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

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\('\).