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The Flavour Puzzle, Discreet Family Symmetries

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The Flavour Puzzle, Discreet Family Symmetries The flavour puzzle, discreet family symmetries 27. 10. 2017 Marek Zrałek Particle Physics and Field Theory Department University of Silesia Outline 1. Some remarks about the history of the flavour problem. 2. Flavour in the Standard Model. 3. Current meaning of the flavour problem? 4. Discrete family symmetries for lepton. 4.1. Abelian symmetries, texture zeros. 4.2. Non-abelian symmetries in the Standard Model and beyond 5. Summary. 1. Some remarks about the history of the flavour problem The flavour problem (History began with the leptons) I.I. Rabi Who ordered that? Discovered by Anderson and Neddermayer, 1936 - Why there is such a duplication in nature? - Is the muon an excited state of the electron? - Great saga of the µ → e γ decay, (Hincks and Pontecorvo, 1948) − − - Muon decay µ → e ν ν , (Tiomno ,Wheeler (1949) and others) - Looking for muon – electron conversion process (Paris, Lagarrigue, Payrou, 1952) Neutrinos and charged leptons Electron neutrino e− 1956r ν e 1897r n p Muon neutrinos 1962r Tau neutrinos − 1936r ν µ µ 2000r n − ντ τ 1977r p n p (Later the same things happen for quark sector) Eightfold Way Murray Gell-Mann and Yuval Ne’eman (1964) Quark Model Murray Gell-Mann and George Zweig (1964) „Young man, if I could remember the names of these particles, I „Had I foreseen that, I would would have been a botanist”, have gone into botany”, Enrico Fermi to advise his student Leon Wofgang Pauli Lederman Flavour - property (quantum numbers) that distinguishes Six flavours of different members in the two groups, quarks and leptons quarks Six Quantum numbers: flavours of leptons - Isospin: I and I3 - Strangeness: S - Charm: C - Bottomness: B’ - Topness: T Quantum numbers: Other quantum numbers: - flavour lepton - Baryon number: B numbers: Lα - Lepton number: L Y -= Hypercharge: B + S + C + B Y′ + T - week isospin T, T3 ' Y = B+ S+ C + B + T 1 - Electric charge: Q e− = L = 1,T = e 3 2 Y Q I 1 = 3 + ν = L = 1,T = − 2 e e 3 2 Standard Model SU(3) ⊗SU(2) ⊗U(1) c L Y „The term flavor was first used in particle physics in the context of the quark model of hadrons. It was coined in 1971 by Murray Gell-Mann and his student at the time, Harald Fritzsch, at a Baskin- Robbins ice-cream store in Pasadena. Just as ice- cream has both color and flavor so do quarks” Browder T.E., el al., Rev. Mod. Physics 81 (2009) 1887 2. Flavour in the Standard Model Standard Model SU(3) ⊗SU(2) ⊗U(1) c L Y ⎛uα ⎞ α α α α L uR α ⎛ν L ⎞ ν R QL = L = ⎜ α ⎟ α L ⎜ α ⎟ α ⎝ dL ⎠ dR ⎝ l L ⎠ l R Fermions kinetic energy and interactions with gauge fields SU(2) ⊗U(1) L Y Only small subgroup of full flavour symmetry is LSM = LGauge + LMatter + LHiggs +LYukawa still valid: θ i ! α 3 α α iϕ α α αβ β α αβ β QL = e QL L! = e L Q! = U Q L! = U L L L R QR R L LL L θ i α iϕ α Flavour α α α αβ β α αβ β u e 3u e! = e e ! !R = R R R u!R = Uu uR ν R = Uν ν R R R θ i symmetry i α ϕ α α α ν! = e ν 6 !α αβ β α αβ β d! = e 3 d R R d = U d e!R = Ue eR R R ⎡U(3)⎤ R dR R R ⎣ ⎦ Conservation of: Barion number Lepton number 1 1 L = − Wi Wiµν − B Bµν Gauge 4 µν 4 µν i i i j k where, Wµν = ∂µ Wν − ∂ν Wµ − gεijkWµ Wν , i = 1,2,3 Bµν = ∂µ Bν − ∂ν Bµ iL (Q D Q L D L − matter = ∑ αL L αL + αL L αL + α +uαR DRuαR + dαR DRdαR + lαR DRlαR +ναR DRναR) µ µ ⎛ ig ! ! ' ⎞ D ≡ γ D ≡ γ ∂ − τW − igYB , for L L Lµ ⎝⎜ µ 2 µ µ ⎠⎟ where, D µ D µ ig'YB , for R R ≡ γ Rµ ≡ γ (∂µ − µ ) And finally: 2 µ ⎛ + 1 2 ⎞ L = D Φ D Φ − λ Φ Φ − v Higgs ( Lµ )( L ) ⎝⎜ 2 ⎠⎟ spontaneous Quark and LHiggs symmetry breaking LYukawa fermion masses 1 ⎛ν⎞ Φ = 2 ⎝⎜ 0⎠⎟ L ⎡u M u u ⎤ ⎡d M d d ⎤ ⎡l M l l ⎤ ⎡ M ν ⎤ h.c Mass = −∑(⎣ αL α ,β βR ⎦ + ⎣ α L α ,β βR ⎦ + ⎣ αL α ,β βR ⎦ + ⎣ναL α ,βνβR ⎦)+ α ,β u * d ν *hl hν u νhα,β d ν hα ,β l α ,β ν ν α ,β M = M = − Mα ,β = − M = α,β α ,β 2 α ,β 2 2 2 Yukawa, and in consequence the mass matrices, are any complex 3 x 3 matrices. So at this stage, the Yukawa Lagrangian depends on ( 4 x 9 x 2) = 72 free parameters. 6 x 2=12 for Majorana neutrinos In what follows we will concentrate on the leptons. Exactly in the same way the quark sector looks like. Mass matrices have to be diagonalized. The leptons flavour states (α) will go to mass eigenstates (i), and leptons obtain their physical mass. For charged leptons For neutrinos ⎛ m 0 0 ⎞ e ⎛ m1 0 0 ⎞ l+ l l l ⎜ ⎟ ν+ ν ν ν ⎜ ⎟ U M U = M = 0 mµ 0 0 0 L R diagonal ⎜ ⎟ UL M UR = M diagonal = ⎜ m2 ⎟ ⎜ 0 0 m ⎟ ⎜ 0 0 ⎟ ⎝ τ ⎠ ⎝ m3 ⎠ The matter Lagrangian will change, e.g. PMNS l+ ν U = UL UL , Pontecorvo, Maki, Nakagawa, Sakata mixing matrix As we need only the left-handed matrices l and ν we first U UL diagonalize: L ⎛ 2 ⎞ ⎛ 2 ⎞ m 0 0 m1 0 0 ⎜ e ⎟ ⎜ ⎟ l+ l l+ l 0 m2 0 Uν + M ν M ν + Uν 2 UL (M M )UL = ⎜ µ ⎟ L ( ) L = ⎜ 0 m2 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 m2 ⎟ ⎜ 0 0 m2 ⎟ ⎝ τ ⎠ ⎝ 3 ⎠ depends only on 9 parameters not 18 2 Matrix l l l + l l l+ UL M M = U L M diag U L depends on 6 ( ) 9 parameters 3 parameters parameters PMNS l+ ν U = U L U L depends on 9 independent parameters, as e.g. in the traditional parameterization: −iδ ⎛ iβ1 ⎞ ⎛ ⎞ e 0 0 c13c12 c13s12 s13e ⎛ ⎞ ⎜ ⎟ 1 0 0 ⎜ iβ ' ⎟ iδ iδ ' 2 ⎜ −iα1 2 ⎟ 0 e 0 ⎜ −s12c23 − c12s23s13e c12c23 − s12s23s13e c13s23 ⎟ 0 e 0 l ⎜ l ⎟ ⎜ ⎟ ν iβ3 ⎜ iδ iδ ⎟ ν −iα 2 i k ⎜ 0 0 k ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎝ e ⎠ ⎝ s12s23 − c12c23s13e −c12s23 − s12c23s13e c13c23 ⎠ ⎝ 0 i 0 e ⎠ Only some elements of Yukawa matrix elements finally appear: ◆ For quark sector: ◆ Lepton sector with Dirac neutrinos: ◆ Lepton sector with Majorana neutrinos: So together Quark + Leptons 20 parameters 22 parameters (Dirac neutrinos) (Majorana neutrinos) + (α, θW , g3, v, λ) 25 27 3. Current meaning of the flavour problem 1. Why are there three families of quarks and leptons? 2. What is the origin of the pattern of quark and lepton masses? 2.1 Why they are so hierarchical for charged leptons? 2.2 Why neutrino masses are so small and probably not hierarchical? 3. Why is the quark mixing so small 4. Why is lepton mixing so large? 5. What is origin of quark and lepton (?) CP violation? 6. Why there are so many free parameters in the Standard Model (25 or 27)? Up to now we don’t know: ◆Whether massive neutrinos are Dirac or Majorana particles. ◆What is the absolute scale of neutrino masses. ◆What is the octant for the atmospheric mixing angle θ23. ◆What are the values of the CP violating phases in the lepton sector. And very important question: ◆How the above mentioned features of leptons are connected to quark properties. In the SM fermion masses and mixing matrix elements are encoded in the mass matrices for charged leptons and for neutrinos, but Number of mass matrix Measured quantities: parameters: 9 x 2 = 18 charged leptons 6 masses 3 + 1 CKM matrix elements 6 x 2 = 12 Majorana neutrinos 6 masses 3 + 3 PMNS matrix elements 30 real parameters 22 measured quantities Several proposals: -- texture zeros, The parameters can be restrict: -- vanishing minors, -- hybrid textures, -- non-abelian discreet symmetries. 4. Discrete flavour symmetries for lepton In particle physics Symmetries play very important role, e.g. Isospin Eightfold way Gauge Symmetry Supersymmetry Grand Unification Superstrings As we know the Yukawa Lagrangian break the flavour symmetry Q! α = U αβQβ L!α = U αβ Lβ R QR R L LL L u!α = U αβ uβ ν!α = U αβν β R uR R R ν R R d!α = U αβ d β e!α = U αβ eβ R dR R R eR R For any matrix transformation in the three dimensional flavour space only baryon and lepton number symmetry remain Maybe it is possible to introduce such set of flavour transformation (given by some symmetry group) that mass matrices will reproduce fermion masses and their mixing matrices ??? So it is reasonable to assume that some symmetry exist which mix the fermion flavour states and Higgs fields (if more than one) and, in consequence, give such relations between Yukawa matrices and vacuum expectation values that: ✴ fermion masses have proper values, ✴ the CKM and PMNS mixing matrices agree with experiments So we assume that: G = Ggauge ⊗ Gflavour Ggauge = SU(3)c × SU(2)L ×U(1)Y Gflavour = ? SU(2)L x SU(2)R or any other gauge ✴ Should be discrete unification group e.g. SU(5), SO(10), ✴ Abelian or not-abelian SU(8), E(8),……… ✴ For non-abelian have 3x3 IR Immediately after the discovery of Two mixing angles were the neutrino oscillations: maximum, the third was zero 0 0 θ13 ≈ 0; θ23 ≈ 45 ; θ12 ≈ 45 Neutrino BI-MAXIMAL mixing ⎛ ⎞ Vissani F., arXiv: hep-ph/9708483; 1/ 2 1/ 2 0 Barger V.
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