Thermodynamics from the S-matrix reloaded: emergent thermal mass

The formalism of Dashen, Ma and Bernstein (DMB) expresses the thermal partition function of a system in terms of the S-matrix operator, roughly $Z(\beta) \propto \int dE\, e^{-\beta E}\,\text{Tr}\,\ln S(E),$ where $S$ denotes the full scattering operator on the asymptotic Fock space -- i.e. including all multi-particle sectors -- defined via the Lippmann-Schwinger equation. Recently we have employed this formalism to compute the free energy of flux tubes (essentially a two-dimensional theory of derivatively coupled scalars) and the two-loop $O(\alpha_s)$ QCD thermal free energy. Moving to higher orders, it is well known that at $O(\alpha_s^2)$ in QCD, or e.g. at $O(\lambda^2)$ in $\lambda\phi^4$ theory, the free energy develops IR divergences. These IR divergences are resolved by the screening Debye mass. However, the DMB formalism expresses the free energy in terms of a trace of the S-matrix operator in the vacuum. How, then, does the Debye mass arise in this framework? In this work we address this question, thereby paving the way for higher-order applications of the DMB formalism in relativistic QFT.