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 $\delta m_h^2/m_h^2 \sim {\cal O}(10^3)$. We find that the one-loop contributions to the Higgs mass are suppressed by four powers of the hypercharge gauge coupling. Naively, the two-loop contributions involving the heavy vector bosons are large. We find that for the vector leptoquarks these two-loop corrections cancel when 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. We show that the two-loop corrections to the Higgs mass involving the new heavy neutral gauge boson are suppressed. We mention a way to relax the lower bound on the vector leptoquark mass and have $\delta m_h^2/m_h^2 \sim {\cal O}(1)$.