Electron and Muon $g-2$ Contributions from the $T'$ Higgs Sector

We study the experimental constraints from electron and muon $g-2$ factors on the Higgs masses and Yukawa couplings in the $T'$ model, and thereby show that the discrepancy between the standard model prediction and experimental value of muon anomalous $g-2$ factor can be easily accommodated.


I. INTRODUCTION
The electron anomalous magnetic moment has been measured to an extremely high precision and agrees with the theoretical prediction calculated from the standard model (SM) [1], with the result ∆a e = |a SM e − a Expt e | < 1 × 10 −10 . (1.1) On the other hand, the most recent theoretical calculation of the muon anomalous magnetic moment gives [2]: a SM µ = (11659183.4 ± 4.9) × 10 −10 , (1.2) where the errors are dominated by the hadronic contribution. The corresponding most updated experimental value is [3]: This implies that a SM µ differs from a Expt µ by 3.1σ, and suggests that a contribution beyond standard model may be required. As we will show, this discrepancy between the theoretical and experimental values can be easily accommodated in the T ′ model [4][5][6] due to the existence of a new and unique Higgs coupling to the muon. While many authors have developed models that resolve this discrepancy [7], only a few have invoked a discrete flavor symmetry.

II. HIGGS CONTRIBUTIONS TO g − 2 FACTORS IN THE T ′ MODEL
The T ′ model [4][5][6] relates quarks and electrons through a discrete flavor symmetry, the binary tetrahedral group T ′ , whose irreducible representations are three singlets, three doublets and a triplet. The renormalizable T ′ model has led to successful predictions of the tribimaximal neutrino mixing matrix as well as the Cabibbo angle [5,6]. More details about the T ′ model, its variants and other related models can be found in the literature [8].
In the T ′ model, electrons and muons couple to the different components of the triplet Higgs H ′ 3 through the interaction terms Y eē H ′ 3, e e and Y µμ H ′ 3, µ µ. To compute the contribution of a virtual Higgs to the electron and muon g − 2 factors, we need to study its contribution to the electron/muon-photon vertex. For f = e, µ, the vertex function is given ( 2.1) whereū(p ′ ) and u(p) are the spinors obeying the equation of motionsū and M H f is the mass of the Higgs which couples to the electron or muon whose mass is denoted by m f .
After some calculations, we obtain where F f (q 2 ) is the form factor associated with the electron or muon, and σ να = i 2 [γ ν , γ α ]. The contributions from the T ′ Higgs sector to electron or muon anomalous magnetic moment is given by For m f ≪ M H f , which is likely to be the case, there is a logarithmic divergence in the above integral as x → 0. This divergence can be extracted by setting 1 − x → 1 and 1 − x 2 → 1 in the integrand. As a result, we obtain Note that for a given value of Y f , ∆a f is strictly decreasing when the ratio M H f /m f increases.
The condition (1.1) implies that any combinations of Y e and M He must be such that where λ e ∼ 3 × 10 −6 is the corresponding electron Yukawa coupling in SM. We required the ratio M He /m e ≫ 1 when we were deriving (2.5), but otherwise a free parameter. To have an assessment on the allowed range of Y e , we need to have some experimental bounds on M He . Apparently, we would have hoped that the LEP [9] bound on Higgs mass may help -due to the non-observation of the "Higgs-strahlung" process e + e − → H Z at LEP, a lower bound has been given to the SM Higgs, namely M H SM ≥ 114.5 GeV. However, in the T ′ model, all the Higgs singlets and triplets couple to Z. Thus, the LEP bound does not apply directly to any of the masses of the Higgs singlets and triplets. If we simply assume that M He 100 GeV, then we require Y e 3.5 in order to satisfy the condition (1.1) . In this case, the upper bound on the Yukawa coupling Y e is very loose and any value of Y e that is perturbatively small would be allowed.
For the muon anomalous magnetic moment, the discrepancy between the theoretical and experimental values can be accounted for easily in the T ′ model if ∆a µ ∼ |a SM µ − a Expt µ | = (24.6 ± 8.0) × 10 −10 , (2.8) leading to the constraint where λ µ ∼ 0.0006 is the corresponding muon Yukawa coupling in SM. It is obvious that Y µ ≫ λ µ , for any choice of M Hµ /m µ ≫ 1. For instance, if we assume that M Hµ 100 GeV, then in order to satisfy (2.9), we require Y µ 0.13.

III. CONCLUSIONS
In this article, we have computed the contributions to electron and muon g − 2 factors from the Higgs sector in the T ′ model. We then used the experimental data to constrain the T ′ model Higgs masses and Yukawa couplings.
coupling Y e would be very loose and any value of Y e consistent with the perturbation theory would be allowed.
Our main result is the demonstration that the discrepancy between the standard model and experimental values of muon anomalous g − 2 factor can be accounted for easily in the T ′ model. Assuming M Hµ 100 GeV, we found that the Yukawa coupling Y µ should be much larger than the corresponding SM value in order to explain the discrepancy.