Can string theory predict the Weinberg angle?

Abstract

We investigate whether the hypercharge assignments in the standard model can be interpreted as a hint at grand unification in the context of heterotic string theory. Our analysis is a follow-up to the recent mini-landscape analysis of the ${\mathbb{Z}}_{6}$-II orbifold of the ${\mathrm{E}}_{8}\ifmmode\times\else\texttimes\fi{}{\mathrm{E}}_{8}^{\ensuremath{'}}$ heterotic string. In that analysis, an intermediate grand unified theory (GUT) was a requirement for finding MSSM-like theories. Nevertheless, about 1% of the models in this mini-landscape were MSSM-like. In this paper we remove this GUT restriction. To this end, we introduce a general method to calculate $\mathrm{U}(1{)}_{Y}$ for any heterotic orbifold and compare our findings to the cases where hypercharge arises from a GUT. Surprisingly, in the overwhelming majority of 3-2 standard models, a nonanomalous hypercharge direction can be defined, for which the spectrum is vectorlike. For these models, we calculate ${sin}^{2}{\ensuremath{\theta}}_{w}$ to see how well it agrees with the standard GUT value. We find that 12% have ${sin}^{2}{\ensuremath{\theta}}_{w}=3/8$, while all others have values which are less. Finally, 89% of the models with ${sin}^{2}{\ensuremath{\theta}}_{w}=3/8$ have $\mathrm{U}(1{)}_{Y}\ensuremath{\subset}\mathrm{SU}(5)$.