Three-loop matching of the dipole operators for b -> s gamma and b -> s g

TL;DR

This work delivers the three-loop matching conditions for the dipole operators governing b -> s gamma and b -> s g, completing the first step toward NNLO QCD corrections for the inclusive B -> Xs gamma rate. Using an EFT where heavy W bosons and the top quark are integrated out, the authors compute off-shell SM 1PI amplitudes up to three loops, perform renormalization with carefully chosen schemes, and extract the finite Wilson coefficients C7 and C8 at O(alpha_s^3) for charm and top sectors. They provide analytic expansions and precise numerical fits across mass-ratio regimes, demonstrating that the NNLO matching contribution is modest (~2%) relative to the total uncertainty, with the remaining steps in progress. The methodology—off-shell matching with multi-scale expansions and cross-checked via dual computational approaches—offers a robust template for complex three-loop matching problems in particle phenomenology.

Abstract

We evaluate the three-loop matching conditions for the dimension-five operators that are relevant for the b -> s gamma decay. Our calculation completes the first out of three steps (matching, mixing and matrix elements) that are necessary for finding the next-to-next-to-leading QCD corrections to this process. All such corrections must be calculated in view of the ongoing accurate measurements of the B -> Xs gamma branching ratio.

Figures (5)

• Figure 1: One of the ${\cal O}(10^3)$ three-loop diagrams that we have calculated.
• Figure 2: One-loop 1PI diagrams for $b \to s \gamma$ in the SM. There is no $W^{\pm}\phi^{\mp}\gamma$ coupling in the background field gauge.
• Figure 3: The coefficients $C_7^{Q(n)}(\mu_0)$ as functions of $y = M_W/m_t(\mu_0)$. The (blue) dot-dashed lines correspond to their expansions in $y$ up to $y^8$. The (red) dashed lines describe the expansions in $(1-y^2)$ up to $(1-y^2)^8$. The (black) solid lines in the one- and two-loop cases correspond to the known exact expressions. The (yellow) vertical strips indicate the experimental range for $y$.
• Figure 4: Same as Fig. \ref{['fig:c7']} but for $C_8^{Q(n)}(\mu_0)$.
• Figure 5: The three-loop top-sector coefficients. The solid lines represent the highest orders we know (as in Figs. \ref{['fig:c7']} and \ref{['fig:c8']}). The dashed and dot-dashed lines show the lower orders.