(${\mathcal{O}}_8$, ${\mathcal{O}}_8$) contribution to $\overline{B}\to X_s\gamma \gamma$ at $O({\alpha }_s)$

In this analysis, we present the contribution associated with the chromomagnetic dipole operator ${\mathcal{O}}_8$ to the double differential decay width $d\mathrm{\Gamma }/(d{s}_{1}d{s}_2)$ for the inclusive process $\overline{B}\to X_s\gamma \gamma$. The kinematical variables $s_1$ and $s_2$ are defined as $s_i=(p_b-q_i{)}^2/m_b^2$, where $p_b$, $q_1$, $q_2$ are the momenta of $b$ quark and two photons. This contribution (taken at tree level) is of order ${\alpha }_s$, like the recently calculated QCD corrections to the contribution of the operator ${\mathcal{O}}_7$. In order to regulate possible collinear singularities of one of the photons with the strange quark, we introduce a nonzero mass $m_s$ for the strange quark. Our results are obtained for exact $m_s$, which we interpret as a constituent mass being varied between 400 and 600 MeV. Numerically it turns out that the effect of the (${\mathcal{O}}_8$, ${\mathcal{O}}_8$) contribution to the branching ratio of $\overline{B}\to X_s\gamma \gamma$ does not exceed $+0.1\%$ for any kinematically allowed value of our physical cutoff parameter $c$, confirming the expected suppression of this contribution relative to the QCD corrections to $d{\mathrm{\Gamma }}_{77}/(d{s}_{1}d{s}_2)$.

Figure 1

On the first line the diagrams defining the ${\mathcal{O}}_8$ contribution to $b\to sg\gamma \gamma$ are shown at the amplitude level. The crosses in the graphs stand for the possible emission places of the gluon (emerging from the operator ${\mathcal{O}}_8$). On the second line the contribution to the decay width corresponding to the interference of diagram 1 with diagram 4 is illustrated. This sample interference diagram gives rise to $\log (m_s/m_b)$ terms due to collinear configurations of one of the photons with the $s$ quark.

Figure 2

$d{\mathrm{\Gamma }}_{88}/(d{s}_{1}d{s}_2)$ [as given in Eqs. (5)–(6)] as a function of $s_1$ for $s_2$ fixed at 0.2, $\mu =m_b/2$ and $m_s$ varied between 400 and 600 MeV. The blue (top), yellow (middle) and red (bottom) lines show the width when choosing $m_s$ to be 400, 500 and 600 MeV, respectively.

Figure 3

Relative shift ($\frac{\mathrm{Br}[\overline{B}\to X_s\gamma \gamma {]}_c^{88}}{\mathrm{Br}[\overline{B}\to X_s\gamma \gamma {]}_c}$) of the branching ratio for $\overline{B}\to X_s\gamma \gamma$ (in percent) due to the (${\mathcal{O}}_8$, ${\mathcal{O}}_8$) contribution as a function of the cut parameter $c$ for $\mu =m_b/2$. The blue (top), yellow (middle) and red (bottom) lines show the relative shifts when setting $m_s=400\mathrm{MeV}$, 500 MeV and 600 MeV, respectively. For other choices of the scale $\mu$ the relative change is even smaller.

Figure 4

The shaded area shows the $(s_1,s_2)$ phase-space region for the case $c^2\ge x_4$.