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( | NOT PEER-REVIEWED | Posted: 19 May 2020 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 May 2020 Peer-reviewed version available at Symmetry 2020, 12, 1000; doi:10.3390/sym12061000 1 INFORMATIONALLY COMPLETE CHARACTERS FOR 2 QUARK AND LEPTON MIXINGS 3 MICHEL PLANATy, RAYMOND ASCHHEIMz, 4 MARCELO M. AMARALz AND KLEE IRWINz Abstract. A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters κ of a fi- nite group G. Here we introduce a d-dimensional Hilbert space with d = cc(G), the number of conjugacy classes of G. Groups under con- sideration should follow two rules, (a) the character table contains both two- and three-dimensional representations with at least one of them faithful and (b) there are minimal informationally complete measure- ments under the action of a d-dimensional Pauli group over the charac- ters of these representations. Groups with small d that satisfy these rules coincide in a large part with viable ones derived so far for reproducing simultaneously the CKM (quark) and PNMS (lepton) mixing matrices. Groups leading to physical CP violation are singled out. 5 PACS: 03.67.-a, 12.15.Ff, 12.15.Hh, 03.65.Fd, 98.80.Cq 6 Keywords: Informationally complete characters, quark and lepton mixings, CP viola- 7 tion, Pauli groups 8 1. Introduction 9 In the standard model of elementary particles and according to the current 10 experiments there exist three generations of matter but we do not know 11 why. The matter particles are fermions of spin 1=2 and comprise the quarks 12 (responsible for the strong interactions) and leptons (responsible for the 13 electroweak interactions as shown in Table 1 and Fig. 1. matter type 1 type 2 type 3 Q T3 YW (1) quarks u c t 2/3 1/2 1/3 d s b -1/3 -1/2 . (2) leptons e µ τ -1 -1/2 -1 νe νµ ντ 0 1/2 . Table 1. (1) The three generations of up-type quarks (up, charm and top) and of down-type quarks (down, strange and bottom), (2) The three generations of leptons (electron, muon and tau) and their partner neutrinos. The symbols Q, T3 and YW are for the charge, the isospin and the weak hy- percharge, respectively. They satisfy the equation Q = T3 + 1 2 YW . 14 In order to explain the CP-violation (the non-invariance of interactions 15 under the combined action of charged-conjugation (C) and parity (P) trans- 16 formations) in quarks, Kobayashi and Maskawa introduced the so-called 1 © 2020 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 May 2020 Peer-reviewed version available at Symmetry 2020, 12, 1000; doi:10.3390/sym12061000 2MICHELPLANATy, RAYMOND ASCHHEIMz, MARCELO M. AMARALz AND KLEE IRWINz Figure 1. An angular picture of the three generations of quarks and leptons. The blue and black pancakes have isospin is 1=2 and −1=2, respectively. The inside and out- 1 side crowns have weak hypercharges 3 and −1, respectively. 17 Cabibbo-Kobayashi-Maskawa unitary matrix (or CKM matrix) that de- 18 scribes the probability of transition from one quark i to another j. These 2 19 transitions are proportional to jVijj where the Vij's are entries in the CKM 20 matrix [1, 2] 0 1 0 1 Vud Vus Vub 0:974 0:225 0:004 UCKM = @Vcd Vcs Vcb A with jUCKM j ≈ @0:225 0:973 0:041A : Vtd Vts Vtb 0:009 0:040 0:999 21 There is a standard parametrization of the CKM matrix with three Euler 22 angles θ12 (the Cabbibo angle), θ23, θ12 and the CP-violating phase δCP . 23 Taking sij = sin(θij) and cij = cos(θij), the CKM matrix reads 0 1 0 −iδ 1 0 1 1 0 0 c13 0 s13e CP c12 s12 0 @0 c23 s23A @ 0 1 0 A @−s12 c12 0A : −iδ 0 −s23 c23 s13e CP 0 c13 0 0 1 24 Similarly, the charged leptons e, µ and τ partner with three generations 25 of flavors of neutrinos νe, νµ and ντ in the charged-current weak interaction. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 May 2020 Peer-reviewed version available at Symmetry 2020, 12, 1000; doi:10.3390/sym12061000 INFORMATIONALLY COMPLETE CHARACTERS FOR QUARK AND LEPTON MIXINGS3 2 26 Neutrino's mass mi can be deduced with probability jUαij where the Uαi's 27 are the amplitudes of mass eigenstates i in flavor α. The so-called Pontecor- 28 voMakiNakagawaSakata unitary matrix (or PMNS matrix) is as follows [3] 0 1 Ue1 Ue2 Ue3 UPMNS = @Uµ1 Uµ2 Uµ3A Uτ1 Uτ2 Uτ3 00:799 ! 0:844 0:516 ! 0:582 0:141 ! 0:1561 with jUPMNSj ≈ @0:242 ! 0:494 0:467 ! 0:678 0:639 ! 0:774A ; 0:284 ! 0:521 0:490 ! 0:695 0:615 ! 0:754 29 where the entries in the matrix mean the range of values allowed by the 30 present day experiments. 31 As for the CKM matrix the three mixing angles are denoted θ12, θ23, θ12 32 and the CP-violating phase is called δCP . 33 The current experimental values of angles for reproducing entries in the 34 CKM and PMNS matrices are in Table 2. angles (in degrees) θ12 θ13 θ23 δCP quark mixings 13.04 0.201 2.38 71 lepton mixings 33.62 8.54 47.2 -90 Table 2. Experimental values of the angles in degrees for mixing patterns of quarks (in the CKM matrix) and leptons (in the PMNS matrix). 35 In the last twenty years or so, a paradigm has emerged that it may exist 36 an underlying discrete symmetry jointly explaining the mixing patterns of 37 quarks and leptons [4, 5]. This assumption follows from the fact that the 38 CKM matrix is found closed to the identity matrix and the entries in the 39 PMNS matrix are found to be of order one except for the almost vanishing 40 Ue3. A puzzling difference between quark and lepton mixing lies in the 41 fact that there is much more neutrino mixing than mixing between the 42 quark flavors. Up and down quark matrices are only slightly misaligned 43 while there exists a strong misalignment of charged leptons with respect to 44 neutrino mass matrices. A valid model should account for these features. 45 The standard model essentially consists of two continuous symmetries, 46 the electroweak symmetry SU(2) × U(1) (that unifies the electromagnetic 47 and weak interactions) and the quantum chromodynamics symmetry SU(3) 48 (that corresponds to strong interactions). There are several puzzles not 49 explained within the standard model including the flavor mixing patterns, 50 the fermion masses, and the CP violations in the quark and lepton sectors. 51 There are astonishing numerical coincidences such as the Koide formula for quark 52 fermion masses [6], the quark-lepton complementarity relations θ12 + lepton quark lepton 53 θ12 ≈ π=4, θ23 ± θ23 ≈ π=4 [7] and efficient first order models 54 such as the tribimaximal model [8]-[11] and the `Golden ratio' model [12, lepton 55 13]. For instance, tribimaximal mixing gives values of angles as θ12 = Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 May 2020 Peer-reviewed version available at Symmetry 2020, 12, 1000; doi:10.3390/sym12061000 4MICHELPLANATy, RAYMOND ASCHHEIMz, MARCELO M. AMARALz AND KLEE IRWINz −1 1 ◦ lepton ◦ lepton 56 sin ( p ) ≈ 35:3 , θ = 45 , θ = 0 and δ = 0, compatible 3 23 13 CP 57 with earlier data. Such a model could be made more realistic by taking two 58 CP-phases instead of one [11]. 59 Currently many discrete models of quark-lepton mixing patterns are based 60 on the representations of finite groups that are both subgroups of U(2) and 61 U(3) [14]-[23]. In the same spirit, we add to this body of knowledge by 62 selecting valid subgroups of unitary groups from a criterion borrowed to the 63 theory of generalized quantum measurements. 64 One needs a quantum state (called a fiducial state) and one also re- 65 quires that such a state is informationally complete under the action of 66 a d-dimensional Pauli group Pd. When such a state is not an eigenstate of a 67 d-dimensional Pauli group it allows to perform universal quantum computa- 68 tion [24]-[26]. In the latter papers, valid states belong to the eigenstates of 69 mutually commuting permutation matrices in a permutation group derived 70 from the coset classes of a free group with relations. From now, the fidu- 71 cial state will have to be selected from the characters κ of a finite group G 72 with the number of conjugacy classes d = cc(G) defining the Hilbert space 73 dimension. Groups under consideration should obey two rules (a) the char- 74 acter table of G contains both two- and three-dimensional representations 75 with at least one of them faithful and (b) there are minimal informationally 76 complete measurements under the action of a d-dimensional Pauli group 77 over the characters of these representations. The first criterion is inspired 78 by the current understanding of quark and lepton mixings (and the stan- 79 dard model) and the second one by the theory of magic states in quantum 80 computing [24]. Since matter particles are spin 1=2 fermions it is entirely 81 consistent to see them under the prism of quantum measurements. 82 In the rest of this introduction we recall what we mean by a minimal 83 informationally complete quantum measurement (or MIC).
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