Toward a NNLO calculation of the [formula omitted] decay rate with a cut on photon energy: I. Two-loop result for the soft function

Abstract

A theoretical analysis of the partial inclusive B¯→Xsγ decay rate with a cut Eγ⩾E0 on photon energy must deal with short-distance contributions associated with three different mass scales: the hard scale mb, an intermediate scale mbΔ, and a soft scale Δ, where Δ=mb−2E0≈1 GeV for E0≈1.8 GeV. The cut-dependent effects are described in terms of two perturbative objects called the jet function and the soft function, which for a next-to-next-to-leading order analysis of the decay rate are required with two-loop accuracy. The two-loop calculation of the soft function is presented here, while that of the jet function will be described in a subsequent paper. As a by-product, we rederive the two-loop anomalous-dimension kernel of the B-meson shape-function.