Electromagnetic dipole operator effect onB¯→XsγatO(αs2)

The flavor-changing electromagnetic dipole operator $O_7$ gives the dominant contribution to the $\overline{B}\to X_s\gamma$ decay rate. We calculate two-loop QCD corrections to its matrix element together with the corresponding bremsstrahlung contributions. The optical theorem is applied, and the relevant imaginary parts of three-loop diagrams are computed following the lines of our recent $t\to X_{b}W$ calculation. The complete result allows us to test the validity of the naive non-Abelianization (NNA) approximation that has been previously applied to estimate the next-to-next-to-leading order QCD correction to $\Gamma (\overline{B}\to X_s\gamma )/\Gamma (\overline{B}\to X_{u}e\overline{\nu })$. When both decay widths are normalized to $m_{b,R}^5$ in the same renormalization scheme $R$, the calculated $\mathcal{O}({\alpha }_s^2)$ correction is sizable ($\sim 6\%$), and the NNA estimate is about $1/3$ too large. On the other hand, when the ratio of the decay widths is written as $S\times m_{b,\overline{\mathrm{MS}}}^2(m_b)/m_{b,\mathrm{pole}}^2$, the calculated $\mathcal{O}({\alpha }_s^2)$ correction to $S$ is at the level of $1\%$ for both the complete and the NNA results.