  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.
