  We investigate the two-loop gap equation for the thermal mass of hot massless
$g^2\phi^4$ theory and find that the gap equation itself has a non-zero finite
imaginary part. This indicates that it is not possible to find the real thermal
mass as a solution of the gap equation beyond $g^2$ order in perturbation
theory. We have solved the gap equation and obtain the real and the imaginary
part of the thermal mass which are correct up to $g^4$ order in perturbation
theory.
