Thermal role of bound states and resonances in scalar QFT

We study the thermal properties of quantum field theories (QFT) with three-leg interaction vertices $g\varphi^{3}$ and $gS\varphi^{2}$ ($\varphi$ and $S$ being scalar fields), which constitute the relativistic counterpart of the Yukawa potential. We follow a non-perturbative unitarized one-loop resummed technique for which the theory is unitary, finite, and well defined for each value of the coupling constant $g$. Using the partial wave decomposition of two-body scattering we calculate the phase shifts, whose derivatives are used to infer the pressure of the system at nonzero temperature. A $\varphi \varphi$ bound state is formed when coupling $g$ is greater than a certain critical value. As one of the main outcomes, we show that this bound state does not count as one state in the thermal gas, since a cancellation with the residual $\varphi \varphi$ interaction typically occurs. The amount of this cancellation depends on the details of the model and its parameters: a variety of possible scenarios is presented. Moreover, even when no bound state occurs, we estimate the role of the interaction (including a resonance in the $gS\varphi^{2}$ theory), which is in general non-negligible. We also show how the overall effect of interaction, including eventual resonances and bound states, can be formally described by a unique expression that makes use of the phase shift derivative below and above the threshold.