Fermion Mixing Angles and the Connection to Non-Trivially Broken Flavor Symmetries

Fermion Mixing ngles and the Connection to Non-Trivially Broken Flavor Symmetries C. Hagedorn hagedorn@mpi-hd.mpg.de Max-Planck-Institut für Kernphysik, Heidelberg, Germany. Blum, CH, M. Lindner numerics:. Blum, CH,. Hohenegger arxiv:79.345 [hep-ph], arxiv:7.56 [hep-ph] Padova, pril 3, 8 p.

Outline Observations General comments on the flavor symmetry G F TBM from non-trivial 4 breaking (ltarelli/feruglio, He/Keum/Volkas, Lam) Study of dihedral groups as flavor symmetry Fermion mixings from non-trivial breaking of groups θ C from D 7 (Lam, our works) θ3 l = π 4 and θl 3 = from D 3 and D 4 (Grimus/Lavoura) Conclusions & Outlook Padova, pril 3, 8 p.

Observations Masses of the charged fermions are strongly hierarchical m u : m c : m t λ 8 : λ 4 :, m d : m s : m b λ 4 : λ :, m e : m µ : m τ λ 4 5 : λ : where λ θ C. Mass hierarchy in the ν sector is milder, ordering and m unknown. Mixing parameters: small mixings for quarks, large mixings for leptons θ q = θ C 3., θ q 3.4 and θ q 3.3 θ l 3 45, θ l 34.4 and θ l 3.5 ( σ) For lepton mixing special structures are allowed: µτ symmetry: sin (θ l 3) =, sin (θ l 3) = tri-bimaximal mixing (TBM): sin (θ l ) = 3, sin (θ l 3) =, sin (θ l 3) = Padova, pril 3, 8 p.

Effect of Flavor Symmetry G F Yukawa couplings in the SM: y u ij Q i H c u c j or y d ij Q i H d c j with y u,d ij Enforce invariance under G F constraints on y u,d ij extension of scalar sector needed: H H k or H H φ k multi-higgs doublets or flavon fields y d ij,k Q i H k d c j or y d ij,k Q i H d c j ( ) φk (M,Λ) renormalizable couplings or in general non-renormalizable Padova, pril 3, 8 p.

Choice of Flavor Symmetry G F The symmetry G F could...... be abelian or non-abelian Padova, pril 3, 8 p.

Choice of Flavor Symmetry G F The symmetry G F could...... be abelian or non-abelian... be continuous or discrete Padova, pril 3, 8 p.

Choice of Flavor Symmetry G F The symmetry G F could...... be abelian or non-abelian... be continuous or discrete... be local or global Padova, pril 3, 8 p.

Choice of Flavor Symmetry G F The symmetry G F could...... be abelian or non-abelian... be continuous or discrete... be local or global... commute with the gauge group or not Padova, pril 3, 8 p.

Choice of Flavor Symmetry G F The symmetry G F could...... be abelian or non-abelian... be continuous or discrete... be local or global... commute with the gauge group or not... be spontaneously broken or explicitly... be broken at low or high energies (low: electroweak scale) (high: seesaw/gut scale) Padova, pril 3, 8 p.

Possible Symmetries Permutation symmetries: symmetric groups S N and alternating groups N with N Æ (only for small N interesting) Dihedral symmetries: single-valued groups and double-valued groups D n with n Æ (all dihedral groups can be interesting) Further double-valued groups: T, O, I,... Subgroups of SU(3): series of (3 n ) and (6 n ) groups with n Æ, as well as finite number of Σ groups (all groups useful) NB: several isomorphisms among the groups, e.g. S 3 D 3 (6) Padova, pril 3, 8 p.

TBM from Non-Trivial 4 Breaking (ltarelli/feruglio, He/Keum/Volkas, Lam) Lepton generations transform according to 4 reps.: l i = @ ν i e i L 3, e c, µ c, τ c MSSM Higgs fields are invariant, i.e. h u,d ( ) ( ) Mass terms: e c l h φk d Λ and Λ l h u l h φk u Λ Flavon fields: ϕ S 3, ξ and ϕ T 3 Superpotential: w l = y e e c (ϕ T l)h d /Λ ( ) n θ + y µ µ c (ϕ T l) h d /Λ Λ + x a ξ (ll)h u h u /Λ + x b (ϕ S ll)h u h u /Λ ( ) n θ + y τ τ c (ϕ T l) h d /Λ Λ dditional symmetry: U() FN (flavor dependent), Z 3 (flavor independent), U() R (flavor independent) Padova, pril 3, 8 p.

TBM from Non-Trivial 4 Breaking ssume vacuum G S : ϕ S = (v S, v S, v S ), ξ = u, G T : ϕ T = (v T,, ). Charged lepton sector: m l = v T Λ h d diag(y e ( θ Λ ) n, y µ ( θ Λ ) n, y τ ) Neutrino sector: m ν = h u Λ with masses: h u Λ a + b/3 b/3 b/3 b/3 b/3 a b/3 b/3 a b/3 b/3 TB mixing! diag(a + b, a, a + b) and a = x a u Λ, b = x b v S Λ Padova, pril 3, 8 p.

TBM from Non-Trivial 4 Breaking Observation: ϕ S = (v S, v S, v S ), ξ = u breaks 4 G S Z in ν sector ϕ T = (v T,, ) breaks 4 G T Z 3 in l sector 4 completely broken in the whole theory Mismatch of different subgroups generates non-trivial (large) mixing and even more predicts exact values of mixing angles (TBM) (independent of choice of parameters, apart from ordering of eigenvalues) Issues in such models maintain vacuum alignment interpretation of mixing keep effects of next-to-leading order corrections under control need of additional symmetries Padova, pril 3, 8 p.

Study of Dihedral Groups as G F large class of groups with similar properties can be scanned ll groups could be relevant as G F, since reps. are one- or twodimensional Hypothesis of non-trivial breaking of G F can be tested by study of further groups apart from 4 allows a systematic study First step: analysis of possible mass matrix structures Result: only a limited number of structures is found pplications: θ C from D 7, θ l 3 maximal and vanishing θ l 3 from D 3 and D 4 Padova, pril 3, 8 p.

Group Theory of and D n is the symmetry group of the regular planar n gon groups are subgroups of SO(3) Order of the group: n Only one- and two-dimensional irreducible representations For n odd:, and,..., n For n even:,..., 4 and,..., n Generator relations for generators and B: n = ½, B = ½, B = B. rep. B for n even rep. B 3 4 rep. B @ e π i n j @ j e n π i j Padova, pril 3, 8 p.

Group Theory of and D n The group is the double covering of the group groups are subgroups of SU() Order of the group: 4 n Irred. reps.:,..., 4 and,..., n Generator relations for generators and B: n = R, B = R, R ½ =, B = B. for n odd rep. B 3 i 4 i for, and 3,4 (n even): see for j even rep. B @ e π i n j @ j e π n i j for j odd rep. B @ e π i n j @ i j e π n i j i Padova, pril 3, 8 p.

Preserved Subgroups of < > < > Z n =< > < 3 > =<,B > < 4 > =<,B > < j > Z j =< n j > (j n) < j > nothing (j n) < j > < j > < j > D j =< n j,b m > (j n; m =,,..., n j ) < j > Z =< B m > (j n; m =,,..., n ) π i m j @ e n @ x y < >+< 3 > Z n =< > (also: < > + < 4 > or < 3 > + < 4 >) < 3 >+< j > Z =< B m > (j n, n j odd; m n j ) (m even for < 3 >; for < 4 >: m odd) Padova, pril 3, 8 p.

Requirements Reveal non-abelian group structure allows the following fermion assignments (indices can, but do not have to, coincide) L ( k, j ) and L c ( i, i, i3 ) or L ( i, j ) and L c ( l, k ) Determinant of mass matrix should be non-zero (reduction of cases) Higgs fields transforming as rep. µ are allowed to have a VEV v, if µ contains/is trivial rep. under subgroup which should be preserved; we assume one Higgs field for each such rep. µ, but no two Higgs fields with exactly the same transformation properties under the group; Higgs fields can be removed from theory by setting VEV to zero (possibly creation of texture zeros or further correlation of matrix elements) For Dirac masses: Higgs fields are assumed to transform as SM SU() L Higgs doublet (all couplings renormalizable; Majorana masses: below) Padova, pril 3, 8 p.

General Results for Groups (Dirac Fermions) The following matrix structures can be found, if the preservation of a non-trivial subgroup is requested (for down-type fermions): a.) diagonal matrix: M = diag(, B, C) b.) semi-diagonal matrix: M = B C c.) block-matrix: M 3 = B C d.) one-zero matrix: M 4 = e.) full matrix: M 5 = B @ B @ Thereby φ = π n m stems from the j D E B C D E C e i φj D e i φ j E e i φ j C C C e i φk B D E B e i φ j E e i φ (j k) D e i φ (j+k) π i m j @ e n NB: Parameters, B, C,... are in some cases correlated. C belonging to generator B m. Padova, pril 3, 8 p.

General Results for Groups (Majorana Fermions) Majorana terms connect fields of same type mass matrix is symmetric some contributions vanish due to anti-symmetry in flavor space results for Dirac fermions only applicable, if L ( i, j ) and L c ( l, k ) with i = l and j = k dditionally we consider cases in which all fermions involved in the mass terms transform as one-dimensional representation under G F matrix structures: diagonal, block form or arbitrary Scalar fields cannot transform as SU() L doublets, but either as SU() L triplets or as gauge singlets in the SM; for ν c also a direct mass term is allowed, if compatible with G F Padova, pril 3, 8 p.

Results for D n Groups Relevant difference: existence of pseudo-real and complex reps. Subgroups can be studied in a similar fashion; structures are very similar to those of the subgroups Mass matrices: no additional structures are found in case of Dirac masses new configuration found in case of Majorana masses, if D n has an odd index n and the representations 3,4 are used; however, structure (semi-diagonal) can be reproduced with group, if 3 + 4 is identified with no new structures found Padova, pril 3, 8 p.

Prediction of θ C (Lam, our works) Take the flavor group D 7 Transformation properties of quarks: Q, (Q, Q 3 ) t, d c, u c, (s c, b c ) t,(c c, t c ) t Higgs fields (SU() L doublets with hypercharge Y = ): H u s, (H u, Hu )t, (h u, hu )t H d s, (Hd, Hd )t, (h d, hd )t Yukawa couplings: L Y = y u Q u c (Hs u)c + y u (Q c c (H u)c + Q t c (H u)c ) + y3 u (Q u c (H u)c + Q 3 u c (H u)c ) + y4 u (Q t c + Q 3 c c )(Hs u ) c + y5 u (Q c c (h u )c + Q 3 t c (h u )c ) + y d Q d c Hs d + yd (Q s c H d + Q b c H d) + yd 3 (Q d c H d + Q 3 d c H d) + y4 d (Q b c + Q 3 s c ) Hs d + y5 d (Q s c h d + Q 3 b c h d ) dditional symmetry: Z (flavor independent), U() FN (flavor dependent) Padova, pril 3, 8 p.

Prediction of θ C ssume the following VEV structure: @ Hu H u = v u e 3 π i 7 @ Hd H d @ e 6 π i 7 = v d @ Hs u, Hs d >,, @ hu h u, @ hd h d = w u e 6 π i 7 = w d @ @ e π i 7, Preserved subgroups: D 7 H u Z = B 3 in the up sector, B 3 = D 7 H d Z = B in the down sector, B = e 6 π i j 7 e 6 π i j 7 «for j! for j e.g. B 3 @ Hu H u = @ e 6 π i 7 e 6 π 7 i @ Hu H u = @ Hu H u Padova, pril 3, 8 p.

Prediction of θ C Mass matrices: M u = and M d = B @ B @ y u Hu s yu v u e 3 π 7 i y u v u e 3 π 7 i y3 u v u e 3 π 7 i y5 u w u e 6 π 7 i y4 u Hu s y3 u v u e 3 π 7 i y4 u Hu s yu 5 w u e 6 π 7 i y d Hd s y d v d y d v d y d 3 v d y d 5 w d y d 4 Hd s y d 3 v d y d 4 Hd s y d 5 w d C. C U u and U d : U u = B @ c u e i β u s u e i β u s u e 6 π 7 i c u su e 6 π 7 i c u e 6 π 7 i C and U d = B @ c d e i β d s d e i β d s d s d c d c d C Padova, pril 3, 8 p.

Prediction of θ C V CKM : CKM = B @ cos( π 4 ) s u ei α c d c u + ( + e 6 π 7 i ) s d s u ei α c u s d ( + e 6 π 7 i ) c d s u cos( 3 π 7 ) cos( π 4 ) s d cos( π 4 ) c d cos( π 4 ) c u ei α c d s u ( + e 6 π 7 i ) c u s d ei α s d s u + ( + e 6 π i 7 ) c d c u Result : with s d,u = sin(θ d,u ), c d,u = cos(θ d,u ) and α = β u β d. ) ( sin 6 π 7 J CP reads: 8 sin( 3 π 7 ) sin( θd ) sin( θ u ) sin ( 3 π 7 + α) V cd = cos( 3 π (exp) 7 ) =.5 compared to V cd =.7 +.. ( ) With general group indices n, j, m u,d : π (mu m cos d ) j Free angles θ d,u and phase α to adjust θ q 3, θq 3 and J CP Mass hierarchy adjusted by U() FN n Padova, pril 3, 8 p.

Predicting θ l 3 = π 4 and θl 3 = (Grimus/Lavoura) Flavor symmetry is D 3 S 3 dditional symmetry: Z (aux) Leptons: D e (,+), (D µ, D τ ) t (,+), e R (, ), (µ R, τ R ) t (,+), ν e R (, ), (ν µ R, ν τ R ) t (, ) Scalars: (φ i are SU() L doublets, χ gauge singlet) φ (, ), φ (,+), φ 3 (,+), (χ, χ ) t (,+) Yukawa couplings: L Y = y D e ν e R φ c + y (D µ ν µ R + D τ ν τ R ) φ c + y 3 D e e R φ + y 4 (D µ µ R + D τ τ R ) φ + y 5 (D µ µ R D τ τ R ) φ 3 + h.c. L νr = M ν e R ν e R + M (ν µ R ν τ R + ν τ R ν µ R ) + y χ (ν e R ν µ R χ + ν e R ν τ R χ) + y χ (ν µ R ν e R χ + ν τ R ν e R χ) + z χ (ν µ R ν µ R χ + ν τ R ν τ R χ ) + h.c. Padova, pril 3, 8 p. 3

Predicting θ l 3 = π 4 and θl 3 = VEV configuration: φ, = v,, φ 3 = v 3, ( χ, χ ) t = (W, W ) t = W (e i α, e i α ) t Preserved subgroups: D 3 φ, φ,3 Z 3 in the charged lepton sector D 3 φ D 3 in the Dirac ν sector D 3 χ, χ Z in the Majorana ν sector for α =, ± π 3, ± π 3, π i.e. @ χ χ = W e i α @ e i α @ {z } α=,±π, @ e 4 π i 3 {z } α= π 3, 3 π, @ e π i 3 {z } α= π 3, 3 π. Padova, pril 3, 8 p. 3

Predicting θ l 3 = π 4 and θl 3 = Mass matrices: Charged leptons: m e = y 3 v, m µ = y 4 v + y 5 v 3, m τ = y 4 v y 5 v 3 Dirac mass for νs: M ν = diag(a, b, b), a = y v, b = y v Majorana mass for ν R s: M RR = B @ M y χ W y χ W y χ W z χ W e 3 i α M y χ W M z χ W e 3 i α C Light ν mass matrix M ν is µτ symmetric, if non-trivial subgroup is preserved, i.e. α =, ± π 3, ±π 3, π (This is enforced by the potential.) Padova, pril 3, 8 p. 3

Predicting θ l 3 = π 4 and θl 3 = (II) (Grimus/Lavoura Flavor symmetry is D 4 dditional symmetry: Z (aux) Leptons: D e ( + +,+), (D µ, D τ ) t (,+), e R ( + +, ), (µ R, τ R ) t (,+), ν e R ( + +, ), (ν µ R, ν τ R ) t (, ). Scalars: (φ i are SU() L doublets, χ i gauge singlets) φ ( + +, ), φ ( + +,+), φ 3 ( + -, +), (χ, χ ) t (,+) Yukawa couplings: L Y = y D e ν e R φ c + y (D µ ν µ R + D τ ν τ R ) φ c + y 3 D e e R φ + y 4 (D µ µ R + D τ τ R ) φ + y 5 (D µ µ R D τ τ R ) φ 3 + h.c. L νr = M ν e R ν e R + M (ν µ R ν µ R + ν τ R ν τ R ) + y χ (ν e R ν µ R χ + ν e R ν τ R χ ) + y χ (ν µ R ν e R χ + ν τ R ν e R χ ) + h.c. Padova, pril 3, 8 p. 3

Predicting θ l 3 = π 4 and θl 3 = (II) VEV configuration: φ, = v,, φ 3 = v 3, ( χ, χ ) t = W (,) t Preserved subgroups: D 4 φ, φ,3 D in the charged lepton sector D 4 φ D 4 in the Dirac ν sector D 4 χ i Z in the Majorana ν sector Mass matrices: charged leptons and M ν same as above; Majorana mass for ν R s: M RR = B @ M M χ M χ M χ M M χ M C with M χ = y χ W Padova, pril 3, 8 p. 3

Conclusions & Outlook Idea : break flavor symmetry to non-trivial subgroup in order to predict fixed mixing angles in the lepton as well as in the quark sector This has been done in specific group 4, here more general study of dihedral groups and D n of arbitrary index n General results: certain matrix structures, several of them can predict maximal mixing pplication to quark sector: prediction of Cabibbo angle θ C in terms of group theoretical quantities with flavor group D 7 Literature: D 3 and D 4 models (Grimus/Lavoura) explain µτ symmetry Realization with flavon fields and mechanism of vacuum alignment are open questions in D 7 model Study of other classes of discrete symmetries Padova, pril 3, 8 p. 3

Thank you. Padova, pril 3, 8 p. 3