Multi-parton contributions to $\bar B \to X_s γ$ at NLO

TL;DR

This work completes the perturbative NLO contributions to the inclusive decay $ar{B} ightarrow X_s \, ext{gamma}$ by computing the four-body $b o s \, q \, ar{q} \, ext{gamma}$ channel at NLO and including the accompanying five-body $b o s \, q \, ar{q} \, g \, ext{gamma}$ bremsstrahlung. The authors implement reverse unitarity, IBP reduction to seven integral topologies, and evaluate master integrals via differential equations, with analytic results expressed in terms of harmonic and Goncharov polylogarithms; photon-energy cuts are incorporated through a detailed phase-space treatment. A careful UV renormalisation and IR subtraction, including a mass-regularised treatment of collinear divergences, yields finite results and reveals collinear logarithms $ obreak \,igl[ obreak m_b^2/m_q^2igr]$ in the final expressions. Numerically, the multi-parton corrections are found to be small (per-mille to sub-percent level) due to cancellations with LO and NLO pieces, consistent across scale and scheme variations, thereby affirming the robustness of the SM prediction at this perturbative order. The results provide the last missing piece to formally complete $ar{B} ightarrow X_s \, ext{gamma}$ at NLO in QCD for leading power, with significant methodological advances in multi-particle phase-space integration and analytic master-integral evaluation.

Abstract

Many contributions to the decay rate of the inclusive radiative $\bar{B}\rightarrow X_s γ$ transition have been calculated to NNLO in QCD during the past decades. However, there are still a few unknown contributions from multi-parton final states which are formally NLO. In the present work, we compute those four-body $b \rightarrow s\, q\, \bar{q}\,γ$ contributions at NLO in QCD which need to be supplemented by the five-body $b \rightarrow s\, q\, \bar{q}\, g\,γ$ bremsstrahlung. This calculation formally completes the purely perturbative contributions to $\bar{B}\rightarrow X_s γ$ at NLO. Our results are obtained by applying modern techniques of integral reduction and evaluation of master integrals. In particular, the analytic integration over the four and five-particle phase space in the presence of a cut on the photon energy turns out to be technically involved. We give our results completely analytically in terms of multiple polylogarithms, including the dependence on the collinear logarithms which arise from the mass-regularisation of collinear divergences. The numerical impact of multi-parton corrections on the $\bar{B}\rightarrow X_s γ$ decay rate turns out to be small, owing to a partial cancellation between LO and NLO contributions.

Figures (11)

• Figure 1: Sample cut-diagrams contributing to LO and NLO four- and five-body contributions. In this work we will focus on contributions from panel (c).
• Figure 2: Sample diagrams of the building blocks from eq. \ref{['eq:Masterformula']}. Panel (a) depicts a contribution to $T_{ij}$, panel (b) to $V_{ij}$, (c) to $R_{ij}$ and (d) to $M_{ij}$. Photons can attach at all places possible and are not shown for clarity. For $T_{ij}$, $V_{ij}$, and $R_{ij}$ the symbols are understood to also contain the convolution of the diagrams without a photon with the LO subtraction kernel $S_0$, as shown in panel (e).
• Figure 3: Sample diagrams for different operator insertions. Cases (a) and (b) only allow for $q=u$, while penguin-penguin insertions allow for $q=u,d,s$ in the final state (cf. panel (c)). Penguin-penguin interferences with $q=s$ allow for an additional insertion (d).
• Figure 4: A sample diagram which fits in topology ${\cal{T}}_2$, together with a four (red) and five (blue) particle cut. As usual, wavy (curly) lines denote photons (gluons), whereas single (double) solid lines stand for massless (massive) quarks. The black squares denote operator insertions from the effective Hamiltonian.
• Figure 5: One-loop four-body master integrals. The dashed lines indicate the cut propagators, the solid light blue single (double) lines indicate massless (massive) propagators. A dot on a line indicates a squared propagator. The dotted red lines connecting lines with momenta $l_1$ and $l_2$ denote numerators $(l_1-l_2)^2-m_1^2-m_2^2$. The orange cross denotes the cut photon propagator with momentum $p_4$. Due to the energy cut, this line breaks the symmetry in the final-state momenta.