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

Abstract: 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 and well-defined for a large range of values 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 by using the so-called phase shift formalism. A $\varphi \varphi$ bound state is formed when the coupling $g$ is larger than a certain critical value. As the main outcomes of 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 is always continuous as a function of the coupling constant $g$ (no sudden jumps occurs when the bound state forms), and we show that the contribution of the bound state to the pressure does not count as \textit{one} state in the thermal gas, since a cancellation with the residual $\varphi \varphi$ interaction occurs. The amount of this cancellation depends on the details of the model and its 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 general non-negligible. We also show how the overall effect of the interaction, including eventual resonances and bound states, can be formally described by a unique expression that makes use of the phase shift continued below the threshold.