Next-To-Leading-Order Matching for the Magnetic Photon-Penguin Operator in the $B \to X_s γ$ Decay

The initial condition at the matching scale $μ_W = O(M_W)$ for the Wilson coefficient of the magnetic photon-penguin operator in the decay $B\to X_s γ$ is calculated in the next-to-leading-order approximation. The technical details of the necessary two-loop calculation in the full theory are described and the matching with the corresponding result in the effective theory is discussed in detail. Our outcome for the initial condition confirms the final results of Adel and Yao and Greub and Hurth. We show that --- contrary to the claims in the second of these papers --- the matching procedure can be properly performed for infrared divergent amplitudes, i.e. independently of contributions from gluon bremsstrahlung.

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