Toward a NNLO calculation of the B-->Xₛ gamma decay rate with a cut on photon energy: I. Two-loop result for the soft function

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A theoretical analysis of the partial inclusive B-->Xₛ gamma decay rate with a cut Egamma>E₀ on photon energy must deal with short-distance contributions associated with three different mass scales: the hard scale mb, an intermediate scale √mb Delta, and a soft scale Delta, where Delta=mb-2E₀=1GeV for E₀=1.8GeV. 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.