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We study lepton flavor violating meson decays induced by box diagramsinvolving a vector leptoquark (LQ) and scalar fields in Pati-Salam (PS)unification with vector-like families. The vector LQ corresponds to the massivegauge boson associated with the PS gauge symmetry breaking and the scalarfields are the physical degrees of freedom of the PS breaking field. The LQgenerally causes the rapid flavor violating decays, such as $K_L \to \mu e$, atthe tree-level unless its mass scale is higher than PeV scale. The vector-likefamilies are introduced to suppress the tree-level contributions mediated bythe LQ and explain the realistic fermion mass matrices. In this paper, we pointout that there are inevitable one-loop contributions to those meson decays fromthe box diagrams mediated by both one LQ and one scalar field, even if thetree-level contributions are suppressed. We consider a concrete model fordemonstration, and find that the vector-like fermion masses have an upper boundfor a given LQ mass when the one-loop induced meson decays are consistent withthe experimental limits. The vector-like fermion mass should be lighter than 3TeV for 20 TeV LQ, if a combination of the couplings does not suppress $K_L \to\mu e$ decay. Our findings would illustrate importance of the box diagramsinvolving both LQ and physical modes of symmetry breaking scalars in genericmodels.

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