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arxiv: 2602.20246 · v1 · pith:OS2QMLRFnew · submitted 2026-02-23 · ✦ hep-ph · hep-th

Supercooled Phase Transitions with Radiative Symmetry Breaking

Alberto Salvio This is my paper

Pith reviewed 2026-05-21 12:18 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords supercooled phase transitionsradiative symmetry breakingfirst-order phase transitionsgravitational wavesprimordial black holesnucleation rate
0
0 comments X

The pith

Theories with radiatively broken symmetries undergo strong supercooling before their first-order phase transitions complete.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper reviews a model-independent approach to phase transitions that occur in field theories where symmetries are broken by radiative effects rather than tree-level terms. Such theories always predict a period of supercooling in which the transition temperature drops well below the scale at which the symmetry-breaking scale is set. A sympathetic reader would care because the resulting transitions can source gravitational waves and primordial black holes, and the approach supplies ready-to-use formulas that apply across many specific models once supercooling is strong enough for perturbation theory to work.

Core claim

In field theories with radiative symmetry breaking, first-order phase transitions become effective only after substantial supercooling, and a model-independent perturbative framework then determines the effective action and nucleation rate, yielding formulas that can be applied directly to any model in this class.

What carries the argument

Model-independent perturbative calculation of the effective action and nucleation rate for strongly supercooled first-order phase transitions triggered by radiative symmetry breaking.

If this is right

  • The transitions produce gravitational waves whose spectrum can be computed from the general formulas.
  • Primordial black holes can form during the supercooled epoch.
  • The same formulas apply without re-derivation to any concrete model that exhibits strong supercooling.
  • Perturbative control holds once the transition temperature lies sufficiently below the symmetry-breaking scale.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The duration of supercooling sets the typical size of the resulting gravitational-wave signal and the mass range of any black holes that form.
  • Models with only mild supercooling would require non-perturbative lattice studies instead of the analytic formulas given here.
  • The same radiative-breaking mechanism could influence the thermal history of the early universe at energies below the symmetry-breaking scale.

Load-bearing premise

Supercooling must be strong enough that perturbative methods reliably fix the effective action and the nucleation rate.

What would settle it

A concrete first-order phase transition observed at a temperature comparable to the symmetry-breaking scale with no extended period of supercooling would contradict the claim.

Figures

Figures reproduced from arXiv: 2602.20246 by Alberto Salvio.

Figure 1
Figure 1. Figure 1: A qualitative picture of the temperature-dependent effective potential corresponding to a FOPT: the two minima associated with the two phases are separated by a potential barrier; Tc is the critical temperature. Figure reproduced from Ref. [39]. Note that J (1) B and J (1) F are obviously positive and negative, respectively. Indeed, computing them explicitly one finds J (1) B = π 2 12 , J(1) F = − π 2 24 .… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The relevant bounce and the corresponding integrand function (divided by 8πLξ2 ) appear￾ing in the bounce action in (5.4) for the effective potential (4.3) and varying λ˜ ≡ λm2/k2 . The maximal height of the curves increases by decreasing λ˜. 5.2.2 Nucleation temperature Inserting Expression (5.12) in Eq. (3.16) for the nucleation temperature Tn leads to a non￾polynomial equation in λ˜: a1 − a2λ˜ = F(λ˜) ≡… view at source ↗
Figure 4
Figure 4. Figure 4: The solution λ˜ n of Eq. (5.13) as a function of a1 and a2 defined in (5.14). The inset in the right plot gives the maximal value of a2 for a given a1 such that the solution λ˜ n exists. Using the definitions of λ˜ and λ in (5.8) and (3.12) one can extract the nucleation temperature. Figure reproduced from Ref. [19]. that was an order-of-magnitude estimate, it remains valid here: indeed, in the improved su… view at source ↗
read the original abstract

First-order phase transitions produce gravitational waves and primordial black holes. They always occur in field theories where symmetries are radiatively broken and masses are correspondingly generated. These theories predict a period of supercooling: phase transitions become effective at temperatures much smaller than the symmetry-breaking scale. This paper reviews a model-independent approach to study phase transitions in this scenario, which can be adopted if supercooling is strong enough. Perturbative methods can be used to determine the effective action and such model-independent approach allows us to obtain ready-to-use formulas that can be applied to any specific model of this sort.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript reviews a model-independent perturbative approach to first-order phase transitions in field theories with radiatively broken symmetries. It argues that strong supercooling (phase transitions effective at T_n ≪ v) permits reliable use of perturbative methods to compute the effective action and nucleation rate, yielding ready-to-use formulas applicable to any model in this class.

Significance. If the applicability regime is properly delimited, the review would provide a practical reference for cosmological studies of gravitational waves and primordial black holes by reducing model-specific calculations to a common framework. The emphasis on ready-to-use formulas is a concrete strength for model-building applications.

major comments (1)
  1. [§3 (Applicability of the perturbative approach)] The central applicability condition—that supercooling must be 'strong enough' for perturbative control of the effective action and nucleation rate at low T—is stated without quantitative bounds (e.g., a limit on the loop-expansion parameter or explicit comparison to non-perturbative results). This renders the regime of validity for the model-independent formulas imprecise and untested against potential IR sensitivities.
minor comments (1)
  1. Notation for the effective potential parameters is introduced without a consolidated table of definitions, which would improve readability when applying the ready-to-use formulas to specific models.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment on the applicability regime. We address the point below and have incorporated revisions to improve the clarity of the validity conditions while preserving the model-independent character of the approach.

read point-by-point responses

  1. Referee: [§3 (Applicability of the perturbative approach)] The central applicability condition—that supercooling must be 'strong enough' for perturbative control of the effective action and nucleation rate at low T—is stated without quantitative bounds (e.g., a limit on the loop-expansion parameter or explicit comparison to non-perturbative results). This renders the regime of validity for the model-independent formulas imprecise and untested against potential IR sensitivities.

    Authors: We agree that the manuscript would benefit from a more explicit discussion of the quantitative aspects of the applicability condition. The central criterion T_n ≪ v is motivated by the requirement that the zero-temperature radiative potential dominates the dynamics, rendering thermal corrections and higher-loop effects small. In the revised manuscript we have expanded the relevant discussion in §3 to include order-of-magnitude estimates for the effective loop-expansion parameter in terms of the supercooling ratio (e.g., showing that corrections remain ≲ 10 % for T_n/v ≲ 0.1 in representative models with O(1) couplings). We have also added references to existing non-perturbative and lattice studies of radiatively broken potentials that support the reliability of the perturbative nucleation rate in the strong-supercooling limit. These additions clarify the regime of validity and address potential IR sensitivities without introducing model-specific assumptions that would undermine the model-independent formulas. revision: yes

Circularity Check

0 steps flagged

Review of established perturbative formulas shows no circularity in derivation chain

full rationale

This is a review paper presenting a model-independent approach for strong-supercooling phase transitions with radiative symmetry breaking. The abstract and context indicate that perturbative methods for the effective action and nucleation rate are adopted when supercooling is strong enough (T_n ≪ v), with ready-to-use formulas offered for application to specific models. These elements are described as drawn from prior literature rather than newly derived here in a self-referential loop. No self-definitional steps appear where a claimed prediction reduces to a fitted input or definition by construction, nor are there load-bearing self-citations that substitute for independent verification. The strong-supercooling condition is framed as an applicability criterion, not a result forced by the paper's own equations. The derivation chain therefore remains self-contained with the burden on externally cited works.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Because the document is a review, the ledger records the background assumptions inherited from the literature being summarized rather than new postulates introduced here. The main modeling choice is the validity of perturbation theory under strong supercooling.

axioms (1)
  • domain assumption Perturbative methods suffice to determine the effective action when supercooling is strong enough.
    Stated in the abstract as the condition under which the model-independent approach can be adopted.

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Forward citations

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