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arxiv: 2606.21380 · v1 · pith:X64Y5RIFnew · submitted 2026-06-19 · ✦ hep-ph · astro-ph.CO· hep-th

Theoretical consistency and phenomenology of supercooled cosmological phase transitions

Maciej Kierkla This is my paper

Pith reviewed 2026-06-26 13:51 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COhep-th
keywords supercooled phase transitiongravitational wavesLISAdimensional reductionbubble nucleationSU(2)cSMeffective field theoryradiative symmetry breaking
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The pith

The SU(2)cSM produces a supercooled phase transition whose gravitational waves are detectable by LISA throughout its parameter space.

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

This paper applies high-temperature dimensional reduction to compute the next-to-leading-order thermal bubble nucleation rate for a supercooled first-order phase transition in the early Universe. It takes the SU(2)cSM, an extension of the conformal Standard Model with an extra SU(2) gauge sector, as its case study and shows that radiative symmetry breaking in this model proceeds through supercooling. The resulting gravitational wave signals remain strong enough for detection by LISA across the entire parameter space examined, which would make the model experimentally testable. The work further demonstrates that higher-order corrections alter both the phase transition dynamics and the predicted signals.

Core claim

The supercooled first-order phase transition in the SU(2)cSM, analyzed for the first time with high-temperature dimensional reduction in a classically scale-invariant model, produces a consistent three-dimensional effective field theory; explicit evaluation of fluctuation determinants then yields the NLO nucleation rate, from which phase transition parameters and gravitational wave spectra are derived that remain detectable by LISA throughout the considered parameter space.

What carries the argument

High-temperature dimensional reduction to a three-dimensional effective field theory for the SU(2)cSM, combined with explicit computation of fluctuation determinants to obtain the NLO thermal bubble nucleation rate.

If this is right

  • The SU(2)cSM becomes experimentally testable through LISA gravitational wave observations.
  • Higher-order corrections must be included because they change both phase transition parameters and gravitational wave predictions.
  • A consistent power-counting scheme exists for applying dimensional reduction to this scale-invariant model in the supercooled regime.
  • The model exhibits radiative symmetry breaking that naturally leads to a supercooled transition.

Where Pith is reading between the lines

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

  • The same dimensional-reduction approach could be applied to other classically scale-invariant extensions to check whether they also produce LISA-detectable signals.
  • If the power-counting scheme holds, the method supplies a template for estimating when supercooling occurs in similar models without full four-dimensional calculations.
  • The demonstrated sensitivity to higher-order terms implies that future gravitational wave data analysis will require comparable NLO treatments to extract model parameters reliably.

Load-bearing premise

The high-temperature dimensional reduction and associated power-counting scheme remain valid for the classically scale-invariant SU(2)cSM when applied to the supercooled regime.

What would settle it

A lattice simulation or direct four-dimensional calculation that shows the three-dimensional effective theory deviates substantially from the true nucleation rate in the supercooled limit, or LISA observations that place an upper limit on gravitational wave amplitude below the predicted signal in the relevant frequency band.

Figures

Figures reproduced from arXiv: 2606.21380 by Maciej Kierkla.

Figure 2.1
Figure 2.1. Figure 2.1: Scheme of ¯h-scaling of example diagrams at tree- (left panel) and one-loop level (right panel). Notice that, here, taking the ¯h → 0 limit isolates the tree-level diagrams computed with Γ[v], but in the end, we obtain the functional W[J] that can generate any correlation function. Hence, tree-level vertices of Γ[v] indeed include the “quantum” corrections to S[ϕ]. Moreover, analogously to the generating… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Diagrammatic representation of Γ[ϕb]. The dot is the zeroth-order term S[ϕb]. Solid lines are propagating ϕ˜ fields, while external lines with empty dots correspond to the background field ϕb. 2.2 Practical uses in real scalar theory 2.2.1 Effective potential Let us now demonstrate how to obtain the 1PI effective action in practice. Following [55], we will consider a simple real scalar theory, defined by… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Potential exhibiting first-order phase transition. The value of global minimum [PITH_FULL_IMAGE:figures/full_fig_p027_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Daisy diagram. Dashed lines correspond to the Matsubara zero mode, while solid [PITH_FULL_IMAGE:figures/full_fig_p030_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Schematic depiction of dimensional reduction in a real scalar theory. [PITH_FULL_IMAGE:figures/full_fig_p035_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Diagrams contributing to the four-point correlation function. Crossed-dot vertex [PITH_FULL_IMAGE:figures/full_fig_p036_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Diagrams contributing to two-point correlation functions up to the 2-loop level. [PITH_FULL_IMAGE:figures/full_fig_p036_3_5.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Thermal scale hierarchy for bubble nucleation. [PITH_FULL_IMAGE:figures/full_fig_p047_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Example plot of a critical bubble (or bounce solution). Vertical grey line on the left [PITH_FULL_IMAGE:figures/full_fig_p048_3_7.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Schematic representation of the effective potential and power counting in SU(2)cSM. [PITH_FULL_IMAGE:figures/full_fig_p053_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Scheme of symmetry breaking pattern in SU(2)cSM. First, the vev [PITH_FULL_IMAGE:figures/full_fig_p054_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Example running of gauge coupling gX. For the curves, we kept a constant mass MX = 10000 GeV, and changed the value of gX defined at MX scale. The argument on the x-axis is defined as t ≡ log µ MZ . Dashed grey line denotes the µ = MZ . Dotted line denotes gX(µ) = 1.15. The black solid line in the left part of the plot denotes QCD scale µ = ΛQCD ≃ 0.1 GeV. In principle, these are the free parameters of t… view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Values of the scalar couplings λφh (denotes as λ2 on left panel) and λφ (denoted as λ3 on right panel) evaluated at the electroweak scale. Grey shaded regions are excluded, from left to right: no electroweak minimum with correct mass and vev of the Higgs exists, perturbativity of gX (see eq. (4.30)). 4.2.5 Parameter space of SU(2)cSM Using the procedure we have just outlined, we are able to study the ava… view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: Values of the new scalar mass MS (left panel) and the vev w (evaluated at µ = MX) (right panel). In the left panel, the thick black line indicates where MS = MH = 125 GeV and across this line, mass ordering between S and H changes (to the left of the line MS < MH, and to the right MH < MS). To the right of the dotted line, ξH becomes numerically equal to 1. The dashed lines indicate a discrepancy between… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Schematic depiction of high- and low-temperature regimes in a model with classical [PITH_FULL_IMAGE:figures/full_fig_p062_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Masses of particles as a function of bubble radius for a benchmark [PITH_FULL_IMAGE:figures/full_fig_p074_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Contributions to the NLO correction to the effective action as a function of radial [PITH_FULL_IMAGE:figures/full_fig_p077_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Spatial gauge contributions in the derivative expansion up to [PITH_FULL_IMAGE:figures/full_fig_p079_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Different approaches for computing ΓT /T 4 for a benchmark point with gX = 0.8 and mX = 104 GeV. Bands illustrate the sensitivity of different approaches to the choice of 4d RG scale at µ4 = πT (solid) and µ4 = 7T (dotted). The [NLO det T 4 ] curve is evaluated at µ4 = 7T. The 3d scale is set to µ3 = T. Convergence of soft-expansion First, one should note the substantial difference between the results ob… view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Schematic depiction of the phase transition in SU(2)cSM. Here, the red circle with [PITH_FULL_IMAGE:figures/full_fig_p090_6_1.png] view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: The values of the critical temperature Tc (left panel) and the temperature at which thermal inflation starts TV (right panel). Figure taken from [1]. before (see the discussion of figure 4.4), and two new shaded regions. The lower left corner (darkest grey) is not analysed because there the PT is sourced by the QCD phase transition, which is beyond the scope of the present work. We will comment on the ph… view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: The difference between the nucleation temperature [PITH_FULL_IMAGE:figures/full_fig_p094_6_3.png] view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: Contour plot of the decimal logarithm of the ratio of the energy transfer rate [PITH_FULL_IMAGE:figures/full_fig_p096_6_4.png] view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: The values of transition strength parameter [PITH_FULL_IMAGE:figures/full_fig_p097_6_5.png] view at source ↗
Figure 6.6
Figure 6.6. Figure 6.6: The values of inverse time scale β/H∗ (left panel) and the length scale of the transition R∗H∗ (right panel). Therefore, as an alternative, we will characterise the PT by its length scale, see below. The plot showing the values of β∗, however, can help us understand the shape of the region excluded by the percolation criterion of eq. (6.16) as it follows the shape of lines of constant β∗. Parameter β∗ is… view at source ↗
Figure 6.7
Figure 6.7. Figure 6.7: Histogram of the product (R∗H∗) × (β∗/H∗). Mean value is given by 4.91. gains mass upon entering the bubble. As a result, the bubbles “feel” an effective pressure acting upon them. While bubbles accelerate at the beginning, they eventually reach terminal velocity when the net pressure difference between the effective driving force acting on the wall and the plasma friction is zero. A crucial question for… view at source ↗
Figure 6.8
Figure 6.8. Figure 6.8: Left panel: the ratio of Lorentz factors for terminal velocity and runaway scenarios. [PITH_FULL_IMAGE:figures/full_fig_p101_6_8.png] view at source ↗
Figure 6.9
Figure 6.9. Figure 6.9: Gravitational wave spectra from the phase transition in SU(2)cSM. Single GW spec [PITH_FULL_IMAGE:figures/full_fig_p104_6_9.png] view at source ↗
Figure 6.10
Figure 6.10. Figure 6.10: Gravitational wave spectra from the phase transition in SU(2)cSM. Example spectra [PITH_FULL_IMAGE:figures/full_fig_p104_6_10.png] view at source ↗
Figure 6.11
Figure 6.11. Figure 6.11: SNR for LISA for SU(2)cSM model calculated with [PITH_FULL_IMAGE:figures/full_fig_p105_6_11.png] view at source ↗
Figure 6.12
Figure 6.12. Figure 6.12: Percolation temperature. Left panel: [daisy] results for Tp. Right panel: relative difference between [NLO det] and [daisy] results for Tp. The orange dashed and dotted lines indicate the excluded regions obtained with [NLO det]. nucleation rate, and then calculate Tp, they need to be RG-evolved to the thermal scale (for details see section 4.2.4). In the large-mass corner, the coupling becomes signific… view at source ↗
Figure 6.13
Figure 6.13. Figure 6.13: Predictions for the Hubble normalised average bubble radius for different approxi [PITH_FULL_IMAGE:figures/full_fig_p108_6_13.png] view at source ↗
Figure 6.14
Figure 6.14. Figure 6.14: The values of the efficiency factor κcol of transferring energy of the PT into GW sourced by bubble collisions obtained in [daisy] approach. The orange dashed and dotted lines indicate the excluded regions obtained with [NLO det]. The red lines indicate contours of κcol = {0.1, 0.5, 0.9} (from above to below) obtained with [NLO det]. –98– [PITH_FULL_IMAGE:figures/full_fig_p108_6_14.png] view at source ↗
Figure 6.15
Figure 6.15. Figure 6.15: Signal-to-noise ratio for LISA for SU(2)cSM model calculated with [daisy] approach. [PITH_FULL_IMAGE:figures/full_fig_p109_6_15.png] view at source ↗
Figure 6.16
Figure 6.16. Figure 6.16: Left panel: Absolute value of the differences in [PITH_FULL_IMAGE:figures/full_fig_p110_6_16.png] view at source ↗
Figure 6.17
Figure 6.17. Figure 6.17: Relative difference between [NLO det] and [NLO ∇] results for Tp. and small gX, where the phase transition is the strongest and the slowest (i.e. the size of bubbles at collision is the largest, see figure 6.6). Throughout the plot, the difference in the percolation temperatures varies from 40% to more than 300%, demonstrating the importance of including the fluctuation determinants. In figure 6.18, we … view at source ↗
Figure 6.18
Figure 6.18. Figure 6.18: Comparison of different approximations to the nucleation rate and the corresponding [PITH_FULL_IMAGE:figures/full_fig_p112_6_18.png] view at source ↗
Figure 6.19
Figure 6.19. Figure 6.19: Percolation temperature as a function of [PITH_FULL_IMAGE:figures/full_fig_p113_6_19.png] view at source ↗
Figure 6.20
Figure 6.20. Figure 6.20: Mean bubble radius at percolation, normalised to Hubble scale, [PITH_FULL_IMAGE:figures/full_fig_p113_6_20.png] view at source ↗
read the original abstract

This dissertation investigates a supercooled phase transition (PT) in the early Universe. Using high-temperature dimensional reduction (DR), we compute the NLO thermal bubble nucleation rate. By explicitly evaluating fluctuation determinants, we provide a state-of-the-art description of thermal bubble nucleation. As a case study, we consider the SU(2)cSM, an extension of the conformal Standard Model with an additional SU(2)$_X$ gauge sector and scalar field that acquires a vev through radiative symmetry breaking. This symmetry breaking proceeds via a supercooled first-order phase transition. The first part of the thesis introduces the theoretical framework, including effective actions, RG improvements, finite-temperature quantum field theory, effective field theory techniques, and thermal bubble nucleation. The second part applies these methods to the SU(2)cSM. We establish a consistent power-counting scheme, construct the leading-order effective potential, analyse symmetry breaking and parameter space, derive an RG-improved potential, and incorporate thermal corrections. We then apply high-temperature DR to a classically scale-invariant model for the first time, derive the corresponding three-dimensional EFT, and compute the NLO nucleation rate. A detailed numerical evaluation of fluctuation determinants enables a comparison of different approximation schemes and their limitations. Finally, we present the phenomenological implications. We determine phase transition parameters, perform parameter scans, and predict the resulting gravitational wave signals. We find that the supercooled phase transition in the SU(2)cSM produces a strong signal detectable by LISA throughout the parameter space considered, making the model experimentally testable. We also demonstrate that higher-order corrections can significantly affect both phase transition dynamics and gravitational wave predictions.

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 / 2 minor

Summary. The manuscript computes the NLO thermal bubble nucleation rate for a supercooled first-order phase transition in the SU(2)cSM (a classically scale-invariant extension of the SM with an additional SU(2)_X gauge sector) by applying high-temperature dimensional reduction to a 3D EFT for the first time in such a model. It explicitly evaluates fluctuation determinants, compares approximation schemes, derives phase-transition parameters via parameter scans, and predicts strong gravitational-wave signals detectable by LISA throughout the considered parameter space, while showing that higher-order corrections can significantly affect the dynamics and GW predictions.

Significance. If the high-T DR remains controlled, the work supplies a state-of-the-art numerical treatment of nucleation rates via explicit determinant evaluation and demonstrates that the SU(2)cSM yields LISA-accessible GW signals, strengthening the model's phenomenological interest. The explicit comparison of schemes and first application to a classically scale-invariant model are concrete strengths.

major comments (1)
  1. [DR application and power-counting section] The central claim that the supercooled PT produces a strong LISA-detectable GW signal throughout parameter space rests on the NLO nucleation rate obtained from high-T DR to the 3D EFT. However, in the supercooled regime T_n ≪ v (with v the radiatively generated vev), the power-counting scheme that assumes T sets the dominant scale for integrating out non-zero Matsubara modes loses its standard justification; the classically scale-invariant nature of the model removes tree-level mass parameters and makes the hierarchy even more sensitive. No explicit verification that the 3D EFT remains a controlled description at the computed T_n is supplied (abstract and the DR-application section).
minor comments (2)
  1. Notation for the 3D EFT parameters and matching conditions could be clarified with an explicit table relating 4D and 3D couplings.
  2. Figure captions for the GW spectra should state the frequency range and the precise definition of the peak amplitude used.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the justification of high-temperature dimensional reduction in the supercooled regime. We address the point below.

read point-by-point responses

  1. Referee: [DR application and power-counting section] The central claim that the supercooled PT produces a strong LISA-detectable GW signal throughout parameter space rests on the NLO nucleation rate obtained from high-T DR to the 3D EFT. However, in the supercooled regime T_n ≪ v (with v the radiatively generated vev), the power-counting scheme that assumes T sets the dominant scale for integrating out non-zero Matsubara modes loses its standard justification; the classically scale-invariant nature of the model removes tree-level mass parameters and makes the hierarchy even more sensitive. No explicit verification that the 3D EFT remains a controlled description at the computed T_n is supplied (abstract and the DR-application section).

    Authors: We appreciate the referee highlighting this subtlety. The manuscript does establish a consistent power-counting scheme adapted to the classically scale-invariant SU(2)cSM in the DR-application section, where the absence of tree-level mass parameters is accounted for by identifying the radiatively generated vev and the thermal scale as the relevant hierarchies for integrating out non-zero Matsubara modes. This scheme is used to derive the 3D EFT and the NLO nucleation rate. We nevertheless agree that an explicit numerical check of the EFT validity (for instance, by evaluating the size of the expansion parameters or estimating higher-dimensional operator contributions at the computed T_n values) is not supplied. We will add such a verification, including a table or discussion of the expansion parameters for representative points in the scanned parameter space, in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained computation

full rationale

The paper's chain proceeds by constructing the 3D EFT from high-T DR, explicitly evaluating fluctuation determinants for the NLO nucleation rate, deriving phase transition parameters, and then scanning to obtain GW predictions. None of these steps reduces a claimed prediction to an input by construction, a fitted parameter renamed as output, or a load-bearing self-citation whose justification collapses to the present work. The LISA-detectability result follows from the downstream numerical evaluation rather than being presupposed in the rate computation or EFT matching.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

Central claim rests on validity of dimensional reduction, power counting, and the existence of a supercooled first-order transition in the chosen parameter space of the SU(2)cSM extension.

free parameters (1)
  • SU(2)cSM model parameters
    Scanned to identify regions realizing the supercooled PT; values chosen to produce the desired radiative breaking.
axioms (2)
  • standard math Standard finite-temperature QFT and effective potential formalism
    Invoked throughout the theoretical framework section.
  • domain assumption Validity of high-temperature dimensional reduction for the classically scale-invariant model
    Central methodological step stated as applied for the first time.
invented entities (1)
  • SU(2)_X gauge sector plus additional scalar no independent evidence
    purpose: Realizes radiative symmetry breaking through a supercooled first-order PT
    The extension is introduced as the case study; no independent evidence supplied beyond the model construction itself.

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Reference graph

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