Chromomagnetic dipole-operator corrections in $\overline{B}\to X_s\gamma$ at $\mathcal{O}({\beta }_0{\alpha }_s^2)$

We calculate the fermionic corrections to the photon-energy spectrum of $\overline{B}\to X_s\gamma$ which arise from the self-interference of the chromomagnetic dipole operator $Q_8$ at next-to-next-to-leading order by applying naive non-Abelianization. The resulting $\mathcal{O}({\beta }_0{\alpha }_s^2)$ correction to the $\overline{B}\to X_s\gamma$ branching ratio amounts to a relative shift of $+0.12\%$ ($+0.27\%$) for a photon-energy cut of 1.6 GeV (1.0 GeV). We also comment on the potential size of resummation and nonperturbative effects related to $Q_8$.

Figure 1

Four-particle cuts of the irreducible bottom-quark self-energy diagrams with a quark bubble contributing to the $b\to s\gamma q\overline{q}$ ($q=u,d,s$) transition at $\mathcal{O}({\alpha }_s^2)$. Left-right reflected diagrams are not shown. The second and third diagrams give rise to collinear logarithms $\ln (m_b^2/m_s^2)$.

Figure 2

Four-particle cuts of the irreducible bottom-quark self-energy diagrams with a light-quark bubble contributing to the $b\to s\gamma u\overline{u}$, $d\overline{d}$, $s\overline{s}$ transition at $\mathcal{O}({\alpha }_s^2)$. Symmetric diagrams are not shown. The shown diagrams give rise to collinear logarithms $\ln (m_b^2/m_{u,d,s}^2)$. In practice, these IR-sensitive terms are regulated by the light-meson masses $M_{\pi }$ and $M_K$.

Figure 3

Three-particle cuts of the irreducible bottom-quark self-energy diagrams with a massive charm and bottom-quark bubble contributing to the $b\to s\gamma g$ transition at $\mathcal{O}({\alpha }_s^2)$. Left-right reflected diagrams are not shown. The second and third diagrams give rise to collinear logarithms $\ln (m_b^2/m_s^2)$.

Figure 4

Left panel: Relative changes in $\mathcal{B}(\overline{B}\to X_s\gamma {)}_{\mathrm{SM}}^{E_{\gamma }>E_0}$ due to the $\mathcal{O}({\beta }_0{\alpha }_s^2)$ contributions to $(Q_8,Q_8)$. The solid (dashed) line shows the fixed-order (resummed) result as a function of the photon-energy cut $E_0$. Right panel: Comparison of perturbative and nonperturbative corrections related to the self-interference $(Q_8,Q_8)$. The solid line indicates the relative shift in $\mathcal{B}(\overline{B}\to X_s\gamma {)}_{\mathrm{SM}}^{E_{\gamma }>E_0}$ due to a resummation of collinear effects, while the dashed and dotted lines illustrate the impact of the nonperturbative components of the photon-fragmentation functions assuming two different models of vector-meson dominance. See text for further details.