Unifying gauge couplings at the string scale

Abstract

Using the current precision electroweak data, we look for the minimal particle content which is necessary to add to the standard model in order to have a complete unification of gauge couplings and gravity at the weakly coupled heterotic string scale. We find that the addition of a vector-like fermion at an intermediate scale and a non-standard hypercharge normalization are in general sufficient to achieve this goal at two-loop level. Requiring the extra matter scale to be below the TeV scale, it is found that the addition of three vector-like fermion doublets with a mass around 700 GeV yields a perfect string-scale unification, provided that the affine levels are $(k_Y, k_2 ,k_3)=(13/3, 1, 2) $, as in the $SU(5) \times SU(5)$ string-GUT. Furthermore, if supersymmetry is broken at the unification scale, the Higgs mass is predicted in the range 125 GeV - 170 GeV, depending on the precise values of the top quark mass and $\tan \beta$ parameter.