Virtual $O(\a_s)$ corrections to the inclusive decay $b \to s γ$

TL;DR

This work computes the O(α_s) virtual corrections to the inclusive decay b → s γ, explicitly including the two-loop O2 contributions and the one-loop corrections to O7 and O8. By employing Mellin-Barnes representations, the O2 two-loop diagrams are solved analytically as an expansion in m_c/m_b, and these results are combined with existing Bremsstrahlung corrections to yield the inclusive B → X_s γ rate. The analysis shows a dramatic reduction in the renormalization-scale dependence of the leading result, elevating the Standard Model prediction’s precision once the Wilson coefficients are updated to next-to-leading order. The study provides explicit expressions for the O2, O7, and O8 virtual corrections and outlines the remaining steps to achieve a complete NLL prediction. This work strengthens the SM baseline for B → X_s γ, tightening constraints on new physics through a more accurate theoretical framework.

Abstract

We present in detail the calculation of the $O(\a_s)$ virtual corrections to the matrix element for $b \to s \g$. Besides the one-loop virtual corrections of the electromagnetic and color dipole operators $O_7$ and $O_8$, we include the important two-loop contribution of the four-Fermi operator $O_2$. By applying the Mellin-Barnes representation to certain internal propagators, the result of the two-loop diagrams is obtained analytically as an expansion in $m_c/m_b$. These results are then combined with existing $O(\a_s)$ Bremsstrahlung corrections in order to obtain the inclusive rate for $B \to X_s \g$. The new contributions drastically reduce the large renormalization scale dependence of the leading logarithmic result. Thus a very precise Standard Model prediction for this inclusive process will become possible once also the corrections to the Wilson coefficients are available.

Figures (12)

• Figure 1: Diagrams 1a, 1b and 1c associated with the operator $O_2$. The fermions ($b$, $s$ and $c$ quark) are represented by solid lines. The wavy (dashed) line represents the photon (gluon).
• Figure 2: Diagrams 2a, 2b and 2c associated with the operator $O_2$.
• Figure 3: Diagrams 3a and 3b associated with the operator $O_2$. We calculate directly their sum and denote it by $M_2(3)$, see text.
• Figure 4: Diagrams 4a and 4b associated with the operator $O_2$. We calculate directly their sum and denote it by $M_2(4)$, see text.
• Figure 5: Building block $I_\beta$ for the diagrams in Figs. (\ref{['fig:1']}) and (\ref{['fig:2']}) with an off-shell gluon.