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

TL;DR

This paper calculates the three-loop QCD matching coefficient for the flavor-changing scalar current involving charm and bottom quarks, providing analytical expressions and precise numerical results crucial for understanding heavy meson formation.

Contribution

It presents the first analytical three-loop expressions for the NRQCD scalar current renormalization and anomalous dimension involving two heavy quarks, with highly accurate numerical coefficients.

Findings

Three-loop correction is significantly large.

Analytical expressions for renormalization constants are obtained.

Precise numerical coefficients with about thirty digits accuracy.

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.