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We investigate a gauge theory realization of non-Abelian discrete flavor symmetries and apply the gauge enhancement mechanism in heterotic orbifold models to field-theoretical model building. Several phenomenologically interesting non-Abelian discrete symmetries are realized effectively from a $U(1)$ gauge theory with a permutation symmetry. We also construct a concrete model for the lepton sector based on a $U(1)^2 rtimes S_3$ symmetry.
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We study the modular symmetry in magnetized D-brane models on $T^2$. Non-Abelian flavor symmetry $D_4$ in the model with magnetic flux $M=2$ (in a certain unit) is a subgroup of the modular symmetry. We also study the modular symmetry in heterotic orbifold models. The $T^2/Z_4$ orbifold model has the same modular symmetry as the magnetized brane model with $M=2$, and its flavor symmetry $D_4$ is a subgroup of the modular symmetry.
A spontaneously broken global discrete symmetry may have pseudo Goldstone modes associated with the spontaneous breaking of the approximate continuous symmetry of the low dimension terms in the Lagrangian. These provide natural candidates for an inflaton that can generate slow roll inflation. We show that, in the case of a non Abelian discrete symmetry, the pseudo Goldstone modes readily couple to further scalar fields in a manner that the end of inflation is determined by these additional scalar fields, generating hybrid inflation. We give a simple parameterisation of the inflationary potential in this case, determine the inflationary parameters resulting, and show that phenomenological successful inflation is possible while keeping the scale of symmetry breaking sub-Plankian. Unlike natural inflation the inflation scale can be very low. We construct two simple hybrid inflation models, one non supersymmetric and one supersymmetric. In the latter case no parameters need be chosen anomalously small.
A detailed description of the method for analytical evaluation of the three-loop contributions to renormalization group functions is presented. This method is employed to calculate the charge renormalization function and anomalous dimensions for non-Abelian gauge theories with fermions in the three-loop approximation. A three-loop expression for the effective charge of QCD is given. Charge renormalization effects in the SU(4)-supersymmetric gauge model is shown to vanish at this level. A complete list of required formulas is given in Appendix. The above-mentioned results of three-loop calculations have been published by the present authors (with A.Yu., Zharkov and L.V., Avdeev) in 1980 in Physics Letters B. The present text, which treats the subject in more details and contains a lot of calculational techniques, has also been published in 1980 as the JINR Communication E2-80-483.
In [1] it was shown how the flavor symmetry A4 (or S4) can arise if the three fermion generations are taken to live on the fixed points of a specific 2-dimensional orbifold. The flavor symmetry is a remnant of the 6-dimensional Poincare symmetry, after it is broken down to the 4-dimensional Poincare symmetry through compactification via orbifolding. This raises the question if there are further non-abelian discrete symmetries that can arise in a similar setup. To this end, we generalize the discussion by considering all possible 2-dimensional orbifolds and the flavor symmetries that arise from them. The symmetries we obtain from these orbifolds are, in addition to S4 and A4, the groups D3, D4 and D6 simeq D3 x Z2 which are all popular groups for flavored model building.
We discuss the possibility of obtaining a non-abelian discrete flavor symmetry from an underlying continuous, possibly gauged, flavor symmetry SU(2) or SU(3) through spontaneous symmetry breaking. We consider all possible cases, where the continuous symmetry is broken by small representations. Small representations are these which couple at leading order to the Standard Model fermions transforming as two- or three-dimensional representations of the flavor group. We find that, given this limited representation content, the only non-abelian discrete group which can arise as a residual symmetry is the quaternion group D_2.
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