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

Abstract

We work out a specific class of three-loop diagrams (of order $\alpha_s^2$) contributing to the decay amplitude for $b \to s \gamma$ associated with the current-current operators $O_1$ and $O_2$ at the physical value of the charm-quark mass $m_c$. For many of the considered diagrams we were able to solve the master integrals using differential equations in the canonical form. For some diagrams we did not find a transformation to canonical form and therefore calculated the corresponding master integral directly as an expansion around $z=m_c^2/m_b^2=0$, retaining power terms up to $z^{10}$ and keeping the accompanying $\log(z)$ terms to all powers. The results for the sum of all considered diagrams are given in tabular form, while contributions of individual diagrams (or combinations thereof) are given in electronic form.