We introduce a new framework for perturbatively computing equilibrium thermodynamic properties of cosmological phase transitions to high loop orders, using the full four-dimensional resummed thermal effective potential and avoiding the limitations of standard high-temperature approximations. By systematically disentangling the physics of hard and soft momentum scales, our approach unifies their treatment within a single expression, enabling consistent handling of both vacuum and thermal divergences across all mass regimes. This core innovation enables the efficient numerical evaluation of massive multiloop thermal sum-integrals, achieved through a finite-temperature generalization of Loop-Tree Duality — an advanced algorithmic technique originally developed to render vacuum Feynman integrals numerically tractable via Monte Carlo methods. As a proof of principle, we apply the framework to a scalar-Yukawa model, presenting a complete two-loop calculation and a novel three-loop extension — the first fully massive three-loop sum-integral computation without relying on high-temperature expansions. Our approach opens the door to precise perturbative predictions of the phase structure in a broad class of beyond-the-Standard-Model scenarios, including those featuring strong first-order phase transitions relevant for gravitational-wave signals, where conventional high-temperature approximations break down.

Finland

Pablo Navarrete

Risto Paatelainen

Kaapo Seppänen

Tuomas Tenkanen

Cosmological phase transitions without high-temperature expansions

We investigate the effect of higher-dimensional marginal operators on the thermodynamics of cosmological phase transitions. Using the Abelian Higgs model as a representative for radiatively-generated one-step transitions, we systematically match these operators, which arise at higher orders in the underlying high-temperature expansion of thermal effective field theory, and use field redefinitions to construct a complete, minimal, and gauge-invariant operator basis. The Abelian Higgs model shares the essential infrared structure of more realistic gauge-Higgs theories at high temperatures, allowing us to test the validity of dimensional reduction in a simplified setting. We argue that for strong transitions, temporal gauge modes, which enhance the transition strength, should be treated on equal footing with spatial ones. Marginal operators are found to weaken the transition and introduce significant uncertainties for strong transitions. For transitions strong enough to produce gravitational waves detectable by LISA, our findings suggest that the high-temperature expansion may break down entirely. This would limit the applicability of effective theory techniques, including their use in non-perturbative lattice studies.

Switzerland

Fabio Bernardo

Germany

Netherlands

Philipp Klose

Philipp Schicho

Higher-dimensional operators at finite temperature affect gravitational-wave predictions

Focusing on supercooled phase transitions in models with classical scale symmetry, we formulate a state-of-the art framework for computing the bubble nucleation rate, accounting for the presence of varying energy scales. In particular, we examine the limitations of derivative expansions in constructing a thermal effective field theory for bubble nucleation. We show that for gauge-field fluctuations, derivative expansions diverge after the two leading orders due to the strong variation in gauge-field masses between the high- and low-temperature phases. By directly computing these contributions using the fluctuation determinant, we capture these effects while also accounting for the dominant contribution at two-loop. Finally, we demonstrate how this approach significantly improves nucleation rate calculations compared to leading-order results, providing a more robust framework for predicting gravitational-wave signals from supercooled phase transitions in models such as the SU(2)cSM.

Poland

Maciej Kierkla

Bogumiła Świeżewska

Jorinde Vis

Finite-temperature bubble nucleation with shifting scale hierarchies

Models with radiative symmetry breaking typically feature strongly supercooled first-order phase transitions, which result in an observable stochastic gravitational wave background. In this work, we analyse the role of higher-order thermal corrections for these transitions, applying high-temperature dimensional reduction to a theory with dimensional transmutation. In particular, we study to what extent high-temperature effective field theories (3D EFT) can be used. We find that despite significant supercooling down from the critical temperature, the high-temperature expansion for the bubble nucleation rate can be applied using the 3D EFT framework, and we point out challenges in the EFT description. We compare our findings to previous studies and find that the next-to-leading order corrections obtained in this work have a significant effect on the predictions for GW observables, motivating a further exploration of higher-order thermal effects.

Sweden

China

Gravitational waves from supercooled phase transitions: dimensional transmutation meets dimensional reduction

Guided by previous non-perturbative lattice simulations of a two-step electroweak phase transition, we reformulate the perturbative analysis of equilibrium thermodynamics for generic cosmological phase transitions in terms of effective field theory (EFT) expansions. Based on thermal scale hierarchies, we argue that the scale of many interesting phase transitions is in-between the soft and ultrasoft energy scales, which have been the focus of studies utilising high-temperature dimensional reduction. The corresponding EFT expansions provide a handle to control the perturbative expansion, and allow us to avoid spurious infrared divergences, imaginary parts, gauge dependence and renormalisation scale dependence that have plagued previous studies. As a direct application, we present a novel approach to two-step electroweak phase transitions, by constructing separate effective descriptions for two consecutive transitions. Our approach provides simple expressions for effective potentials separately in different phases, a numerically inexpensive method to determine thermodynamics, and significantly improves agreement with the non-perturbative lattice simulations.

United Kingdom

Oliver Gould

Perturbative effective field theory expansions for cosmological phase transitions

For computing thermodynamics of the electroweak phase transition, we discuss a minimal approach that reconciles both gauge invariance and thermal resummation. Such a minimal setup consists of a two-loop dimensional reduction to three-dimensional effective theory, a one-loop computation of the effective potential and its expansion around the leading-order minima within the effective theory. This approach is tractable and provides formulae for resummation that are arguably no more complicated than those that appear in standard techniques ubiquitous in the literature. In particular, we implement renormalisation group improvement related to the hard thermal scale. Despite its generic nature, we present this approach for the complex singlet extension of the Standard Model which has interesting prospects for high energy collider phenomenology and dark matter predictions. The presented expressions can be used in future studies of phase transition thermodynamics and gravitational wave production in this model.

Japan

Graham White

Combining thermal resummation and gauge invariance for electroweak phase transition

The energy budget for gravitational waves of a cosmological first order phase transitions depends on the speed of sound in the thermal plasma in both phases around the bubble wall. Working in the real-singlet augmented Standard Model, which admits a strong two-step electroweak phase transition, we compute higher order corrections to the pressure and sound speed. We compare our result to lower-order approximations to the sound speed and the energy budget and investigate the impact on the gravitational wave signal. We find that deviations in the speed of sound from   <math> <msubsup> <mi>c</mi> <mi>s</mi> <mn>2</mn> </msubsup> </math>  $$ {c}_s^2 $$  = 1/3 are enhanced up to   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (5%) in our higher-order computation. This results in a suppression in the energy budget of up to   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (50%) compared to approximations assuming   <math> <msubsup> <mi>c</mi> <mi>s</mi> <mn>2</mn> </msubsup> </math>  $$ {c}_s^2 $$  = 1/3. The effect is most significant for hybrid and detonation solutions. We generalise our discussion to the case of multiple inert scalars and the case of a reduced number of fermion families in order to mimic hypothetical dark sector phase transitions. In this sector with modified field content, the sound speed can receive significant suppression, with potential order-of-magnitude impact on the gravitational wave amplitude.

Speed of sound in cosmological phase transitions and effect on gravitational waves

A gauge-invariant framework for computing bubble nucleation rates at finite temperature in the presence of radiative barriers was presented and advocated for model-building and phenomenological studies in an accompanying article [1]. Here, we detail this computation using the Abelian Higgs Model as an illustrative example. Subsequently, we recast this approach in the dimensionally-reduced high-temperature effective field theory for nucleation. This allows for including several higher order thermal resummations and furthermore delineate clearly the approach’s limits of validity. This approach provides for robust perturbative treatments of bubble nucleation during possible first-order cosmic phase transitions, with implications for electroweak baryogenesis and production of a stochastic gravitational wave background. Furthermore, it yields a sound comparison between results of perturbative and non-perturbative computations.

Joonas Hirvonen

Johan Löfgren

United States

Michael Ramsey-Musolf

Computing the gauge-invariant bubble nucleation rate in finite temperature effective field theory

Perturbation theory alone fails to describe thermodynamics of the electroweak phase transition. We review a technique combining perturbative and non-perturbative methods to overcome this challenge. Accordingly, the principal theme is a tutorial of high­temperature dimensional reduction. We present an explicit derivation with a real singlet scalar and compute the thermal effective potential at two-loop order. In particular, we detail the dimensional reduction for a real-singlet extended Standard Model. The resulting effective theory will impact future non-perturbative studies based on lattice simulations as well as purely perturbative investigations.

Juuso Österman

Robust approach to thermal resummation: Standard Model meets a singlet

We revisit the perturbative expansion at high temperature and investigate its convergence by inspecting the renormalisation scale dependence of the effective potential. Although at zero temperature the renormalisation group improved effective potential is scale independent at one-loop, we show how this breaks down at high temperature, due to the misalignment of loop and coupling expansions. Following this, we show how one can recover renormalisation scale independence at high temperature, and that it requires computations at two-loop order. We demonstrate how this resolves some of the huge theoretical uncertainties in the gravitational wave signal of first-order phase transitions, though uncertainties remain stemming from the computation of the bubble nucleation rate.