Flavor structures of charged fermions and massive neutrinos Zhi-zhong Xinga,b,c aInstitute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China bSchool of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China cCenter of High Energy Physics, Peking University, Beijing 100871, China Abstract Most of the free parameters in the Standard Model (SM) — a quantum field theory which has successfully elucidated the behaviors of strong, weak and electromagnetic interactions of all the known fundamental particles, come from the lepton and quark flavors. The discovery of neutrino oscillations has proved that the SM is incomplete, at least in its lepton sector; and thus the door of opportunity is opened to exploring new physics beyond the SM and solving a number of flavor puzzles. In this review article we give an overview of important progress made in understanding the mass spectra, flavor mixing patterns, CP-violating effects and underlying flavor structures of charged leptons, neutrinos and quarks in the past twenty years. After introducing the standard pictures of fermion mass generation, flavor mixing and CP violation in the SM extended with the presence of massive Dirac or Majorana neutrinos, we briefly summarize current experimental knowledge about the flavor parameters of quarks and leptons. Various ways of describing flavor mixing and CP violation are discussed, the renormalization-group evolution of flavor parameters is illuminated, and the matter effects on neutrino oscillations are interpreted. Taking account of possible extra neutrino species, we propose a standard parametrization of the 6 × 6 flavor mix- ing matrix and comment on the phenomenological aspects of heavy, keV-scale and light sterile neutrinos. We pay particular attention to those novel and essentially model-independent ideas or approaches regarding how to determine the Yukawa textures of Dirac fermions and the effective mass matrix of Majorana neutrinos, including simple discrete and continuous flavor symmetries. An outlook to the future development in unraveling the mysteries of flavor structures is also given. Keywords: lepton, quark, neutrino oscillation, fermion mass, flavor mixing, CP violation, lepton number violation, sterile neutrino, Yukawa texture, flavor symmetry PACS: 11.10.Hi, 12.15.-y, 12.15.Ff, 12.60.-i, 14.60.Pq, 14.60.St, 23.40.-s, 23.40.Bw, 95.35.+d arXiv:1909.09610v4 [hep-ph] 24 Apr 2020 Contents 1 Introduction4 1.1 A brief history of lepton and quark flavors . .4 1.2 A short list of the unsolved flavor puzzles . .8 Email address: [email protected] (Zhi-zhong Xing ) Preprint submitted to Physics Reports July 27, 2021 2 The standard picture of fermion mass generation 15 2.1 The masses of charged leptons and quarks . 15 2.1.1 The electroweak interactions of fermions . 15 2.1.2 Yukawa interactions and quark flavor mixing . 17 2.2 Dirac and Majorana neutrino mass terms . 19 2.2.1 Dirac neutrinos and lepton flavor violation . 19 2.2.2 Majorana neutrinos and lepton number violation . 21 2.2.3 The canonical seesaw mechanism and others . 23 2.3 A diagnosis of the origin of CP violation . 27 2.3.1 The Kobayashi-Maskawa mechanism . 27 2.3.2 Baryogenesis via thermal leptogenesis . 31 2.3.3 Strong CP violation in a nutshell . 35 3 Current knowledge about the flavor parameters 39 3.1 Running masses of charged leptons and quarks . 39 3.1.1 On the concepts of fermion masses . 39 3.1.2 Running masses of three charged leptons . 40 3.1.3 Running masses of six quarks . 42 3.2 The CKM quark flavor mixing parameters . 44 3.2.1 Determination of the CKM matrix elements . 44 3.2.2 The Wolfenstein parameters and CP violation . 47 3.3 Constraints on the neutrino masses . 49 3.3.1 Some basics of neutrino oscillations . 49 3.3.2 Neutrino mass-squared differences . 51 3.3.3 The absolute neutrino mass scale . 54 3.4 The PMNS lepton flavor mixing parameters . 59 3.4.1 Flavor mixing angles and CP-violating phases . 59 3.4.2 The global-fit results and their implications . 63 3.4.3 Some constant lepton flavor mixing patterns . 65 4 Descriptions of flavor mixing and CP violation 68 4.1 Rephasing invariants and commutators . 68 4.1.1 The Jarlskog invariants of CP violation . 68 4.1.2 Commutators of fermion mass matrices . 70 4.2 Unitarity triangles of leptons and quarks . 72 4.2.1 The CKM unitarity triangles of quarks . 72 4.2.2 The PMNS unitarity triangles of leptons . 74 4.3 Euler-like parametrizations of U and V ....................... 75 4.3.1 Nine distinct Euler-like parametrizations . 75 4.3.2 Which parametrization is favored? . 80 4.4 The effective PMNS matrix in matter . 82 4.4.1 Sum rules and asymptotic behaviors . 82 4.4.2 Differential equations of Ue ......................... 87 2 4.5 Effects of renormalization-group evolution . 89 4.5.1 RGEs for the Yukawa coupling matrices . 89 4.5.2 Running behaviors of quark flavors . 93 4.5.3 Running behaviors of massive neutrinos . 97 5 Flavor mixing between active and sterile neutrinos 101 5.1 A parametrization of the 6 × 6 flavor mixing matrix . 101 5.1.1 The interplay between active and sterile neutrinos . 101 5.1.2 The Jarlskog invariants for active neutrinos . 105 5.1.3 On the (3+2) and (3+1) flavor mixing scenarios . 107 5.2 The seesaw-motivated heavy Majorana neutrinos . 109 5.2.1 Naturalness and testability of seesaw mechanisms . 109 5.2.2 Reconstruction of the neutrino mass matrices . 111 5.2.3 On lepton flavor violation of charged leptons . 113 5.3 keV-scale sterile neutrinos as warm dark matter . 115 5.3.1 On the keV-scale sterile neutrino species . 115 5.3.2 A possibility to detect keV-scale sterile neutrinos . 117 5.4 Anomaly-motivated light sterile neutrinos . 120 5.4.1 The anomalies hinting at light sterile neutrinos . 120 5.4.2 Some possible phenomenological consequences . 122 6 Possible Yukawa textures of quark flavors 124 6.1 Quark flavor mixing in the quark mass limits . 124 6.1.1 Quark mass matrices in two extreme cases . 124 6.1.2 Some salient features of the CKM matrix . 125 6.2 Quark flavor democracy and its breaking effects . 127 6.2.1 S3 and S3L × S3R flavor symmetry limits . 127 6.2.2 Breaking of the quark flavor democracy . 129 6.2.3 Comments on the Friedberg-Lee symmetry . 131 6.3 Texture zeros of quark mass matrices . 133 6.3.1 Where do texture zeros come from? . 133 6.3.2 Four- and five-zero quark flavor textures . 135 6.3.3 Comments on the stability of texture zeros . 138 6.4 Towards building a realistic flavor model . 139 6.4.1 Hierarchies and U(1) flavor symmetries . 139 6.4.2 Model building based on A4 flavor symmetry . 141 7 Possible charged-lepton and neutrino flavor textures 144 7.1 Reconstruction of the lepton flavor textures . 144 7.1.1 Charged leptons and Dirac neutrinos . 144 7.1.2 The Majorana neutrino mass matrix . 149 7.1.3 Breaking of µ-τ reflection symmetry . 152 7.2 Zero textures of massive Majorana neutrinos . 157 3 7.2.1 Two- and one-zero flavor textures of Mν .................. 157 7.2.2 The Fritzsch texture on the seesaw . 160 7.2.3 Seesaw mirroring between Mν and MR ................... 163 7.3 Simplified versions of seesaw mechanisms . 165 7.3.1 The minimal seesaw mechanism . 165 7.3.2 The minimal type-(I+II) seesaw scenario . 168 7.3.3 The minimal inverse seesaw scenario . 170 7.4 Flavor symmetries and model-building approaches . 172 7.4.1 Leptonic flavor democracy and S3 symmetry . 172 7.4.2 Examples of A4 and S4 flavor symmetries . 174 7.4.3 Generalized CP and modular symmetries . 176 8 Summary and outlook 181 1. Introduction 1.1. A brief history of lepton and quark flavors The history of particle physics can be traced back to the discovery of the electron by Joseph Thomson in 1897 [1]. Since then particle physicists have been trying to answer an age-old but fundamentally important question posed by Gottfried Leibniz in 1714: Why is there something rather than nothing? Although a perfect answer to this question has not been found out, great progress has been made in understanding what the Universe is made of and how it works, both microscopically and macroscopically. Among many milestones in this connection, the biggest and most marvelous one is certainly the Standard Model (SM) of particle physics. The SM is a renormalizable quantum field theory consisting of two vital parts: the electroweak part which unifies electromagnetic and weak interactions based on the SU(2)L × U(1)Y gauge groups [2,3,4], and the quantum chromodynamics (QCD) part which describes the behaviors of strong interactions based on the SU(3)c gauge group [5,6,7]. Besides the peculiar spin-zero Higgs boson and some spin-one force-mediating particles — the photon, gluons, W± and Z0 bosons, the SM contains a number of spin-half matter particles — three charged leptons (e, µ, τ), three neutrinos (νe, νµ, ντ), six quarks (u, c, t and d, s, b), and their antiparticles. Fig.1 provides a schematic illustration of these elementary particles and their interactions allowed by the SM, in which each of the fermions is usually referred to as a “flavor”, an intriguing term inspired by and borrowed from different flavors of ice cream 1. It is straightforward to see • why the photon, gluons and neutrinos are massless.