QFT treatment of a bound state in a thermal gas

Abstract: We investigate how to include bound states in a thermal gas in the context of quantum field theory (QFT). To this end, we use for definiteness a scalar QFT with a $\varphi^{4}$ interaction, where the field $\varphi$ represents a particle with mass $m$. A bound state of the $\varphi$-$\varphi$ type is created when the coupling constant is negative and its modulus is larger than a certain critical value. We investigate the contribution of this bound state to the pressure of the thermal gas of the system by using the $S$-matrix formalism involving the derivative of the phase-shift scattering. Our analysis, which is based on an unitarized one-loop resumed approach which renders the theory finite and well-defined for each value of the coupling constant, leads to following main results: (i) We generalize the phase-shift formula in order to take into account within a unique formal approach the two-particle interaction as well as the bound state (if existent). (ii) \textit{On the one hand}, the number density of the bound state in the system at a certain temperature $T$ is obtained by the standard thermal integral; this is the case for any binding energy, even if it is much smaller than the temperature of the thermal gas. (iii) \textit{On the other hand}, the contribution of the bound state to the total pressure is partly -- but not completely -- canceled by the two-particle interaction contribution to the pressure. (iv) The pressure as function of the coupling constant is \textit{continuous} also at the critical coupling for the bound state formation: the jump in pressure due to the sudden appearance of the bound state is exactly canceled by an analogous jump (but with opposite sign) of the interaction contribution to the pressure.