  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.
