Three-loop QCD matching of the flavor-changing scalar current involving the heavy charm and bottom quark

We compute the matching coefficient between the quantum chromodynamics (QCD) and the non-relativistic QCD (NRQCD) for the flavor-changing scalar current involving the heavy charm and bottom quark, up to the three-loop order within the NRQCD factorization. For the first time, we obtain the analytical expressions for the three-loop renormalization constant Z̃_s(x,R_f) and the corresponding anomalous dimension γ̃_s(x,R_f) for the NRQCD scalar current with the two heavy bottom and charm quark. We present the precise numerical results for those relevant coefficients (C_FF(x_0), … , C_FBB(x_0)) with an accuracy of about thirty digits. The three-loop QCD correction turns out to be significantly large. The obtained matching coefficient C_s(μ _f,μ ,m_b,m_c) is helpful to analyze the threshold behaviours when two different heavy quarks are close to each other and form the double heavy B_c meson family.