The photon-energy spectrum in $B\to X_sγ$ to N$^3$LO: light-fermion and large-$N_{\rm c}$ corrections

Abstract

We calculate the photon-energy spectrum of the inclusive radiative decay $B\to X_sγ$, induced by the electromagnetic dipole operator $O_7$, to next-to-next-to-next-to-leading order and consider the complete corrections for light fermions, for the contributions with two closed massive fermion loops, and for the limit of large QCD colour factors $N_{\rm c}$ in the remaining part. We discuss the total decay rate both without and with a cut on the photon energy. In addition to the on-shell renormalization of the bottom-quark mass, we also consider the kinetic mass and the MSR mass schemes. The latter two lead to an improved perturbative behaviour of the decay rate.

Figures (6)

• Figure 1: Sample Feynman diagrams contributing to the forward scattering amplitude of $b \gamma \to b \gamma$. Their imaginary parts provide the real emission contribution to the photon energy spectrum in $B \to X_s \gamma$. Thin (bold) lines denote the strange (bottom) quarks.
• Figure 2: Normalized spectrum as a function of $x=2E_\gamma/m_b$ in the on-shell scheme. For this plot a fixed value $\alpha_s=0.22$ has been chosen.
• Figure 3: The total partonic decay rate in the kinetic mass scheme as a function of the renormalization scale $\mu$.
• Figure 4: The total partonic decay rate with $m_b$ in the MSR scheme as a function of the renormalization scale $\mu$.
• Figure 5: The $O_7$ contribution to the rate $\Gamma$ as a function of on the lower cut on the photon energy $E_\mathrm{cut}$ in the kinetic scheme (top) and the MSR scheme (bottom).