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

TL;DR

This paper explores the possibility of string unification within the SU(5) x SU(5) gauge framework, identifying specific intermediate particles and conditions necessary for unification, with implications for collider physics and proton decay.

Contribution

It demonstrates that only certain SU(5) x SU(5) models with specific intermediate particles can achieve string unification, highlighting the role of hypercharge decomposition and additional scalar fields.

Findings

Only models with adjoint components at intermediate scales unify.

Triplet particles at TeV scale could be detected at colliders.

Additional scalar triplet may induce proton decay at high energies.

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.