String unification scale and the hyper-charge Kac-Moody level in the non-supersymmetric standard model

The string theory predicts the unification of the gauge couplings and gravity. The minimal supersymmetric Standard Model, however, gives the unification scale ∼2×1016 GeV which is significantly smaller than the string scale ∼5×1017 GeV of the weak coupling heterotic string theory. We study the unification scale of the non-supersymmetric minimal Standard Model quantitatively at the two-loop level. We find that the unification scale should be at most ∼4×1016 GeV and the desired Kac-Moody level of the hyper-charge coupling should be 1.33≲kY≲1.35.