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 $$\tilde{Z}_s(x,R_f)$$ 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.

Decay constants of $$ c\overline{b} $$ mesons involving the ten heavy flavor-changing currents at N3LO QCD

“…[108,109]. Two-loop corrections to the matching coefficients for the heavy flavor-changing pseudo-scalar, spatial vector and temporal axial-vector currents are available in the literature [55,56,105,106,110,111,115]. At the N 3 LO of α s , the matching coefficients have been numerically computed for the seven heavy flavor-changing currents: scalar [56,115], pseudo-scalar [56,115], spatial vector [56,106,115], temporal axial-vector [56,105,115], spatial axial-vector [56], spatial-temporal tensor [112] and spatial-spatial axial-tensor [112].…”

“…In this section, we will present the details of our calculations for the hard QCD vertex functions Γ J with the heavy flavor-changing temporal-vector (v, 0), spatial-spatial tensor (t, ij) and spatial-temporal axial-tensor (t5, i0) currents, which are coupled to the P -wave c b mesons B * c0 and B c1 . And the calculation details for the hard QCD vertex functions with the other seven currents can be found in our previous research [56,112,115].…”

“…With the aid of numerical fitting techniques such as the PSLQ algorithm [182], we have individually reconstructed the analytic expressions of Z J for all ten heavy flavor-changing currents, where Z J for J ∈ {(v, 0), (t, ij), (t5, i0)} are new while Z J for the remaining seven currents are already known in previous literature [56,105,106,112,115]. Our results verify Z s ≡ Z v,0 and the other two relations for Z J as described in eq.…”

“…where n l is the number of massless flavors. The corresponding NRQCD heavy flavor-changing current anomalous dimension γJ is related to Z J by [56,112,115,146,[183][184][185][186][187]]…”

“…In QCD we consider n l massless quarks, n b bottom quarks with mass m b and n c charm quarks with mass m c possibly appearing in the quark loop, while the heavy charm and bottom JHEP06(2024)012 quarks are decoupled in the NRQCD. For matching between QCD and NRQCD with the same perturbative expansion parameter α s , we employ both the coupling running [56,115,188] and the decoupling relation [56,115,142,187,[189][190][191][192][193][194][195]…”