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REVIEW 2 major objections 4 minor 135 references

Singlet-driven and doublet-driven strong first-order electroweak phase transitions leave opposite experimental footprints: gravitational waves or large Higgs self-coupling deviations.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · grok-4.5

2026-07-13 22:23 UTC pith:72S3RUET

load-bearing objection Clean, usable map of two disjoint SFOEWPT regimes in the RxSM, with consistent one-loop trilinear couplings; the LISA/collider split is real but only semi-quantitative because of fixed wall velocity. the 2 major comments →

arxiv 2603.18799 v2 pith:72S3RUET submitted 2026-03-19 hep-ph

Investigating a strong first-order electroweak phase transition in the RxSM at future linear e^+e^- colliders and LISA

Johannes Braathen , Sven Heinemeyer , Carlos Pulido Boatella , Alain Verduras Schaeidt This is my paper
classification hep-ph
keywords RxSMstrong first-order electroweak phase transitiongravitational wavesLISAtrilinear Higgs couplingdi-Higgs productionILC1000electroweak baryogenesis
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper maps how a strong first-order electroweak phase transition can occur inside the real-singlet extension of the Standard Model and shows that the transition falls into two disjoint classes with opposite experimental signatures. When the transition is driven mainly by the singlet field, the 125 GeV Higgs boson stays essentially Standard-Model-like, so colliders see almost nothing, yet the latent heat and duration of the transition produce a stochastic gravitational-wave background that LISA can detect with signal-to-noise ratio greater than 10. When the transition is instead driven by the electroweak doublet, the same barrier that makes the transition strong forces the trilinear Higgs self-coupling to be 35–70 percent larger than its Standard-Model value; that deviation produces clear excesses or deficits in di-Higgs production at a 1 TeV electron-positron collider, while the gravitational-wave signal remains marginal. Because both calculations consistently include one-loop corrections to the same trilinear couplings, the two probes become complementary: a positive LISA signal with no collider deviation points to the singlet-driven class, while a large measured self-coupling with a weak gravitational-wave signal points to the doublet-driven class. The result therefore supplies a concrete experimental strategy for reconstructing which shape of the Higgs potential was realised in the early Universe.

Core claim

In the RxSM the parameter regions that realise a strong first-order electroweak phase transition split into two disjoint phenomenological classes: singlet-driven transitions produce LISA signal-to-noise ratios above 10 while the one-loop trilinear Higgs coupling modifier stays within a few percent of unity, whereas doublet-driven transitions force that modifier into the range 1.35–1.7 (and correspondingly large deviations in e+e- di-Higgs rates) while the LISA signal-to-noise ratio stays below 10 for almost the entire plane.

What carries the argument

Two benchmark planes that isolate the two driving directions of the transition, with all relevant trilinear scalar couplings evaluated at full one-loop order both for the finite-temperature effective potential and for the collider cross sections.

Load-bearing premise

The claimed split between “detectable” and “undetectable” gravitational-wave signals rests on fixing the bubble-wall velocity at the pessimistic value 0.95; a lower velocity would raise the signal-to-noise ratio by a large factor and could erase the dichotomy.

What would settle it

A LISA detection with SNR greater than 10 together with a measured trilinear Higgs coupling still consistent with the Standard Model within a few percent would confirm the singlet-driven class; a measured coupling modifier between 1.35 and 1.7 with no corresponding LISA signal would confirm the doublet-driven class; either opposite combination would falsify the claimed complementarity.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A positive LISA signal with SM-like Higgs couplings would select the singlet-driven region of the RxSM and rule out a pure doublet-driven barrier.
  • A measured trilinear coupling modifier of order 1.5 at ILC1000, even without a LISA signal, would select the doublet-driven region and indicate that the electroweak barrier itself drove the transition.
  • Differential m_hh distributions at a 1 TeV e+e- collider can separately resolve the resonant heavy-Higgs peak and the continuum shift induced by the modified self-coupling, giving two independent handles on the same potential.
  • Because the same one-loop couplings enter both the phase-transition calculation and the collider rates, a joint measurement would reconstruct which field direction dominated the early-Universe transition.

Where Pith is reading between the lines

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

  • If future lattice or real-time simulations fix the bubble-wall velocity well below 0.95, the “GW-only” window of the singlet-driven plane may shrink and the two classes could begin to overlap in SNR space.
  • The same complementarity logic should apply to any multi-scalar model in which the barrier can be generated either by a singlet or by the electroweak doublet itself; the RxSM merely supplies the cleanest laboratory.
  • A null result at both LISA and ILC1000 would not kill the possibility of a strong first-order transition, but would force it into a narrow intermediate strip where both signals are just below threshold.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript studies the general real-singlet extension of the SM (RxSM) as a minimal setting that can realise a strong first-order electroweak phase transition (SFOEWPT). Using BSMPT 3.1.1 (with the RxSM newly implemented) the authors map the thermal histories of the scalar potential and identify six distinct patterns, three of which produce SFOEWPTs. Two representative benchmark planes are constructed: BP1 (singlet-driven) and BP2 (doublet-driven). One-loop corrections to the trilinear couplings λ_hhh and λ_hhH are evaluated consistently with anyH3 in a fully on-shell scheme and are fed both into the thermal analysis and into MadGraph predictions for e^{+}e^{-} o Zhh and e^{+}e^{-} o u u hh at ILC1000. The central claim is that the two classes of SFOEWPT yield complementary experimental signatures: BP1 produces LISA SNRs ≳ 10 while κ_λ remains SM-like; BP2 forces 1.35 ≲ κ_λ^{(1)} ≲ 1.7 (and correspondingly large deviations in the di-Higgs rates) while the LISA SNR stays below 10 for almost the entire plane.

Significance. If the reported dichotomy holds, the work supplies a concrete, falsifiable illustration of the complementarity between future e^{+}e^{-} colliders and LISA for reconstructing the shape of the Higgs potential. The consistent inclusion of one-loop trilinear couplings in both the thermal and collider calculations, the public-code implementation of the RxSM in BSMPT, and the explicit differential distributions with experimental cuts and polarisations constitute genuine technical advances over earlier RxSM studies. The two benchmark planes cleanly separate the phenomenological regimes and therefore provide useful targets for both communities.

major comments (2)
  1. The SNR values that underpin the claimed “detectable vs. undetectable” split between BP1 and BP2 are computed exclusively with the fixed, pessimistic wall velocity v_w = 0.95 (and three years of LISA observation). The text itself notes (around the discussion of Fig. 4 and Fig. 6) that a lower velocity v_w ∼ 0.6 can raise the SNR by a large factor. Because v_w is not derived from the model parameters, the quantitative complementarity is only semi-quantitative; a short sensitivity scan or an explicit statement of the range of v_w over which the dichotomy survives is needed to make the central claim robust.
  2. The linear relations that define BP1 (Eq. (1)) and BP2 (Eq. (2)) are chosen so as to maximise either the GW signal or the κ_λ deviation. While the choices are transparent, the paper should demonstrate that the qualitative split (SM-like κ_λ + large SNR versus large κ_λ + small SNR) persists for nearby, less optimised slices of parameter space; otherwise the complementarity could be an artefact of the particular hyperplanes selected.
minor comments (4)
  1. Fig. 1 captions and the accompanying text use the labels A–F for thermal histories; it would help the reader if the same letters were printed directly on the panels.
  2. The units of κ_SH in the right-hand panel of Fig. 6 and in the caption of Fig. 7 are occasionally written as “TeV” while the plotted values are in GeV; a uniform convention would avoid confusion.
  3. A brief remark on the numerical stability of the one-loop effective potential near the vacuum-trapping boundary (region F of BP2) would reassure the reader that the sharp transition from SFOEWPT to trapping is not a lattice artefact of BSMPT.
  4. The polarisation choice (−80 %, +30 %) and the integrated luminosity 3200 fb^{-1} for ILC1000 are stated only in the caption of Fig. 7; they should also appear in the main text of Sec. 5.

Circularity Check

1 steps flagged

No load-bearing circularity; only minor explicit parameter tuning of benchmark planes to illustrate high-SNR regions already identified in the scan.

specific steps
  1. fitted input called prediction [Section 4 (Benchmark plane 1), eq. (1) and surrounding text; also left panel of Fig. 4]
    "Here it should be kept in mind that the choice of κ_S H and v_S that defines BP1, see eq. (1) was made to ensure that large parts of this parameter plane exhibit high values of the SNR (the maximal values of the SNR that are reached are however not particularly dependent on this choice)."

    The linear relations that fix κ_SH(v_S) in terms of cos α are deliberately chosen so that the subsequent BSMPT evaluation yields SNR ≳ 10 over a sizable fraction of the displayed plane. Reporting “strong GW signals” inside that plane is therefore partly by construction of the slice, even though the existence of high-SNR points was first established in the unrestricted scan of Fig. 2 and the paper notes that peak SNR values are robust. This is a mild, acknowledged selection effect rather than a definitional identity or a data-driven fit renamed as a prediction.

full rationale

The paper's central claims (two disjoint classes of SFOEWPT thermal histories in the RxSM, with complementary GW vs. collider signatures) are obtained by scanning the seven-dimensional parameter space with the public tool BSMPT (for nucleation/percolation temperatures, ξ_n, α, β/H_*, and LISA SNR) and anyH3 (for one-loop on-shell trilinear couplings λ_hhh and λ_hhH), then feeding the latter into MadGraph for e⁺e⁻ di-Higgs rates. These computations are independent of one another and of external data fits. The two benchmark planes are defined after the scan (eqs. (1)–(2)) by fixing linear relations among κ_S, κ_SH, v_S and cos α that populate the high-ξ_n regions already found; the paper itself states that the choice for BP1 “was made to ensure that large parts of this parameter plane exhibit high values of the SNR” while noting that the maximal SNR values “are however not particularly dependent on this choice.” This is ordinary illustrative selection, not a self-definitional or fitted-input “prediction.” Self-citations (to the authors’ prior anyH3/renormalisation papers and to the full arXiv:2510.12569 version of the same work) supply the computational framework but are not load-bearing uniqueness theorems or unverified premises; the thermal histories, κ_λ ranges and SNR maps are new numerical results. The fixed v_w = 0.95 assumption affects quantitative SNR thresholds but is an external dynamical input, not a circular reduction. Hence the derivation chain is self-contained against the paper’s own equations and tools; score remains at the low end of the 0–2 band.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The central claim rests on the standard RxSM Lagrangian, the one-loop finite-temperature effective potential as implemented in BSMPT, the on-shell renormalisation of trilinear couplings from the authors’ earlier work, and two hand-chosen linear relations that define the benchmark planes. No new particles or forces are postulated; the free parameters are the usual seven of the RxSM plus the fixed wall velocity used for SNR estimates.

free parameters (3)
  • bubble wall velocity vw
    Fixed by hand to the pessimistic value 0.95 for all SNR calculations; the paper notes that lower values can change the SNR by large factors, so the detectability claim depends on this choice.
  • κ_S, κ_SH, v_S linear relations that define BP1 and BP2
    Chosen by the authors so that large regions of each plane exhibit either high SNR or large κ_λ; they are not derived from a more fundamental principle.
  • seven RxSM input parameters (m_h, m_H, α, v, v_S, κ_S, κ_SH)
    Scanned or fixed within ranges allowed by theory and experimental constraints; the existence of SFOEWPT regions is a numerical output of these inputs.
axioms (3)
  • domain assumption The one-loop finite-temperature effective potential (with the daisy resummation scheme of BSMPT) correctly captures the nucleation and percolation temperatures of the RxSM.
    Standard working assumption of the thermal-history literature; higher-order corrections can shift the strength of the transition by O(10–30 %).
  • domain assumption The fully on-shell renormalisation scheme of Ref. [46] for the trilinear couplings λ_hhh and λ_hhH is the appropriate scheme for both the thermal potential and the collider matrix elements.
    Taken from the authors’ previous paper; scheme dependence of the one-loop corrections is not re-evaluated here.
  • domain assumption Perturbative unitarity, boundedness-from-below and HiggsTools constraints exhaust the relevant theoretical and experimental bounds on the RxSM parameter space.
    Standard checklist; additional constraints (e.g. from electroweak precision or dark-matter searches if the singlet is stable) are not re-derived.

pith-pipeline@v1.1.0-grok45 · 20597 in / 3017 out tokens · 25614 ms · 2026-07-13T22:23:23.202277+00:00 · methodology

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read the original abstract

The general real singlet extension of the Standard Model (SM), the RxSM, is one of the simplest theories Beyond-the-Standard Model (BSM) that can accommodate a strong first-order electroweak phase transition (SFOEWPT). We investigate the possible thermal histories of the scalar potential in the RxSM, and the regions of the model parameter space in which SFOEWPT can be realised. We then explore complementary avenues to probe such scenarios experimentally: either using searches for a stochastic background of gravitational waves (GWs), or using searches for di-Higgs production processes at future collider experiments, focusing on the case of a high-energy $e^+e^-$ collider. An important aspect of our work is that one-loop corrections to all relevant trilinear scalar couplings are consistently included both in the calculation of dynamics of the electroweak phase transition (EWPT) and in collider processes. We find entirely different phenomenological signatures for different parts of the RxSM parameter space giving rise to SFOEWPTs. On the one hand, if the SFOEWPT is driven by the singlet field, the 125 GeV Higgs boson is very SM-like and signs of BSM physics would be difficult to find at colliders, but strong GW signals could be produced. On the other hand, in scenarios where a SFOEWPT is driven by the doublet field, BSM deviations in properties of the detected Higgs boson, particularly in its trilinear self-coupling, typically lead to observable signals at colliders, while detectable GW signals are much more challenging to achieve. This work highlights the complementarity of collider experiments and cosmological observations to determine the dynamics of the EWPT and reconstruct the shape of the Higgs potential realised in Nature.

Figures

Figures reproduced from arXiv: 2603.18799 by Alain Verduras Schaeidt, Carlos Pulido Boatella, Johannes Braathen, Sven Heinemeyer.

Figure 1
Figure 1. Figure 1: Tracing of the different minima of the potential as a function of the temperature for the six thermal histories possible in the RxSM. Blue lines represent the minima of the EW doublet and orange lines those of the singlet field, while red vertical lines indicate the critical temperature Tc . The solid lines represent the path followed by the Universe. 400 600 800 mH [GeV] 0.980 0.985 0.990 0.995 1.000 cos … view at source ↗
Figure 2
Figure 2. Figure 2: RxSM parameter scan results for ξn ≡ vn/Tn. Left: {mH, cos α} plane; right: {vS , mH} plane. SFOEWPT, differing by whether the initial (high-temperature) value of vS is negative (case D) or positive (case E). The difference in the high-temperature value of vS has a strong impact on which field dominates dynamics of the EWPT and in turn on the associated phenomenol￾ogy, as we will see below. Concerning the … view at source ↗
Figure 3
Figure 3. Figure 3: Parameter scan results for BP1, with ξn indicated by the colour of the scatter points. The different thermal histories are labelled following fig. 1. Coloured regions are excluded perturbative unitarity (light blue), NLO stability of the EW vacuum (grey), and direct searches for heavy Higgs bosons [126] (light red). In region B, ξn ∈ [0, 1]. 260 280 300 320 mH [GeV] 0.985 0.990 0.995 cos α 0 10 20 30 40 50… view at source ↗
Figure 4
Figure 4. Figure 4: Parameter scan results for points in benchmark plane 1 with a SFOEWPT (regions C and D of fig. 3). Left: SNR at LISA (assuming vw = 0.95 and three years of observation time) in the plane {mH, cos α}. The red line indicates the region with SNR > 10. Right: ξn in the plane {mH, κ (1) λ } for points with SNR > 10. 5 Benchmark plane 2: a doublet-driven EWPT We turn now to the second plane (BP2), related to the… view at source ↗
Figure 5
Figure 5. Figure 5: Left: parameter scan results for benchmark plane 2, with ξn shown by the colour coding of the scatter points. The labelling of different thermal histories follows that of fig. 1. Coloured regions are excluded by vacuum trapping (pink), NLO stability of the potential (grey), or experimental searches (red by Ref. [134], and orange by Ref. [135]). Right: Temperature-dependent one-loop effective potential at t… view at source ↗
Figure 6
Figure 6. Figure 6: Parameter scan results for points in benchmark plane 2 with a SFOEWPT (region E of fig. 5). Left: SNR at LISA (assuming vw = 0.95 and three years of observation time) in the plane {mH, κS H}; right: ξn in the plane {mH, κ (1) λ }. in the most optimistic case, a detection of primordial GW from the SFOEWPT would only be possible for a borderline region of BP2. The right panel of fig. 6 shows for the same sca… view at source ↗
Figure 7
Figure 7. Figure 7: Differential polarised di-Higgs production cross-section distributions as a function of the di-Higgs invariant mass mhh, at the ILC1000. Left: for the process e + e − → Zhh → Zbbb¯ b¯; right: for the process e + e − → νν¯hh → νν¯bbb¯ b¯. The right axis indicates the sum of event numbers for the two polarisations (∓80%, ±30%) of the electrons and positrons, considering Lint = 3200 fb−1 each. Values of λhhH … view at source ↗

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