Scale dependence improvement of the quartic scalar field thermal effective potential in the optimized perturbation theory

TL;DR

The paper tackles the persistent renormalization-scale dependence in finite-temperature perturbation theory by introducing the Variational Renormalization Group (VRG), a hybrid approach that combines Renormalization Group Improvement (RGI) with Optimized Perturbation Theory (OPT). By applying VRG to the massive lambda-phi^4 theory, the authors show substantially improved scale stability for the finite-temperature effective potential, pressure, and critical temperature in both symmetric and broken phases. The method preserves the universality class of the model (second-order transition in the broken phase) and demonstrates convergence between perturbative orders, outperforming OPT alone and offering competitive comparisons with other nonperturbative techniques. The work suggests that VRG can be a robust tool for precise thermal physics in cosmology and condensed matter, with potential extensions to other resummation schemes such as HTLpt.

Abstract

Perturbation theory, as well as most thermal field resummation methods widely used to study finite-temperature quantum field theories, presents a non-negligible renormalization scale dependence. To address this limitation, we propose an alternative method that combines the Renormalization Group Improvement (RGI) prescription for the thermal effective potential with the Optimized Perturbation Theory (OPT) variational resummation technique. Here, we apply this new framework, termed Variational Renormalization Group (VRG), to evaluate the effective potential of the scalar $λφ^4$ theory at finite temperatures, which represents a benchmark model for phase transition studies. We show that the proposed approach significantly improves scale stability, compared to the use of OPT alone, across key thermodynamic quantities, including the effective potential, critical temperature, and pressure. These results establish the VRG as a robust alternative tool for precision studies of thermal phase transitions, with direct implications for cosmological applications (e.g., early-universe thermodynamics) and condensed matter systems.

Figures (8)

• Figure 1: Feynman diagrams contributing to the effective potential for the $\lambda \phi^4$ model in the OPT case at NLO. The black circle represents a quartic vertex proportional to $\delta \lambda$, a black square is a (new) quadratic vertex proportional to $\delta \eta^2$ while the crossed circle represents renormalization counterterms. The external lines represent the background scalar field $\varphi$ in the effective potential.
• Figure 2: The pressure subtracted by the constant vacuum term, $\Delta P = P-P_{\rm vacuum}$, normalized by the ideal gas result (panel a) and the optimal mass parameter $\overline \eta$ (panel b) as functions of $T/m_0$ for the OPT and VRG methods at orders $\delta$ and $\delta^2$. In both cases, the coupling value is fixed at the representative value $\lambda_0 = 12.25$ and the scale dependence range is given by $\pi T \le \mu \le 4\pi T$. In the OPT up to $\delta$ and $\delta^2$ order, and VRG up to $\delta^2$, the upper curve in the bands corresponds to the value $\mu=4\pi T$, while the lower curve in the bands corresponds to $\mu=\pi T$. Otherwise, the VRG up to $\delta$ order gets inverted. This pattern is repeated in the other figures.
• Figure 3: Similar to Fig. \ref{['fig2']} but showing the normalized pressure (panel a) and the optimal mass parameter $\overline \eta$ (panel b) as a function of the renormalized coupling, with the temperature fixed at $T= 20 m_0$.
• Figure 4: The pressure subtracted by the constant vacuum term, $\Delta P = P-P_{\rm vacuum}$, normalized by the ideal gas result, as a function of the coupling, for the (symmetric) massless case. In panel (a), the OPT, VRG, and RGOPT results are compared for $\pi T \le \mu \le 4\pi T$. In panel (b), the OPT, VRG, FRG, and 2PI predictions are compared for the central scale, $\mu = 2\pi T$. The renormalized coupling is taken at the central value for the scale, $\lambda\equiv\lambda(2\pi T)$ and the temperature is fixed at $T = \mu_0$.
• Figure 5: Subtracted effective potential, $\Delta V = V_{\rm eff}(\varphi,T) - V_{\rm eff}(\varphi=0,T)$, in units of mass as a function of the field for fixed coupling $\lambda_0 =12.25$ and temperature ($T= 20 m_0$). The scale dependence range is given by $\pi T \le \mu \le 4\pi T$. The figure shows the results obtained with the OPT and VRG at perturbative orders $\delta$ and $\delta^2$.