Self Consistent Thermal Resummation: A Case Study of the Phase Transition in 2HDM

TL;DR

The paper tackles the reliability of finite-temperature resummations for predicting FOEWPTs in the 2HDM by implementing the self-consistent Partial Dressing scheme and comparing it to AE and Parwani. It demonstrates that PD yields self-consistent thermal masses and a Boltzmann-suppressed decoupling of heavy states, removing symmetry non-restoration seen with AE and modifying GW forecasts substantially. The study maps the FOEWPT-favored regions under PD and contrasts them with those from the other schemes, showing meaningful differences in $T_c$, $T_n$, and the stochastic GW spectrum. It also discusses the experimental prospects for testing these regions via HL-LHC measurements of Higgs properties and di-Higgs signals, as well as forthcoming GW detectors, highlighting how resummation choices introduce notable theoretical uncertainties that must be addressed for robust phenomenology.

Abstract

An accurate description of the scalar potential at finite temperature is crucial for studying cosmological first-order phase transitions (FOPT) in the early Universe. At finite temperatures, a precise treatment of thermal resummations is essential, as bosonic fields encounter significant infrared issues that can compromise standard perturbative approaches. The Partial Dressing (or the tadpole resummation) method provides a self consistent resummation of higher order corrections, allowing the computation of thermal masses and the effective potential including the proper Boltzmann suppression factors and without relying on any high-temperature approximation. We systematically compare the Partial dressing resummation scheme results with the Parwani and Arnold Espinosa (AE) ones to investigate the thermal phase transition dynamics in the Two-Higgs-Doublet Model (2HDM). Our findings reveal that different resummation prescriptions can significantly alter the nature of the phase transition within the same region of parameter space, confirming the differences that have already been noticed between the Parwani and AE schemes. Notably, the more refined resummation prescription, the Partial Dressing scheme, does not support symmetry non-restoration in 2HDM at high temperatures observed using the AE prescription. Furthermore, we quantify the uncertainties in the stochastic gravitational wave (GW) spectrum from an FOPT due to variations in resummation methods, illustrating their role in shaping theoretical predictions for upcoming GW experiments. Finally, we discuss the capability of the High-Luminosity LHC and proposed GW experiments to probe the FOEWPT-favored region of the parameter space.

Figures (11)

• Figure 1: Left: Convergence of the thermal mass obtained from the gap equation as a function of the number of iterations. The thermal mass at each iteration is normalized to its final converged value. The lines represent only the convergent cases from a random sample of $h_1$, $h_2$, and $T$ values ranging from 0 to 300 GeV. Right: Convergent thermal mass points in the $h_1$-$h_2$ plane. Some points along the Higgs direction fail to converge.
• Figure 2: Thermal masses for each scalar dof obtained from the full gap equation after the iteration procedure. The dashed lines represent the high-temperature approximation for comparison.
• Figure 3: Thermal mass of the scalar dof as a function of the light Higgs field direction $h$, with all other scalar field directions set to zero. The three panels correspond to increasing temperatures. The solid lines are the thermal masses obtained by iterating the gap equation, while the dashed line is the truncated thermal mass in the high-temperature approximation. At $T=0$, the black dots indicate the physical masses at $v=246\mathop{\mathrm{\text{GeV}}}\nolimits$ for this benchmark. The wiggles in the plot are artifacts from numerical resolution, not real physical features.
• Figure 4: Phase transition behavior in the $m_H-m_A$ plane under the PD resummation scheme. The blue region indicates a second-order phase transition. The transition becomes first order in the yellow to red region with the strength $v_c/T_c$. The black region is excluded as the electroweak minima is metastable/unstable with the global minima at the origin. We remove the points above the dashed line of $\lambda_3 > 4\pi$ where perturbativity of the theory breaks down and below $\lambda_3 <-\sqrt{\lambda_1 \lambda_2}$ where the potential no longer remains bounded from below. The parameter scan covers 130 points in $(m_H, m_A)$, 30 values of $T$ for each $(m_H,m_A)$, 270 points in $(h_1, h_2)$ for each $(T,m_H,m_A)$, and $N_{\text{iter}}$, the number of iterations of the gap equation, varying between 12 and 40.
• Figure 5: The strength of the FOEWPT, measured by $v_n/T_n$, in the $m_H-m_A$ plane of the 2HDM. In the top panel, we show the results using the PD resummation scheme. The bottom panels present results for the AE (left) and Parwani (right) schemes. In the AE scheme, EWSNR appears as an effect of the resummation procedure rather than a negative thermal mass. Regions labeled "Vacuum trapped" indicate parameter space where the system does not tunnel to the EW minima.