Calculable Violation of Gauge-Yukawa Universality and Top Quark Mass in the Gauge-Higgs Unification

We find that the one-loop correction to the ratio of Yukawa coupling and gauge coupling in the gauge-Higgs unification, “gauge-Yukawa universality violation”, is finite and calculable in any space-time dimension. Applying this result to the ratio of top quark and W-boson masses, we show that the order one correction required to generate a viable top quark mass is indeed possible if the fermion embedding top quark belongs to the large dimensional representation of the gauge group and a vacuum expectation value of Higgs scalar field is very small comparing to the compactification scale. Gauge-Higgs unification [1, 2, 3] is one of the attractive scenarios solving the hierarchy problem without invoking supersymmetry. In this scenario, Higgs doublet in the Standard Model (SM) is identified with the extra spatial components of the higher dimensional gauge fields. One of the remarkable features is that quantum corrections to Higgs mass is insensitive to the cutoff scale of the theory and calculable regardless of the nonrenormalizability of higher dimensional gauge theory. The reason is that the Higgs mass term as a local operator is forbidden by the higher dimensional gauge invariance. Then, the finite mass term is generated radiatively and expressed by the Wilson line phase as a non-local operator. This fact has opened up a new avenue to the solution of the hierarchy problem [4]. Since then, much attention has been paid to the gauge-Higgs unification and many interesting works have been done from various points of view [5]-[25]. The finiteness of Higgs mass has been studied and verified in various models and in various types of compactification at one-loop level [26]-[29] and even at two loop level [31]. It is natural to ask whether any other finite physical observables exist in the gaugeHiggs unification. The naive guess is that such observables are in the gauge-Higgs sector of the theory if they ever exist. The present authors studied the structure of divergences for S and T parameters in the gauge-Higgs unification since such parameters are described by higher dimensional gauge invariant operators with respect to gauge and Higgs fields, and are expected to be finite by virtue of the higher dimensional gauge symmetry. The result is that both parameters are divergent (convergent) more than (in) five dimensions as expected from the power counting argument. However, a nontrivial prediction we have found, specific to the gauge-Higgs unification, is that some linear combination of S and T parameters is finite even in six dimensions [32]. One of the authors (N.M.) also has shown that the gluon fusion amplitude and the amplitude of two photon decay of Higgs boson, which are very important processes at LHC, are finite in any space-time dimension in the gauge-Higgs unification [33]. This is a new and only known calculable physical observable other than the Higgs mass. Although the gauge-Higgs unification is very predictive in the gauge-Higgs sector of the standard model as mentioned above, the matter sector is too restrictive to generate a desirable flavor structure since Yukawa coupling is given by the gauge coupling, to start with. This immediately leads to the fact that fermion masses become W-boson mass and Yukawa hierarchy cannot be explained, unless some suitable mechanism is adopted. Obtaining light fermion masses is easily realized by introducing the Z2-odd bulk mass because the zero mode wave functions for fermions with different chiralities are localized at different fixed points along the extra dimension, which naturally yields a small Yukawa For the case of gravity-gauge-Higgs unification, see [30] In a toy model of QED compactified on a circle, the anomalous magnetic moment was shown to be finite in arbitrary space-time dimensions [34]. Recently, the cancellation mechanism of the ultraviolet (UV) divergence in the magnetic moment was further clarified in a realistic SU(3) model on an orbifold S1/Z2 [35]. 1 coupling due to the small overlap integral of the zero mode wave functions. However, getting the top quark mass is nontrivial task since we need an enhancement factor of roughly 2, as mt ≃ 2mW . In flat space gauge-Higgs unification model, this enhancement factor can be obtained from the group theoretical factor of large dimensional representation which a bulk fermion embedding top quark belongs to [15]. In warped space case, it is known that the enhancement factor comes from the product of curvature scale and compactification radius [20]. In this paper, we propose an alternative mechanism to generate a viable top quark mass by taking into account one-loop corrections to Yukawa coupling in the flat space gauge-Higgs unification. Naively thinking, this seems to be clearly impossible because the loop corrections are always suppressed. However, this is not necessarily the case in the case of gauge-Higgs unification. As will be shown later, the one-loop correction effects have additional factor of Dynkin index for the representation which the matter fermion belongs to and a logarhythmic factor of Higgs vacuum expectation value (VEV) other than the one-loop factor. If we consider the fermion belonging to large dimensional representation, we can have a large Dynkin index. We further note that the logarhythmic factor of Higgs VEV is likely to be large since the Higgs VEV should be tiny compared to the compactification scale, typically around O(10−2), to realize the correct pattern of electroweak symmetry breaking and obtain a Higgs mass satisfying the experimental data. Combining these effects, we can expect that the one-loop correction to Yukawa coupling becomes O(1). Quantum correction to Yukawa coupling is generally a cutoff scale dependent, especially in the nonrenormalizable higher dimensional theories, and is independent of the quantum correction to the gauge coupling. Thus, the quantum correction seems to have no definite prediction. In the gauge-Higgs unification, however, Yukawa and gauge coupling are identical, to start with, being described by the same covariant derivative, i.e. “gauge-Yukawa universality” holds. Hence, even if the universality is violated at the quantum level, the violation should be finite and calculable. It is interesting to note that a similar situation happens to MSSM, where Higgs self-coupling is provided by the gauge interaction, D-term, and the deviation of the Higgs mass from the gauge boson masses is finite, even at the quantum level [36]. According to this line of argument, we calculate here one-loop corrections to the ratio of Yukawa coupling and the gauge coupling, which will be shown to be independent of the cutoff scale of the theory, namely calculable and finite. Since Yukawa coupling is provided by a part of the gauge coupling gψ̄AMΓ ψ, the renormalized Yukawa coupling is obtained by taking into account the wave function renormalization factors of extra component of the gauge field Ay = √ ZyA bare y , a fermion