Two-photon decays of hadronic molecules

In many calculations of the two-photon decay of hadronic molecules, the decay matrix element is estimated using the wave function at the origin prescription, in analogy to the two-photon decay of parapositronium. We question the applicability of this procedure to the two-photon decay of hadronic molecules for it introduces an uncontrolled model dependence into the calculation. As an alternative approach, we propose an explicit evaluation of the hadron loop. For shallow bound states, this can be done as an expansion in powers of the range of the molecule binding force $1/\beta$. In the leading order one gets the well-known pointlike limit answer. We estimate, in a self-consistent and gauge-invariant way, the leading range corrections for the two-photon decay width of weakly bound hadronic molecules emerging from kaon loops. We find them to be small, of order $\mathcal{O}(m\epsilon /{\beta }^2)$, where $m$ and $\epsilon$ denote the mass of the constituents and the binding energy, respectively. The role of possible short-ranged operators and of the width of the scalars remains to be investigated.