Calculation of Matsubara-sector mass gaps at weak coupling

Derive a detailed weak-coupling calculation establishing the behavior of the mass gaps m_1-2\pi T and m_2-4\pi T in the first two nonzero Matsubara sectors of the finite-temperature theory, which are conjectured to be parametrically of order gT.

Background

The feasibility of the lattice method depends on the exponential decay of the Euclidean correlator outpacing the exponentially growing spatial kernel. This requires the lightest masses in the relevant Matsubara sectors to exceed the corresponding Matsubara frequencies by a substantial amount. The authors state that, at weak coupling, the differences for the n=1 and n=2 sectors are presumably of order gT, but explicitly note that no detailed calculation supporting this expectation is known to them.