Computing the Weak Mixing Angle from Anomaly Cancellation

I remark that the weak mixing angle in the standard model may be computed even in the absence of a grand unification symmetry. In particular, if there is an additional gauged $U(1)$ symmetry at some large scale which can be made anomaly-free only by a Green-Schwarz (GS) mechanism, this typically results in a prediction for the weak angle. In the case of the standard model one can see that the standard Peccei-Quinn symmetry may be gauged and the anomalies cancelled through a GS mechanism. Remarkably enough, cancelation of anomalies works only for the `canonical' value $sin^2\theta _W=3/8$. In the case of the supersymmetric standard model one can also find $U(1)$ currents which may be made anomaly-free through a GS but the canonical value is only obtained in the absence of any Higgs multiplet. If the analysis is extended to include $U(1)$ R-symmetries, there is a unique class of $U(1)$s which give rise to the canonical value. The R-symmetry is only anomaly-free for $sin^2\theta _W=(4N_g-3)/(10N_g-3N_D-3)$, where $N_g,N_D$ are the number of generations and Higgs pairs. The natural context in which the above scenario may naturally arise is string theory. I also emphasize other interesting possibilities offered by the GS mechanism to model-building.