Theorem on vanishing contributions to $\sin^2\theta_W$ and intermediate mass scale in Grand Unified Theories with trinification symmetry

Abstract

We prove that the values of the electroweak mixing angle $\sin^2\theta_W$ and intermediate mass scale $M_I$ have vanishing contributions due to one-loop, two-loop and gravitational corrections in Grand Unified Theories which accommodate an intermediate trinification symmetry ($G_{333D}$) invoked with spontaneous D-parity mechanism operative at mass scale greater than $M_I$. The proof of theorem is robust and we verify the results numerically using supersymmetric as well as non-supersymmetric version of $E_6$-GUT.