Deciphering the Long-Distance Penguin Contribution to ${\overline{B}}_{d,s}\to \gamma \gamma$ Decays

We compute for the first time the long-distance penguin contribution to the double radiative $B$-meson decays by applying the perturbative factorization theorem. The numerically dominant penguin amplitude arises from the soft-gluon radiation off the light up-quark loop rather than the counterpart charm-loop effect. Importantly, the long-distance up-quark penguin contribution brings about the substantial cancellation of the known factorizable power correction, thus enabling $B_{d,s}\to \gamma \gamma$ to become new benchmark probes of physics beyond the standard model.

Figure 1

Diagrammatical representation of the soft-gluon emission from the factorizable quark loop in ${\overline{B}}_q\to \gamma \gamma$, where the symmetric diagram due to exchanging two on shell photons is not shown. The quark and photon fields marked with red color carry the anti-hard-collinear momenta, while the off shell quark in the fermion loop and the other photon marked with blue color correspond to the hard-collinear fields.

Figure 2

Theory predictions for the $CP$-averaged branching fractions of ${\overline{B}}_d\to \gamma \gamma$ [top] and ${\overline{B}}_s\to \gamma \gamma$ [down] obtained from including only the leading-power contributions [gray bands], from taking into account both the leading-power contributions and the various subleading-power corrections estimated in Ref. [3] [blue bands], and from adding further the newly determined long-distance penguin corrections [red bands].