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We study the thermal properties of quantum field theories (QFT) withthree-leg interaction vertices $g\varphi^{3}$ and $gS\varphi^{2}$ ($\varphi$and $S$ being scalar fields), which constitute the relativistic counterpart ofthe Yukawa potential. We follow a non-perturbative unitarized one-loop resummedtechnique for which the theory is unitary and well-defined for a large range ofvalues of the coupling constant $g$. Using the partial wave decomposition oftwo-body scattering we calculate the phase shifts, whose derivatives are usedto infer the pressure of the system at nonzero temperature by using theso-called phase shift formalism. A $\varphi \varphi$ bound state is formed whenthe coupling $g$ is larger than a certain critical value. As the main outcomesof this work, we estimate the influence of particle interaction on the pressure(both without and with the bound state), and we demonstrate that the latter isalways continuous as a function of the coupling constant $g$ (no sudden jumpsoccurs when the bound state forms), and we show that the contribution of thebound state to the pressure does not count as \textit{one} state in the thermalgas, since a cancellation with the residual $\varphi \varphi$ interactionoccurs. The amount of this cancellation depends on the details of the model andits parameters and 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 generalnon-negligible. We also show how the overall effect of the interaction,including eventual resonances and bound states, can be formally described by aunique expression that makes use of the phase shift continued below thethreshold.

Role of bound states and resonances in scalar QFT at nonzero temperature