Direct renormalization of flavor-changing currents in vNRQCD for $B_c^{(*)}$ decays

Abstract

We present a direct vNRQCD calculation of the two-loop anomalous dimensions of the leading flavor-changing nonrelativistic currents that govern the leptonic decays of the $B_c$ and $B_c^*$ mesons. The calculation is performed at the matching scale $ν=1$, where the anomalous dimensions are the ultraviolet counterterms required to cancel the residual infrared poles of the full-QCD short-distance coefficients. In contrast to the conventional extraction from those infrared poles, we determine the current renormalization constants directly in the effective theory. The pseudoscalar and vector channels are obtained from one spin-unified formula, with the spin dependence entering through $S(S+1)$. We also clarify the role of ultrasoft-induced mixing operators in the unequal-mass theory: they are essential for the full velocity running below $ν=1$, but their Wilson coefficients vanish at the matching point and do not alter the fixed-order current anomalous dimensions reported here. The results agree with the known matching calculations and reduce smoothly to the equal-mass quarkonium limits.