Phenomenological study of the double radiative decay $B ->K\gamma\gamma$

Using the operator product expansion (OPE) technique, we study the rare double radiative decay $B\to K \gamma\gamma$ in the Standard Model (SM) and beyond. We estimate the short-distance (SD) contribution to the decay amplitude in a region of the phase space which is around the point where all decay products have energy $\sim m_b/3$ in the rest frame of the $B$-meson. At lowest order in $1/m_b$, the $B\to K \gamma\gamma$ matrix element is then expressed in terms of the usual $B\to K$ form factors known from semileptonic rare decays. The integrated SD branching ratio in the SM in the OPE region turns out to be $\Delta {\cal{B}}(B \to K \gamma \gamma)_{SM}^{OPE} \simeq 1 \times 10^{-9}$. We work out the di-photon invariant mass distribution with and without the resonant background through $B\to K \{\eta_c,\chi_{c0}\}\to K\gamma \gamma$. In the SM, the resonance contribution is dominant in the region of phase space where the OPE is valid. On the other hand, the present experimental upper limit on $B_s \to \tau^+ \tau^-$ decays, leaves considerable room for New Physics (NP) in the one-particle-irreducible contribution to $B\to K \gamma \gamma$ decays. In this case, we find that the SD $B\to K \gamma \gamma$ branching ratio can be enhanced by one order of magnitude with respect to its SM value and the SD contribution can lie outside of the resonance peaks.