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We study first-order electroweak phase transitions in the real-singlet extended Standard Model, for which non-zero mixing between the Higgs and the singlet can efficiently strengthen the transitions. We perform large-scale parameter space scans of the model using two-loop effective potential at next-to-next-to leading order in the high-temperature expansion, greatly improving description of phase transition thermodynamics over existing one-loop studies. We find that 1) two-loop corrections to the effective potential lead to narrower regions of strong first-order transitions and significantly smaller critical temperatures, 2) transitions involving a discontinuity in the singlet expectation value are significantly stronger at two-loop order, 3) high-temperature expansion is accurate for a wide range of parameter space that allows strong transitions, although it is less reliable for the very strongest transitions. These findings suggest revisiting past studies that connect the possibility of a first-order electroweak phase transition with future collider phenomenology., Comment: 26 pages, 11 figures

We complete the perturbative program for equilibrium thermodynamics of cosmological first-order phase transitions by determining the finite-temperature effective potential of gauge-Higgs theories at next-to-next-to-next-to-next-to-leading order (N$^4$LO). The computation of the three-loop effective potential required to reach this order is extended to generic models in dimensionally reduced effective theories in a companion article. Our N$^4$LO result is the last perturbative order before confinement renders electroweak gauge-Higgs theories non-perturbative at four loops. By contrasting our analysis with non-perturbative lattice results, we find a remarkable agreement. As a direct application for predictions of gravitational waves produced by a first-order transition, our computation provides the final fully perturbative results for the phase transition strength and speed of sound., Comment: 20 pages; 6 figures; 4 tables

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., Comment: Revised version, published in JHEP. 45 pages, 9 figures

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., Comment: 58 pages, 14 figures

Beyond the Standard Model physics is required to explain both dark matter and the baryon asymmetry of the universe, the latter possibly generated during a strong first-order electroweak phase transition. While many proposed models tackle these problems independently, it is interesting to inquire whether the same model can explain both. In this context, we link state-of-the-art perturbative assessments of the phase transition thermodynamics with the extraction of the dark matter energy density. These techniques are applied to a next-to-minimal dark matter model containing an inert Majorana fermion that is coupled to Standard Model leptons via a scalar mediator, where the mediator interacts directly with the Higgs boson. For dark matter masses 180 GeV $ < M_\chi <$ 300 GeV, we discern regions of the model parameter space that reproduce the observed dark matter energy density and allow for a first-order phase transition, while evading the most stringent collider constraints., Comment: 54 pages, 12 figures, 1 table; v2: Journal version, extended section 3.2

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 $c_s^2 = 1/3$ are enhanced up to $\mathcal O(5\%)$ in our higher-order computation. This results in a suppression in the energy budget of up to $\mathcal O (50\%)$ compared to approximations assuming $c_s^2 = 1/3$. 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., Comment: 43 pages, 10 figures

DRalgo is an algorithmic implementation that constructs an effective, dimensionally reduced, high-temperature field theory for generic models. The corresponding Mathematica package automatically performs the matching to next-to-leading order. This includes two-loop thermal corrections to scalar and Debye masses as well as one-loop thermal corrections to couplings. DRalgo also allows for integrating out additional heavy scalars. Along the way, the package provides leading-order beta functions for general gauge-charges and fermion-families; both in the fundamental and in the effective theory. Finally, the package computes the finite-temperature effective potential within the effective theory. The article explains the theory of the underlying algorithm while introducing the software on a pedagogical level., Comment: 46 pages, 4 figures

We provide a roadmap for analyzing the interplay between hypothetical future collider observations and the detection of a gravitational wave signal produced by a strong first order electroweak phase transition in beyond the Standard Model (BSM) theories. A cornerstone of this roadmap is a combination of a dimensionally reduced, three-dimensional effective field theory and results of both perturbation theory and non-perturbative lattice simulations. For the first time we apply these state-of-the-art methods to a comprehensive parameter space scan of a BSM theory. Concretely, we study an extension with the real scalar triplet, which admits a possible two-step electroweak symmetry-breaking thermal history. We find that (1) a first order transition during the second step could generate a signal accessible to LISA generation detectors and (2) the gravitational wave signal displays a strong sensitivity to the portal coupling between the new scalar and the Higgs boson, and (3) the ability for future experiments to detect the produced gravitational waves depends decisively on the wall velocity of the bubbles produced during the phase transition. We illustrate how a combination of direct and indirect measurements of the new scalar properties, in combination with the presence or absence of a gravitational wave detection, could test the model and identify the values of the model parameters., Comment: 14 pages, 5 figures

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., Comment: 38 pages, 5 figures, no gauge dependence

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 arXiv:2112.05472. 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., Comment: 64 pages, 3 figures, 2 tables; v2: Journal version, added figures 1 and 2, extended sections 1 and 3.4

We present a gauge-invariant framework for bubble nucleation in theories with radiative symmetry breaking at high temperature. As a procedure, this perturbative framework establishes a practical, gauge-invariant computation of the leading order nucleation rate, based on a consistent power counting in the high-temperature expansion. In model building and particle phenomenology, this framework has applications such as the computation of the bubble nucleation temperature and the rate for electroweak baryogenesis and gravitational wave signals from cosmic phase transitions., Comment: 6 pages, 1 figure, updated funding information for v2

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., Comment: 36 pages, 7 figures

We investigate the electroweak phase transition in the real-singlet extension of the Standard Model at two-loop level, building upon existing one-loop studies. We calculate the effective potential in the high-temperature approximation and detail the required resummations at two-loop order. In typical strong-transition scenarios, we find deviations of order $20\% - 50\%$ from one-loop results in transition strength and critical temperature for both one- and two-step phase transitions. For extremely strong transitions, the discrepancy with one-loop predictions is even larger, presumably due to sizable scalar couplings in the tree-level potential. Along the way, we obtain a dimensionally-reduced effective theory applicable for non-perturbative lattice studies of the model., Comment: 26 pages, 4 figures

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 high-temperature 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., Comment: 86 pages, 2 figures and 141 Feynman diagrams

We critically examine the magnitude of theoretical uncertainties in perturbative calculations of first-order phase transitions, using the Standard Model effective field theory as our guide. In the usual daisy-resummed approach, we find large uncertainties due to renormalisation scale dependence, which amount to two to three orders-of-magnitude uncertainty in the peak gravitational wave amplitude, relevant to experiments such as LISA. Alternatively, utilising dimensional reduction in a more sophisticated perturbative approach drastically reduces this scale dependence, pushing it to higher orders. Further, this approach resolves other thorny problems with daisy resummation: it is gauge invariant which is explicitly demonstrated for the Standard Model, and avoids an uncontrolled derivative expansion in the bubble nucleation rate., Comment: 103 pages, 11 figures

New field content beyond that of the Standard Model of particle physics can alter the thermal history of electroweak symmetry breaking in the early universe. In particular, the symmetry breaking may have occurred through a sequence of successive phase transitions. We study the thermodynamics of such scenario in a real triplet extension of the Standard Model, using nonperturbative lattice simulations. Two-step electroweak phase transition is found to occur in a narrow region of allowed parameter space with the second transition always being first order. The first transition into the phase of non-vanishing triplet vacuum expectation value is first order in a non-negligible portion of the two-step parameter space. A comparison with 2-loop perturbative calculation is provided and significant discrepancies with the nonperturbative results are identified.

Making use of a dimensionally-reduced effective theory at high temperature, we perform a nonperturbative study of the electroweak phase transition in the Two Higgs Doublet model. We focus on two phenomenologically allowed points in the parameter space, carrying out dynamical lattice simulations to determine the equilibrium properties of the transition. We discuss the shortcomings of conventional perturbative approaches based on the resummed effective potential - regarding the insufficient handling of infrared resummation but also the need to account for corrections beyond 1-loop order in the presence of large scalar couplings - and demonstrate that greater accuracy can be achieved with perturbative methods within the effective theory. We find that in the presence of very large scalar couplings, strong phase transitions cannot be reliably studied with any of the methods.

We present the first end-to-end nonperturbative analysis of the gravitational wave power spectrum from a thermal first-order electroweak phase transition (EWPT), using the framework of dimensionally reduced effective field theory and pre-existing nonperturbative simulation results. We are able to show that a first-order EWPT in any beyond the Standard Model (BSM) scenario that can be described by a Standard Model-like effective theory at long distances will produce gravitational wave signatures too weak to be observed at existing and planned detectors. This implies that colliders are likely to provide the best chance of exploring the phase structure of such theories, while transitions strong enough to be detected at gravitational wave experiments require either previously neglected higher-dimension operators or light BSM fields to be included in the dimensionally reduced effective theory and therefore necessitate dedicated nonperturbative studies. As a concrete application, we analyze the real singlet-extended Standard Model and identify regions of parameter space with single-step first-order transitions, comparing our findings to those obtained using a fully perturbative method. We discuss the prospects for exploring the electroweak phase diagram in this model at collider and gravitational wave experiments in light of our nonperturbative results., Comment: 18 pages + references, 6 figures

In a series of two papers, we make a comparative analysis of the performance of conventional perturbation theory to analyze electroweak phase transition in the real triplet extension of Standard Model ($\Sigma$SM). In Part I (this paper), we derive and present the high-$T$ dimensionally reduced effective theory that is suitable for numerical simulation on the lattice. In the sequel (Part II), we will present results of the numerical simulation and benchmark the performance of conventional perturbation theory. Under the assumption that $\Sigma$ is heavy, the resulting effective theory takes the same form as that derived from the minimal standard model. By recasting the existing non-perturbative results, we map out the phase diagram of the model in the plane of triplet mass $M_\Sigma$ and Higgs portal coupling $a_2$. Contrary to conventional perturbation theory, we find regions of parameter space where the phase transition may be first order, second order, or crossover. We comment on prospects for prospective future colliders to probe the region where the electroweak phase transition is first order by a precise measurement of the $h\rightarrow\gamma\gamma$ partial width., Comment: 16 pages, 5 figures

Due to the infrared problem of high-temperature field theory, a robust study of the electroweak phase transition (EWPT) requires use of non-perturbative methods. We apply the method of high-temperature dimensional reduction to the two Higgs doublet model (2HDM) to obtain three-dimensional effective theories that can be used for non-perturbative simulations. A detailed derivation of the mapping between the full four-dimensional and the effective three-dimensional theories is presented. The results will be used in future lattice studies of the 2HDM. In the limit of large mass mixing between the doublets, existing lattice results can be recycled. The results of such a study are presented in a companion paper., Comment: 36 pages, no figures

We perform a nonperturbative study of the electroweak phase transition (EWPT) in the two Higgs doublet model (2HDM) by deriving a dimensionally reduced high-temperature effective theory for the model, and matching to known results for the phase diagram of the effective theory. We find regions of the parameter space where the theory exhibits a first-order phase transition. In particular, our findings are consistent with previous perturbative results suggesting that the primary signature of a first-order EWPT in the 2HDM is $m_{A_0} > m_{H_0} + m_Z$., Comment: 13 pages, v2: references added, two loop scan results and discussion of dimension 6 operators added; v3: corrected minor errors in scanning, added discussion of model constraints; v4: title changed, further minor changes to body, added plot of effective potential, published in PRL

We derive an effective dimensionally reduced theory for the Standard Model augmented by a real singlet scalar. We treat the singlet as a superheavy field and integrate it out, leaving an effective theory involving only the Higgs and $\mathrm{SU}(2)_\mathrm{L} \times \mathrm{U}(1)_Y$ gauge fields, identical to the one studied previously for the Standard Model. This opens up the possibility of efficiently computing the order and strength of the electroweak phase transition, numerically and nonperturbatively, in this extension of the Standard Model. Understanding the phase diagram is crucial for models of electroweak baryogenesis and for studying the production of gravitational waves at thermal phase transitions., Comment: 52 pages, 3 figures; v2: references added, additional discussion of temporal gluons, discussion extended with some parameter space scans, published in JHEP

In real-time lattice simulations of cosmic strings in the Abelian Higgs model, the broken translational invariance introduces lattice artefacts; relativistic strings therefore decelerate and radiate. We introduce two different methods to construct a moving string on the lattice, and study in detail the lattice effects on moving strings. We find that there are two types of lattice artefact: there is an effective maximum speed with which a moving string can be placed on the lattice, and a moving string also slows down, with the deceleration approximately proportional to the exponential of the velocity. To mitigate this, we introduce and study an improved discretisation, based on the tree-level L\"{u}scher-Weisz action, which is found to reduce the deceleration by an order of magnitude, and to increase the string speed limit by an amount equivalent to halving the lattice spacing. The improved algorithm is expected to be very useful for 3D simulations of cosmic strings in the early universe, where one wishes to simulate as large a volume as possible., Comment: 14 pages, 10 figures; v2: minor changes, accepted for publication in PRD; v3: added erratum noting that improvement is not fully O(a^4), methods and results unchanged