Four-body contributions to B -> Xs gamma at NLO

TL;DR

The paper computes the NLO four-body contributions to $\bar{B}\to X_s\gamma$ arising from $b\to s\bar{q} q\gamma$, providing a complete treatment of the four-body phase space, renormalization, and collinear logarithms. By exploiting operator identities and diagrammatic symmetries, the calculation reduces to a manageable set of insertions, with phase-space integration performed via Cutkosky rules and a splitting-function approach to regulate collinear divergences. The resulting interference matrices $G_{ij}^{(0)}(\delta)$ and $G_{ij}^{(1)}(\mu,\delta)$ yield a four-body contribution that is positive and at most about 1% of the SM rate for realistic parameters and photon-energy cuts. The analysis confirms LO four-body results, highlights the importance of collinear logs, and reinforces current perturbative uncertainty estimates, while noting a few remaining three-body NLO pieces for future work. Overall, the work tightens the SM prediction for $\bar{B}\to X_s\gamma$ at the percent level and strengthens the reliability of theoretical error estimates used in precision flavor tests.

Abstract

Ongoing efforts to reduce the perturbative uncertainty in the B -> Xs gamma decay rate have resulted in a theory estimate to NNLO in QCD. However, a few contributions from multi-parton final states which are formally NLO are still unknown. These are parametrically small and included in the estimated error from higher order corrections, but must be computed if one is to claim complete knowledge of the B -> Xs gamma rate to NLO. A major part of these unknown pieces are four-body contributions corresponding to the partonic process b -> s qbar q gamma. We compute these NLO four-body contributions to B -> Xs gamma, and confirm the corresponding tree-level leading-order results. While the NLO contributions arise from tree-level and one-loop Feynman diagrams, the four-body phase-space integrations make the computation non-trivial. The decay rate contains collinear logarithms arising from the mass regularization of collinear divergences. We perform an exhaustive numerical analysis, and find that these contributions are positive and amount to no more than 1% of the total rate in the Standard Model, thus confirming previous estimates of the perturbative uncertainty.

Figures (8)

• Figure 1: Sample cut-diagrams contributing to LO and NLO four-body matrix elements. In our calculation we include all contributions but those from panel (f). See the text for details.
• Figure 2: Diagrams of type (i). Crosses denote alternative insertions of the photon vertex (always one vertex at each side of the cut). Circle-cross denotes the alternative insertion of the gluon vertex.
• Figure 3: Diagrams of type (ii). Crosses denote alternative insertions of the photon vertex (always one vertex at each side of the cut). Circle-cross denotes the alternative insertion of the gluon vertex.
• Figure 4: Diagrams of type (iii). Crosses denote alternative insertions of the photon vertex (always one vertex at each side of the cut.
• Figure 5: Tree-level counterterm diagrams. Crosses denote alternative insertions of the photon vertex (always one vertex at each side of the cut). These diagrams can be classified in types (i), (ii), (iii) as done for the loop diagrams.