The photon energy spectrum in $B\to X_s\gamma$ at N$^3$LL$'$

TL;DR

This paper provides high-precision predictions for the photon energy spectrum in inclusive B meson decays to strange quarks and photons, using advanced resummation techniques and addressing theoretical uncertainties at N$^3$LL$'$.

Contribution

It introduces a comprehensive N$^3$LL$'$ calculation for the spectrum, including a novel method to estimate unknown higher-order corrections and an analysis of the impact of different b-quark mass schemes.

Findings

Stable results with good convergence using MSR mass scheme.

Development of a method to parametrize unknown $ ext{O}(\alpha_s^3)$ corrections.

Identification of the limitations of the 1$S$ mass scheme at 3-loop order.

Abstract

We present predictions for the photon energy spectrum in inclusive $B\to X_s\gamma$ decays mediated by the electromagnetic penguin operator $O_7$ to N$^3$LL$'$. We use soft-collinear effective theory (SCET) to resum the singular contributions in the peak region at large photon energy. In the tail region the resummed predictions are matched to fixed order at N$^3$LO, where we include the known fixed-order contributions for $O_7$ up to $\mathcal{O}(\alpha_s^2)$. We develop a method to suitably parametrize the still unknown $\mathcal{O}(\alpha_s^3)$ nonsingular corrections in terms of theory nuisance parameters, whose variations provide an estimate of the associated theory uncertainty. In this context, we also study different ways to treat higher-order cross terms in the matching. Another important aspect of our analysis is the short-distance scheme used for the $b$-quark mass $m_b$. We find that in the present context, the 1$S$ mass scheme, which was previously used up to 2-loop order, fails to work at 3-loop order, because the mass scheme enters at a soft scale much smaller than $m_b$ here, for which the 1$S$ scheme was not devised. Using instead the MSR mass scheme with $R\sim 1\,\mathrm{GeV}$, we obtain stable results with good perturbative convergence up to N$^3$LL$'$.