777-785
+(2017),
+[arXiv:1706.00410v2 [hep-ph]].
+[37] Syuhei
+Iguro,
+Michihisa
+Takeuchi
+and
+Ryoutaro
+Watanabe,Eur. Phys. J. C 81,
+no. 5,
+406 (2021),
+[arXiv2011.02486v4 [hep-ph]].
+

L9E0T4oBgHgl3EQfSwBA/content/tmp_files/load_file.txt

@@ -0,0 +1,437 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf,len=436
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='02226v1  [hep-ph]  5 Jan 2023  Resolving RD and RD∗ Anomalies in Adjoint SU(5)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Ismael1,2 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Khalil2  1Physics Department, Faculty of Science, Ain Shams University, Cairo 11566, Egypt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' and  2Center for Fundamental Physics, Zewail City of Science and Technology, 6th of October City, Giza 12578, Egypt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (Dated: January 6, 2023)  We investigate the RD and RD∗ anomalies in the context of non-minimal SU(5), where Higgs  sector is extended by adjoint 45-dimensional multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' One of the light spectrum of this model  could be the scalar triplet leptoquark that is contained in this multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' We demonstrate that  this particular scalar leptogquark mediation of the transition b → cτν is capable of simultaneously  accounting for both RD and RD∗ anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' We further emphasize that another Yukawa coupling  controls its contribution to b → sℓ+ℓ−, ensuring that RK and RK∗ remain consistent with the  standard model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  INTRODUCTION  Semileptonic decays B → {D, D∗}τν have received a  lot of attention in recent years because they provide a  good opportunity to test the Standard Model (SM) and  look for possible new physics beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Recent intrigu-  ing measurements of RD,D∗ by BaBar [1, 2], Belle [3–6],  and LHCb collaborations [7] are significant hints of new  physics that violate lepton flavor universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' The ratios  RD,D∗ are defined by  RD∗,D ≡ BR(Bq → {D∗, D}τν)  BR(Bq → {D∗, D}lν) ,  (1)  where l = e, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' The current experimental averages of RD  and RD∗ are given by [8]  RD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='339 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='026 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='014 ,  (2)  RD∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='295 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='010 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (3)  However, the SM predictions are given as follows: [9–11]  RSM  D  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='298 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='004 ,  (4)  RSM  D∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='254 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='005 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (5)  This shows that the measured RD and RD∗ results devi-  ate from the SM expectations by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='9σ and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='2σ, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' On the other hand, the LHCb recently announced  new results for the ratios  RK = BR(B+ → K+µ+µ−)  BR(B+ → K+e+e−) ,  (6)  RK∗ = BR(B0 → K∗0µ+µ−)  BR(B0 → K∗0e+e−) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (7)  It has been reported that RK and RK∗ are given for two  dilepton invariant mass-squared bins by [12, 13]  Low − q2  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  RK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='994 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='09  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='082 (stat) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='027  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='029 (syst)  RK∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='927 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='0933  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='087 (stat) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='034  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='033 (syst)  (8)  Central − q2  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  RK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='949 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='042  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='041 (stat) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='023  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='023 (syst)  RK∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='027 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='072  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='068 (stat) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='027  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='027 (syst)  These measurements are consistent with the SM predic-  tions: RK,K∗ ≃ 1 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' As a result, they would impose  sever constraints on any new physics contributions that  could lead to lepton flavor non-universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  In this paper, we argue that the scalar triplet lepto-  quark within the adjoint SU(5) framework can account  for the discrepancy between RD,D∗ experimental results  and SM expectations, while preserving RSM  K,K∗ results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  The Adjoint SU(5) is the simplest extension of minimal  SU(5) Grand Unified Theory (GUT), in which the Higgs  sector is extended by a 45-dimensional multiplet (45H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  As is well known, minimal SU(5) has a number of se-  rious problems, such as the incorrect prediction for the  fermion mass relation: mµ(e) = ms(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' One possible so-  lution to some of these flaws is to introduce an extra  45H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' The scalar triplet is one of the 45H components,  with the following (3∗, 2, −7/6) representation under the  SM gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Because of its special interactions with  quarks and leptons, this scalar triplet, which is a lepto-  quark type particle, does not contribute to proton de-  cay, as explained in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' This distinguishes SU(5) scalar  triplet from previous leptoquark scenarios discussed in  2  the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' [16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Although the scalar letptoquark  contributes to the semileptonic decays b → cτν at the  tree level, it is still subdominant because the leptoquark’s  mass is quite heavy of order TeV, which is sufficient to  account for the given ∼ 10% discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Controlling the  contribution of scalar leptoquarks to the b → sℓ+ℓ− can  be accomplished by constraining one of the free Yukawa  couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' In section 2 we in-  troduce the SU(5) scalar leptoquark and its associated  interactions, emphasizing that it does not contribute to  proton decay but can play important role in the following  decays: b → cτν and b → sℓ+ℓ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Section 3 is devoted to  anlayzing the new contribution of our scalar leptoquark  to RD,D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' RK,K∗ analysis is discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Fi-  nally our conclusions and prospects are give in section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  SCALAR LEPTOQUARK IN ADJOINT SU(5)  As previously advocated, extending the Higgs sector of  SU(5) by 45H helps to solve some of the problems that  this simple example of GUT model faces [20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' The  45H transforms under the SM gauge as  45H = (8, 2)1/2 ⊕ (1, 2)1/2 ⊕ (3, 1)−1/3 ⊕ (3, 3)−1/3  ⊕ (6∗, 1)−1/3 ⊕ (3∗, 2)−7/6 ⊕ (3∗, 1)4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (9)  It also satisfies the following constraints: 45αβ  γ  = −45βα  γ  and �5  α(45)αβ  α  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Through non-vanishing Vacuum  Expectation Values (VEVs) of 5H and 45H:  ⟨5H⟩ =  v5, ⟨45H⟩15  1  = ⟨45H⟩25  2  = ⟨45H⟩35  3  = v45, ⟨45H⟩45  4  =  −3v45, the electroweak symmetry SU(2)L × U(1)Y is  spontaneously broken into U(1)em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  The 45H scalar triplets are defined as:  (3∗, 2)ij  c −7/6 ≡ (45H)ij  c ≡ Φij  c ,  (10)  (3∗, 1)ab  k 4/3 ≡ (45H)ab  k ≡ Φab  k ,  [(3, 1)ib  c ⊕ (3, 3)ib  c ]−1/3 ≡ (45H)ib  c ≡ Φib  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  It has been emphasized [15] that while the scalar triplets  Φab  k  and Φib  c  contribute to the proton decay and they  must be superheavy, the scalar triplet Φij  c does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' It  has no interaction terms that would cause proton decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  By writing Φij  c as (φi  1, φi  2)T , one can demonstrate that  the scalar triplet has the following peculiar interactions:  L=2Y 2  ABeT  ACuc  Biφi1∗+4(Y 4  AB−Y 4  BA)uiT  A Cec  Bφi1  −2Y 2  ABνT  ACuc  Biφi2∗+4(Y 4  AB−Y 4  BA)diT  A Cec  Bφi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' (11)  The first two interaction terms would imply the decay of  b → sℓ+ℓ− through scalar triplet leptoquark φi1 media-  tion, while the last two interaction terms clearly account  for the decay b → cτν via scalar triplet leptoquark φi2  mediation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' These terms can be written as  L = 2Y 2  AB¯uBiPLνAφi2∗ − 4Y 4′  AB¯eBPLdi  Aφi2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=',  (12)  where we used CT = −C and ¯Ψ = ΨcT  L , and define  Y 4′  AB ≡ (Y 4  AB −Y 4  BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' In the mass eignestate basis, where  dA → V CKM  AB  dB, νA → V PMNS  AB  νB, uA → uA, eA → eA,  the above Lagrangian takes the form:  L = 2Y 2  AB ¯u′BiPLV PMNS  AK  ν′  kφi2∗ − 4Y 4′  AB ¯e′BPLV CKM  AK  d′  Kφi2  +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (13)  In this regards, the amplitude of b → cτν transition is  given by  M=−8Y 4′  13V CKM  13  M 2  φ  �1  2(¯uτPLvντ )(¯uCPLub)  + 1  8(¯uτσµνPLvντ )(¯uCPLσµνub) ×  �  Y 2  12V PMNS  13  (14)  +Y 2  22V PMNS  23  +Y 2  32V PMNS  33  ��  +  �  Y 4′  13 V CKM  13  →Y 4′  23 V CKM  23  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Because V CKM  13  and V CKM  23  are so small (10−3 and 10−2,  respectively), the amplitude of b → cτν is essentially  determined by the leptoquark masses Mφ, Y 2  22, Y 2  32, and  Y 4′  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  SU(5) LEPTOQUARK CONTRIBUTION TO  RD,D∗  The general expression of the effective Hamiltonian for  b → cl ¯νl can be written as [24]  Heff = 4GF Vcb  √  2  �  (1 + gVL)[¯cγµPLb][¯lγµPLνl]  + gVR[¯cγµPRb][¯lγµPLνl] + gSL[¯cPLb][¯lPLνl]  + gSR[¯cPRb][¯lPLνl] + gT [¯cσµντ PLb][¯lσµνPLνl]  �  ,(15)  where GF is the Fermi coupling constant, Vcb is the  Cabibbo-Kobayashi-Maskawa (CKM) matrix element be-  tween charm and bottom quarks while PL/R = (1 ∓  3  γ5)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Here, gi is defined as gi = CNP  i  /CSM, i ≡  VL, VR, SL, SR, T , with CSM =  4GF Vcb  √  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 15 shows  that gVL = gVR = gSR = 0, whereas gSL and gT are given  by  gSL = −  √  2Z  M 2  φGF  ,  gST = gSL  4  = −  Z  2  √  2M 2  φGF  , (16)  with  Z =  �  Y 2  12V PMNS  13  + Y 2  22V PMNS  23  + Y 2  32V PMNS  33  �  �  Y 4′  13V CKM  13  V CKM  23  + Y 4′  23  �  (17)  Substituting with the SM parameters as well as the  form factors involved in the definition of the matrix ele-  ments to their central values, one finds [25]  R(D) = R(D)SM�  1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='02|gSL|2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='9|gT|2  + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='49 Re[g∗  SL] + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='14 Re[g∗  T ]  �  ,  (18)  R(D∗) = R(D∗)SM�  1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='04|gSL|2 + 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='07|gT|2  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='11 Re[g∗  SL] − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='12 Re[g∗  T ]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (19)  A few remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' First, the gSL and gT can  be complex due to non-zero phases in U PMNS as well as  complex values of the Yukawa couplings Y 2 and Y 4′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Sec-  ond, because the tree-level scalar leptoquark contributes  to the branching ratio of the tauonic decay B−  c → τ −¯ντ,  experimental constraints from this decay must be in-  cluded in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  The modified branching ratio  BR(B−  c → τ −¯ντ) is given by [25–27]  BR(B−  c →τ −¯ντ)=BR(B−  c →τ −¯ντ)SM|1−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='065gSL|2, (20)  with BR(B−  c  → τ −¯ντ)SM = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='21) × 10−2 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  The experimental bound on BR(B−  c → τ −¯ντ) varies from  ≤ 10% to ≤ 60% [28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Third, it is also worth noting  that our type of scalar leptoquarks would not contribute  to lepton flavor violation, like τ → µγ or B − ¯B mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Fourth, we impose the constraints of the D∗ and τ lon-  gitudinal polarizations that come from Belle experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Their expressions depend on the same Wilson coefficients  affecting RD and RD∗, which are written as [25, 27]  F D∗  L  F D∗  L,SM  =  � RD∗  RSM  D∗  �−1�  1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='08|gSL|2 + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='02|gT|2  − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='24 Re[g∗  SL] − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='37 Re[g∗  T]  �  (21)  P D∗  τ  P D∗  τ,SM  =  � RD∗  RSM  D∗  �−1�  1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='07|gSL|2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='86|gT|2  + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='22 Re[g∗  SL] − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='37 Re[g∗  T]  �  (22)  The experimental values of F D  L and P D∗  τ  are given by  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='035 [32] and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='51+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='21  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='16 [4, 5, 33]  respectively, whereas their SM predictions are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='04  [34] and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='497 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='013 [35] Finally, running the coeffi-  cients gSL and gT from the scale µ = 1T eV to the scale  mb = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='2GeV implies that [36, 37]:  �  gSL  gT  �  =  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='71 0  0  1  � �  gSL(µ = 1T eV )  gT (µ = 1T eV )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (23)  In the presence of the aforementioned experimental  constraints, we performed a numerical analysis of RD  and RD∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 1, we show the dependence of RD and  RD∗ on the most relevant parameters, which are the mass  of leptoquark Mφ (left panel) and the real and imag-  inary parts of the Yukawa coupling Y 4′  23 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  The other parameters in these plots were set as follows:  Y 2  12 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='5, Y 2  22 = Y 2  32 = Y 4′  13 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Furthermore, the  coupling Y 4′  23 is fixed with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='48 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='1i in the plot of RD  and RD∗ versus Mφ (left panel), whereas in the 3D plot  of RD and RD∗ versus real and imaginary parts of Y 4′  23  (right panel), the mass Mφ varies along the [800, 1500]  GeV, while real and imaginary parts of Y 4′  23 vary along  the [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='5] and [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='5], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  The correlation between RD and RD∗ is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  2, left-panel, and the correlation between the constraints  on the BR(B−  c → τ −¯ντ) and RD and RD∗ is highlighted  in the right-panel of this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' The parameters are set in  the same way as in the previous plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  These plots show that in this class of models, both RD  and RD∗ can be significantly enhanced and lie within  one sigma of the recent experimental limits, with scalar  leptoquark masses of order one TeV, which is consistent  with experimental constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  4  1000  1050  1100  1150  1200  1250  1300  1350  1400  1450  M  (GeV)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='42  RD & R D *  RD RD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='2  RD&R D *  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='36  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content="38  (Y 23  4' )  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content="4  (Y 23  4' )  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='42  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='4  RD RD  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' RD and RD∗ as function of the Letoquark mass and and real and imaginary parts of the Yukawa coupling Y23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' The  other parameters are fixed as mentioned in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='42  RD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='32  RD *  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='25  BR (B  )  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='42  RD & R D *  RD RD  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' The correlation between RD and RD∗ (left) and between both RD and RD∗ and BR(B−  c → τ −¯ντ) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' The scan  is conducted over the regions of parameter space mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  SU(5) LEPTOQUARK CONTRIBUTION TO  RK,K∗  In this section, we show that, while the scalar lep-  toquark causes non-universality of lepton flavor in the  process B → Dℓν, it does not necessarily cause non-  universality in the process B → Kℓ+ℓ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' The Lagrangian  that generates the b → sℓ+ℓ− transition is given by  L=−4Y 4′  AB ¯e′BPLV CKM  AK  di′  Kφi2−4Y 4′  AB ¯d′iKV CKM∗  AK  PR e  ′  Bφ∗  i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (24)  Thus, for b → s µ µ+, the Lagrangian is given as  L ⊃ −4Y 4′  32 ¯µ′PLbi′φi2 − 4Y 4′∗  12  ¯  Si′V CKM∗  12  PR µ  ′φ∗  i2  − 4Y 4′∗  32  ¯  Si′V CKM∗  32  PR µ  ′φ∗  i2,  (25)  where V CKM  13  ≈ 0 and V CKM  33  ≈ 1 are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Also,  we may neglect V CKM  32  respect V CKM  12  (although we in-  clude all terms in our numerical calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Thus, the  amplitude of this process is given by  M = 8Y 4′  32Y 4′∗  12 V CKM∗  12  M 2  φ  � ¯UsγµPLUb  �� ¯UµγµPLVν  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' (26)  We used the Fierz transformation identity  � ¯UsPRVµ  �� ¯  UµPLUb  �  = 1  2  � ¯UsγµPLUb  �� ¯UµγµPLvµ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (27)  As a result, the Wilson coefficient Cµ  9 for b → s µ µ+  process is written as  Cµ  9 (Λ) = 8Y 4′  32Y 4′∗  12 V CKM∗  12  M 2  φ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (28)  5  where the scale Λ ≈ 1TeV, and Cµ  10(Λ) = −Cµ  9 (Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' On  the other hand, the Lagrangian that generates the pro-  cess b → s e e+ is given by  L = − 4Y 4′  31 ¯e  ′PLbi′φi2 − 4Y 4′∗  21  ¯  Si′PRe  ′φ∗  i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (29)  After applying Fierz identity, the amplitude of b → s e e+  is given by  M = 8Y 4′  31Y 4′∗  21  M 2  φ  � ¯UsγµPLUb  �� ¯UeγµPLVe  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (30)  Hence, the Wilson coefficient Ce  9(Λ) for b → s e e+ will  be  Ce  9(Λ) = 8Y 4′  31Y 4′∗  21  M 2  φ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (31)  Moreover, Ce  10(Λ) = −Ce  9(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' The effective Hamiltonian  Heff for RK process is given by  Heff =  �  i  �  Ci(µb)Oi(µb) + ˜Ci(µb) ˜Oi(µb)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  (32)  Through renormalization group equation (RGE), we ob-  tain  Ce,µ  9,10(Λ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='2 Ce,µ  9,10(µb),  (33)  where Oi(µb) are ∆B = 1 transition operator, which is  evaluated at the mb scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' ˜Ci(µb), ˜Oi(µb) are obtained by  replacing L ↔ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' The relevant operators that describe  the Rk and Rk∗ in our model are  O9 =  �  ¯sγµPLb  ��¯lγµl),  O10 =  �  ¯sγµPLb  ��¯lγµγ5l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' (34)  The Rk and Rk∗ expressions are written as  Rk ≈1 + ∆+,  (35)  Rk∗ ≈1 + ∆+ + p(∆+ − ∆−),  (36)  where p is a function of q2  min and q2  max and ∆± is given  by  ∆± =  2  |CSM  9  |2 + |CSM  10 |2  �  ℜ  �  CSM  9  (CNP,µ  9  ± ˜  C9  NP,µ)∗�  + ℜ  �  CSM  10 (CNP,µ  10  ± ˜  C10  NP,µ)∗�  − (µ ↔ e)  �  (37)  For our model, ˜CNP  9,10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Therefore, we obtain  ∆+ = ∆− = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='4  �  CSM  9  − CSM  10  �  |CSM  9  |2 + |CSM  10 |2 ℜ  �  CNP,µ  9  (µb)−CNP,e  9  (µb)  �∗  (38)  It is worth mentioning that, whereas RK,K∗ is essen-  tially dependent on the couplings Y 4′  21 and Y 4′  32, RD,D∗ is  dependent on Y 2  22, Y 2  33 and Y 4′  23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' As a result, it is entirely  possible to keep RK,K∗ equal to the SM expectation, con-  sistently with the new LHCb results, while leaving RD,D∗  intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' To make RK,K∗ close to one, ∆+ should be very  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' This can be accomplished by having Y 4′  12 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  CONCLUSIONS  In this paper we have demonstrated that, in the pres-  ence of experimental constraints on flavor and lepton  violation observables, measured values of RD and RD∗  within 1σ can be explained in non-minimal SU(5) with  adjoint 45-dimensional Higgs multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Enhancements  for both RD and RD∗ are made possible by a tree level  transition of b → cτν, which is mediated by the associ-  ated scalar leptoquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' We also emphasized that even  though this leptoquark may contribute to RK and RK∗,  they remain independent of RD and RD∗ enhancements  because they are given in terms of different Yukawa cou-  plings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' As a result, their contributions can be easily sup-  pressed, and RK and RK∗ continue to be identical to SM  predictions, which are consistent with the most recent  LHCb data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work is partially supported by Science, Tech-  nology & Innovation Funding Authority (STDF) under  grant number 37272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  6  REFERENCES  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Lees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' [BaBar Collaboration], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  109, 101802 (2012) [arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='5442 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Lees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' [BaBar Collaboration], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D 88,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 7, 072012 (2013) [arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='0571 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Huschle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' [Belle Collaboration], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D 92,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 7, 072014 (2015) [arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='03233 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Hirose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' [Belle Collaboration], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  118, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 21, 211801 (2017) [arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='00529 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Hirose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' [Belle Collaboration], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D 97,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 1, 012004 (2018) [arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='00129 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Caria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' [Belle Collaboration], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  124, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 16, 161803 (2020) [arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='05864 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [7] LHCb Collaboration, https://indico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='ch/event/1187939/  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [8] https://hflav-eos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='ch/hflav-  eos/semi/spring21/html/RDsDsstar/RDRDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='html  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Bigi and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Gambino, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D 94, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 7, 094008  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' [arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='08030 [hep-ph]]  [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Gambino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Jung and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Schacht, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' B 795,  386-390 (2019) [arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='08209 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Bordone, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Jung and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' van Dyk, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' C  80, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 2, 74 (2020) [arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='09398 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [12] LHCb collaboration, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='09152 [hep-ex] (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [13] LHCb collaboration, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='09153 [hep-ex] (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [14] Marzia Bordone, Gino Isidori and Andrea Pattori,Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' C 76, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 8, 440 (2016) [arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='07633 [hep-  ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [15] Ilja Dorsner, Svjetlana Fajfer, Nejc Kosnik, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D  86, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 1, 015013 (2012) [ arXiv:1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='0674v2 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [16] Syuhei Iguro, Michihisa Takeuchi and Ryoutaro Watan-  abe,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  C  81,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  5,  406  (2021)  [arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='02486 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [17] David Marzocca and Sokratis Trifinopoulos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 127, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 6, 061803 (2021) [arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='05730 [hep-  ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [18] Arvind Bhaskar and Anirudhan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Madathil and Tanu-  moy Mandal and Subhadip Mitra, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D 106, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  11, 115009 (2022) [arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='09031 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [19] Peter  Cox,  Alexander  Kusenko,  Olcyr  Sumensari  and  Tsutomu  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Yanagida,  JHEP  05,  35  (2017)  [arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='03923 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [20] Bartosz Fornal and Benjam´ın Grinstein, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  119, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 24, 241801 (2017) [ arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='08535 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [21] Ilja Dorsner, Pavel Fileviez Perez, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' B 642, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  03, 248-252 (2006) [arXiv:0606062 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Khalil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Salem, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Allam, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D 89, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 09,  095011 (2015) [arXiv:1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='1482v2 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Khalil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Salem, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' B 876, 473 (2013)  [arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='3689 [hep-ph]]  [24] Minoru Tanaka and Ryoutaro Watanabe, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D  87, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 3, 034028 (2013) [arXiv:1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='1878 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [25] Syuhei Iguro, Teppei Kitahara, Yuji Omura, Ryoutaro  Watanabe and Kei Yamamoto,JHEP 02, 194 (2019)  [arXiv 1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='08899 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [26] Pouya Asadi, Matthew R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Buckley and David Shih,  JHEP 09, 10 (2018) [arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='04135 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [27] Pouya Asadi, Matthew R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Buckley and David Shih, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D 99, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 3, 035015 (2019) [arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='06597 [hep-  ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [28] Robert  Fleischer,  Ruben  Jaarsma  and  Gilberto  Tetlalmatzi-Xolocotzi,  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  C  81,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='7,  658 (2021) [arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='04023 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [29] Debjyoti Bardhan and Diptimoy Ghosh, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D  100, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 1, 011701 (2019) [arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='10432 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [30] Monika Blanke,  Andreas Crivellin,  Stefan de Boer,  Teppei Kitahara, Marta Moscati, Ulrich Nierste and  Ivan Nisandzic,Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D 99, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='7, 075006 (2019)  [arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='09603 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [31] Monika Blanke, Andreas Crivellin, Teppei Kitahara,  Marta Moscati, Ulrich Nierste and Ivan Nisandzic,Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D 100, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 3, 035035 (2019) [arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='08253 [hep-  ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Abdesselam  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [Belle  Collaboration],  arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='03102 [hep-ex] (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [33] Karol Adamczyk, arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='06380 [hep-ex] (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [34] Ashutosh Kumar Alok, Dinesh Kumar, Suman Kumb-  hakar and S Uma Sankar, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D 95, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 11,  115038 (2017), [arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='03164 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [35] Minoru Tanaka and Ryoutaro Watanabe, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' D  87, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' 3, 034028 (2013), [arXiv:1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='1878v3 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [36] Mart´ın Gonz´alez-Alonso, Jorge Martin Camalich and  Kin  Mimouni,Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  B  772,  777-785  (2017),  [arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='00410v2 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content='  [37] Syuhei  Iguro,  Michihisa  Takeuchi  and  Ryoutaro  Watanabe,Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfSwBA/content/2301.02226v1.pdf'}
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