diff --git "a/MtE2T4oBgHgl3EQfBQa7/content/tmp_files/load_file.txt" "b/MtE2T4oBgHgl3EQfBQa7/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/MtE2T4oBgHgl3EQfBQa7/content/tmp_files/load_file.txt" @@ -0,0 +1,1195 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf,len=1194 +page_content='Quark-lepton Yukawa ratios and nucleon decay in SU(5) GUTs with type-III seesaw Stefan Antusch, Kevin Hinze, and Shaikh Saad Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland E-mail: stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='antusch@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='ch, kevin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='hinze@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='ch, shaikh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='saad@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='ch Abstract: We consider an extension of the Georgi-Glashow SU(5) GUT model by a 45-dimensional scalar and a 24-dimensional fermionic representation, where the latter leads to the generation of the observed light neutrino masses via a combination of a type I and a type III seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Within this scenario, we investigate the viability of predictions for the ratios between the charged lepton and down-type quark Yukawa couplings, focusing on the second and third family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Such predictions can emerge when the relevant entries of the Yukawa matrices are generated from single joint GUT operators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' under the condition of single operator dominance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We show that three combinations are viable, (i) yτ/yb = 3/2, yµ/ys = 9/2, (ii) yτ/yb = 2, yµ/ys = 9/2, and (iii) yτ/yb = 2, yµ/ys = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We extend these possibilities to three toy models, accounting also for the first family masses, and calculate their predictions for various nucleon decay rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We also analyse how the requirement of gauge coupling unification constrains the masses of potentially light relic states testable at colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='03601v1 [hep-ph] 9 Jan 2023 Contents 1 Introduction 1 2 GUT scenario 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Particle content 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Neutrino masses 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Quark-lepton Yukawa ratios 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 Toy models 5 3 Numerical procedure 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Implementation of the charged fermion Yukawa sector 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Implementation of the neutrino sector 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 GUT scale parameters and low energy observables 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 Fitting procedure 7 4 Results 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Benchmark points 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Highest posterior densities 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Quark-lepton mass ratios 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Intermediate-scale particle masses 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Nucleon decay width and GUT scale 12 5 Conclusion 12 Appendices 12 A Definition of new Yukawa couplings 12 B Renormalization group equations 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Gauge couplings 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Yukawa matrices 16 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Effective neutrino mass operator 20 1 Introduction Grand Unified Theories (GUTs) [1–6] are arguably one of the most appealing extensions of the Standard Model (SM) of particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In 1974, a simple and elegant GUT based on the unifying gauge group SU(5) was proposed by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Georgi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Glashow (GG model) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' However, this model is incompatible with the current experimental data for three main reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Firstly, the GG model does not allow for gauge coupling unification, which – 1 – is a necessary condition for a GUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Secondly, it predicts massless neutrinos, which is in conflict with neutrino oscillation experiments requiring that at least two neutrino should be massive [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Thirdly, since the SM Higgs doublet is embedded into a 5-dimensional Higgs representation of SU(5), the GG model predicts the GUT scale relation between the charged lepton and down-type quark Yukawa matrices Ye = Y T d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) This relation in particular implies a GUT scale unification of the tau and bottom Yukawa couplings yτ = yb, as well as a unification of the muon and strange Yukawa couplings yµ = ys, which disagrees with the low energy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The first shortcoming requires extending the particle content of the minimal model by additional GUT representations and suitably splitting the masses of their component fields such that the running gauge couplings meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The second shortcoming can be addressed by introducing SU(5) representations that allow neutrino mass generation at the tree level [8–14] or at the loop level [15–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' the third shortcoming can for instance be resolved by generating the Yukawa couplings from linear combinations of the renormalisable and higher dimensional non-renormalisable operators [21],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' or at the renormalizable level by either introducing a 45-dimensional Higgs field and considering linear combinations of couplings between the SM fermions and both the 5- as well as the 45-dimensional Higgs field [22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' or by introducing vector-like fermions which mix with the SM fermions [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' However, historically a first and very aesthetic solution for the third problem was proposed by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Georgi and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Jarlskog (GJ model) in 1979 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In their model, the particle content of the GG model is extended by a 45-dimensional Higgs field (as well as by two 5-dimensional Higgs fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' If the 45-dimensional Higgs field couples to the SM fermions this gives rise to the GUT scale relation Ye = −3Y T d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) Considering a linear combination of the operators giving the relations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) would, on the one hand, solve the shortcoming (as already mentioned above), but, on the other hand, predictivity in the Yukawa sector would be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Predictivity is however maintained if it is ensured that different generations of charged leptons and down-type quarks couple to different Higgs fields (which can, for example, be achieved when a family symmetry is introduced on top of the gauge symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' To achieve predictivity, without referring to any particular family symmetry, the GJ model hypothesizes the following textures of the Yukawa coupling matrices, Yd = � � � 0 B 0 A C 0 0 0 D � � � , Y T e = � � � 0 B 0 A −3C 0 0 0 D � � � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3) implying the GUT scale relations yτ/yb = 1, yµ/ys = −3, ye/yd = −1/3 which were at that time compatible with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' However, the current data suggests (taking – 2 – only the known SM particles into account in the renormalization group (RG) evolution) that other ratios such as yτ/yb = 3/2, yµ/ys = 9/2 are better suited (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Interestingly, these latter ratios can be obtained from higher dimensional operators [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' With these higher dimensional operators at hand, models similar to the GJ model can be build if the following two conditions are satisfied: (i) the Yukawa matrices should be hierarchical, (ii) the 22- and 33- entry should be dominated by a single GUT operator, a concept which is referred to as single operator dominance [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Following this approach, non-SUSY GUT scenarios in which neutrino masses are gen- erated by a type I or a type II seesaw have been investigated in [27], respectively [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For GUT scenarios with a type I seesaw it was shown that the GUT scale ratios yτ/yb = 3/2 and yµ/ys = 9/2 are compatible with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, for GUT scenarios in which neutrino masses are generated by a type II seesaw it was found, that two combi- nations of GUT scale relations are viable, namely (i) yτ/yb = 3/2 and yµ/ys = 9/2 and (ii) yτ/yb = 2 and yµ/ys = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In this paper we will investigate the viability of such GUT scale ratios for the case that neutrino masses stem from a combination of a type I [46–50] and a type III [51] seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In this regard, we will consider a GUT scenario in which the particle content of the GG model is extended by a fermionic adjoint representation as well as by a 45-dimensional Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 The former representation is needed to generate neutrino masses, while the latter gives rise to operators yielding potentially viable GUT scale Yukawa ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, both of these representations help to allow for gauge coupling unification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Using the Mathematica package ProtonDecay [52] and extending the above scenario to “toy models” we also compute the nucleon decay widths for various decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Finally, we compute the masses of the added fermion and scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The paper is organized as follows: While the GUT scenario as well as the toy models are introduced in Section 2, the procedure for the numerical analysis is explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Section 4 the results are presented and discussed, before concluding in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Appendix A, definitions of the newly introduced Yukawa couplings are given, while all relevant RGEs that we have derived are listed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 2 GUT scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Particle content The SM fermions are embedded as usual into three generations of 5F i and 10F i 5F i = dc i(3, 1, 1 3) ⊕ ℓi(1, 2, −1 2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) 10F i = qi(3, 2, 1 6) ⊕ uc i(3, 1, −2 3) ⊕ ec i(1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) In the considered scenario, neutrino masses are generated via a combination of a type I and a type III seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The corresponding fermionic singlet Σc and triplet Σb (under 1For models in which the concept of single operator dominance has been applied, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [31–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 2A non-supersymmetric SU(5) GUT with this particle content was first considered in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' However, so far it has not been studied under the assumption of single operator dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 3 – SU(2)L) are contained in an adjoint fermionic representation 24F = Σa(8, 1, 0) ⊕ Σb(1, 3, 0) ⊕ Σc(1, 1, 0) ⊕ Σd(3, 2, −5 6) ⊕ Σe(3, 2, 5 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3) Moreover, the GUT Higgs fields decompose under the SM gauge group as 24H = Φa(8, 1, 0) ⊕ Φb(1, 3, 0) ⊕ Φc(1, 1, 0) ⊕ Φd(3, 2, −5 6) ⊕ Φe(3, 2, 5 6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4) 5H = Ta(3, 1, −1 3) ⊕ Ha(1, 2, 1 2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5) 45H = φa(8, 2, 1 2) ⊕ φb(6, 1, −1 3) ⊕ φc(3, 3, −1 3) ⊕ φd(3, 2, −7 6) ⊕ φe(3, 1, −4 3) ⊕ Tb(3, 1, −1 3) ⊕ Hb(1, 2, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6) After the SU(5) breaking, the color triplets Ta and Tb mix to yield the mass eigenstates t1 = cos(α)Ta +sin(α)Tb and t2 = − sin(α)Ta +cos(α)Tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Similarly, Ha and Hb mix to form the mass eigenstates h1 = cos(β)Ha + sin(β)Hb and h⊥ 2 = − sin(β)Ha + cos(β)Hb, where h1 is the SM Higgs doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Neutrino masses At tree-level the relevant GUT operators for neutrino mass generation read3 L ⊃ YA 5F 24F 5H + YB 5F 24F 45H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7) After the GUT symmetry breaking the following relevant terms emerge L ⊃ −Y2ℓΣbHa − Y8ℓΣbHb − Y4ℓΣcHa − Y13ℓΣcHb − mΣbΣbΣb − mΣcΣcΣc, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8) where mΣb and mΣb are the respective masses of Σb and Σc, and where the GUT scale relations Y2 = − � 3 10 YA, Y4 = YA, Y8 = √ 5 4 YB, and Y13 = √ 3 4 YB (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' After the SU(2) triplet Σb and SU(2) singlet Σc have been integrated out and the two Higgs fields Ha and Hb have taken their vacuum expectation values (vevs) va and vb, where v2 a + v2 b = v2 = (246 GeV)2, and where va = v cos(β) and vb = v sin(β), the neutrino mass matrix mν reads mij ν = −(Y i 2 va + Y i 8 vb)(Y j 2 va + Y j 8 vb) 4mΣb − (Y i 4 va + Y i 13 vb)(Y j 4 va + Y j 13 vb) 4mΣc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='10) Since the neutrino mass matrix mν is of rank two, two massive and one massless neutrino are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3After the GUT symmetry breaking these two GUT operators decompose into 19 SM Yukawa interac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For details see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 4 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Quark-lepton Yukawa ratios With X and Y representing one or multiple Higgs fields, the charged fermion masses stem from GUT operators of the form Y ij 5 : 10F i5F jX ⊃ Y ij d , Y ij e (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='11) Y ij 10 : 10F i10F jY ⊃ Y ij u , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='12) where Yu, Yd and Ye denote the usual SM charged fermion Yukawa matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Assuming in the charged fermion Yukawa sector the concept of single operator dominance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' that each Yukawa entry is dominated by a singlet GUT operator, allows to connect the down- type with the charged lepton Yukawa matrix via group theoretical Clebsch-Gordan (CG) factors cij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In SU(5) GUTs, and considering up to dimension five operators, the potentially viable CG factors are |cij| ∈ {1/6, 1/2, 2/3, 1, 3/2, 2, 3, 9/2, 6, 9, 18}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The possible GUT operators yielding these ratios are given in [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, if the matrix Y5 is assumed to be of hierarchical nature and dominated by its diagonal entries, then the second and third family down-type quark and charged lepton masses stem dominantly from the GUT operators O2 and O3 dominating the 22 and 33 positions in Y5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Depending on which operators are chosen for O2 and O3, different GUT scale Yukawa ratios yτ/yb and yµ/ys are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Our numerical analysis (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Section 4) shows that there are only two possible choices for the GUT scale ratio yτ/yb, namely 3/2 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The former CG factor can be complemented by a factor 9/2 for the second family, while for the latter CG factor two different completions, yµ/ys = 6 or yµ/ys = 9/2, are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 Toy models We now extend the above motivated scenarios to three toy models which also include the first family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For simplicity, we chose the matrix Y5 to be of diagonal nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The double ratio (yµyd)/(yeys) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9, which is nearly constant under renormalization group running (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [53]), suggests, that the the ratio yµ/ys = 9/2 is best complemented by a ratio ye/yd = 4/9, while the best completion of the ratio yµ/ys = 6 is given by ye/yd = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Utilizing these ratios, our three toy models relate the down-type with the charged lepton Yukawa matrix via Model 1: Ye = diag �4 9, 9 2, 3 2 � Y T d , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='13) Model 2: Ye = diag �4 9, 9 2, 2 � Y T d , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='14) Model 3: Ye = diag �1 2, 6, 2 � Y T d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15) Moreover, for simplicity4 we assume in each toy model that Y10 is dominated by the operator 10F 10F 5H in all entries, yielding a symmetric up-type Yukawa matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Yu = 4We might consider higher-dimensional operators also for Y10, for example to explain the mass hierarchy, however since no Yukawa ratio predictions arise from this sector, we stick to the simplest case in our toy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 5 – Y T u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Finally, in all toy models neutrino masses stem from a linear combination of the operators 5F 24F 5H and 5F 24F 45H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3 Numerical procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Implementation of the charged fermion Yukawa sector We implement all three toy models at the GUT scale as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In all three models the down-type Yukawa matrix Yd is simply implemented as Yd = diag(yd 1, yd 2, yd 3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) while the charged lepton Yukawa matrix Ye is implemented according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='14), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Since Yu is symmetric we use a Takagi decomposition and imple- ment it as Yu = U † uY diag u U ∗ u, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) where5 Uu = � � � 1 0 0 0 cu 23 su 23 0 −su 23 cu 23 � � � � � � cu 13 0 su 13e−iδu 0 1 0 −su 13eiδu 0 cu 13 � � � � � � cu 12 su 12 0 −su 12 cu 12 0 0 0 1 � � � � � � eiβu 1 0 0 0 eiβu 2 0 0 0 1 � � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3) and where Y diag u = diag(yu 1, yu 2, yu 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Implementation of the neutrino sector In order to simplify the analysis we assume in the neutrino sector that the Yukawa matrices Y5 and Y6 (for the definitions of these couplings, see Appendix A) are of the form Y5 = z1 � � � 0 1 1 � � � , Y6 = z2 � � � 1 1 3 � � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4) where z1 and z2 are real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Furthermore, we denote the relative phase difference between mΣb and mΣc by γ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' γ = arg(mΣb/mΣc)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' This structure is motivated by CSD3 [56] which in the case of type I seesaw has been shown to correctly describe the low-scale neutrino observables together with a normal neutrino mass hierarchy (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [27] for a recent work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 GUT scale parameters and low energy observables Each toy model contains 33 input parameters which decompose into the GUT scale MGUT, the SU(5) gauge coupling gGUT, the masses of the added particles,6 mΦa, mΦb, mφa, mφb, 5Here we have dropped three unphysical parameters but kept the GUT phases βu 1 and βu 2 which effect the nucleon decay widths [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 6Note that mΣd = mΣe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 6 – mφc, mφd, mφe, mΣa, mΣb, mΣc, mΣd, mt1, mt2, mh2, the singular values yu 1, yu 2, yu 3, yd 1, yd 2, yd 3 and angles θu 12, θu 13, θu 23 as well as phases δu, βu 1 , βu 2 of the charged fermion Yukawa matrices, the parameters of the neutrino Yukawa couplings z1, z2, and γ, and the eigenstate mixing angles α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The respective ranges of these input parameters are given by7 MGUT < MPl, mΦa, mΦb, mφa, mφb, mφc, mφd, mφe, mΣa, mΣb, mΣc, mΣd, mh2 ∈ [1 TeV, MGUT], mt1, mt2 ∈ [1011 GeV, MGUT], gGUT, yu 1, yu 2, yu 3, yd 1, yd 2, yd 3 ∈ [0, 1], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5) θu 12, θu 13, θu 23, α, β ∈ [0, π/2], δu, βu 1 , βu 2 , γ ∈ [−π, π), z1, z2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' These input parameters are fitted to the 22 low-scale observables (listed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6)) and the nucleon decay widths of thirteen decay channels (listed in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' g1, g2, g3, yu, yc, yt, yd, ys, yb, θCKM 12 , θCKM 13 , θCKM 23 , δCKM, ye, yµ, yτ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6) ∆m2 21, ∆m2 31, θPMNS 12 , θPMNS 13 , θPMNS 23 , δPMNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For the SM gauge couplings and Yukawa observables we take the experimental values from [53], while the values for the neutrino sector are taken from NuFIT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 Fitting procedure After implementing the input parameters given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5) at the GUT scale we compute the RG evolution to the Z scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For the gauge couplings we use a 2-loop running, while we compute the running of the Yukawa matrices and the effective neutrino mass operator at 1-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The nucleon decay widths are computed using the Mathematica package Proton Decay [52] (for a description of the calculation see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Taking into account all observ- ables we compute at the low scale the χ2-function which we minimize using a differential evolution algorithm giving us a benchmark point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' With a flat prior distribution we calcu- late 4 × 106 data points performing a Markov-chain-Monte-Carlo (MCMC) analysis using an adaptive Metropolis-Hastings algorithm [65] which we start from this benchmark point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' These data points are finally used to compute the highest posterior density (HPD) ranges of various quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4 Results The results of our numerical analysis are presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We are in particular interested in the nucleon decay predictions, the intermediate-scale particle masses as well 7Note that although we do not put any perturbativity constraints on the neutrino Yukawa couplings z1 and z2 the fit automatically choses them to be below 1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 7 – decay channel τ/B [year] Γpartial [GeV] Reference Proton: p → π0 e+ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 · 1034 < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 · 10−67 [58] p → π0 µ+ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 · 1034 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 · 10−66 [58] p → η0 e+ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 · 1034 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 · 10−66 [59] p → η0 µ+ > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 · 1033 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 · 10−66 [59] p → K0 e+ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 · 1033 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 · 10−65 [60] p → K0 µ+ > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 · 1033 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8 · 10−66 [61] p → π+ ν > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 · 1032 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 · 10−65 [62] p → K+ ν > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 · 1033 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 · 10−66 [63] Neutron: n → π− e+ > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 · 1033 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 · 10−66 [59] n → π− µ+ > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 · 1033 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 · 10−66 [59] n → π0 ν > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 · 1033 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 · 10−65 [62] n → η0 ν > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 · 1032 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 · 10−65 [60] n → K0 ν > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 · 1032 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 · 10−64 [60] Table I: Current experimental bounds on the decay widths Γpartial, respectively lifetime τ/B at 90 % confidence level, where B is the branching ratio for the decay channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' See also [64] for future projections and sensitivities of various upcoming detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' as the low scale predictions for the charged lepton and down-type quark mass ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 we show the results of our minimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Starting an MCMC analysis from these benchmark points allows us to obtain the HPD ranges of various quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The results of this analysis is presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Benchmark points We obtain for all three models benchmark points through a minimization of the χ2-function as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Table II the input parameters for the respective benchmark points are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, the dominant pulls χ2 i are presented in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' All three models can be very well fitted to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The strongest (though quite small) pull is given by the first and second family down-type quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The biggest difference between the three models is the respectively favored GUT scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For Models 2 and 3 a GUT scale above 1017 GeV is favored, while for the benchmark point of Model 1 a GUT scale below 1016 GeV is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' This also results in different results for the predicted nucleon decay rates (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Another difference is the preferred choice of some of the intermediate-scale particle masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In the presented benchmark point the mass of the fermionic field Σa is obtained to be at the GUT scale for Model 1, at the intermediate scale for Model 2 and at the relatively low scale (23 TeV) for Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, a mass of the leptoquark φc of 1 – 8 – TeV, respectively 4 TeV is obtained for Model 3, respectively Model 2, whereas for Model 1 the mass of this field is above 106 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For the HPD results of these particle masses confer the subsequent section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Model 1 Model 2 Model 3 gGUT / 10−1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='94 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='33 log10(MGUT / GeV) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 log10(mφa / GeV) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='43 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 log10(mφc / GeV) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 log10(mΣa / GeV) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='36 log10(mΣb / GeV) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 log10(mΣc / GeV) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 log10(mΣd / GeV) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 yu 1 / 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='99 yu 2 / 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='18 yu 3 / 10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='54 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='65 yd 1 / 10−6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='46 yd 2 / 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='99 yd 3 / 10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='64 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='36 z1 / 10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='50 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='42 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='86 z2 / 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='50 γ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='68 α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='50 Table II: The GUT scale input parameters of the benchmark points for all three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' χ2 χ2 yd χ2 ys χ2 yb χ2 yµ χ2 yτ χ2 Γ(p→π0e+) Model 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='44 Model 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 Model 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 Table III: The total χ2 as well as the dominant pulls χ2 i for the benchmark points of all three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Highest posterior densities As described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 we vary the input parameters listed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 around their benchmark points (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Table II) using an MCMC analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' From these generated points we then compute the HPD intervals of various parameters and observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 9 – Model 1 Model 2 Model 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='19 ye/yd HPD intervals for ye/yd Model 1 Model 2 Model 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 yμ/ys HPD intervals for yμ/ys Model 1 Model 2 Model 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='63 yτ/yb HPD intervals for yτ/yb Figure 1: Low scale (MZ) HPD intervals for charged lepton and down-type quark Yukawa ratios of all three families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The 1σ (2σ) HDP intervals are colored dark (light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Figures 1 – 4 we use the following color coding: For Model 1, 2, and 3 the HPD intervals of various quantities are colored red, green, and blue, respectively, while the 1σ (2σ) HPD intervals are colored dark (light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Quark-lepton mass ratios The HPD results for the low scale charged lepton and down-type quark mass ratios are presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The horizontal dashed line represents the current experimental central value, whereas the white region shows the current experimental 1σ range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Clearly, all three models are capable of reproducing viable mass ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' This strengthens the results of the benchmark points in the previous subsection (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Tables II and III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Compared to Model 2 and 3, Model 1 gives a bit smaller predictions for the mass ratios for all three generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Intermediate-scale particle masses Figure 2 shows the predicted HPD intervals of the intermediate-scale particle masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Most of the masses are predicted to be out of the reach of current and future colliders, because they would either produce too much proton decay, spoil gauge coupling unification or be- cause of the fit of the fermion masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' But interestingly, the fields Φb, φc and Σa are not only potentially within the reach of future searches, but can also be used to distinguish be- tween the different models: An observation of the one of the fields Φb or Σa would strongly hint towards Model 3, while an observation of the field φc would disfavor Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In fact, the most promising lookout could be for the leptoquark φc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The upper bound of the HPD 1σ range is predicted to be 23 TeV (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8 TeV) in Model 2 (3), whereas the upper bound of the 2σ intervals is 175 TeV (17 TeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In the following, we briefly state the current collider bounds on these particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The scalar triplet, Φb, with zero hypercharge, residing in the 24H multiplet is expected to be light in Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Note that Φb contains a neutral Φ0 b and a pair of singly charged Φ± b states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In the low-energy effective theory, a term of the form h† 1Φ2 bh1 is allowed, where – 10 – mΦa mΦb mTa mTb m ϕa m ϕb m ϕc m ϕd m ϕe mΣa mΣb mΣc mΣd Model 1 Model 2 Model 3 2 4 6 8 10 12 14 16 18 log10(μ/GeV) HPD intervals for intermediate-scale particle masses Figure 2: HPD intervals of the intermediate-scale particle masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The 1σ (2σ) HDP intervals are colored dark (light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' h1 is the SM Higgs doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' As a result of this coupling, the SM Higgs can decay into two photons h0 → γγ via a one-loop diagram mediated by the Φ± b states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Consistency with the LHC data requires these charged states to have masses above 250 GeV [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The scalar leptoquark φc, which is a triplet of SU(2)L, resides around the TeV scale in Models 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In both models, its coupling to the SM fermions is dominated by the third-generation quark and lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Hence, within our scenarios, its decay branching fraction is dominated by a bτ final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Since leptoquarks carry color, they are efficiently produced at the LHC through gluon-initiated as well as quark-initiated processes [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' LHC searches of pp → bbττ from pair-produced leptoquarks rule out leptoquark masses below 1400 GeV [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' As can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2), the color octet fermion Σa, which is expected to be light in Model 3, couples, for example, to a singlet down-quark (lepton doublet) and a super-heavy colored triplet (octet) scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Consequently, the lifetime of a TeV scale Σa is expected to be large, and it behaves like a long-lived gluino that typically arises in split- supersymmetric scenarios [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Long-lived colored particles would hadronize, forming so-called R-hadrons [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' These bound states are comprised of the long-lived state and light SM quarks or gluons, and interact with the detector material, typically inside the calorimeters, via hadronic interactions of the light-quark constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Motivated by split- supersymmetric models, R-hadrons are extensively searched for at the LHC [73, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Non- observation of any deviations of the signal from the expected background puts to a lower – 11 – limit on the mass of the long-lived Σa fermion of 2000 GeV [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Nucleon decay width and GUT scale Figure 3 shows the predictions for the HPD intervals of the GUT scale MGUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, the predicted HPD ranges for the nucleon decay widths of the various decay channels are presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The blue line segments in the latter picture indicate the current experimental bounds at 90 % confidence level (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, the future constraints on the decay widths for the decay channels p → π0e+ and n → π−e+ which will be provided by Hyper-Kamiokande [75] are presented by orange line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Figure 3 it can be seen that Model 1 clearly predicts the GUT scale to be below 1016 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' On the other hand, a much larger GUT scale is preferred by the Models 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Since the nucleon decay width is inversely proportional to the forth power of the GUT scale in the case of gauge boson mediated nucleon decay, this also results in strongly different prediction for the nucleon decay widths of the various channels as it can be seen in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The nucleon decay predictions for Model 1 are very close to the current bounds, the 1σ HPD interval of the proton decay channel p → π0e+ will be fully probed by Hyper-Kamiokande.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, Hyper-Kamiokande will probe most of the 1σ HPD interval of the neutron decay channel n → π−e+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' On the other hand, the gauge boson mediated nucleon decay is highly suppressed in Models 2 and 3 and cannot be probed by any planed experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Therefore, observation of nucleon decay in the decay channels p → π0e+ and n → π−e+ would clearly favour Model 1 over the Models 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 5 Conclusion In this paper we considered an extension of the Georgi-Glashow SU(5) GUT scenario by a 45-dimensional scalar and a 24-dimensional fermionic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Neutrino masses in this scenario are generated by a combination of a type I and a type III seesaw mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Assuming the concept of single operator dominance we investigated which GUT scale charged lepton and down-type quark Yukawa ratios can be viable for the second and third family and found that three combinations work: (i) yτ/yb = 3/2, yµ/ys = 9/2, (ii) yτ/yb = 2, yµ/ys = 9/2, and (iii) yτ/yb = 2, yµ/ys = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Also taking into account the origin of the first family masses we extended these possibilities to three toy models and analyzed various of their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We showed that experimental discrimination between these models could be possible since they predict different nucleon decay rates as well as distinct light relics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Appendices A Definition of new Yukawa couplings The Lagrangian density contains the two terms L ⊃ YA 5i F 24F 5H + YB 5i F 24F 45H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) – 12 – Model 1 Model 2 Model 3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 MGUT HPD intervals for MGUT Figure 3: Predicted HPD intervals of the GUT scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The 1σ (2σ) HDP intervals are colored dark (light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' p → π0e+ p → π0 μ+ p → η0e+ p → η0 μ+ p → K0e+ p → K0 μ+ p → π+ν p → K+ν n → π-e+ n → π- μ+ n → π0ν n → η0ν n → K0ν Model 1 Model 2 Model 3 85 80 75 70 65 log10(Γ/GeV) HPD intervals for decay widths Γ of different nucleon decay channels Figure 4: Predicted HPD intervals of the nucleon decay widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The 1σ (2σ) HDP intervals are colored dark (light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For each decay channel the blue line segments represent the current experimental constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The future Hyper-Kamiokande constraints for the decay channels p → π0e+ and n → π−e+ are indicated by orange line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='– 13 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='After the GUT symmetry breaking they decompose into ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='L = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15YA dcΣcTa − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='10YA ℓΣcHa + YA dcΣaTa + YA ℓΣbHa+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='YA dcΣdHa + YA ℓΣeTa + ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB dcΣbφc+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 YB ℓΣbHb − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB dcΣdφa − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6YB dcΣdHb + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB ℓΣdφe− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB dcΣeφd + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB ℓΣeTb − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB ℓΣeφc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='≡ Y1 dcΣcTa + Y2 ℓΣcHa + Y3 dcΣaTa + Y4 ℓΣbHa+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='Y5 dcΣdHa + Y6 ℓΣeTa + Y7 dcΣcTb + Y8 ℓΣcHb+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='Y9 dcΣaTb + Y10 dcΣaφb + Y11 ℓΣaφa + Y12 dcΣbφc+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='Y13 ℓΣbHb + Y14 dcΣdφa + Y15 dcΣdHb + Y16 ℓΣdφe+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='Y17 dcΣeφd + Y18 ℓΣeTb + Y19 ℓΣeφc ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) where we defined the Yukawa matrices YN, with N = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' , 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' B Renormalization group equations Here the RGEs for the gauge and Yukawa couplings as well as for the effective neutrino mass operator are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We have used the Mathematica package SARAH [76, 77] to obtain the RGEs for the gauge and Yukawa couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The SM contribution for the RGE of the effective neutrino mass operator is taken from [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In order to compute the new contribution for this RGE we have used the method described therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We use the following definition for the Heaviside-Theta function H(µ, m) = � 1, for µ ≥ m, 0, for µ < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Gauge couplings The RGEs for gauge couplings (i, k = 1 − 3) are given by µdgi dµ = βgi 1−loop 16π2 + βgi 2−loop (16π2)2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) where βgi 1−loop is the 1-loop and βgi 2−loop is the 2-loop contribution given by βgi 1−loop = � aSM i + H(µ, m)∆ai � g3 i (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3) βgi 2−loop = � k bSM ik g2 k + � k ∆bikg2 k H(µ, m) + βY,SM i + ∆βY i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4) – 14 – Here, aSM i , bSM ik and βY,SM i are the well known SM 1-loop and 2-loop coefficients as well as Yukawa contributions [79, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, the ∆βY i are given by ∆βY i = g3 i � k cikY T k Y ∗ k H2 k, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5) where we introduced the abbreviation H2 k = H(µ, mF )H(µ, mH) associated to each of the Yukawa interactions, where, F and H refer to the BSM fermion and scalar appearing in that interaction, respectively, and where the cik are given by c1k = − �1 5, 3 10, 8 15, 9 20, 29 10, 17 5 , 1 5, 3 10, 8 15, 8 15, 12 5 , 3 5, 9 20, 116 15 , 29 10, 17 5 , 29 5 , 17 5 , 51 10 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6) c2k = − � 0, 1 2, 0, 11 4 , 3 2, 3, 0, 1 2, 0, 0, 4, 6, 11 4 , 4, 3 2, 3, 3, 3, 9 2 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7) c3k = − �1 2, 0, 13 3 , 0, 2, 1, 1 2, 0, 13 3 , 13 3 , 6, 3 2, 0, 16 3 , 2, 1, 4, 1, 3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8) Finally, the ∆ai and ∆bi are given as a sum over the 1-loop and 2-loop coefficients of the BSM fermions and scalars, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆ai = � I ∆aI i , ∆bi = � I ∆bI i , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9) where I runs over all BSM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The 1-loop coefficients are then given by ∆aφa i = �4 5, 4 3, 2 � , ∆aφb i = � 2 15, 0, 5 6 � , ∆aφc i = �1 5, 2, 1 2 � , ∆aφd i = �49 30, 1 2, 1 3 � , ∆aφe i = �16 15, 0, 1 6 � , ∆aΦa i = {0, 0, 1 2}, ∆aΦb i = � 0, 1 3, 0 � , ∆aΣa i = {0, 0, 2}, ∆aΣb i = � 0, 4 3, 0 � , ∆aΣd,e i = �5 3, 1, 2 3 � , ∆ah⊥ i = � 1 10, 1 6, 0 � , ∆at,t⊥ i = � 1 15, 0, 1 6 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='10) whereas the 2-loop coefficients read ∆bφa ik = � � � 36 25 36 5 144 5 12 5 52 3 48 18 5 18 84 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bφb ik = � � � 8 75 0 16 3 0 0 0 2 3 0 115 3 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bφc ik = � � � 4 25 24 5 16 5 8 5 56 32 2 5 12 11 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bφd ik = � � � 2401 150 147 10 392 15 49 10 13 2 8 49 15 3 22 3 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bφe ik = � � � 1024 75 0 256 15 0 0 0 32 15 0 11 3 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bΦa ik = � � � 0 0 0 0 0 0 0 0 21 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bΦb ik = � � � 0 0 0 0 28 3 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bΣa ik = � � � 0 0 0 0 0 0 0 0 48 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bΣb ik = � � � 0 0 0 0 64 3 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bΣd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e ik = � � � 25 12 15 4 20 3 5 4 49 4 4 5 6 3 2 38 3 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bh⊥ ik = � � � 9 50 9 10 0 3 10 13 6 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='t⊥ ik = � � � 4 75 0 16 15 0 0 0 2 15 0 11 3 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='11) – 15 – B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Yukawa matrices The RGEs of the Yukawa matrices read µdYf dµ = βf 16π2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='12) where f = {u, d, e, k} and (k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' , 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For the SM Yukawa matrices Yu, Yd and Ye (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' f = u, d, e) the beta functions are given by βf = βSM f + δβf, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='13) where βSM f is the SM beta function [79, 80], and where δβf = YfT1 + � k af k(Yk)j(Y T d Y ∗ k )i H2 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='14) Here, we have defined T1 as T1 = Y T 2 Y ∗ 2 H2 2 + 3 2Y T 4 Y ∗ 4 H2 4 + 3Y T 5 Y ∗ 5 H2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15) while the af k are given by au k = � 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='16) ad k = �1 2, 0, 4 3, 0, 3, 0, 1 2, 0, 4 3, 4 3, 0, 3 2, 0, 8 3, 1, 0, 2, 0, 0 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='17) ae k = � 0, −3 2, 0, 15 4 , 0, 3 2, 0, 1 2, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 3 2, 9 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='18) In order to simplify the notation, from hereon, associated to each Yukawa Yi → Yi H2 i must be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The beta function of the Yukawa matrices Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' , Y19 then read β1 = Y1 � −1 5g2 1 − 4g2 3 + � k a1 kY T k Y ∗ k � + � YdY † d � Y1 + � w b1 w � Y T 1 Y ∗ w � Yw + 8 3 � Y T 3 Y ∗ 9 � Y7 + 2 � Y T 6 Y ∗ 18 � Y7, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='19) β2 = Y2 � − 9 20g2 1 − 9 4g2 2 + � k a2 kY T k Y ∗ k + T � + � −3 2Y T e Y ∗ e � Y2 + � w b2 w � Y T 2 Y ∗ w � Yw + 3 2 � Y T 4 Y ∗ 13 � Y8 + 3 � Y T 5 Y ∗ 15 � Y8, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='20) β3 = Y3 � −1 5g2 1 − 13g2 3 + � k a3 kY T k Y ∗ k � + � YdY † d � Y3 + � w b3 w � Y T 3 Y ∗ w � Yw + � Y T 1 Y ∗ 7 � Y9 + 2 � Y T 6 Y ∗ 18 � Y9, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='21) β4 = Y4 � − 9 20g2 1 − 33 4 g2 2 + � k a4 kY T k Y ∗ k + T � + �5 2Y T e Y ∗ e � Y4 + � w b4 w � Y T 4 Y ∗ w � Yw – 16 – + � Y T 2 Y ∗ 8 � Y13 + 3 � Y T 5 Y ∗ 15 � Y13, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='22) β5 = Y5 � −29 20g2 1 − 9 4g2 2 − 8g2 3 + � k a5 kY T k Y ∗ k + T � + � 3YdY † d � Y5 + � w b5 w � Y T 5 Y ∗ w � Yw + � Y T 2 Y ∗ 8 � Y15 + 3 2 � Y T 4 Y ∗ 13 � Y15, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='23) β6 = Y6 � −17 10g2 1 − 9 2g2 2 − 4g2 3 + � k a6 kY T k Y ∗ k � + �1 2Y T e Y ∗ e � Y6 + � w b6 w � Y T 6 Y ∗ w � Yw + � Y T 1 Y ∗ 7 � Y18 + 8 3 � Y T 3 Y ∗ 9 � Y18, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='24) β7 = Y7 � −1 5g2 1 − 4g2 3 + � k a7 kY T k Y ∗ k � + � YdY † d � Y7 + � w b7 w � Y T 7 Y ∗ w � Yw + � 2Y T 18Y ∗ 6 � Y1 + 8 3 � Y T 9 Y ∗ 3 � Y1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='25) β8 = Y8 � − 9 20g2 1 − 9 4g2 2 + � k a8 kY T k Y ∗ k � + � Y T e Y ∗ e � Y8 + � w b8 w � Y T 8 Y ∗ w � Yw + 3 2 � Y T 13Y ∗ 4 � Y2 + 3 � Y T 15Y ∗ 5 � Y2, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='26) β9 = Y9 � −1 5g2 1 − 13g2 3 + � k a9 kY T k Y ∗ k � + � YdY † d � Y9 + � w b9 w � Y T 9 Y ∗ w � Yw + 2 � Y T 18Y ∗ 6 � Y3 + � Y T 7 Y ∗ 1 � Y3, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='27) β10 = Y10 � −1 5g2 1 − 13g2 3 + � k a10 k Y T k10Y ∗ k � + � YdY † d � Y10 + � w b10 w � Y T 10Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='28) β11 = Y11 � − 9 20g2 1 − 9 4g2 2 − 9g2 3 + � k a11 k Y T k Y ∗ k � + � Y T e Y ∗ e � Y11 + � w b11 w � Y T 11Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='29) β12 = Y12 � −1 5g2 1 − 6g2 2 − 4g2 3 + � k a12 k Y T k Y ∗ k � + � YdY † d � Y12 + � w b12 w � Y T 12Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='30) β13 = Y13 � − 9 20g2 1 − 33 4 g2 2 + � k a13 k Y T k Y ∗ k � + �1 2Y T e Y ∗ e � Y13 + � w b13 w � Y T 13Y ∗ w � Yw + 3 � Y T 15Y ∗ 5 � Y4 + � Y T 8 Y ∗ 2 � Y4, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='31) β14 = Y14 � −29 20g2 1 − 9 4g2 2 − 8g2 3 + � k a14 k Y T k Y ∗ k � + � YdY † d � Y14 + � w b14 w � Y T 14Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='32) β15 = Y15 � −29 20g2 1 − 9 4g2 2 − 8g2 3 + � k a15 k Y T k Y ∗ k � + � YdY † d � Y15 + � w b15 w � Y T 15Y ∗ w � Yw – 17 – + 3 2 � Y T 13Y ∗ 4 � Y5 + � Y T 8 Y ∗ 2 � Y5, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='33) β16 = Y16 � −17 10g2 1 − 9 2g2 2 ��� 4g2 3 + � k a16 k Y T k Y ∗ k � + �1 2Y T e Y ∗ e � Y16 + � w b16 w � Y T 16Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='34) β17 = Y17 � −29 20g2 1 − 9 4g2 2 − 8g2 3 + � k a17 k Y T k Y ∗ k � + � YdY † d � Y17 + � w b17 w � Y T 17Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='35) β18 = Y18 � −17 10g2 1 − 9 2g2 2 − 4g2 3 + � k a18 k Y T k Y ∗ k � + �1 2Y T e Y ∗ e � Y18 + � w b18 w � Y T 18Y ∗ w � Yw + � Y T 7 Y ∗ 1 � Y6 + 8 3 � Y T 9 Y ∗ 3 � Y6, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='36) β19 = Y19 � −17 20g2 1 − 9 2g2 2 − 4g2 3 + � k a19 k Y T k Y ∗ k � + �1 2Y T e Y ∗ e � Y19 + � w b19 w � Y T 19Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='37) where the coefficients af k are given by a1 k = {3, 1, 8 3, 0, 0, 2, 3 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='38) a2 k = {3 2, 5 2, 0, 3 2, 3, 0, 3 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='39) a3 k = {1, 0, 9 2, 0, 0, 2, 0, 0, 1 2, 1 2, 1, 0, 0, 0, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='40) a4 k = {0, 1, 0, 11 4 , 3, 0, 0, 0, 0, 0, 0, 3 2, 1 2, 0, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='41) a5 k = {0, 1, 0, 3 2, 9 2, 0, 0, 0, 0, 0, 0, 0, 0, 4 3, 1 2, 1 2, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='42) a6 k = {1, 0, 8 3, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 2, 3 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='43) a7 k = {3 2, 1, 0, 0, 0, 0, 3, 1, 8 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='44) a8 k = {3 2, 1, 0, 0, 0, 0, 3 2, 5 2, 0, 0, 0, 0, 3 2, 0, 3, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='45) a9 k = {0, 0, 1 2, 0, 0, 0, 1, 0, 9 2, 1 2, 1, 0, 0, 0, 0, 0, 0, 2, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='46) a10 k = {0, 0, 1 2, 0, 0, 0, 0, 0, 1 2, 19 6 , 1, 0, 0, 0, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='47) a11 k = {0, 0, 1 2, 0, 0, 0, 0, 0, 1 2, 1 2, 6, 0, 0, 1, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='48) a12 k = {0, 0, 0, 1 2, 0, 0, 0, 0, 0, 0, 0, 4, 1 2, 0, 0, 0, 0, 0, 1}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='49) a13 k = {0, 0, 0, 1 2, 0, 0, 0, 1, 0, 0, 0, 3 2, 11 4 , 0, 3, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='50) a14 k = {0, 0, 0, 0, 1 2, 0, 0, 0, 0, 0, 1, 0, 0, 5, 1 2, 1 2, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='51) – 18 – a15 k = {0, 0, 0, 0, 1 2, 0, 0, 1, 0, 0, 0, 0, 3 2, 4 3, 9 2, 1 2, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='52) a16 k = {0, 0, 0, 0, 1 2, 0, 0, 0, 0, 0, 0, 0, 0, 4 3, 1 2, 4, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='53) a17 k = {0, 0, 0, 0, 0, 1 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 1 2, 3 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='54) a18 k = {0, 0, 0, 0, 0, 1 2, 1, 0, 8 3, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='55) a19 k = {0, 0, 0, 0, 0, 1 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1 2, 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='56) while the coefficients bf k read b1 w = {0, 0, 4 3, 0, 1, 0, 3 2, 0, 0, 4 3, 0, 3 2, 0, 8 3, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='57) b2 w = {0, 0, 0, 3 4, 0, 3 2, 0, 3 2, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='58) b3 w = {1 2, 0, 0, 0, 1, 0, 1 2, 0, 0, 4 3, 0, 3 2, 0, 8 3, 1, 0, 2, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='59) b4 w = {0, 1 2, 0, 0, 0, 3 2, 0, 1 2, 0, 0, 4, 0, 9 4, 0, 0, 3 2, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='60) b5 w = {1 2, 0, 4 3, 0, 0, 0, 1 2, 0, 4 3, 4 3, 0, 3 2, 0, 8 3, 4, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='61) b6 w = {0, 1 2, 0, 3 4, 0, 0, 0, 1 2, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 7 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='62) b7 w = {3 2, 0, 4 3, 0, 1, 0, 0, 0, 4 3, 4 3, 0, 3 2, 0, 8 3, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='63) b8 w = {0, 3 2, 0, 3 4, 0, 3 2, 0, 0, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='64) b9 w = {1 2, 0, 4, 0, 1, 0, 1 2, 0, 0, 4 3, 0, 3 2, 0, 8 3, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='65) b10 w = {1 2, 0, 4 3, 0, 1, 0, 1 2, 0, 4 3, 0, 0, 3 2, 0, 8 3, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='66) b11 w = {0, 1 2, 0, 3 4, 0, 3 2, 0, 1 2, 0, 0, 0, 0, 3 4, 0, 0, 3 2, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='67) b12 w = {1 2, 0, 4 3, 0, 1, 0, 1 2, 0, 4 3, 4 3, 0, 0, 0, 8 3, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='68) b13 w = {0, 1 2, 0, 9 4, 0, 3 2, 0, 1 2, 0, 0, 4, 0, 0, 0, 0, 3 2, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='69) b14 w = {1 2, 0, 4 3, 0, 1, 0, 1 2, 0, 4 3, 4 3, 0, 3 2, 0, 0, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='70) b15 w = {1 2, 0, 4 3, 0, 4, 0, 1 2, 0, 4 3, 4 3, 0, 3 2, 0, 8 3, 0, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='71) b16 w = {0, 1 2, 0, 3 4, 0, 3 2, 0, 1 2, 0, 0, 4, 0, 3 4, 0, 0, 0, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='72) b17 w = {1 2, 0, 4 3, 0, 1, 0, 1 2, 0, 4 3, 4 3, 0, 3 2, 0, 8 3, 1, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='73) b18 w = {0, 1 2, 0, 3 4, 0, 7 2, 0, 1 2, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 0, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='74) b19 w = {0, 1 2, 0, 3 4, 0, 3 2, 0, 1 2, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 3 2, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='75) – 19 – B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Effective neutrino mass operator The RGE for the effective neutrino mass operator reads 16π2µdκ dµ = βSM κ + ∆βκ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='76) where βSM κ is the SM contribution as given in [78] and ∆βκ is the correction due to the added BSM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For ∆βκ we find8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='∆βκ = κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='�1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2Y ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Y T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4Y ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 Y T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2Y ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 Y T ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='13 + 6Y T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15Y ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='77) References [1] J.' metadata={'source': 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661–664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Pati and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Salam, “Lepton Number as the Fourth Color,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' D 10 (1974) 275–289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='D 11, 703–703 (1975)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Georgi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Glashow, “Unity 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