$\mathcal{PT}$-symmetric Field Theories at Finite Temperature

Abstract: We investigate the thermal properties of $\mathcal{PT}$-symmetric scalar field theories with purely imaginary couplings. The free energy governs the asymptotic density of states, providing an effective measure of the number of degrees of freedom, while thermal masses and one-point functions provide predictions for operator dimensions and three-point functions in the corresponding $d=2$ Conformal Field Theories. Naive finite-temperature perturbation theory near upper critical dimensions is spoiled by infrared divergences. To remove these divergences, we introduce a ''thermal normal-ordering'' scheme that resums these contributions and yields a systematic $ε$-expansion. This framework allows us to compute the free energy, thermal masses, and one-point functions in the cubic and quintic $O(N)$ models. We compare the thermal free energy density, thermal masses, and one-point function in two dimensions with exact results derived from the proposed Ginzburg-Landau descriptions of the non-unitary minimal models $M(2,5)$ and $M(3,8)_D$. Eventually, we employ two-sided Padé extrapolations to obtain estimates for the thermal free energy in $d=3,4,5$.