Importance of vector leptoquark-scalar box diagrams in Pati-Salam unification with vector-like families

We study lepton flavor violation (LFV) induced by one-loop box diagrams in Pati-Salam (PS) unification with vector-like families. The vector leptoquark (LQ) associated with the PS gauge symmetry breaking generally causes various LFV processes such as K$_L$ → μe and μ → e conversion at the tree-level, thereby driving its mass scale to be higher than PeV scale. The vector-like families are introduced to suppress such tree-level LFV processes, allowing the LQ to have TeV scale mass. In this paper, we point out that there are inevitable one-loop contributions to those LFV processes from the box diagrams mediated by both one LQ and one scalar field, even if the tree-level contributions are suppressed. We consider a concrete model for demonstration, and show that the vector-like fermion masses have an upper bound for a given LQ mass when the one-loop induced processes are consistent with the experimental limits. The vector-like fermion mass should be lighter than 3 TeV for 20 TeV LQ, if a combination of the couplings does not suppress K$_L$ → μe decay. Our findings would illustrate importance of the box diagrams involving both LQ and physical modes of symmetry breaking scalars in TeV scale vector LQ models.