diff --git "a/GNE3T4oBgHgl3EQftgtb/content/tmp_files/load_file.txt" "b/GNE3T4oBgHgl3EQftgtb/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/GNE3T4oBgHgl3EQftgtb/content/tmp_files/load_file.txt" @@ -0,0 +1,820 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf,len=819 +page_content='CFTP/23-001 A viable A4 3HDM theory of quark mass matrices Iris Brée∗, Sérgio Carrôlo †, Jorge C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Romão ‡, João P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Silva §, CFTP, Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal January 13, 2023 Abstract It is known that a three Higgs doublet model (3HDM) symmetric under an exact A4 sym- metry is not compatible with nonzero quark masses and/or non-block-diagonal CKM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We show that a 3HDM with softly broken A4 terms in the scalar potential does allow for a fit of quark mass matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Moreover, the result is consistent with mh = 125GeV and the h → WW, ZZ signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We also checked numerically that, for each point that passes all the constraints, the minimum is a global minimum of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 1 Introduction The observation in 2012 of a scalar particle with 125GeV by the ATLAS and CMS collaborations [1, 2] has incentivized experimental searches for beyond the Standard Model (SM) particles at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' On par with these experimental endeavors, theoretical efforts in the search for extra scalar particles have been strengthened since this discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' A promising framework is found in N-Higgs doublet models (NHDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Such models have many free parameters, which are often curtailed by imposing some discrete family symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Here, we focus on the implementation of A4 in a three Higgs doublet model (3HDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The A4 group is the group of even permutations on 4 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' It is the smallest discrete group to contain a three-dimensional irreducible representation (irrep), which is ideal for describing the three families of quarks with a minimal number of independent Yukawa couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Thus, NHDM supplemented by the A4 discrete symmetry has long been of interest in flavour physics research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' A number of early articles include: [3], mainly devoted to the leptonic sector and where the solution to the quark sector is briefly mentioned to include a fourth Higgs doublet and all quark fields in singlets (which is effectively the same as the Standard Model quark sector);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' [4], where A4 is broken by dimension four Yukawa couplings, thus rendering the theory non-renormalizable or, alternatively, the low energy limit of a broader complete theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' [5], which requires three Higgs doublets in the down-type quark sector and a further two in the up-type quark sector, consisting ∗iris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='bree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='silva@tecnico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='ulisboa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='pt †sergio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='carrolo@tecnico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='ulisboa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='pt ‡jorge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='romao@tecnico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='ulisboa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='pt §jpsilva@cftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='utl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='pt 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='04676v1 [hep-ph] 11 Jan 2023 of a 5HDM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' and [6], which is devoted to the leptonic sector, but has the interesting side query that it might be possible to recover a realistic CKM matrix through soft-breaking of A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Quark mass matrices in the context of a 3HDM with Higgs doublets in the triplet representation of A4 were studied in [7] and [8], with the vacuum expectation value (vev) structure (eiα, e−iα, r), where α and r are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Unfortunately, Degee, Ivanov and Keus [9] proved in 2013 that such a vacuum can never be the global minimum of the A4 symmetric 3HDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In this beautiful paper, geometric techniques were used in order to identify all possible global minima (thus, all possible viable vacua) of the A4 symmetric 3HDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Immediately thereafter, those minima were used to show that all assignments of the quark fields into irreps of A4, when combined with the possible vevs for the exact A4 potential, yield vanishing quark masses and/or a CP conserving CKM matrix, both of which are forbidden by experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This is in fact a consequence of a much broader theorem, proved in [10, 11]: given any flavour symmetry group, one can obtain a physical CKM mixing matrix and, simultaneously, non-degenerate and non-zero quark masses only if the vevs of the Higgs fields break completely the full flavour group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The idea is that a symmetry will reduce the number of redundant Yukawa couplings present in the SM, and it might even predict relations among observables which turn out to be consistent with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' When studying in detail the extensions of A4 to the quark sector found by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' [12], we noticed that, in some of them, if it weren’t for the particular form of the vevs allowed by the exact A4 3HDM potential, the Yukawa matrices could allow for massive quarks, and for a realistic CKM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Since the A4 symmetric potential doesn’t allow for minima other than those shown in [9], here we consider the case where the A4 symmetry is softly broken by the addition of quadratic terms to the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Such terms do not spoil the theory’s renormalizability, but break the A4 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Our article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We define the notation for the scalar potential in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1, discuss the Yukawa Lagrangian and the form of the possible mass matrices in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2, giving all the expressions needed for the fit in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3 we present our fit to the quarks mass matrices, while in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 4 we discuss the viability of the vacuum found in the fit in terms of the scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 5 is devoted to the implementation of the theoretical constraints to be imposed, and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 6 we briefly discuss the constraints coming from the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The results and conclusions are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 7 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The Appendices contain some additional expressions that are needed for the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2 Parameterization for the softly-broken A4 3HDM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 Potential and candidates for local minimum The softly-broken potential of the 3HDM with an A4 symmetry is given by VH = V4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' A4 + M2 ij � φ† iφj � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (1) where V4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' A4 is the quartic potential for the A4 symmetric three Higgs doublet model (3HDM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' which is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' in the notation of [9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2 V4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' A4 =Λ0 3 � φ† 1φ1 + φ† 2φ2 + φ† 3φ3 �2 + Λ1 �� Re � φ† 1φ2 ��2 + � Re � φ† 2φ3 ��2 + � Re � φ† 3φ1 ��2� + Λ2 �� Im � φ† 1φ2 ��2 + � Im � φ† 2φ3 ��2 + � Im � φ† 3φ1 ��2� + Λ3 3 � (φ† 1φ1)2 + (φ† 2φ2)2 + (φ† 3φ3)2 − (φ† 1φ1)(φ† 2φ2) − (φ† 2��2)(φ† 3φ3) − (φ† 3φ3)(φ† 1φ1) � + Λ4 � Re � φ† 1φ2 � Im � φ† 1φ2 � + Re � φ† 2φ3 � Im � φ† 2φ3 � + Re � φ† 3φ1 � Im � φ† 3φ1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (2) The matrix M2 ij is a general hermitian matrix, which can be parameterized by (M2 ij) = � � � m2 11 m2 12eiθ12 m2 13eiθ13 m2 12e−iθ12 m2 22 m2 23eiθ23 m2 13e−iθ13 m2 23e−iθ23 m2 33 � � � , (3) where m2 ij are real parameters with the dimension of mass squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 Additionally, in the notation of [13], the exact A4 potential can be written as VA4 =r1 + 2r4 3 � (φ† 1φ1) + (φ† 2φ2) + (φ† 3φ3) �2 + 2(r1 − r4) 3 � (φ† 1φ1)2 + (φ† 2φ2)2 +(φ† 3φ3)2 − (φ† 1φ1)(φ† 2φ2) − (φ† 2φ2)(φ† 3φ3) − (φ† 3φ3)(φ† 1φ1) � + 2r7 � |φ† 1φ2|2 + |φ† 2φ3|2 + |φ† 3φ1|2� + � c3 � (φ† 1φ2)2 + (φ† 2φ3)2 + (φ† 3φ1)2� + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (4) The relation between the two notations is r1 = 1 3(Λ0 + Λ3) , r4 = 1 6(2Λ0 − Λ3) , r7 = 1 4(Λ1 + Λ2) , Re(c3) = 1 4(Λ1 − Λ2) , Im(c3) = −1 4Λ4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (5) We consider that the scalar fields can take complex vacuum expectation values (vevs), to be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Thus, we write, φi = � ϕ+ i |vi|eiρi √ 2 + 1 √ 2 (xi + ixi+3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (6) Because CP is spontaneously violated, the unrotated neutral fields have no definite CP, and for convenience we label them xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' , 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We can also use the gauge freedom to absorb one of the phases in the vevs, that we choose to be ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Therefore we have the vector of vevs defined as ⃗v = (|v1|, |v2|eiρ2, |v3|eiρ3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (7) 1In the quadratic terms, the combination − M0 √ 3 � φ† 1φ1 + φ† 2φ2 + φ† 3φ3 � is also invariant under A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' But, since we are keeping all soft-breaking terms, we find the notation in (3) more convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3 This vev contributes with four free parameters to our model, because one of the parameters is constrained by the mass of the gauge bosons to match the observed SM values, |v1|2 + |v2|2 + |v3|2 ≡ v2 ≃ (246GeV)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (8) The vev can also be parameterised as ⃗v = v � cos(β1) cos(β2), cos(β2) sin(β1)eip2, sin(β2)eip3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (9) Of the quantities arising out of the scalar potential, the vevs are the only relevant to the quark mass matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This leads many authors to just proclaim some vevs, without checking whether they can indeed be the global minima of a realistic Higgs potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We will perform this crucial verification below, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 Yukawa Lagrangian As in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' [7, 12], we consider that the Higgs doublets are in the 3 of A4 as well as the three left-handed SU(2) doublets QLj of hypercharge 1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' There are three right-handed SU(2) singlets nR,j of hypercharge −1/3 and three right-handed SU(2) singlets pR,j of hypercharge 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Our assignments for the singlets are as follows nR1, pR1 → 1, nR2, pR2 → 1′, nR3, pR3 → 1′′ of A4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (10) Then, the A4 transformations on the fields are generated by [7, 12] T : � � � � � � � � � � � φ1 → φ2 → φ3 → φ1, QL1 → QL2 → QL3 → QL1, nR1 → nR1, nR2 → ωnR2, nR3 → ω2nR3, pR1 → nR1, pR2 → ωpR2, pR3 → ω2pR3, (11) and S : �φ1 → φ1, φ2 → −φ2, φ3 → −φ3, QL1 → QL1, QL2 → −QL2, QL3 → −QL3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (12) One can easily verify that the scalar potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (4) is invariant under the previous transfor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Now we write the A4 invariant Yukawa Lagrangian for quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We have −LYukawa = √ 2 ˆa � QL1φ1 + QL2φ2 + QL3φ3 � nR1 + √ 2ˆb � QL1φ1 + ω QL2φ2 + ω2 QL3φ3 � nR2 + √ 2 ˆc � QL1φ1 + ω2 QL2φ2 + ω QL3φ3 � nR3 + √ 2 ˆa′ � QL1 ˜φ1 + QL2 ˜φ2 + QL3 ˜φ3 � pR1 + √ 2ˆb′ � QL1 ˜φ1 + ω QL2 ˜φ2 + ω2 QL3 ˜φ3 � pR2 + √ 2 ˆc′ � QL1 ˜φ1 + ω2 QL2 ˜φ2 + ω QL3 ˜φ3 � pR3 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=', (13) 4 where, as usual, ˜φj ≡ i σ2φ∗ j , (14) and we define ˆa = aei α, ˆb = bei β, ˆc = cei γ, ˆa′ = a′ei α′, ˆb′ = b′ei β′, ˆc′ = c′ei γ′ , (15) where a, b, c, a′, b′, c′ are real and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This choice of invariant Lagrangian corresponds to the case I identified in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' [12] (see the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 Yukawa matrices, masses and CKM We aim to fit six quark masses and four CKM matrix elements to the currently accepted SM values for these observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Therefore, we’re interested in softly-broken A4 symmetric models with up to ten parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' [12] has studied all of the possible extensions of A4 to the fermion sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Using their results, we can check which of them can accommodate non-vanishing quark masses, CKM mixing angles and CP violation by considering a general vev ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We take the Jarlskog invariant as a measure of CP violation [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Out of all possibilities, we are left with five of them, which we list in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' There, A are real constants, Ω are constants in the [0, 2π[ interval, ω = ei 2π 3 (ω3 = 1) and T is the transpose of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Case Md Mu I � � � aeiαv1 beiβv1 ceiγv1 aeiαv2 ωbeiβv2 ω2ceiγv2 aeiαv3 ω2beiβv3 ωceiγv3 � � � � � � A → A′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' A ∈ {a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' c} Ω → Ω′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Ω ∈ {α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' γ} vi → v∗ i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3} � � � II IT d IT u III � � � 0 (aeiα − beiβ)v3 (aeiα + beiβ)v2 (aeiα + beiβ)v3 0 (aeiα − beiβ)v1 (aeiα − beiβ)v2 (aeiα + beiβ)v1 0 � � � � � � A → A′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' A ∈ {a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' b} Ω → Ω′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Ω ∈ {α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' β} vi → v∗ i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3} � � � IV Id IIIu V IIId Iu Table 1: Extensions of A4 to the Yukawa sector with non-vanishing determinant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' and non-zero J for general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' complex valued,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' vevs (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' v3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In the Table, Id stands for the matrix Md for case I and similarly for the other entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In the Yukawa sector, there are ten observables, six masses, three mixing angles and one Jarlskog invariant, therefore, we would prefer to look for a case with ten parameters, or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' All possible neutral vevs of the 3HDM are consistent with the parameterization in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (9), which consists of four free parameters that we can fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' two angles, and two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Looking at the cases in Table 1, we will see that it is possible to reduce the number of free parameters by performing both basis transformations to right-handed quarks and global U(1)Y rephasings, both of which have no effect on the physical predictions of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' For case I, the down quark mass matrices read 5 Md = � � � aeiαv1 beiβv1 ceiγv1 aeiαv2 ωbeiβv2 ω2ceiγv2 aeiαv3 ω2beiβv3 ωceiγv3 � � � = DvWDaDα, (16) where (remember that the vi are complex) Dv = diag(v1, v2, v3) , Da = diag(a, b, c) , Dα = diag(eiα, eiβ, eiγ) , W = � � � 1 1 1 1 ω ω2 1 ω2 ω � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (17) We see that we can perform a unitary transformation to the right-handed quarks that removes all three phases α, β, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The same holds for Mu, by performing the substitution A → A′, Ω → Ω′ and vi → v∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' For case II, the quark mass matrices have the form Md = DaDαWDv and Mu = Da′Dα′WDv∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (18) Thus, we can remove the phases of the vev through an appropriate transformation of the right- handed quarks, but we can only remove α through a global rephasing QL → Q′ L = eiαQL of the left-handed quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This rephasing, however, doesn’t remove the phase α′ in the matrix Mu, because, in general, it will be different from α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Now, for case III, we can only remove α and α′ 2 by performing the corresponding rephasings of the right-handed quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Finally, for the case IV (V), we can remove, as we did for case I, all the phases from the Yukawa couplings α, β, γ (corresponding primes), and only one phase α′ (corresponding unprimed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' To summarise, we can classify the cases above according to their number of free-parameters, which we show in the Table below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' case #θ #p #A #α #A′ α′ Total I 2 2 3 3 − 3 = 0 3 3 − 3 = 0 10 II 2 2 − 2 = 0 3 3 − 1 = 2 3 3 13 III 2 2 2 2 − 1 = 1 2 2 − 1 = 1 10 IV 2 2 3 3 − 3 = 0 2 2 − 1 = 1 10 V 2 2 2 2 − 1 = 1 3 3 − 3 = 0 10 Table 2: Number of parameters for each case of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' θ and p are the angles and phases of the vev, A(A′) are the Yukawa couplings of the down (up) quarks, and α(α′) are the Yukawa phases of the down (up) quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The minus signs correspond to the parameters that we can remove by a basis transformation of the quark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In this work, we study case I, that corresponds to the Lagrangian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Then, given that DαD† α = 1 and DaD† a = Da2 = diag(a2, b2, c2), we find Hd ≡ MdM† d = DvSdD† v , Hu ≡ MuM† u = D† vSuDv , (19) 2We could have chosen to remove β and β′ instead, this is just one possible choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 6 where Sd = WDa2W † and a2 → a′2 for the up quark case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This matrix can now be explicitly written out using appropriate parameters as Sd = � � � Σd Zdeiφd Zde−iφd Zde−iφd Σd Zdeiφd Zdeiφd Zde−iφd Σd � � � , (20) where Σd and Zd are real, and Σd ≡ a2 + b2 + c2 , Zd eiφd ≡ a2 + ω2b2 + ωc2 , (21) with corresponding primes for the up case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' For completeness, the specific forms for Hd and Hu found after using the parameterizations in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (9) and (21) are written in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The eigenvalues of the matrices Hd and Hu will be fitted for the (square of the) quark masses, (m2 d, m2 s, m2 b) and (m2 u, m2 c, m2 t ), respectively We now turn to the Cabibbo-Kobayashi-Maskawa (CKM) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' As found by Branco and Lavoura [15], the absolute values of the CKM matrix can be obtained through calculating the traces of appropriate powers of the matrices Hu and Hd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' They observe that Tr � Ha uHb d � ≡ Lab = � k,i Uki(Da u)kk(Db d)ii , (22) where Uki = |Vki|2 and V is the CKM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The CKM matrix is unitary and therefore U only has four independent entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Consequently, in order to compute U, it is only necessary to resort to L11 =Uki(Du)kk(Dd)ii , L12 =Uki(Du)kk(D2 d)ii , L21 =Uki(D2 u)kk(Dd)ii , L22 =Uki(D2 u)kk(D2 d)ii .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (23) These equations are linear in Uik and are, therefore, invertible for this variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Thus, by picking U11, U21, U13, and U23 (respectively, Uud, Ucd, Uub, and Ucb), we are able to obtain a unique solution for the magnitudes of the CKM elements as a function of Lab and the quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' U11 = � mb2 − ms2� � mc2 − mt2� a11 det ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' U21 = � mb2 − ms2� � mu2 − mt2� a21 det ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' U13 = � md2 − ms2� � mc2 − mt2� a13 det ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' U23 = � md2 − ms2� � mu2 − mt2� a23 det ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (24) where a11 = L11 � mb2 + ms2� � mc2 + mt2� − L12 � mc2 + mt2� − L21 � mb2 + ms2� + L22 7 +m2 b � −m2 cm2 t � m2 d + m2 s � − m2 sm2 u � m2 c + m2 t � + m2 sm4 u � + m2 cm2 dm2 t � m2 d − m2 s � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='(25) a21 = −L11 � mb2 + ms2� � mt2 + mu2� + L12 � mu2 + mt2� + L21 � mb2 + ms2� − L22 +m2 b � m2 cm2 s � m2 t + m2 u − m2 c � + m2 t m2 u � m2 d + m2 s �� + m2 dm2 t m2 u � m2 s − m2 d � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (26) a13 = −L11 � md2 + ms2� � mt2 + mc2� + L12 � mc2 + mt2� + L21 � md2 + ms2� − L22 +m2 bm2 cm2 t � m2 d + m2 s − m2 b � + m2 dm2 s � m2 c � m2 t + m2 u � + m2 u � m2 t − mu2�� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (27) a23 = L11 � md2 + ms2� � mt2 + mu2� − L12 � mu2 + mt2� − L21 � md2 + ms2� + L22 +m2 t m2 u � m4 b − m2 b � m2 d + m2 s � − m2 dm2 s � + m4 cm2 dm2 s − m2 cm2 dm2 s � m2 t + m2 u � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (28) and det = � mb2 − md2� � mc2 − mu2� � md2 − ms2� � mu2 − mt2� � mb2 − ms2� � mc2 − mt2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (29) In these equations, the Lij are obtained by evaluating the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Finally, we note that knowing these four CKM magnitudes, we can determine the Jarslkog invariant [14], up to its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Thus, given some phase convention, we are also able to determine the phases of all CKM matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3 The fit to the quark mass matrices 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 The fitting procedure We have implemented a χ2 analysis of the model, through a minimization performed using the CERN Minuit library [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The observables employed in this analysis, labeled by i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=', 11 are specified in Table 3, where Xi represents the experimental mean value of the observable Xi and σi is the experimental error, which, when both left and right bounds are stated, is assumed to be the largest of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The data on the quark masses as well as for the CKM matrix elements and Observable Experimental value Model prediction mu [MeV] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='15 mc [MeV] 1270 ± 20 1271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='9 mt [GeV] 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='30 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='69 md [MeV] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='66 ms [MeV] 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='08 mb [MeV] 4180 ± 30 4179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='74 |V11| 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='97435 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='00016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='97434 |V21| 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='22486 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='00067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='22479 |V13| 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='00369 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='00011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='00369 |V23| 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='04182 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='00085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='04175 J (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='15) × 10−5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='09 × 10−5 Table 3: Experimental values and fit results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' the Jarlskog invariant experimental values were obtained from [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' As mentioned, |J| is fixed by |V11|, |V21|, |V13|, and |V23|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' However, using it in the fit speeds the numerical convergence onto a good solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 8 The χ2 function depends on the 10 parameters of our model, β1, β2, ρ2, ρ3, Σd, Σu, Zd, Zu, φd, φu (30) and is written as χ2(p) = 11 � i=1 � Pi(p) − Xi σi �2 , (31) where Pi(p) is our model’s prediction for each of the 11 (10 + J) observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The fit is complicated by the fact that the masses (squared) are obtained from the eigenvalues of Hd, Hu but the elements of the CKM also depend on the masses, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' So, we start by calculating the eigenvalues of Hd and Hu, which depend only on the parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Then, we evaluate the Lij from the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (22), and finally the CKM elements are obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In Appendix A we give the explicit expressions for the matrices Hd and Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 Results of the fit We have found an excellent fit of our model to the data, given in the second column of Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This fit results in χ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='058, for the parameters β1 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='42608 radians , β2 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='54243 radians , ρ2 =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='27865 radians , ρ3 =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='37039 radians , Σd =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='288824 × 10−3 , Σu =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='492828 , Zd =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='181571 × 10−3 , Zu =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='475911 , φd = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='73226 radians , φu =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='206453 × 10−3 radians .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (32) This fit also leads to the data in the third column of Table 3, as well as to the vevs |vi| = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='00625, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='90462, 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='901) (GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (33) We notice that the vevs obey v1 < v2 << v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This hierarchy of vevs is related to the hierarchy of the quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This was also obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' [7], although their model is not consistent, as their vev structure is not that of [9] for the symmetric A4 potential they consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 4 Viability of the vacuum found in the fit We start by defining the three doublets as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Next we define the physical eigenstates for the charged Higgs as (G+, S+ 2 , S2 3)T , and for the neutral states we have (G0, S0 2, S0 3, S0 4, S0 5, S0 6)T , identifying the would-be Goldstone bosons G+ ≡ S+ 1 and G0 ≡ S0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' With these conventions, and following the definitions in [18], we define the 3 × 3 matrix ˜U by ϕ+ i ≡ 3 � j=1 ˜UijS+ j , (34) 9 and the 3 × 6 matrix ˜V by xi + ixi+3 = 6 � j=1 ˜VijS0 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (35) These matrices3 are then related to the diagonalization matrices of the charged and neutral scalars, to which we now turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 The minimization of the potential In our procedure we already know the values of the vevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' So, we use the stationarity equations to solve for the soft parameters, and leave the quartic parameters of the potential Λi as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In this way we can solve for m2 11, m2 22, m2 33 as well as for Im(m2 12), Im(m2 13), leaving as free parameters the Λi and Re(m2 12), Re(m2 13), Re(m2 23), Im(m2 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' When evaluating the scalar mass matrices (see below) the conditions have to be applied to ensure that we are at the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' For completeness we write these conditions in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 The charged mass matrix The charged mass matrix is obtained from the second derivatives at the minimum, M2 C = ∂2VH ∂ϕ+ i ∂ϕ− j ����� Min .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (36) The matrix M2 C is an hermitian matrix, with real eigenvalues and satisfying, with our usual conventions, RchM2 CR† ch = diag(0, m2 S+ 2 , m2 S+ 3 ) ≡ M2 Dch , (37) where Rch is an unitary matrix that satisfies, S+ i = 3 � j=1 (Rch)ij ϕ+ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (38) This can be seen from Lmass = − ϕ− i � M2 C � ij ϕ+ j = −ϕ− i � R† chRchM2 CR† chRch � ij ϕ+ j = −ϕ− i � R† chM2 DchRch � ij ϕ+ j = − S− i � M2 Dch � ij S+ j , (39) where we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We have checked both algebraically and numerically that we have a zero eigenvalue corre- sponding to G+ and we require that all other masses squared are positive, a condition for a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3From the point of view of a simultaneous fit of the Yukawa and scalar sectors, it is a pity that these matrices ˜V and ˜U have in the literature the same notation as the CKM matrix V and Uki = |Vki|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 The neutral mass matrix Since in our case CP is not conserved, we denote the unrotated neutral scalars by xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' , 6, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We therefore obtain the neutral mass matrix as, M2 N = ∂2VH ∂xi∂xj ����� Min .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (40) This is a symmetric real matrix diagonalized by an orthogonal 6 × 6 matrix, RneuM2 NRT neu = diag(0, m2 S0 2, m2 S0 3, m2 S0 4, m2 S0 5, m2 S0 6) ≡ M2 Dneu , (41) with S0 i = 6 � j=1 (Rneu)ij xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (42) As for the case of the charged scalars, we have checked both algebraically and numerically that we have a zero eigenvalue corresponding to G0 and we require that all other masses squared are positive, a condition for a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 5 Theoretical Constraints After having shown that a solution exists for the vevs and parameters in the Yukawa sector that correctly fits the quarks masses and the CKM entries, we have to show that this is compatible with the scalar potential analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In particular we have to show that the vevs correspond to a local minimum of the potential and that both the theoretical constraints as well as those coming from LHC are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In this section we analyze the theoretical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 Perturbative Unitarity This problem was already solved in [13], so we take the potential in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' From Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' [13] we have the following expression for the eigenvalues λi4 λ1 =2 (2Re(c3) + r1) (43) λ2 =2 �√ 3 |Im(c3)| − Re(c3) + r1 � (44) λ3 =2 � − √ 3 |Im(c3)| − Re(c3) + r1 � (45) λ4 =2(r4 + r7) (46) λ5 =2(r4 − r7) (47) λ6 =2(r1 + 2r7) (48) λ7 =2(r1 − r7) (49) λ8 =2(r4 + |c3|) (50) λ9 =2(r4 − |c3|) (51) λ10 =6r1 + 8r4 + 4r7 (52) λ11 =6r1 − 2(2r4 + r7) (53) λ12 =6|c3| + 2r4 + 4r7 (54) 4We use λi instead of Λi, in order to not confuse with the notation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 11 λ13 = − 6|c3| + 2r4 + 4r7 (55) Perturbative unitarity is satisfied if |λi| < 8π, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (56) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 The BFB conditions For the A4 symmetric potential, the conditions for boundedness from below along the neutral directions (BFB-n) have been conjectured in [19], and proved to hold in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' These are Λ0 + Λ3 ≥ 0 , (57) 4 3(Λ0 + Λ3) + 1 2(Λ1 + Λ2) − Λ3 − 1 2 � (Λ1 − Λ2)2 + Λ2 4 ≥ 0 , (58) Λ0 + 1 2(Λ1 + Λ2) + 1 2(Λ1 − Λ2) cos (2kπ/3) + 1 2Λ4 sin (2kπ/3) ≥ 0 (k = 1, 2, 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (59) However, as shown in [21, 19], a potential which is BFB-n is not necessarily BFB along the charge breaking directions (BFB-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Necessary BFB-c conditions have yet to be found for the A4 3HDM, but sufficient conditions have been proposed in [22] following the technique developed in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' They are, Ad ≥ 0 , Ao ≥ −Ad/2 , (60) where Ad =a = 2 3(Λ0 + Λ3) , Ao =b + min(0, c) − d =1 3(2Λ0 − Λ3) + 1 2(Λ1 + Λ2) + min(0, −1 2(Λ1 + Λ2)) − 1 2 � (Λ1 − Λ2)2 + Λ2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (61) It is important to remark that, since these are sufficient, but not necessary, conditions, some good points in parameter space may be excluded by this restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 The oblique parameters S, T, U For this we use the notation and results from [18], which require the matrices ˜U and ˜V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (38) with the definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (34), we conclude that ˜U = R† ch , (62) where the matrix Rch is obtained from the numerical diagonalization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Similarly, comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (42) with the definition of ˜V in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (35), we get, ˜Vij = � RT neu � ij + i � RT neu � i+3,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (63) Having ˜U and ˜V , we can construct the needed matrices Im � ˜V † ˜V � , ˜U† ˜U, ˜V † ˜V and ˜U† ˜V , and implement the procedure of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4 Global minimum After finding a set of mi,j and Λi which reproduce the vevs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (33) necessary for a good fit of the quark mass matrices, and after performing the previous theoretical checks on the scalar potential, we must still ensure that our minimum is indeed the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This step is almost never taken in studies of quark mass matrices, since there are no exact analytical formulae for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Moreover, one must check that there are no lower minima both along the neutral directions and along the charge breaking directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We follow the strategy discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Take a specific set of m2 ij and Λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Then we parameterize the scalar doublets as [21, 22], ⟨φ1⟩ = √r1 � 0 1 � , ⟨φ2⟩ = √r2 � sin(α2) cos(α2)eiβ2 � , ⟨φ3⟩ = √r3eiγ � sin(α3) cos(α3)eiβ3 � , (64) where we have already used the gauge freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Now we let the vevs run free, for both charge conserving and charge violating directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We give one seed point and perform a minimization of the potential using the CERN Minuit library [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We obtain not only the value of the potential at the minimum, but also the values of ri, α2, β2, α3, β3 and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Then, we take one more (randomly generated) seed point and repeat the minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Finally, we take the minimum as the global one if it is found as the global minimum in each of 200 searches with randomly generated seed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We have done this verification for every point that passed all the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In all cases, we found that the local minimum was also a global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In particular we always found that sin(α2) = sin(α3) = 0, (65) showing that we do not have charged breaking directions5 and, comparing with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (6), we verified numerically that, |vi| √ 2 = √ri, ei ρ2 = cos(α2) ei β2, ei ρ3 = cos(α3) ei (β3+γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (66) 6 Simple LHC Constraints Up to now we have implemented the theoretical constraints on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The next step is to implement the LHC constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' To do this completely one would have to implement all the decays of the neutral and charged Higgs as well as their branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' One would also have to worry about the electric dipole moments (EDM) and the flavour-changing neutral couplings (FCNC), as the model does not have a structure of couplings of the Higgs to the fermions that automatically ensures vanishing FCNC [24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This lies beyond the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Nonetheless, we can implement easily the constraints that come from h → WW/ZZ in the κ formalism, where the deviation from the coupling of the SM Higgs boson to a pair of W’s (or Z’s) is measured by κV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In our model, κV = Rneu 21 v1 + Rneu 22 v2 cos(ρ2) + Rneu 23 v3 cos(ρ3) + Rneu 25 v2 sin(ρ2) + Rneu 26 v3 sin(ρ3), (67) where Rneu is matrix defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We take the experimental constraint from ATLAS [27], κW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='0206 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='05172 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='05087, κZ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='99 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='06136 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='05214 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (68) 5To cross check our numerical procedure we also considered points that violated the BFB conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' And, indeed for these points, our algorithm showed that the potential was not BFB and could have charge breaking directions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 13 7 Results In this section we present the results of the analysis of the scalar potential after imposing that we have a good solution for the fit of the quarks masses and CKM entries, as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 Scanning strategy We start by imposing the vevs obtained in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' v1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='00625 (GeV), v2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='90462 ei 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='27865 (GeV), v3 = 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='901 ei 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='37039 (GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (69) Now we vary the free parameters of the potential in the following ranges, log10 |Λi| ∈ [−3, 1], log10 |Im(m2 23)| ∈ [−1, 7]GeV2, log10 |Re(m2 ij)| ∈ [−1, 7]GeV2, (70) where in the last equation we use m2 ij ∈ � m2 12, m2 13, m2 23 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (71) We randomly scan as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (70), and then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Apply the theoretical constraints that only depend on the Λi, that is BFB and perturbative unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Then obtain the eigenvalues for the charged and neutral scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Verify that all the masses squared are positive, and that we have a zero eigenvalue corresponding to the Goldstone bosons, G0 and G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Verify the S, T and U oblique parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Apply the LHC constraint on κV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Check numerically that the vev is indeed a global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 The scalar spectrum We found that there is a strong correlation in the scalar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' If we denote the masses of the neutral scalars by (mG0 = 0, mS0 2, mS0 3, mS0 4, mS0 5, mS0 6), and (mG+ = 0, mH+ 1 , mH+ 2 ) for the charged scalars, we find numerically that mS0 3 ≃ mS0 4 ≃ mH+ 1 , mS0 5 ≃ mS0 6 ≃ mH+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (72) This is true even if we do not require mS0 2 = 125 GeV, and specially true after implementing the constraints of perturbative unitarity, BFB and STU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' But, as we want to reproduce the LHC results, we also required that [17] mS0 2 = 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='17 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (73) In the following figures we show the correlation among the masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Included in red are the points generated before the theoretical cuts were applied, and in green the points remaining after the constraints were implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 14 Figure 1: Left panel: Relation between mS0 3 and mS0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Right panel: Relation between mS0 3 and mH+ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Color conventions: No cuts (red);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' with cuts (green) Figure 2: Left panel: Relation between mS0 5 and mS0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Right panel: Relation between mS0 5 and mH+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Color conventions: No cuts (red);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' with cuts (green) 15 2000 1500 1000 500 0 500 1000 1500 2000 ms (GeV)2000 1500 mHt (GeV) 1000 500 0 500 1000 1500 2000 ms (GeV)2000 1500 1000 500 0 500 1000 1500 2000 msg (GeV)2000 1500 (GeV) 1000 500 0 500 1000 1500 2000 msg (GeV)Figure 3: Left panel: Relation between mS0 3 and mS0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Right panel: Relation between mH+ 1 and mH+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Color conventions: No cuts (red);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' with cuts (green) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 The κV constraint We can now implement the κV constraint on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In the following figures, in red are points without cuts, in green with cuts but no κV constraint, and finally in blue points remaining after this constraint is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We took the ATLAS result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' (68) at 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' While the theoretical constraints cut around 99% of the points, the κV constraint only cuts 9% of the remaining points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 4 we show the relation between κV and Λ0,4 for the three sets of points as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Figure 4: Left panel: Relation between κV and Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Right panel: Relation between κV and Λ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Color conventions: No cuts (red), with theoretical cuts (green), and after the κV constraint (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In fact it is not obvious from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 4 that the κV constraint only cuts about 9% of the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This is because there is a very large number of points with |κV | ≲ 1, even without theoretical cuts, and this is even more so after imposing the theoretical cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In this figure, we have 157810 points in the green region, but from these 145074 are in the blue region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' That is, after theoretical 16 2000 1500 1000 500 500 1000 1500 2000 ms (GeV)2000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='. 1500 (GeV) 1000 H 500 500 1000 1500 2000 mHt (GeV)10 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='8 [Ky]10 4 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='8 [Kvlcuts, 91% of the points also satisfy the κV constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 5 we show the relation between Λ0 and Λ3,4 for the same sets of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Figure 5: Left panel: Relation between Λ0 and Λ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Right panel: Relation between Λ0 and Λ3 Color conventions: No cuts (red), with theoretical cuts (green), and after the κV constraint (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We see that, while for (Λ0, Λ4) there is not much difference before and after the κV constraint, the same is not true for (Λ0, Λ3), where the constraints impose a linear relation between those two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 8 Conclusions It is known that the 3HDM symmetric under an exact A4 symmetry is not compatible with non- zero quark masses and/or non-block-diagonal CKM matrix [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' In this work, we studied a 3HDM with A4 softly broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This allows us to evade the above result, by enlarging the structure of the possible vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We obtained an excellent fit of the quarks mass matrices, including the CP-violating Jarlskog invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' This leads to a unique solution for the vevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We showed that, with the solution for the vevs obtained from the fit, it is possible to have a local minimum of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We enforce this by imposing that all squared masses are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' As in our scheme the scalar masses are not input parameters, we have to restrict one of the neutral scalars to have the mass of the known Higgs boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We have implemented the BFB, perturbative unitarity and the oblique parameters S, T, U theoretical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' From LHC, we have considered the observed Higgs mass and the κV constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='6 After imposing the other constraints, we found that most of the points are close to the alignment required to respect the experimental κV constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' We have discovered a strong correlation among the masses of the scalars, even before applying the theoretical constraints, especially for moderate to large scalar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' One important point is that we have numerically checked for all the points that pass our constraints, that for a given set of parameters of the potential, our minimum is the true global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 6The detailed study of other LHC constraints as well as those coming from FCNC and the EDM lies beyond the scope of the present work, and is left for a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='6 No0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4 NoAcknowledgments This work is supported in part by FCT (Fundação para a Ciência e Tecnologia) under Contracts CERN/FIS-PAR/0002/2021, CERN/FIS-PAR/0008/2019, UIDB/00777/2020, and UIDP/00777/2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' these projects are partially funded through POCTI (FEDER), COMPETE, QREN, and the EU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' The work of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' was supported by a CFTP fellowship with reference BL210/2022-IST-ID and the work of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' by a CFTP fellowship with reference BL255/2022-IST-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' A The matrices Hd and Hu Hd(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 1) =Σdv2 cos2(β1) cos2(β2) (74) Hd(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2) =v2Zd cos(β1) cos2(β2) cos(ρ2 − φd) sin(β1) − i v2Zd cos(β1) cos2(β2) sin(β1) sin(ρ2 − φd) (75) Hd(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3) =v2Zd cos(β1) cos(β2) cos(ρ3 + φd) sin(β2) − i v2Zd cos(β1) cos(β2) sin(β2) sin(ρ3 + φd) (76) Hd(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 1) =(Hd(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2))∗ (77) Hd(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2) =Σdv2 cos2(β2) sin2(β1) (78) Hd(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3) =v2Zd cos(β2) cos(ρ2 − ρ3 + φd) sin(β1) sin(β2) + i v2Zd cos(β2) sin(β1) sin(β2) sin(ρ2 − ρ3 + φd) (79) Hd(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 1) =(Hd(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3))∗ (80) Hd(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2) =(Hd(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3))∗ (81) Hd(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3) =Σdv2 sin2(β2) (82) Hu(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 1) =Σuv2 cos2(β1) cos2(β2) (83) Hu(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2) =v2Zu cos(β1) cos2(β2) cos(ρ2 − φu) sin(β1) − i v2Zu cos(β1) cos2(β2) sin(β1) sin(ρ2 − φu) (84) Hu(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3) =v2Zu cos(β1) cos(β2) cos(ρ3 + φu) sin(β2) − i v2Zu cos(β1) cos(β2) sin(β2) sin(ρ3 + φu) (85) Hu(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 1) =(Hu(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2))∗ (86) Hu(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2) =Σuv2 cos2(β2) sin2(β1) (87) Hu(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3) =v2Zu cos(β2) cos(ρ2 − ρ3 + φu) sin(β1) sin(β2) + i v2Zu cos(β2) sin(β1) sin(β2) sin(ρ2 − ρ3 + φu) (88) Hu(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 1) =(Hu(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3))∗ (89) Hu(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 2) =(Hu(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3))∗ (90) Hu(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' 3) =Σuv2 sin2(β2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='(91) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='The minimization conditions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='11 = − sec(ρ2) sec(ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='24v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='−12Im(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='23)v2v3 sin(2(ρ2 − ρ3)) + cos(ρ2 − ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4Λ0v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+6Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 + 6Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 + 3Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 − 3Λ2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 + 2Λ3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 − v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 − v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+4Λ0v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 cos(ρ2 + ρ3) + 4Λ0v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 cos(ρ2 + ρ3) + 4Λ0v4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 cos(ρ2 + ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+6Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 cos(ρ2 + ρ3) + 6Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 cos(ρ2 + ρ3) − 3Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 cos(3(ρ2 − ρ3)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+3Λ2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 cos(3(ρ2 − ρ3)) − 2Λ3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 cos(ρ2 + ρ3) − 2Λ3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 cos(ρ2 + ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+4Λ3v4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 cos(ρ2 + ρ3) + 3Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 sin(ρ2 − ρ3) + 3Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 sin(ρ2 + ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+3Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 sin(ρ2 − ρ3) − 3Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 sin(ρ2 + ρ3) − 3Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 sin(ρ2 − ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+3Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 sin(3(ρ2 − ρ3)) − 12Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='23)v2v3 cos(2(ρ2 − ρ3)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+24Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='13)v1v3 cos(ρ2) + 24Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='12)v1v2 cos(ρ3) + 12Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='23)v2v3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='(92) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='22 = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='12v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 sec(ρ2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='−4Im(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='23)v3 sin(ρ3) + v2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3(Λ1 − Λ2) cos(ρ2 − 2ρ3) − Λ4v2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 sin(ρ2 − 2ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+4Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='23)v3 cos(ρ3) + 4Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='12)v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+ v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4Λ0v2 + 6Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 + 3Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 + 3Λ2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='−2Λ3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 + 4Λ3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 − 2Λ3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 + 3Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 tan(ρ2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='(93) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='33 = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='12v3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 sec(ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4Im(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='23)v2 sin(ρ2) + v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v3(Λ1 − Λ2) cos(2ρ2 − ρ3) − Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v3 sin(2ρ2 − ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+4Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='23)v2 cos(ρ2) + 4Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='13)v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+ v3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4Λ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 + v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 + v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+ 6Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 + 3Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+3Λ2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 − 2Λ3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 − 2Λ3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2 + 4Λ3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 − 3Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1 tan(ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='(94) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='Im(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='12) = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='sec(ρ2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4Im(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='23)v3 cos(ρ2 − ρ3) − Λ1v2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 sin(2(ρ2 − ρ3)) − Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 sin(2ρ2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+Λ2v2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 sin(2(ρ2 − ρ3)) + Λ2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 sin(2ρ2) − Λ4v2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='3 cos(2(ρ2 − ρ3)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v2 cos(2ρ2) − 4Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='23)v3 sin(ρ2 − ρ3) − 4Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='12)v1 sin(ρ2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='(95) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='Im(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='13) = − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='sec(ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='4Im(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='23)v2 cos(ρ2 − ρ3) − Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v3 sin(2(ρ2 − ρ3)) + Λ1v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v3 sin(2ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+Λ2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v3 sin(2(ρ2 − ρ3)) − Λ2v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v3 sin(2ρ3) − Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='2v3 cos(2(ρ2 − ρ3)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='+Λ4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='1v3 cos(2ρ3) − 4Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='23)v2 sin(ρ2 − ρ3) + 4Re(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='13)v1 sin(ρ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='(96) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='References ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content='[1] ATLAS collaboration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE3T4oBgHgl3EQftgtb/content/2301.04676v1.pdf'} +page_content=' Phys.' metadata={'source': 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