The double radiative annihilation of the heavy-light fermion bound states

Abstract

We consider the double-radiative decays of heavy-light QED and QCD atoms, $\mu^+ e^- \to \gamma\gamma$ and $\bar{B}^{0}_s \to \gamma\gamma$. Especially, we take under scrutiny contributions coming from operators that vanish on the free-quark mass shell. We show that by field redefinitions these operators are converted into contact terms attached to the bound state dynamics. A net off-shell contribution is suppressed with respect to the effect of the well known flavour-changing magnetic-moment operator by the bound-state binding factor. The negligible off-shellness of the weakly bound QED atoms becomes more relevant for strongly bound QCD atoms. We analyze this off-shellness in model-approaches to QCD, one of them enabling us to keep close contact to the related effect in QED. We also comment on the off-shell effect in the corresponding process $\bar{B}_d \to K^* \gamma$, and discuss possible hindering of the claimed beyond-standard-model discovery in this decay mode.