The soft-energy region in the radiative decay of bound states

The orthopositronium decay to three photons is studied in the phase-space region where one of the photons has an energy comparable to the relative three-momentum of the e+e− system (ω∼mα). The NRQED computation in this regime shows that the dominant contribution arises from distances ∼1/mω, which allows to treat the Coulomb interaction perturbatively. The small-photon energy expansion of the 1-loop decay spectrum from full QED yields the same result as the effective theory. By doing the threshold expansion of the 1-loop QED amplitude we confirm that the leading term arises from a loop-momentum region where q0∼q2/m∼ω. This corresponds to a new non-relativistic loop-momentum region, which has to be taken into account for the description of a non-relativistic particle-antiparticle system that decays through soft photon emission.