Three-loop matching coefficients for heavy flavor-changing currents and the phenomenological applications

Within the framework of non-relativistic QCD (NRQCD) factorization, we compute the matching coefficients between full Quantum Chromodynamics (QCD) and NRQCD for the heavy flavor-changing vector, axial-vector, scalar and pseudo-scalar currents up to next-to-next-to-next-to-leading order (N$^3$LO). We accomplish the analytical expressions for the three-loop renormalization constants and the corresponding anomalous dimensions for all of the four NRQCD currents with two different heavy flavors. The three-loop QCD corrections to the matching coefficients turn out to be significantly larger than lower order corrections. By employing the scale relation, we obtain the N$^3$LO corrections to the wave functions at the origin for the vector $B_c^*$ meson and the pseudo-scalar $B_c$ meson from the known result for the equal-mass heavy quarkonium in potential NRQCD (pNRQCD). We find large cancellations at the third order between the matching coefficients and the wave functions at the origin, and obtain the convergent decay constants of $B^*_{c}$ and $B_{c}$ up to N$^3$LO. We present the complete perturbative QCD predictions for the decay constants, leptonic decay widths, and branching ratios of the beauty-charmed mesons.