Decay constants of $c\bar{b}$ mesons involving the ten heavy flavor-changing currents at N${}^3$LO QCD

Abstract

Within the nonrelativistic QCD (NRQCD) framework, we complete the three-loop calculations of the NRQCD renormalization constants and the matching coefficients, for the heavy flavor-changing temporal vector, spatial-spatial tensor, spatial-temporal axial-tensor currents, which are coupled to the $P$-wave $c\bar{b}$ mesons. We further study the ten decay constants for the $c\bar{b}$ mesons ($B_c,B_c^{\ast },B_{c0}^{\ast },B_{c1}$) coupled with the ten heavy flavor-changing currents involving (pseudo-)scalar, (axial-)vector and (axial-)tensor up to the next-to-next-to-next-to-leading order (N${}^3$LO) of ${\alpha }_s$. We obtain the six ratios of decay constants by approximating them to the corresponding ratios of matching coefficients. We find the N${}^3$LO QCD corrections to the six ratios of decay constants have good convergence and weak scale-dependence. We finally predict the hierarchical relationship among the ten decay constants for the $c\bar{b}$ mesons coupled with the ten currents.