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Vector Leptoquarks Beyond Tree Level II: $\mathcal{O}(α_s)$ Corrections and Radial Modes

We clarify open issues in relating low- and high-energy observables, at next-to-leading order accuracy, in models with a massive leptoquark embedded in a flavor non-universal $SU(4) \times SU(3) \times SU(2) \times U(1)$ gauge group. Extending previous work on this subject, we present a complete analysis of the $\mathcal{O}(α_s)$ corrections to the matching conditions of semileptonic operators at the high scale. These corrections are not negligible, but they do not exceed the 10% level and are subleading compared to the $\mathcal{O}(α_4)$ corrections proportional to the leading leptoquark coupling, which is expected to be much larger than the QCD coupling in the parameter space region of phenomenological interest. We further analyze the impact of radial modes, both at $\mathcal{O}(α_4)$ and at $\mathcal{O}(α_s)$ accuracy, highlighting their role in the renormalization of the theory.