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. Using the potential matching coefficients at the boundary $\nu=1$, we determine the current ultraviolet counterterms directly in the effective theory. They provide the ${\cal O}(\alpha_s^2)$ ordinary factorization-scale RGE required to cancel the residual infrared poles of the full-QCD short-distance coefficients. 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 distinct role of the velocity RGE: the ultrasoft-induced mixing coefficients vanish at $\nu=1$ and become essential for the full velocity running below the matching boundary. The results agree with the known matching calculations and reduce smoothly to the equal-mass quarkonium limits. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.