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Strong first-order phase transitions in the NMSSM --- a comprehensive survey

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The NMSSM can host strong first-order electroweak phase transitions in two of its four phase histories.

desk verdict A comprehensive, honestly hedged NMSSM phase-structure survey with a new tool and classification; the main caveat is gauge dependence that could shift some categorical claims. read the letter →

arxiv 1908.11847 v3 pith:4ZHIYVC5 submitted 2019-08-30 hep-ph

Peter Athron , Csaba Balazs , Andrew Fowlie , Giancarlo Pozzo , Graham White , Yang Zhang This is my paper · ORCID
classification hep-ph
keywords NMSSMelectroweakphasetransitionfirst-orderbaryogenesisgravitationalwavessingletextensionHiggssectorfinite-temperatureeffectivepotential
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks which symmetry-breaking histories in the NMSSM can produce the strongly first-order electroweak phase transition needed for electroweak baryogenesis. By tracing the minima of the finite-temperature effective potential over a scan of millions of parameter points, it identifies four distinct phase histories and shows that only two of them, Type-H-and-S and Type-Only-S(maintain), can yield completing strong transitions with strength above one that end in the observed vacuum. The other two histories, Type-Only-H and Type-Only-S(flip), contain strong-looking transitions that fail to nucleate. If the classification is right, it narrows the viable NMSSM parameter space and sharpens predictions for Higgs searches and gravitational wave experiments.

What carries the argument

The machinery is the temperature-traced minima of the one-loop finite-temperature effective potential, including daisy resummation, for the NMSSM matched to a two-Higgs-doublet-plus-singlet effective theory. The central object is the order parameter γ_EW, the jump in the Higgs-field combination divided by the critical temperature; the paper uses phase tracing to identify which pairs of minima become degenerate, then checks bubble nucleation for benchmark points to decide whether a transition actually completes in the early Universe.

What would settle it

Recompute the paper's benchmark points using a gauge-invariant effective potential, such as a full two-loop or dimensionally reduced calculation: if any Type-H-and-S or Type-Only-S(maintain) point loses its first-order character or drops below γ_EW = 1, or if any Type-Only-H or Type-Only-S(flip) point nucleates successfully, the central viability dichotomy would fail.

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Extended reading notes

Core claim

The central claim is that the NMSSM's electroweak phase structure is classified by the first symmetry-breaking transition: Type-H-and-S, where Higgs and singlet VEVs appear together; Type-Only-H, where only Higgs VEVs appear first; and Type-Only-S, where the singlet gets a VEV first and during the strongest transition either maintains its sign or flips it. Across the scanned parameter space, only Type-H-and-S and Type-Only-S(maintain) contain strong first-order transitions with γ_EW > 1 that complete and end in the observed electroweak vacuum. The Type-Only-H histories require an intermediate transition that temporarily restores electroweak symmetry, and that transition never nucleates; the Type-Only-S(flip) histories have a strongest transition whose tunneling action is too large to complete. The scan also produces sharp phenomenological predictions: quartic couplings near λ ≈ 0.6 and κ ≈ 0.2, small tan β, an effective μ parameter below about 300 GeV, and the 125 GeV Higgs often being the second-lightest CP-even scalar rather than the lightest.

Load-bearing premise

The classification assumes that the one-loop finite-temperature effective potential with daisy resummation reliably decides the order, strength, and completion of every transition, despite the authors' own note that the critical temperature and order parameter are gauge dependent at this truncation.

Editorial extensions

If this is right

  • Only Type-H-and-S and Type-Only-S(maintain) histories can support electroweak baryogenesis among the scanned NMSSM points.
  • Type-Only-H and Type-Only-S(flip) histories should not be counted as baryogenesis candidates, because their necessary transitions fail to nucleate.
  • Viable strong transitions concentrate in a narrow parameter window: λ ≈ 0.6, κ ≈ 0.2, small tan β, and light higgsino mass parameter.
  • The 125 GeV Higgs is usually the second-lightest CP-even Higgs, not the lightest, in samples that pass collider constraints.
  • Type-Only-S(maintain) histories can provide two-step transitions that may yield observable gravitational wave signatures.

Reading between the lines

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

  • The same four-way classification logic likely applies to any singlet-extended Higgs model, and the paper's THDMS matching means its results carry over to a subspace of non-supersymmetric models.
  • Earlier scans that located only the strongest transition may have overcounted viable baryogenesis points, since the paper shows that a strong transition can be unreachable in the actual cosmological history.
  • The preference for the 125 GeV Higgs being the second-lightest state gives a concrete target for searches for an additional light CP-even scalar with suppressed couplings.
  • A gauge-invariant or higher-order calculation could shift individual benchmarks between the categories, so the viability boundary should be rechecked with such methods before relying on the precise parameter limits.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper studies the thermal phase structure of the Z3-symmetric NMSSM matched to a THDMS effective field theory, with all superpartners integrated out. Using a one-loop effective potential with daisy resummation in Feynman gauge, the authors trace the temperature dependence of the minima with their new code PhaseTracer, cross-checked against CosmoTransitions for benchmarks. From a MultiNest scan over eight NMSSM parameters, subject to LHC Higgs constraints, LEP chargino bounds, and a sampling penalty that targets gamma_EW ~ 1, they classify the phase histories into Type-H-and-S, Type-Only-H, and Type-Only-S(maintain/flip). They report that Type-H-and-S and Type-Only-S(maintain) benchmarks can contain completing strong first-order electroweak phase transitions ending in the observed vacuum, whereas their Type-Only-H and Type-Only-S(flip) benchmarks fail to nucleate. They also characterize the associated parameter regions, including lambda ~ 0.6, kappa ~ 0.2, small tan beta, and the frequent occurrence of the SM-like Higgs as the second-lightest CP-even state.

Significance. If the results hold, this is a useful step beyond earlier NMSSM phase-transition studies: it maps multi-phase histories rather than assuming a single transition between the symmetric and broken minima, shows that not every candidate transition actually completes, and identifies which phase histories can in principle support electroweak baryogenesis or gravitational-wave signals. The paper ships model files and uses public spectrum and sampling tools, and the benchmark results are independently cross-checked with CosmoTransitions. The quantitative predictions for lambda, kappa, tan beta, and the Higgs mass ordering are falsifiable by future collider and gravitational-wave observations. The main limitation is that the central classification and viability verdicts rest on a one-loop, daisy-resummed effective potential whose gauge dependence is acknowledged but only qualitatively checked; this is the key risk to the strength of the conclusions.

major comments (3)
  1. [Sec. 4 and Sec. 5.3] The gauge dependence of the one-loop potential is load-bearing, not a side remark. The authors state in Sec. 4 that at this truncation the critical temperature is gauge dependent and that the order parameter gamma_EW defined in Eq. (33) always depends on the gauge parameter xi. The distinction between Type-Only-S(maintain) and Type-Only-S(flip) is defined by which transition has the largest gamma_EW, so a moderate xi-shift can move a sample between categories. Likewise, the no-nucleation verdicts for the Type-Only-H and Type-Only-S(flip) benchmarks are based on bounce actions computed from the same gauge-dependent potential, so an O(1) change in SE/TN in Eq. (34) could turn a 'cannot complete' benchmark into a completing one. The Landau-gauge check reported in Sec. 5.3 is described only as 'typically mild' and is not tabulated per transition. Please provide a quantitative xi = 1 versus xi = 0 comparison of TC, gamma_EW, and SE/TN for all four benchmark points, and show explicitly that the strongest-transition label and the completion outcomes are unchanged. Without this, the benchmark-level viability conclusions are not yet demonstrated to be gauge independent.
  2. [Sec. 5.5] The paper describes the work as a 'comprehensive survey' and uses the scan to make statements about relative rarity, such as Type-Only-H being by far the rarest scenario and the absence of negative-mu samples in some categories. However, the final paragraph of Sec. 5.5 concedes that the sampling coverage at large m_h3 is inadequate and that large masses 'may just be rare with our sampling strategy.' Since MultiNest is an importance-sampling algorithm with a targeted penalty chi2_SFOPT of Eq. (37), the raw number of saved points does not by itself quantify coverage of the eight-dimensional parameter space. Please provide a coverage or convergence diagnostic per category, or rephrase the frequency and absence claims as upper limits. This does not invalidate the four benchmark analyses, but it is needed to support the 'comprehensive' and 'whole parameter space' claims in the abstract and Sec. 5.2.
  3. [Secs. 5.2 and 5.4] The classification labels each sample by its 'strongest FOPT' even when that transition is not on the realized cosmological history, as in Type-Only-H and in the samples where the strongest FOPT does not end in the SM vacuum. The text explains this in Sec. 5.2, but the scatter plots in Figs. 3, 5, 6, 8, 11, and 12 do not distinguish points whose strongest transition is part of a completing history from those for which it is not. Since the paper's headline conclusion is about which scenarios could in principle help provide a viable theory of electroweak baryogenesis, please mark in the figures which samples pass the nucleation check, or clearly state in the figure captions and conclusions that the displayed samples are only candidate transitions pending nucleation calculations.
minor comments (4)
  1. [Sec. 4] The sentence 'The Nielsen identities in (20) imply that the critical temperature is gauge independent' is potentially misleading in its immediate context, because the next paragraph explains that the one-loop truncation makes the critical temperature gauge dependent. Please rephrase to make explicit that gauge independence holds for the full effective potential at exact extrema, not for the truncated one-loop result used here.
  2. [Table 2] The entry 'N/A' for the first transition in the Type-Only-H column is unclear: the text says this transition is second order, but the table label '2nd at T = 155' is ambiguous. Please state explicitly that the first EW-breaking transition is a crossover and therefore has no critical temperature or nucleation temperature.
  3. [Table 1] The table lists ranges for |kappa|, |A_lambda|, |A_kappa|, |A_t|, and |v_S|, while the text says both positive and negative values are considered. Please clarify in the table caption that the scan covers both signs with the stated magnitude ranges, or change the table entries to signed ranges.
  4. [Abstract and Sec. 6] The abstract and conclusions state that the paper 'checked which scenarios could in principle help provide a viable theory of EW baryogenesis.' Given that only four benchmark points receive nucleation checks and the scan samples are not all checked for completion, I suggest adding a qualifier such as 'for benchmark points' to these summary statements, so that the scope is not overstated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the phase-structure classification and SFOPT samples are emergent outputs of an ab initio scan, with benchmark nucleation independently checked in CosmoTransitions.

full rationale

The paper's central claims are emergent outputs of a parameter scan, not identities built into the inputs. The inputs are the NMSSM/THDMS Lagrangian parameters, matching conditions, and running in FlexibleSUSY; the outputs are minimum trajectories, critical temperatures, the order parameter gamma_EW, and nucleation actions computed with PhaseTracer and CosmoTransitions. The sampling penalty chi2_SFOPT in Eq. (37) targets gamma_EW approximately 1, but it does not encode the four phase-history categories, the sign-maintain/sign-flip distinction, the strongest-transition assignment, or the no-nucleation verdicts; these arise from tracing the finite-temperature potential. Benchmark points are independently checked with CosmoTransitions, and the nucleation condition SE/TN ~ 140 is applied to those benchmarks, so no fitted parameter is renamed as a prediction. The gauge dependence of the one-loop potential is explicitly acknowledged in Sec. 4 and checked in Landau gauge in Sec. 5.3, which is an honest limitation and a correctness risk rather than a circular step. Self-citations to FlexibleSUSY and the thermal-function implementation are tool citations backed by external code, not load-bearing uniqueness theorems invoked to forbid alternatives. The claimed features such as lambda around 0.6 and kappa around 0.2, and the tendency for the SM-like Higgs to be the second-lightest, are conditional properties of the surviving samples after the LEP/LHC and SFOPT selection, not restatements of the sampling penalty or of any fitted parameter.

Assumptions & free parameters 9 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the standard NMSSM/THDMS setup with eight scanned parameters (ranges chosen by hand), a one-loop effective potential with daisies, and a phase-tracing algorithm. No new physical entities are introduced. The sampling penalty centered on gamma_EW about 1 guides the scan but does not by itself create the parameter-space structure.

free parameters (9)
  • lambda = scanned flat over [0, pi/2]
    NMSSM singlet-Higgs quartic coupling; range chosen to remain perturbative.
  • kappa = scanned |kappa| flat over [0, pi/2] with sign
    Singlet self-coupling; sign sampled.
  • A_lambda = scanned |A_lambda| in [0, 10 TeV] with hybrid metric
    Soft trilinear term associated with lambda.
  • A_kappa = scanned |A_kappa| in [0, 10 TeV] with hybrid metric
    Soft trilinear term associated with kappa.
  • A_t = scanned |A_t| in [0, 10 TeV] with hybrid metric
    Stop trilinear coupling entering the threshold correction to lambda_2.
  • m_SUSY = scanned log-uniform over [1, 10 TeV]
    Geometric mean of stop soft masses; sets the EFT matching scale.
  • v_S = scanned |v_S| in [0, 10 TeV] with hybrid metric and sign
    Singlet VEV input at the top mass scale.
  • tan beta = scanned log-uniform over [1, 60]
    Ratio of the two Higgs doublet VEVs.
  • sigma_SFOPT = 0.2
    Width of the Gaussian penalty in Eq. (37) used to focus the scan on gamma_EW around 1; chosen by hand and shapes the distribution of transition strengths in the sample.
assumptions (7)
  • domain assumption One-loop effective potential with Arnold-Espinosa daisy resummation determines the phase structure.
    The entire classification of first- vs. second-order transitions and the order parameter gamma_EW are computed from this potential (Sec. 3.2).
  • domain assumption All superpartners are heavy enough to be integrated out at m_SUSY; the NMSSM is matched to the THDMS.
    The scan uses the THDMS EFT below m_SUSY; if stops were light, thermal contributions would change the transition (Sec. 2.1).
  • domain assumption CP violation is assumed to enter radiatively from outside the Higgs sector; the Higgs potential is treated as real and CP-conserving.
    The authors state this simplification restricts the study to PTs between CP-even vacua (Sec. 1, Sec. 2.1).
  • domain assumption No charge- or CP-breaking minima are considered.
    Field-dependent masses set charged and CP-odd Higgs fields to zero, and the VEVs are assumed real (Sec. 3.1, footnote 4).
  • standard math Bubble nucleation occurs when S_E(T_N)/T_N is about 140.
    Used to decide whether a transition completes for the benchmark points (Eq. 34).
  • domain assumption The PhaseTracer phase-tracing algorithm correctly identifies all relevant minima and their temperature evolution.
    The classification of phase histories relies on this numerical method (App. B); it traces phases from T=0 and T=1 TeV minima, so intermediate phases could in principle be missed.
  • domain assumption The Z3 domain wall problem can be avoided without affecting phenomenology, following Refs. [78-80].
    Invoked to justify the Z3-symmetric NMSSM (Sec. 2).

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Cite this review

Pith. "Pith review of Strong first-order phase transitions in the NMSSM --- a comprehensive survey." pith.science (2026). https://pith.science/paper/4ZHIYVC5

@misc{pith2026190811847,
  author       = {Pith},
  title        = {Pith review of: Strong first-order phase transitions in the NMSSM --- a comprehensive survey},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ZHIYVC5}},
  note         = {Machine review of arXiv:1908.11847}
}
read the original abstract

Motivated by the fact that the Next-to-Minimal Supersymmetric Standard Model is one of the most plausible models that can accommodate electroweak baryogenesis, we analyze its phase structure by tracing the temperature dependence of the minima of the effective potential. Our results reveal rich patterns of phase structure that end in the observed electroweak symmetry breaking vacuum. We classify these patterns according to the first transition in their history and show the strong first-order phase transitions that may be possible in each type of pattern. These could allow for the generation of the matter-antimatter asymmetry or potentially observable gravitational waves. For a selection of benchmark points, we checked that the phase transitions completed and calculated the nucleation temperatures. We furthermore present samples that feature strong first-order phase transitions from an extensive scan of the whole parameter space. We highlight common features of our samples, including the fact that the Standard Model like Higgs is often not the lightest Higgs in the model.

Figures

Figures reproduced from arXiv: 1908.11847 by the authors.

Figure 1
Figure 1. Phase structures for typical points in the categories Type-H-and-S (upper left), Type-Only-H (upper right), Type-Only-S(maintain) (lower left) and Type-Only-S (flip) (lower right). The lines show the field values at a particular minimum as a function of temperature. The arrows indicate that at that temperature the two phases linked by the arrows are degenerate and thus that a FOPT could occur in the direction of the… view at source ↗
Figure 2
Figure 2. The Higgs and singlet field values at the true and false minima at the critical temperature of the strongest FOPT for samples for which the strongest FOPT ends in the SM vacuum. The lower panels of [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. The critical temperature and order parameter for the strongest PTs for samples for which the strongest FOPT ends in the SM vacuum. The points are colored by the effective µ-parameter. temperature and the strength of the PT. We now delineate the regions of the NMSSM parameter space in which our four scenarios occur. We checked that in all scenarios the stops were truly decoupled by checking stop mixing, Xt = At−µeff … view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: The parameters (µeff,tan β) for samples for which the strongest FOPT ends in the SM vacuum. The points are colored by the γEW of the strongest FOPT. sensitive to our models. Samples with µeff < 0 were extremely rare in the Type-H-and-S and Type-Only-S(maintain) scenari…
Figure 5
Figure 5. Figure 5: The quartics (λ, κ) for samples for which the strongest FOPT ends in the SM vacuum. The points are colored by the γEW of the strongest FOPT. positive Aκ with negative κ, as well as a horizontal slice of points at Aκ ≈ 10 GeV for Type-H-and-S and Type-Only-S(maintain). …
Figure 6
Figure 6. Figure 6: The trilinears (Aλ, Aκ) for samples for which the strongest FOPT ends in the SM vacuum. The points are colored by the parameter κ. 5.4.2 The strongest FOPT does not end in the SM vacuum Other than the scenario discussed above, we have plenty of samples in which the str…
Figure 7
Figure 7. Figure 7: The Higgs and singlet fields at the critical temperature of the strongest FOPT for samples for which the strongest FOPT does not end in the SM vacuum. for each sample. Let us stress that strictly speaking, we count the number of temperatures at which two vacua are dege…
Figure 8
Figure 8. Figure 8: The critical temperature and order parameter for the strongest PTs for samples for which the strongest FOPT does not end in the SM vacuum. The points are colored by the effective µ-parameter. not end in the SM vacuum could still potentially explain the observed baryon …
Figure 9
Figure 9. Figure 9: Number of FOPTs with γEW & 1 per point, for points for which the strongest FOPT ends in the SM vacuum (left panel) and does not end in the SM vacuum (right panel). for the strongest FOPT, displayed in the lower left panel of [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: (µeff,tan β) for samples for which the strongest FOPT does not end in the SM vacuum. The points are colored by the γEW of the strongest FOPT. with masses below 600 GeV in the samples from our scan, with the SM-like Higgs being either h1 or h2. In [PITH_FULL_IMAGE:fig…
Figure 11
Figure 11. Figure 11: (λ, κ) for samples for which the strongest FOPT does not end in the SM vacuum. The points are colored by the γEW of the strongest FOPT. The reason we see so few samples where the SM-like Higgs is the lightest state for the categories mentioned above seems to be the co…
Figure 12
Figure 12. Figure 12: (Aλ, Aκ) for samples for which the strongest FOPT does not end in the SM vacuum. The points are colored by the κ-parameter. Lastly, we note that many of the panels in [PITH_FULL_IMAGE:figures/full_fig_p027_12.png]
Figure 13
Figure 13. Figure 13: Masses of the non-SM-like Higgs bosons in our four scenarios, for points for which the strongest FOPT ends in the SM vacuum (left block of four plots) and does not end in the SM vacuum (right block of four plots). We show points satisfying µ > 100 GeV and γEW > 1 (gra…

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