Two-loop calculation of the anomalous dimensions of four-fermion operators with static heavy quarks

Abstract

We compute in the heavy-quark effective theory the two-loop anomalous dimension of the effective four-fermion operator O ΔB = 2 whose matrix element determines the B-parameter of the B-meson. Dimensional reduction is the regularization method chosen in this calculation, which is performed using the techniques already developed by us to treat the two-loop renormalization in standard dimensional regularization of the heavy-quark axial current. For N = 3, we find γ O ΔB = 2 (2) = − 1 36 [202+26ξ(2)−16n F ] , with ξ(2) = 1 6 π 2 . This anomalous dimension turns out to be equal to the one in standard dimensional regularization. We employ γ O ΔB = 2 (2) to derive the complete scheme-independent order α s correction to therelation between O ΔB = 2 in QCD and its counterpart O ΔB = 2 in the effective theory. For the heavy-quark axial current, we recalculate the corresponding analogous order- α s relation in dimensional reduction and the 't Hooft-Veltman dimensional regularization scheme. We show explicitly that this is renormalization-scheme independent, which completes the study carried out by us in our previous work. Moreover, these relations are used to obtain the renormalization constants for the lattice values of the B-meson decay constant, ƒ B, and B-parameter, B B, whose product enters the theoretical prediction of the B 0− B 0 mixing. As an extra result, the theoretic subtleties of dimensional reduction applied to the heavy-quark effective theory are exposed and resolved.