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

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${}^{\prime }$. 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${}^{\prime }$.

12 figures, 35 pages + appendix

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Cited by in corpus (2)

• Robust estimates of theoretical uncertainties at fixed-order in perturbation theory
• The Q_{1,2}-Q_7 interference contributions to b -> s gamma at O(alpha_s^2) for the physical value of m_c