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arXiv:13mm.nnnn [hep-ph] Higgs couplings to t,b, and τ with Flavor Symmetry 3 1 0 2 n Paul H. Frampton∗ a J 3 ] h p - p Department of Physics and Astronomy, UNC-Chapel Hill, NC 27599. e h [ 1 v 4 6 4 0 . 1 Abstract 0 3 Experimantal study at LHC of the possible Higgs boson should soon provide 1 v: accurate estimates of the Yukawa couplings YHf¯f for f′ = t,b,τ. In the presence of a non-abelian discrete flavor symmetry G (e.g. T ) the usual prediction that i F X r YHf¯f ∝ mf should be inexact, and departures therefrom will provide through GF a valuable input to an improved derivation of the quark and lepton mixing matrices. ∗ [email protected] The 2012 discovery of a resonance at 125 GeV at the LHC [1,2], strongly sugges- ∼ tive of the Higgs boson, has naturally caused intense interest. It preliminary properties are consistent within significant errors with the Higgs particle predicted by the minimal standard model. The two-body decays which can be measured accurately in the near future include ¯ H γγ, H bb, and H τ¯τ. These are respectively sensitive to the Yukawa couplings → → → Y (through the top triangle contribution which competes with the W-loop), Y , and Ht¯t H¯bb Y . Hτ¯τ In the minimal standard model, the Yukawa couplings Y appear in the simple form Hf¯f ¯ Y ffH (1) Hf¯f so that they are proportional to the masses Y m (2) Hf¯f f ∝ with proportionality constant < H >−1 where < H > is the vacuum expectation value of the Higgs field. In a renormalizable model with a non-trivial flavor symmetry G , which we will take F here to be non-abelianand discrete, there must be several Higgs and the Yukawa couplings of the lightest one H will generally deviate fromthe simple proportionality of Eq.(2). Such deviations may likely be small but crucial to understanding how the group G operates. F One may even say that if the conventional prediction of Eq. (2) would hold exactly at high precision then renormalizable G models would be disfavored. F These statements are true for general renormalizable G models. To illustrate them, F ′ we focus on the choice G = T [3] and the minimal model discussed in [4]. The flavor F ′ group is (T Z ), and we shall concentrate only on the third-generation couplings Y 2 Hf¯f × for f = t,b,τ. ′ The leptons are assigned under (T Z ) as 2 × ν τ − τ (cid:18)(cid:18)µννµ−(cid:19)(cid:19)LLLL(3,+1) µτeR−R−−R (((111312,,,−−−111))) NNNR(RR((321)))(((111321,,,+++111))). (3) e − e  (cid:18) (cid:19)L   Imposing strict renormalizability on the lepton lagrangian allows as nontrivial terms for  the τ mass only ′ Y (L τ H )+h.c. (4) τ L R 3 1 ′ ′ where H transforms as H (3, 1). 3 3 − ′ The left-handed quark doublets (t,b) ,(c,d) ,(u,d) are assigned under (T Z ) to L L L 2 × t b QL (11,+1) (cid:18) (cid:19)L c (5) s (cid:18)u(cid:19)L QL (21,+1)   d  (cid:18) (cid:19)L and the six right-handed quarks as   tR (11,+1) bR (12,+1) c uR CR (23,−1) (6) R (cid:27) s dR SR (22,+1) R (cid:27) We must two new scalars H (1 ,+1) and H (1 ,+1) whose VEVs 11 1 13 3 < H >= m /Y < H >= m /Y (7) 11 t t 13 b b ′ provide the (t,b) masses. In particular, no T doublet (2 ,2 ,2 ) scalars have been added. 1 2 3 This allows a non-zero value only for Θ . The other angles vanish making the third family 12 stable The Yukawa couplings to the third family of quarks are contained in (quarks) LY = Yt({QL}11{tR}11H11) +Yb({QL}11{bR}12H13) +h.c. (8) ′ The use of T singlets and doublets permits the third family to differ from the first two and thus make plausible the mass hierarchies m m , m > m and m > m as t b b c,u b s,d ≫ outlined in [3]. Such a model leads to the formula [4] for the Cabibbo angle √2 tan2Θ = (9) 12 3 ! or equivalently sinΘ = 0.218.. close to the experimental value sinΘ 0.227. 12 12 ≃ 2 It can also lead to the successful relationship between neutrino micing angles θ ij π θ = (√2)−1 θ (10) 13 23 4 − (cid:12) (cid:12) (cid:12) (cid:12) which is also in excellent agreement with the la(cid:12)test exp(cid:12)eriments [5]. In such a model, the lightest Higgs H is a linear combination ′ H = aH +bH +cH +... (11) 11 13 3 and the consequent Yukawa couplings are Y = a−1Y , Y = b−1Y , Y = c−1Y (12) Ht¯t t H¯bb b Hτ¯τ τ ′ ′ The VEV < H > is shared between the < H > (α = 1 ,1 ,3,...) irreps of T and α 1 3 there is no reason to expect a = b = c = ... = 1 so that the proprtionality of Eq.(2) will be lost. In fact, if Eq.(2) remained exact, the only solution would be a trivial one where ′ all states transform as 1 of T and the G is inapplicable. The successes in [4] and [5] 1 F would, in such a case, be accidental. On the other hand, if Eq.(2) is inexact, the evaluations of the coefficients a,b,c,... can then be used to understand more perspicuously the derivations of mixing angles for quarks and leptons given respectively in [4] and [5], in a first clear departure from the minimal standard model. 3 Acknowledgements This work should have been supported by U.S. Department of Energy grant number DE- FG02-06ER41418. References [1] J. Incalada, talk given at CERN on July 4, 2012. CMS Collaboration. CMS-PAS- HIG-12-020. [2] F. Gionetti, talk given at CERN on July 4, 2012. ATLAS Collaboration. ATLAS- CONF-2012-093. [3] P.H. Frampton and T.W. Kephart, Int. J. Math. Phys. A10, 4689 (1995). hep-ph/9409330. [4] P.H. Frampton, T.W. Kephart, and S. Matsuzaki, Phys. Rev. D78, 073004 (2008). arXiv:0807.4713[hep-ph] [5] D.A.Eby and P.H. Frampton, Phys. Rev. D (in press). arXiv:1112.2675[hep-ph] 4