Question for SU(5) x SU(5) string unification

Abstract: Here we ask if it is possible to have string unification in $\mathsf{SU(5)\times SU(5)}$ gauge group. We specifically investigate the weakly coupled heterotic string unification for the four couplings in this framework. We show that only a limited versions of $\mathsf{SU(5)\times SU(5)}$ with adjoint representation components at intermediated scales, between $M_Z$ and $\Lambda$ (unification), are allowed. This is essentially due to the limitation in the parameter space to the gauge coupling constant $\alpha_1^{-1}$ related to hypercharge. Indeed, only the vanishing hypercharge decompositions of $\mathsf{SU(5)L}$ subgroup can help to this unification, namely, the fermion and boson triplets, $\Sigma_3\sim(1,3)_0$, and the fermion and boson octets, $\Sigma_8\sim(8,1)_0$. Thus, these intermediate particles are compatible with the so-called Adjoint $\mathsf{SU(5)}$. The triplets must live in TeV region and could be accessible at colliders while the octets must alive in very high energy scales. We also show that the non-SUSY unification scenario requires the introduction of an additional $\mathsf{SU(2)_L}$ scalar color triplet, $\eta \sim (3,3){-{1}/{3}}$, at relatively low energies ($\sim 10^{5-11}$GeV), and it may induce the proton decay.