A simple criterion for effects beyond hard thermal loops

Thermal amplitudes with ultrasoft momenta, which are not accessible by standard methods of perturbation theory, have recently attracted a lot of interest. However, the comparison of external momenta with the ultrasoft scale $g^2T\ln(1/g)$ is a too crude criterion, since amplitudes with hard external momenta can also be non perturbative, if these momenta are close enough to the light-cone. In this letter, I give a more refined criterion to decide if an amplitude is non perturbative, that applies to all situations. In physical terms, this condition states that non-perturbative effects appear if the particles running in loops have to travel distances larger than their mean free path.