Weak mixing angle and grand unified gauge theories

Abstract

We discuss the role of ${{sin}^2\theta }_W$ (${\theta }_W=\mathrm{weak}\mathrm{mixing}\mathrm{angle}$) in the Weinberg-Salam SU(2)×U(1) model. A definition of the renormalized value of ${{sin}^2\theta }_W$ is given, the effects of radiative corrections are considered, and the possibility of precisely determining this parameter experimentally is examined. Within the framework of grand unified gauge thories, we derive an expression for the effective quantity ${{sin}^2\theta }_W\left(M_W\right)$ which can be compared with experiment. Our results are illustrated for the Georgi-Glashow SU(5) model and their implications regarding the predicted lifetime of the proton ${\tau }_p$ are discussed. For ${{sin}^2\theta }_W\left(M_W\right)\simeq 0.21-0.20$, values consistent with neutral-current measurements, we estimate that the SU(5) model predicts ${\tau }_p\simeq {10}^{30}-{10}^{33}$ yr, a range that is not very far from the present experimental bound. We also show that by increasing the number of scalar multiplets, the model can accommodate larger values for ${{sin}^2\theta }_W\left(M_W\right)$.