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Leading logarithms of the $b$ quark mass in inclusive $B\to X_sγ$ decay

Part of the order $α_s$ correction to the inclusive $B\to X_s\,γ$ photon spectrum is enhanced by $\log(m_b^2/m_s^2)$. We discuss its origin and sum the corrections proportional to $[α_s\log(m_b^2/μ^2)]^n$ to all orders. These are the calculable leading logarithms in the parton fragmentation functions of a quark or gluon into a photon. Although the gluon fragmentation into a photon starts only at order $α_s^2$, its contribution is of the same order as the $s$ quark's in the leading log sum. For not too small values of the photon energy, the resummation yields a moderate suppression. In the standard model, the coefficient of the operator whose matrix element gives rise to such terms is small. A measurement of the photon spectrum around $1\,$GeV would provide a theoretically clean determination of $C_8$, the Wilson coefficient of the $b\to s\,g$ operator.