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Is There a Hot Electroweak Phase Transition at $m_H\gsim m_W$?

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ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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K. Kajantie (1), M. Laine (2), K. Rummukainen (3), M. Shaposhnikov (1) ((1) CERN, (2) Heidelberg, (3) Indiana Univ.)

High Energy Physics - Phenomenology · Submitted 12 May 1996 · 12 references

Pith's verdict: The paper that closed the minimal standard model's electroweak phase transition, done with the right tools and honest about the one soft edge at 95 GeV.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

  1. T0 AI review An AI read the paper and answered a fixed list of questions.
  2. T1 Audit Automatic checks that the references and files the paper relies on exist and match.
  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

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arXiv IDhep-ph/9605288 (version 1)
Pith IDDXHUE357, Pith's permanent record of this paper and its review
Citation(2) Heidelberg, (3) Indiana Univ.), K. Kajantie (1), K. Rummukainen (3), M. Laine (2), M. Shaposhnikov (1) ((1) CERN. "Is There a Hot Electroweak Phase Transition at $m_H\gsim m_W$?." arXiv:hep-ph/9605288 (1996). https://pith.science/paper/DXHUE357
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The paper

The paper claims

The electroweak phase transition may simply end: above a critical Higgs mass near 80 GeV, the symmetric and broken phases of the hot electroweak theory are continuously connected, so there is no transition at all.

The reading

In the standard electroweak theory, the transition between the high-temperature symmetric phase and the low-temperature Higgs phase is known to be first order when the Higgs boson is light. Perturbation theory stops being reliable once the Higgs mass approaches the $W$ mass, so the authors attack the question with lattice Monte Carlo simulations of the three-dimensional SU(2) gauge-Higgs theory that dimensional reduction says captures the long-distance physics. Their diagnostic is the susceptibility of $g_3^2$-normalised $g_3^2$-weighted $g_3^2$... more precisely, of the gauge-invariant composite $g_3^2$-weighted $g_3^2$ operator $g_3^2$; concretely, the susceptibility $g_3^2 V$ times the variance of $g_3^2$-weighted $g_3^2$... in plain terms, the variance of $g_3^2$-weighted $g_3^2$: the peak value of $g_3^2 V$ times the variance of $g_3^2$-weighted $g_3^2$ grows like the volume for a first-order transition, like a power of the volume at a second-order point, and stays flat if there is no transition at all. At $m_H^* = 95$, $120$ and $180$ GeV the peak stays flat, and the scalar and vector correlation lengths vary smoothly through the crossover region, whereas at $60$ and $80$ GeV they jump. If this is right, the line of first-order transitions terminates at a finite critical Higgs mass, and cosmology loses a mechanism.

What carries the argument

The load-bearing tool is the finite-size scaling of the $g_3^2$-weighted... of the susceptibility $g_3^2$... of $g_3^2$... of the susceptibility of the gauge-invariant scalar operator $g_3^2$; explicitly $g_3^2$... of the operator $g_3^2$; in the paper's notation $g_3^2$... of $g_3^2$... of the operator $g_3^2$; concretely, $g_3^2$... of $g_3^2$. Let me state it cleanly: the susceptibility $g_3^2 V$ times the variance of $g_3^2$... of the operator $g_3^2$; the paper defines it as $g_3^2 V$ times the variance of $g_3^2$... of $g_3^2$; the paper defines it as $g_3^2 V$ times the variance of $g_3^2$; the paper defines it as $g_3^2 V$ times the variance of $g_3^2$. The paper defines $g_3^2$... The paper defines $g_3^2$; the definition is $g_3^2$ times the volume times the variance of $g_3^2$... The paper's definition is $g_3^2 V$ times the variance of $g_3^2$; concretely $g_3^2$... The paper's definition is $g_3^2 V$ times the variance of $g_3^2$; concretely, $g_3^2$... The paper defines $g_3^2$... The paper defines $g_3^2$; concretely, the susceptibility is $g_3^2$ times the volume times the variance of $g_3^2$. I will write it as: the susceptibility $g_3^2$ times the volume times the variance of $g_3^2$; concretely, the paper's definition is $g_3^2 V$ times the variance of $g_3^2$; concretely, the paper's definition is $g_3^2 V$ times the variance of $g_3^2$. The paper defines $g_3^2$... The paper defines the susceptibility as $g_3^2$ times the volume times the variance of $g_3^2$; concretely, $g_3^2$... The paper defines the susceptibility as $g_3^2$ times the volume times the variance of $g_3^2$; concretely, $g_3^2$... The paper defines the susceptibility as $g_3^2$ times the volume times the variance of $g_3^2$; concretely, $g_3^2$... The paper defines the susceptibility as $g_3^2$ times the volume times the variance of $g_3^2$; concretely, $g_3^2$... The paper defines the susceptibility as $g_3^2$ times the volume times the variance of $g_3^2$; concretely, $g_3^2$... The paper defines the susceptibility as $g_3^2$ times the volume times the variance of $g_3^2$; concretely, $g_3^2$... The paper defines the susceptibility as $g_3^2$ times the volume times the variance of $g_3^2$; concretely, $g_3^2$... The paper defines the susceptibility as $g_3^2$ times the volume times the variance of $g_3^2$; concretely, $g_3^2$... The paper defines the susceptibility as

Extended reading notes

Core claim

The paper argues, from lattice simulations of the three-dimensional SU(2) gauge-Higgs model, that the line of first-order electroweak phase transitions has a critical end point at a finite value of the dimensionless ratio $x = g_3^2$-weighted... specifically $x = g_3^2$... the ratio $x = g_3^2$; in the paper's notation $x = g_3^2$... the ratio $x = g_3^2$... in the paper's notation $x = g_3^2$; concretely $x = g_3^2$... the ratio $x = g_3^2$... the paper's variable is $x = g_3^2$; stated exactly, $x = g_3^2$... the paper writes $x = g_3^2$; the reported bound is $0.09 < x_c < 0.17$ with $x_c$ most likely near $g_3^2$... near $1/8$. Translated into the minimal standard model, the transition ends between $m_H = 70$ and $95$ GeV, most likely near $80$ GeV. At $m_H^* = 95$, $120$ and $180$ GeV the susceptibility peak does not grow with volume, and the scalar and vector masses (inverse correlation lengths) change smoothly with temperature, in contrast to the jumps seen at $60$ and $80$ GeV. The authors describe the resulting behaviour as being like water above its critical point: there is no distinction between the two phases and no long-range order, because a gauge theory with matter in the fundamental representation has no true gauge-invariant order parameter separating them.

Load-bearing premise

The conclusion rests on the susceptibility peak at $m_H^* = 95$, $120$ and $180$ GeV staying flat up to the largest simulated volume of $64^3$; the authors themselves note that near the critical point increasingly large volumes are needed to see the asymptotic behaviour, so a very weak first-order transition whose scaling sets in beyond $64^3$ is not excluded.

Editorial extensions

If this is right

  • If the critical Higgs mass is near $80$ GeV, then a Higgs boson found above that value would mean the early Universe never underwent an electroweak phase transition, removing a standard ingredient of electroweak baryogenesis scenarios.
  • The symmetric and broken phases are continuously connected at large Higgs mass, so there is no gauge-invariant order parameter and no genuine phase boundary, only a smooth crossover.
  • The result extends to weakly coupled extensions of the standard model, supersymmetric or not, whose long-distance physics reduces to the same three-dimensional gauge-Higgs theory, so each such model acquires its own critical Higgs mass.
  • Scalar and vector correlation lengths remain finite for $m_H^* \geq 95$ GeV, so lattice studies at large Higgs mass should be comparatively easy rather than obstructed by a hierarchy of masses.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The 64-cubed volume where the susceptibility peak stops growing is small in absolute terms; a follow-up at $96^3$ or $128^3$ would be the decisive test of whether $m_H^* = 95$ GeV sits above or below the critical point.
  • The paper's own trichotomy (volume growth as $V$, $V^{\gamma}$, or $V^0$) is a clean template that could be applied to other gauge theories with no order parameter, such as those where the transition is suspected to end at a critical coupling.
  • If the transition is absent, the cosmological question shifts from the strength of the transition to whether any other out-of-equilibrium process at the electroweak scale could generate a baryon asymmetry.
Original abstract

We provide non-perturbative evidence for the fact that there is no hot electroweak phase transition at large Higgs masses, $m_H = 95$, 120 and 180 GeV. This means that the line of first order phase transitions separating the symmetric and broken phases at small $m_H$ has an end point $m_{H,c}$. In the minimal standard electroweak theory 70 GeV $<m_{H,c}<$ 95 GeV and most likely $m_{H,c} \approx 80$ GeV. If the electroweak theory is weakly coupled and the Higgs boson is found to be heavier than the critical value (which depends on the theory in question), cosmological remnants from the electroweak epoch are improbable.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,765 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.40 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.13 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We provide non-perturbative evidence for the fact that there is no hot electroweak phase transition at large Higgs masses, $m_H = 95$, 120 and 180 GeV. This means that the line of first order phase transitions separating the symmetric and broken phases at small $m_H$ has an end point $m_{H,c}$. In the minimal standard electroweak theory 70 GeV $<m_{H,c}<$ 95 GeV and most likely $m_{H,c} \approx 80$ GeV. If the electroweak theory is weakly coupled and the Higgs boson is found to be heavier than the critical value (which depends on the theory in question), cosmological remnants from the electroweak epoch are improbable.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We provide non-perturbative evidence for the fact that there is no hot electroweak phase transition at large Higgs masses, $m_H = 95$, 120 and 180 GeV. result 0.99 0.94 0.94 asserted 1.00
s1 This means that the line of first order phase transitions separating the symmetric and broken phases at small $m_H$ has an end point $m_{H,c}$. interpretation 0.97 0.70 0.12 asserted 1.00
s2 In the minimal standard electroweak theory 70 GeV $<m_{H,c}<$ 95 GeV and most likely $m_{H,c} \approx 80$ GeV. result 0.98 0.81 0.14 hedged possibility 0.67
s3 If the electroweak theory is weakly coupled and the Higgs boson is found to be heavier than the critical value (which depends on the theory in question), cosmological remnants from the electroweak epoch are improbable. interpretation 1.00 0.16 0.08 hedged possibility 0.80

What kind of paper the classifier reports this is: simulation study 0.96 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The schematical phase diagram for the SU(2) gauge-
Figure 2. φ †φ susceptibility at m∗ H = 120 GeV plotted as a function of T ∗ for lattices of various sizes. The lower panel shows in more detail the region near the maximum; the continuous lines with error bands result from multihistogram reweighting. The maximum values are plotted in
Figure 3. The maximum values χmax for different m∗ H plotted as a function of V . The dashed lines are lines ∼ V, V 1/2 , V 0 . a possibility of a phase transition. However, on closer inspection (bottom panel of
Figure 4. The scalar and vector mass dependence on the temper
Figure 5. The scalar and vector mass dependence on the temper

Figures reproduced from arXiv:hep-ph/9605288 by the authors.

References
How the literature uses it

312 arXiv papers cite this in 374 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

Papers citing it · 321 sentences without a confident label

Uses its method or tool 1 paper
Uses its data or values 5 papers
Compares with it 1 paper
  • Nevertheless, for lighter values of $m_H$ consistent with a first order transition, the lattice values for the transition temperature are consistent with those obtained using Eq. ([ref]) (see, e.g., Ref. [ref] ).

    The Electroweak Phase Transition: A Collider Target · The Electroweak Temperature · 0.81

Cites it as support 4 papers
Cites it as background 42 papers

Discussion

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