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Renormalization group and 3-3-1 model with the discrete flavour symmetries

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Renormalization group equations of the 3-3-1 models with A4 and S4 flavor symmetries as the only intermediate gauge group between the standard model and the scale of unification of the three coupling constants are presented. We shall assume that there is no necessarily a group of grand unification at the scale of convergence of the couplings. | Communications in Physics, Vol. 22, No. 1 (2012), pp. 1-6 RENORMALIZATION GROUP AND 3-3-1 MODEL WITH THE DISCRETE FLAVOUR SYMMETRIES HOANG NGOC LONG Institute of Physics, VAST NGUYEN THI KIM NGAN Department of Physics, Can Tho University Abstract. Renormalization group equations of the 3-3-1 models with A4 and S4 flavor symmetries as the only intermediate gauge group between the standard model and the scale of unification of the three coupling constants are presented. We shall assume that there is no necessarily a group of grand unification at the scale of convergence of the couplings. I. INTRODUCTION Since the birth of the Standard Model (SM) many attempts have been done to go beyond it, and solve some of the problems of the model such as the unification of coupling constants. In looking for unification of the coupling constant by passing through a 3-3-1 models [1,2], we shall assume that 1) The 3-3-1 gauge group is the only extension of the SM before the unification of the running coupling constants. 2) The hypercharge associated with the 3-3-1 gauge group is adequately normalized such that the three gauge couplings unify at certain scale MU . and 3) There is no necessarily a unified gauge group at the scale of convergence of the couplings MU . In the absence of a grand unified group, there are no restriction on MU coming from proton decay. If the unification came from a grand unified symmetry group G, the normalization of the hypercharge Y would be determined by the group structure. However, under our assumptions, this normalization factor is free and the problem could be addressed the opposite way, since the values obtained for a could in turn suggest possible groups of grand unification in which the 3-3-1 group is embedded, we shall explore this possibility as well. I.1. RGE analysis Renormalization group equations are 2 HOANG NGOC LONG AND NGUYEN THI KIM NGAN α−1 U α−1 U α−1 U 4b2 −1 α2L (MZ )−1 = α (M ) − EM Z 2 3 a2 − 4b3 ) 2 bY − 4b3 .