Three-loop contributions to $b\to s\gamma$ associated with the current-current operators

Abstract

In a recent work, we calculated all three-loop diagrams contributing to the decay amplitude for $b \to s \gamma$ where none of the gluons touch the $b$-leg. In the present paper, we complete the calculation by working out all remaining three-loop diagrams (of order $\alpha_s^2$) associated with the current-current operators $O_1$ and $O_2$ at the physical value of the charm-quark mass $m_c$. Using the programs AMFlow and DiffExp to solve the differential equations for the master integrals, we obtained precise numerical results at 23 values for $z=m_c^2/m_b^2$, ranging from $z=1/1000$ to $z=1/5$, along with asymptotic expansions around $z=0$. For certain diagrams, the asymptotic expansion breaks down in the physical $z$-range, necessitating a Taylor expansion (which we do around $z=1/10$). In all expansions, we retained power terms up to $z^{20}$ and included the accompanying $\log(z)$ terms to all powers for asymptotic expansions. Numerical results for the sum of all diagrams (including those calculated in the previous paper) are presented in tabular form, while the mentioned expansions of individual diagram classes are provided electronically. We note that our results for the asymptotic expansions around $z=0$ are in good agreement with those recently published by Fael et al. and Czaja et al..