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

Abstract

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 ${\tilde{Z}}_s(x,R_f)$ and the corresponding anomalous dimension ${\tilde{\gamma }}_s(x,R_f)$ for the NRQCD scalar current with the two heavy bottom and charm quark. We present the precise numerical results for those relevant coefficients $(C_{FF}(x_0),\cdots ,C_{FBB}(x_0))$ with an accuracy of about thirty digits. The three-loop QCD correction turns out to be significantly large. The obtained matching coefficient $C_s({\mu }_f,\mu ,m_b,m_c)$ is helpful to analyze the threshold behaviours when two different heavy quarks are close to each other and form the double heavy $B_c$ mesons.

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