The authors calculate the rates and lepton ({ell}) invariant mass distributions for decays of the form 0{sup {minus}+} {r_arrow} {ell}{sup +}{ell}{sup {minus}}{gamma}, which are important radiative corrections to the purely leptonic decays 0{sup {minus}+} {r_arrow} {ell}{sup +}{ell}{sup {minus}}. The approach uses the loop diagrams which arise by including the two photon intermediate state and they retain the imaginary parts of the loops--a radiative extension of the unitarity bound for the process. These results are compared with those obtained using a model in which the meson couples directly to the leptons.
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