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

Abstract

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 $$\tilde{Z}_s(x,R_f)$$ Z ~ s ( x , R f ) and the corresponding anomalous dimension $$\tilde{\gamma }_s(x,R_f)$$ γ ~ 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 $$(C_{FF}(x_0), \ldots , C_{FBB}(x_0))$$ ( 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 $$C_s(\mu _f,\mu ,m_b,m_c)$$ 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 $$B_c$$ B c meson family.