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

TL;DR

<3-5 sentence high-level summary> The paper tackles achieving NNLO precision for the partial inclusive $\bar{B} \to X_s \gamma$ decay rate with a photon-energy cut by evaluating the soft function that encodes cut-related effects. It develops a two-loop calculation by rewriting the delta-operator as a discontinuity of a light-cone propagator, mapping to on-shell heavy-quark/light-cone integrals, and reducing to four master integrals. The authors obtain the bare soft function, perform renormalization with a convolutional $Z$, extract the two-loop non-logarithmic coefficient $c_0^{(2)}$, and verify compatibility with known cusp and non-cusp anomalous dimensions. They find sizable two-loop corrections in the soft function, though physical predictions are stabilized by mass-scheme changes and the soft function’s universality implies relevance for end-point spectra of other inclusive heavy-to-light decays.

Abstract

A theoretical analysis of the partial inclusive B-->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 m_b, an intermediate scale \sqrt{m_b Delta}, and a soft scale Delta, where Delta=m_b-2E_0=1GeV for E_0=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.

Figures (2)

• Figure 1: Two-loop graphs contributing to the soft function. Double lines denote heavy-quark propagators, while crosses denote possible insertions of the operator $(\omega+in\cdot D+i0)^{-1}$.
• Figure 2: One- and two-loop corrections to the soft functions $s(L,\mu)$ and $\widetilde{s}(L,\mu)$ evaluated at $\alpha_s(\mu)=0.45$. The dashed lines show the one-loop results results, while the solid lines give the complete two-loop results derived in the present work. The gray lines are obtained if only the $\beta_0\alpha_s^2$ terms are kept in the two-loop contributions.