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Finite temperature field theory and phase transitions

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Mariano Quiros

High Energy Physics - Phenomenology · Submitted 14 Jan 1999 · 73 references

Pith's verdict: Solid, honest lecture notes on finite-temperature field theory; the physics is standard and correctly worked out, and the Higgs-mass washout bounds are era-specific numbers rather than new results.

T0 · AI review

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arXiv IDhep-ph/9901312 (version 1)
Pith ID5RNTGOHM, Pith's permanent record of this paper and its review
CitationMariano Quiros. "Finite temperature field theory and phase transitions." arXiv:hep-ph/9901312 (1999). https://pith.science/paper/5RNTGOHM
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The paper

The paper claims

The paper translates the requirement that sphalerons not erase the baryon asymmetry into upper bounds on the Higgs mass: about 42 GeV in the Standard Model and roughly 105 GeV in the MSSM.

The reading

This set of lecture notes develops the machinery of field theory at finite temperature and uses it to ask whether the electroweak phase transition could have produced the universe's baryon asymmetry. The central move is to turn the requirement that sphalerons not erase a previously generated asymmetry into a bound on the order parameter of the transition, $��(T_c)/T_c$, and then into an upper bound on the Higgs mass. In the Standard Model the resulting ceiling is about $42��{GeV}$, already below the experimental lower limit; in the Minimal Supersymmetric Standard Model with a light right-handed stop the ceiling can rise to roughly $105��{GeV}$. The payoff is a collider-testable link between baryogenesis and Higgs-sector masses.

What carries the argument

The load-bearing object is the finite-temperature sphaleron suppression factor $��exp[-E_{��rm sph}(T)/T]$, which controls how fast baryon-number-violating transitions proceed after the phase transition. The paper combines the zero-temperature sphaleron energy $E_{��rm sph}=(2m_W/��_W)B(��/g^2)$ with the finite-temperature scaling law $E_{��rm sph}(T)=E_{��rm sph}��langle��(T)��rangle/��langle��(0)��rangle$, then integrates the washout equation to get an exponent $X��sim 10^{10}��kappa��zeta^7 e^{-��zeta}$ with $��zeta=E_{��rm sph}(T_c)/T_c$. Requiring $X��lesssim 10$ yields $��zeta��gtrsim 45$, and the chain closes through the cubic term $E T��^3$ in the effective potential, which fixes $��(T_c)/T_c$ and hence the Higgs mass.

Extended reading notes

Core claim

The paper's central claim is that the condition of not washing out any previously generated baryon asymmetry can be translated, in both the Standard Model and the MSSM, into upper bounds on the Higgs mass. The chain runs through the finite-temperature sphaleron energy: requiring $E_{��rm sph}(T_c)/T_c ��gtrsim 45$ implies $��(T_c)/T_c ��gtrsim 1.3$, and using $��(T_c)/T_c ��sim 4 E v^2/m_h^2$ gives $m_h ��lesssim 42��{GeV}$ in the Standard Model. Because that value is already excluded, the Standard Model cannot preserve a pre-existing baryon asymmetry; the MSSM, with a light right-handed stop enhancing the cubic term in the effective potential, can push the bound to about $105��{GeV}$ while remaining consistent with the absence of color-breaking minima in a wide parameter region. The review presents this as a way to convert baryogenesis from a cosmological story into a constraint on the Higgs sector.

Load-bearing premise

The load-bearing premise is that the finite-temperature sphaleron energy scales linearly with the Higgs vacuum expectation value, $E_{��rm sph}(T)=E_{��rm sph}(0)��langle��(T)��rangle/��langle��(0)��rangle$, an approximation imported from Refs. 49 and 50 rather than tested here.

Editorial extensions

If this is right

  • In the Standard Model the washout bound, about $42��{GeV}$, lies below the experimental Higgs-mass limit, so electroweak baryogenesis is ruled out in that model.
  • In the MSSM with a light right-handed stop, the same logic leaves a viable window with $m_h$ between roughly $90$ and $105��{GeV}$ and stop masses of order $100$ to $165��{GeV}$, making the scenario a collider target.
  • A strong first-order transition requires $��(T_c)/T_c ��gtrsim 1$, not merely a nonzero jump in the order parameter.
  • Color-breaking minima in the stop direction constrain the allowed MSSM region; the paper separates absolute stability, metastability, and two-step transitions.
  • Two-loop corrections can strengthen the transition for small right-handed stop masses, so the one-loop Higgs-mass ceiling is not the final word.

Reading between the lines

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

  • If the linear sphaleron scaling is only approximate, the numerical Higgs-mass ceilings should be recomputed with a temperature-dependent sphaleron profile; a shift of a few GeV would move the MSSM window relative to collider reach.
  • The same washout argument gives a model-independent design target for any extension of the Standard Model: new light bosons coupled to the Higgs that generate a large cubic term while avoiding color-breaking or charge-breaking minima.
  • A future collider that measures a Higgs mass above about $110��{GeV}$ would not by itself kill all MSSM baryogenesis, but it would exclude the minimal light-stop window described here unless additional states or a two-step transition reopen it.
  • The prefactor $��kappa$ in the washout rate is the main numerical soft spot; an improved determinant calculation could move the $��zeta��gtrsim 45$ threshold and hence the Higgs-mass bound by several GeV.
Original abstract

We review different aspects of field theory at zero and finite temperature, related to the theory of phase transitions. We discuss different renormalization conditions for the effective potential at zero temperature, emphasizing in particular the MS-bar renormalization scheme. Finite temperature field theory is discussed in the real and imaginary time formalisms, showing their equivalence in simple examples. Bubble nucleation by thermal tunneling, and the subsequent development of the phase transition is described in some detail. Some attention is also devoted to the breakdown of the perturbative expansion and the infrared problem in the finite temperature field theory. Finally the application to baryogenesis at the electroweak phase transition is done in the Standard Model and in the Minimal Supersymmetric Standard Model. In all cases we have translated the condition of not washing out any previously generated baryon asymmetry by upper bounds on the Higgs mass.

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, 5,506 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.28 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.41 · compares with prior results 0.12 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We review different aspects of field theory at zero and finite temperature, related to the theory of phase transitions. We discuss different renormalization conditions for the effective potential at zero temperature, emphasizing in particular the MS-bar renormalization scheme. Finite temperature field theory is discussed in the real and imaginary time formalisms, showing their equivalence in simple examples. Bubble nucleation by thermal tunneling, and the subsequent development of the phase transition is described in some detail. Some attention is also devoted to the breakdown of the perturbative expansion and the infrared problem in the finite temperature field theory. Finally the application to baryogenesis at the electroweak phase transition is done in the Standard Model and in the Minimal Supersymmetric Standard Model. In all cases we have translated the condition of not washing out any previously generated baryon asymmetry by upper bounds on the Higgs mass.

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 review different aspects of field theory at zero and finite temperature, related to the theory of phase transitions. contribution 0.68 0.51 0.06 asserted 0.67
s1 We discuss different renormalization conditions for the effective potential at zero temperature, emphasizing in particular the MS-bar renormalization scheme. contribution 0.62 0.48 0.03 asserted 0.89
s2 Finite temperature field theory is discussed in the real and imaginary time formalisms, showing their equivalence in simple examples. contribution 0.52 0.48 0.04 asserted 1.00
s3 Bubble nucleation by thermal tunneling, and the subsequent development of the phase transition is described in some detail. contribution 0.80 0.37 0.05 asserted 0.96
s4 Some attention is also devoted to the breakdown of the perturbative expansion and the infrared problem in the finite temperature field theory. contribution 0.55 0.26 0.03 asserted 0.64
s5 Finally the application to baryogenesis at the electroweak phase transition is done in the Standard Model and in the Minimal Supersymmetric Standard Model. contribution 0.97 0.83 0.11 asserted 0.99
s6 In all cases we have translated the condition of not washing out any previously generated baryon asymmetry by upper bounds on the Higgs mass. result 0.87 0.86 0.41 asserted 0.99

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.78.

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. 1PI diagrams contributing to the one-loop effectiv
Figure 2. 1PI diagrams contributing to the one-loop effectiv
Figure 3. 1PI diagrams contributing to the one-loop effectiv
Figure 4. Contour used in the real time formalism.
Figure 5. One-loop contribution to the self-energy for the s
Figure 6. Two-loop contribution to the self-energy for the s
Figure 7. Daisy (n+1)-loop contribution to the self-energy
Figure 8. Non-daisy (n+2)-loop contribution to the self-en
Figure 9. Values of mh, mt˜ for which v(Tc)/Tc = 1 (solid line), T U c = Tc (dashed line), me U = me c U (short-dashed line), for mQ = 1 TeV and Aet = 0, 200, 300 GeV. The region on the left of the solid line is consistent with a strongly first order phase transition. A two step phase transition may occur in the regions on the left of the dashed line, while on the left of the short-dashed line, the physical vacuum at T = 0 becomes metastable. The region on the left of both the dashed and short-dashed lines leads to a stable color breaking vacuum state at zero temperature and is hence physica

Figures reproduced from arXiv:hep-ph/9901312 by the authors.

References
How later papers used it

348 later papers cite it

Evidence: substantial. 343 of the citing papers share no author with it, and 96 of those carry a confident label from Pith's citation classifier.

Pith read 39 of the 566 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 22 of 87 papers with this label
Uses its data or values 3 of 8 papers with this label
How the literature uses it

357 arXiv papers cite this in 584 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 citing it · 365 sentences without a confident label

Uses its method or tool 91 papers
Uses its data or values 8 papers
Compares with it 2 papers
Cites it as support 2 papers
Cites it as background 89 papers

Discussion

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