Neutrino Masses & Flavor Mixing Zhi-zhong Xing 邢志忠 (IHEP, Beijing) @Schladming Winter School 2010, Styria, Austria Lecture B
Lepton Flavors & Nobel Prize 2 1975 1936 = 1936 1897 = 39 Positron: Predicted by P.A.M. Dirac 1928; Discovered by C.D. Anderson 1932; Nobel Prize 1936
Sarma-Xing Theorem? 3 In 1995, it was the Indian physicist Sarma who first discovered the 39-year gap of charged leptons. 1975 + 39 = 2014 My contribution: corrected 2114 to 2014, a discovery would be possible 100 years earlier!
Flavor Mixing & CP Violation 4 Flavor mixing: a mismatch between weak/flavor eigenstates and mass eigenstates of fermions due to coexistence of 2 types of interactions. Weak eigenstates: members of weak isospin doublets that transform into each other via the interaction with the W boson; Mass eigenstates: states of definite masses which are created by the interaction with the Higgs boson (Yukawa interactions). CP violation: matter and antimatter, or a reaction and its CP-conjugate process, are distinguishable ---- coexistence of 2 types of interactions. P C Charge-conjugation T 1957: P violation 1964: CP violation
Known and Unknown 5 Three quark flavor mixing angles and the CP-violating phase have been measured to a good degree of accuracy, in particular at B factories. Two lepton flavor mixing angles have been measured from atmospheric and solar neutrino oscillations, the third one is restricted to be small. Leptonic CP violation: why not? How to probe it? (W. Winter s lectures) Outline of Lecture B In this lecture we shall discuss the following topics: --- How to diagnose CP violation in the SM + 3 light massive neutrinos? --- Parametrizations of the MNS matrix and leptonic unitarity triangles. --- Comparison between CKM and MNS matrices --- why tri-bimaximal? --- Neutrino mixing and CP violation: an overview from heaven to hell. For simplicity, our discussions will try to avoid any model dependence.
Part A Diagnosis of CP Violation 6 In the minimal SM (SM + 3 right-handed s), the Kobayashi-Maskawa mechanism is responsible for CP violation. Nobel Prize 2008 The strategy of diagnosis: given proper CP transformations of the gauge, Higgs and fermion fields, one may prove that the 1 st, 2 nd and 3 rd terms are formally invariant, and the 4 th term can be invariant only if the corresponding Yukawa coupling matrices are real. (spontaneous symmetry breaking doesn t affect CP.)
Part A Gauge fields: CP Transformations 7 Higgs fields: Lepton or quark fields: Spinor bilinears:
Part A CP Violation 8 The Yukawa interactions of leptons & quarks are formally invariant under CP if and only if If the effective Majorana mass term is added into the SM, then the Yukawa interactions of leptons can be formally invariant under CP if If the flavor eigenstates are transformed into the mass eigenstates, the source of CP violation will show up in the charged-current interactions: quarks leptons Comment A: CP violation is present because fermions interact with both the gauge bosons and the Higgs boson. Comment B: the charged-current interactions have been experimentally verified, but the Yukawa interactions have never been verified. Comment C: in the SM the CKM matrix U must be unitary. But whether the MNS matrix V is unitary or not depends on the -mass models.
Part A Parametrization 9 The 3 3 unitary matrix V can be parametrized as a product of 3 unitary rotation matrices in the complex (1, 2), (2, 3) and (3, 1) planes: Category A: 3 independent possibilities. V O O O i j i ( i j) Category B: 6 independent possibilities. V O O O i j k ( i j k) After the unitarity conditions are imposed on V, it apparently includes 3 Euler angles and 9 phases. Some of the 9 phases can always be rotated away by redefining the phases of 3 left-handed charged lepton fields & 3 left-handed neutrino fields, but the mass term must keep unchanged. The latter depends on the nature of neutrinos (Dirac or Majorana).
Part A Phases 10 For instance, the so-called standard parametrization is given as follows: V
Part A Physical Phases 11 Because charge leptons are Dirac particles, the phases a, b and c can all be rotated away by redefining the phases of their independent lefthanded and right-handed fields. If neutrinos are Dirac particles, the phases x, y and z can be removed for the same reason. Then the Dirac neutrino mixing matrix is If neutrinos are Majorana particles, their left- and right-handed fields are correlated. Hence only a common phase of three left-handed fields can be redefined (e.g., z = 0). The Majorana neutrino mixing matrix is
Part A Rephasing Invariants 12 Rephasing invariants are the quantities independent of the redefinition of the phases of charged-lepton and neutrino fields: Jarlskog parameter is a rephasing-invariant measure of CP / T violation in neutrino oscillations: V i V j V j V i Off-diagonal asymmetries of V describe the structural characters of the MNS matrix V (i.e., wheter it is symmetric or not):
Part A Unitarity Triangles 13
Part B Flavor Puzzle 14? (H. Fritzsch s lecture) Tiny neutrino masses should have a different origin --- seesaws?
Part B CKM Matrix 15 Cabibbo(1963)-Kobayashi-Maskawa(1973) quark mixing matrix: Flavor mixing measures a mismatch between the mass and interaction bases. Unitarity is the only but powerful constraint, imposed by the SM itself, on the CKM matrix. Small quark mixing angles may be due to large quark mass hierarchies: A big CP-violating phase in the CKM matrix V is seen.
Part B PMNS Matrix 16 Pontecorvo-Maki-Nakagawa-Sakata(1962) lepton mixing matrix: mixing Flavor mixing measures a mismatch between the mass and interaction bases. Whether the MNS matrix is unitary or not remains an open question, relying on the origin of -masses. Large lepton mixing angles might imply a tiny neutrino mass hierarchy? CP violation?
Part B Constant Mixing Patterns? 17 Democratic Neutrino Mixing Pattern (H. Fritzsch, Z.Z. Xing 1996) Tri-bimaximal Neutrino Mixing Pattern (P. Harrison, D. Perkins, W. Scott 2002; Z.Z. Xing 2002) Simple numbers in V may imply possible underlying flavor symmetries, such as the mu-tau symmetry. 23 45, 13 0
Part B Why Tri-bimaximal? 18 ---- The phases in V can be redefined; ---- The democratic pattern equals a bi-trimaximal mixing pattern. tri-maximal mixing bi-maximal mixing Quark-lepton Complementarity: an accident or something behind it? 12 23 (MNS) (MNS) 12 23 (CKM) (CKM) / / 4 4 The smallest (1,3) mixing angle appears in both quark & lepton sectors --- small perturbations?
Part B Flavor Symmetries Guiding Principle 19 Flavor Symmetry S 3, S 4, A 4, Z 2, U(1) F, SU(2) F, Symmetry Breaking Observed patterns of fermion masses and flavor mixing
Part B Example: A New Symmetry 20 The mass operator of Dirac neutrinos in the diagonal e/ / basis: Invariant under the FL translation, where z is a spacetime-independent Grassmann number. z The -mass matrix (R. Friedberg, T.D. Lee, 2006): a, b, c are all real Predictions: one neutrino to be massless & nearly tri-bimaximal mixing. This symmetry must be broken to give correct masses to 3 s.
Part C Neutrino Mixing in Heaven 21 A mechanism of neutrino mass generation most likely works at a super-high energy scale. After integrating out the heavy degrees of freedom, one is left with the unique Weinberg dimension-5 operator: New Physics Scale RGEs = Cable Car To compare a theory with experiments, one must run the RGEs of the effective -mass matrix down to low energies: RGE = renormalization-group equation Electroweak Scale
Part C One-loop RGEs 22 The one-loop RGE: SM + 3 s: MSSM + 3 s: The -mass matrix: A good parametrization of neutrino mixing which can make the RGEs of neutrino masses and flavor mixing parameters greatly simpler: Fritzsch Xing 97 & 98 A similar parametrization is good for understanding quark flavor mixing.
Some comments: 1) Running behaviors of 3 -masses are similar; 2) is most sensitive to the RGE corrections; 3) 3 CP-violating phases are entangled in their RGE evolution: one is radiatively generated? 4) or could also be radiatively generated.
Part D Neutrino Mixing in Hell 24 The effective Hamiltonian for 3-flavor neutrinos in vacuum & in matter: A 2 2 G F N e E (W. Winter s lecture) The effective neutrino mass-squared differences in matter: here we assume uniform medium density.
Part D Matter Effects 25 The effective neutrino mixing matrix elements in matter (i j k): When anti-neutrinos are concerned in matter, we can get similar formulas by the replacements: A A, V V These vacuum-matter relations help extract the genuine neutrino mass and mixing parameters from some realistic long-baseline experiments.
Part D Sum and Product Rules 26 The sum rules for neutrino mixing parameters in vacuum and in matter: equal traces and equal off-diagonal elements lead to two sum rules: [Xing (2001)] CP violation in vacuum and in matter: Naumov relation (1992) Toshev relation (1991)
Part D Unitarity Triangles in Matter 27 The complex sides of 3 unitarity triangles in matter can be solved from 3 linear equations for 3 parameters: Example:
Non-unitarity Neutrino Mixing? 28 Example A: light sterile neutrinos --- no good TH / EX motivation today. Example B: heavy Majorana neutrinos --- popular seesaw mechanisms. Example C: whole tower of KK states --- models with extra dimensions. The scheme of Minimal Unitarity Violation (Antusch et al 07): -- Only 3 light neutrino species are considered; -- Sources of non-unitarity are allowed only in those terms of the SM Lagrangian which involve neutrinos. Unitarity of the MNS matrix: good at an 1% level & bad at an 1% level. Constraint on the 3 3 -mixing matrix V ---- data on -oscillations, W and Z decays, rare LFV modes and lepton universality tests,... (Antusch et al 07):
Summary of Lecture B 29 (A) Flavor mixing & CP violation in the SM + 3 massive neutrinos are due to the coexistence of 2 types of interactions. (B) The unitary 3 3 neutrino mixing matrix contains 1 (Dirac) or 3 (Majorana) CP-violating phases, & has 6 unitarity triangles with the same area in the complex plane. (C) It is possible that the 3 3 mixing matrix of light neutrinos is not exactly unitary, but non-unitarity effects must be small. (D) Flavor puzzles of leptons & quarks: similarities & differences ---- Hierarchies of quark masses & mixing angles correlated? ---- Flavor symmetries & constant neutrino mixing patterns? (E) RGE running behaviors of 3 neutrino masses, 3 flavor mixing angles & 3 CP-violating phases from superhigh to low scales. (F) Matter effects on neutrino masses, mixing & CP violation. References on renormalization-group equations of massive neutrinos: ---- S. Antusch et al., NPB 674 (2003) 401; JHEP 0503 (2005) 024; ---- J.W. Mei, PRD 71 (2005) 073012.