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

TL;DR

This work completes the two-loop calculation of the jet function relevant for the partial rate of $\bar{B}\to X_s\gamma$ with a photon-energy cut, introducing $j(L,\mu)$ and its RG evolution. By reducing two-loop diagrams to four master integrals, employing Mellin-Barnes techniques, and performing rigorous renormalization, the authors extract the two-loop jet anomalous dimension $\gamma_1^J$ and the constant $b_0^{(2)}$, revealing substantial two-loop corrections to $j$ and quantifying their phenomenological impact. The results, together with the previously computed soft function, enable a full NNLO, RG-improved prediction and highlight the universality of the jet function across hard QCD processes. The findings lay the groundwork for precise phenomenology of inclusive radiative $B$ decays and jet-related observables in high-energy QCD.

Abstract

The complete two-loop expression for the jet function J(p^2,mu) of soft-collinear effective theory is presented, including non-logarithmic terms. Combined with our previous calculation of the soft function S(omega,mu), this result provides the basis for a calculation of the effect of a photon-energy cut in the measurement of the B-->X_s+gamma decay rate at next-to-next-to-leading order in renormalization-group improved perturbation theory. The jet function is also relevant to the resummation of Sudakov logarithms in other hard QCD processes.

Figures (2)

• Figure 1: Two-loop diagrams contributing to the jet function in QCD. Gluons emitted from the crossed circles originate from the Wilson lines. Not shown are additional diagrams resulting from mirror images in which the two external points are exchanged. The first diagram is the full fermion two-point function, not just the one-particle irreducible part.
• Figure 2: One- and two-loop predictions for the jet functions $j(L,\mu)$ and $\widetilde{j}(L,\mu)$ evaluated at $\alpha_s(\mu)=0.3$. The dashed lines show the one-loop 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.