Predictions for $B \to K γγ$ decays

We present a phenomenological study of the rare double radiative decay $B\to K γγ$ in the Standard Model (SM) and beyond. Using the operator product expansion (OPE) technique, 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/Q, where $Q$ is of order $m_b$, the $B\to K γγ$ 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 $Δ{\cal{B}}(B \to K γγ)_{SM}^{OPE} \simeq 1 \times 10^{-9}$. We work out the di-photon invariant mass distribution with and without the resonant background through $B\to K \{η_c,χ_{c0}\}\to Kγγ$. In the SM, the resonance contribution is dominant in the region of phase space where the OPE is valid. The present experimental upper limit on $B_s \to τ^+ τ^-$ decays, which constrains the scalar/pseudoscalar Four-Fermi operators with $τ^+ τ^-$, leaves considerable room for new physics in the one-particle-irreducible contribution to $B\to K γγ$ decays. In this case, we find that the SD $B\to K γγ$ 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.

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