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 ($B_c,B_c^,B_{c0}^,B_{c1}$) 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.