Finite Naturalness and Quark-Lepton Unification

We study the implications of finite naturalness in Pati-Salam models where $SU(3{)}_C$ is embedded in $SU(4)$. For the minimal realization at low-scale of quark-lepton unification, which employs the inverse seesaw for neutrino masses, we find that radiative corrections to the Higgs boson mass are at least $\delta m_h^2\mathrm{/}{m}_h^2\sim \mathcal{O}({10}^4)$. The one-loop contributions to the Higgs mass are suppressed by four powers of the hypercharge gauge coupling. We find that for the vector leptoquarks the naively leading part of the two-loop corrections cancel. We assume the Dirac Yukawa couplings for neutrinos are equal to the up-type quark Yukawa couplings as predicted in the minimal theory for quark-lepton unification. Despite these findings, the two-loop corrections still dominate the finite naturalness bound. We mention a way to relax the lower bound on the vector leptoquark mass and have $\delta m_h^2\mathrm{/}{m}_h^2\sim \mathcal{O}({10}^2)$.