Toward a NNLO calculation of the (B)over-bar → Xsγ decay rate with a cut on photon energy:: I.: Two-loop result for the soft function

A theoretical analysis of the partial inclusive (B) over bar -> X(S)gamma decay rate with a cut E gamma >= E-0 on photon energy must deal with short-distance contributions associated with three different mass scales: the hard scale root m(b)Delta, ail intermediate scale and a soft scale Delta, where Delta = m(b) - 2E(0) approximate to 1 GeV for E-0 approximate to 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. (c) 2006 Elsevier B.V. All rights reserved.