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Complementarity of gravitational wave analyses and di-Higgs production in the exploration of the Electroweak Phase Transition dynamics in the RxSM

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper claims that in the real-singlet extension of the Standard Model, the field-space direction that drives a strong first-order electroweak phase transition determines whether the transition is visible in gravitational waves or in di-

desk verdict A solid, carefully done RxSM study with a plausible singlet/doublet complementarity claim, but the claim rests on two tailored benchmark planes and one region-description inconsistency. read the letter →

arxiv 2510.12569 v2 pith:OEACCBCR submitted 2025-10-14 hep-ph

Johannes Braathen , Sven Heinemeyer , Carlos Pulido Boatella , Alain Verduras Schaeidt This is my paper
classification hep-ph
keywords electroweakphasetransitionstrongfirst-orderrealsingletextensiongravitationalwavesdi-HiggsproductionHiggstrilinearcouplingLISAfuturecolliders
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 studies the simplest extension of the Standard Model with a real singlet scalar, which can make the electroweak phase transition strongly first order. It claims that the way the transition proceeds determines which experimental probe will see it. If the transition is driven by the singlet field, the Higgs boson stays Standard-Model-like and di-Higgs production shows nothing, but bubble dynamics produce gravitational waves detectable by a space-based observatory in a substantial region. If the transition is driven by the Higgs doublet, the trilinear Higgs coupling deviates strongly and di-Higgs distributions at future colliders should show clear deviations, while gravitational waves are detectable only in a narrow strip. This complementarity matters because no single experiment can cover the full parameter space of a minimal Higgs-sector extension.

What carries the argument

The central organizing object is the direction of the tunnelling path in the (doublet, singlet) field space during the SFOEWPT, identified by the sign and size of the singlet VEV at nucleation. The quantitative machinery is the one-loop, temperature-dependent effective potential (with thermal resummation and an on-shell-like renormalization) used to compute the bounce action, the nucleation temperature Tn, the phase-transition strength ξn = vn/Tn, and the gravitational-wave power spectrum; consistency requires using the same one-loop order for the trilinear couplings λhhh and λhhH that enter di-Higgs production. The key output relation is the anticorrelation: singlet-driven transitions give

What would settle it

A scan over the full five-dimensional RxSM parameter space that locates a singlet-driven SFOEWPT with a sizeable deviation in the Higgs trilinear coupling, or a doublet-driven SFOEWPT with a LISA-detectable gravitational-wave signal while κλ remains near one, would falsify the claimed dichotomy; conversely, observing both a LISA background and clear di-Higgs deviations for the same parameters would also break it.

Watch

Extended reading notes

Core claim

The paper establishes that the RxSM's strong first-order electroweak phase transition (SFOEWPT) comes in two phenomenologically disjoint varieties. In the first, the transition is driven by the singlet field direction: tunnelling starts from a negative singlet VEV, the nucleation temperature is low (Tn ≲ 70 GeV), and the resulting stochastic gravitational-wave background is detectable at LISA with signal-to-noise ratios above 10 for a conservative wall velocity vw = 0.95 across a large part of the plane; at the same points the trilinear couplings λhhh and λhhH are essentially SM-like, so di-Higgs production is indistinguishable from the Standard Model. In the second, the transition is driven

Load-bearing premise

The paper's central separation is demonstrated only on two specially chosen two-dimensional slices of the five-dimensional parameter space, and the paper itself calls the analysis a proof of concept; if those slices are not representative of the full space, the clean split between gravitational-wave-visible and collider-visible scenarios could be an artifact of the selection.

Editorial extensions

If this is right

  • If a space-based gravitational-wave observatory sees a stochastic background matching an SFOEWPT, and di-Higgs measurements at the HL-LHC and a 1 TeV e+e− collider remain SM-like, the transition in the RxSM was almost certainly singlet-driven.
  • If a future collider sees a di-Higgs deviation consistent with κλ around 1.5–1.7 in the mhh distribution, the same model would predict little or no LISA signal for the doublet-driven transition; the two observations should not be expected together.
  • Total di-Higgs cross-sections are not a reliable probe on their own: cancellations between enhanced λhhh and resonant H contributions can restore SM values, so only differential mhh distributions, and channels like e+e−→ννhh for mH ≲ 650 GeV, can separate model from SM.
  • A singlet-driven SFOEWPT that is visible at LISA would leave essentially no trace in di-Higgs searches; collider null results would not constrain these scenarios, so gravitational-wave observatories are the only way to access them.
  • The complementarity reverses the usual intuition: the stronger the gravitational-wave signal, the more Standard-Model-like the Higgs sector appears at colliders.

Reading between the lines

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

  • If this direction-based dichotomy is generic, it suggests a selection rule for model-building: extensions that use a singlet-like field to strengthen the transition will hide from colliders but shine in gravitational-wave observatories, while doublet-driven models are the natural targets of precision Higgs programs; neither search alone maps the full SFOEWPT parameter space.
  • The paper's benchmark planes are constructed to maximize ξn; a full five-dimensional scan or a random sample would test whether the clean separation persists away from these slices. Until then, the 'significant parts of parameter space' claim for singlet-driven gravitational waves should be read as conditional on this selection.
  • Since the sign of the singlet VEV at nucleation appears to control the delay of the transition and hence the gravitational-wave strength, a natural next step is to map the phenomenology as a function of vS(Tn) rather than the fixed input vS; this may uncover additional singlet-driven regions in the full parameter space.
  • A dedicated computation of the bubble wall velocity, which the paper treats as an input, could shift the boundary of the doublet-driven gravitational-wave region; if vw turns out systematically below 0.6, the 'narrow strip' could become a substantial discovery region.
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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. The paper studies the real singlet extension of the Standard Model (RxSM) and investigates whether scenarios with a strong first-order electroweak phase transition (SFOEWPT) can be probed complementarily by future gravitational-wave (GW) observations, specifically LISA, and by di-Higgs production at the HL-LHC and a 1 TeV e+e− collider. The authors implement the RxSM in BSMPTv3, compute the one-loop finite-temperature effective potential, identify six thermal histories, and map regions with ξn = vn/Tn > 1. From a five-dimensional parameter scan they isolate two benchmark planes, constructed to maximize ξn: plane 1 (Eq. 37) features a singlet-driven SFOEWPT with strong GW signals and SM-like di-Higgs production; plane 2 (Eq. 39) features a doublet-driven SFOEWPT with substantial deviations in κλ and di-Higgs rates but only a narrow GW-observable region. The central claim is that singlet-driven SFOEWPTs are GW-loud and collider-quiet, while doublet-driven SFOEWPTs are collider-visible but GW-quiet.

Significance. If the qualitative complementarity holds across the RxSM parameter space, the result would be an important guide for future experimental strategy: it would show that no single probe can cover all SFOEWPT scenarios, and that singlet-driven transitions may be invisible to colliders while being accessible to LISA. The paper has clear strengths: it uses and extends public tools (BSMPTv3, anyH3, HPAIR, MadGraph5), includes one-loop corrections consistently in both the phase-transition and di-Higgs computations, applies existing experimental constraints via HiggsBounds/HiggsSignals, and provides concrete benchmark points with tabulated couplings and significances. These features make the analysis reproducible in principle. However, the central qualitative claim is currently demonstrated only on two specially engineered two-dimensional planes, not on the full five-dimensional parameter space, and the paper itself labels the study a 'proof of concept' in Sec. 5. The strength of the abstract's 'significant parts of the parameter space' is therefore not yet fully supported.

major comments (3)
  1. [Sec. 4.2, Eqs. (37) and (39), Figs. 2-10] The central complementarity claim is a statement about the RxSM parameter space, but it is demonstrated only on two two-dimensional benchmark planes that are specifically constructed to maximize ξn. The full five-dimensional scan in Fig. 2 is not used to compute SNR or κλ, and no quantitative, systematic classification into 'singlet-driven' versus 'doublet-driven' transitions is applied across the allowed region. If, for example, some singlet-driven points outside these planes have sizeable κλ, or some doublet-driven points have SNR > 10 at realistic vw, the dichotomy would be an artifact of the slice selection rather than a property of the model. The paper's own caveat that the work is a 'proof of concept' (Sec. 5) tempers the claim, but the abstract's 'significant parts of the parameter space' needs either a full-volume demonstration or a weakened formulation.
  2. [Sec. 4.2 vs. Sec. 4.3.1, Eq. (37)] The description of the first SFOEWPT region in Sec. 4.2 states that the strongest transitions occur for κS ≳ −300 GeV, yet benchmark plane 1 fixes κS = −900 GeV. This places the plane outside the stated region and, together with the fitted relations for κSH(cosα) and vS(cosα), makes the plane's claimed representativeness unclear. The authors should clarify how Eq. (37) was selected, and ideally show the location of the plane relative to the regions identified in Fig. 2 (e.g., in the {κS, κSH} plane).
  3. [Sec. 4.3.3, Figs. 9 and 10] The observability conclusions are sensitive to the assumed bubble-wall velocity vw, which is not computed. In benchmark plane 2, no point reaches SNR > 10 for vw = 0.95, but sizeable regions become observable for vw = 0.6. Since the central asymmetry relies on identifying one class as 'GW-quiet', this conclusion is conditional on an external parameter. The authors do discuss this dependence and state that vw = 0.95 is conservative, but the abstract's categorical phrasing should be qualified so that the reader does not overinterpret the dichotomy without a full wall-velocity determination or a more robust scan.
minor comments (4)
  1. [Sec. 6, first paragraph] Typo: 'we have explored the the dynamics' should read 'we have explored the dynamics'.
  2. [Sec. 4.3.3] The text says 'a maximum is reached in all four BPs for vw ∼ 7'; this should presumably be 'vw ∼ 0.7'.
  3. [Sec. 5, first paragraph] The renormalisation-scheme consistency check is performed for only four representative benchmark points. This is a useful spot check, but the statement that the two schemes agree 'within the renormalisation scale dependence of the BSMPT predictions' should be presented as a limited check, not a general proof, especially since the full benchmark planes are not tested.
  4. [Sec. 3.3.3] The SNR threshold for observability is set to 10 without an explicit justification or reference to a LISA detection criterion. Please provide a reference or a brief motivation for this threshold.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: GW/di-Higgs predictions are forward computations from the RxSM Lagrangian; the ξn-maximising benchmark planes are parameter-space choices, not fitted proxies for the claimed observables.

full rationale

The derivation chain is a forward computation: Lagrangian (Eq. 2) -> one-loop thermal potential (Eq. 14) in BSMPTv3 -> bounce action and nucleation (Eqs. 25-28) -> ξn (Eq. 24), transition parameters (Eqs. 29-30), and LISA SNR (Eq. 35); on the collider side, tree-level trilinears (Eq. 11) -> one-loop trilinears via the public code anyH3 with the OS scheme of Ref. [64] -> cross-sections with HPAIR/MadGraph. The abstract's complementarity statement is obtained by evaluating these observables on two benchmark planes (Eqs. 37 and 39). Those planes are selected using the 5D scan to maximise ξn, but this is parameter-space selection, not a fit to the predicted observables: the SNR and di-Higgs rates are then computed, not imposed. No relation such as 'plane 1 is GW-loud because it was chosen to be GW-loud' appears; the planes are chosen for phase-transition strength only. Self-citations to Refs. [64,74] provide the renormalisation scheme and the di-Higgs framework; they are public, code-based, and not used to forbid alternatives, so they are not load-bearing circularity. The manuscript itself flags its scope: 'we emphasise that our work is meant to be a proof of concept...' (Sec. 5) and 'our study only takes into account parts of the experimental effects and uncertainties' (Sec. 6); these limit generality but are not circular. A separate consistency issue (Sec. 4.2 describes the first strong-transition region with κS ≳ −300 GeV, while plane 1 in Eq. (37) uses κS = −900 GeV) is a correctness/representativeness concern, not a circular step.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The RxSM has five free BSM parameters, which are scanned; the central complementarity conclusions, however, rest on hand-picked benchmark-plane relations (Eqs. 37 and 39) that maximize the transition strength, and on an uncomputed bubble wall velocity. These are selections, not fits to external data.

free parameters (4)
  • bubble wall velocity vw = 0.95 (assumed), 0.6 (alternative)
    Not computed; directly controls GW SNR (Fig. 9).
  • benchmark plane 1 relation coefficients (κSH, vS as functions of cosα) = κSH=5662.9 cosα−5688.4 GeV; vS=4239.5 cosα−4067.6 GeV; κS=-900 GeV
    Chosen to maximise ξn in the low-vS region (Eq. 37); defines the plane on which the singlet-driven conclusions rest.
  • benchmark plane 2 fixed parameters = cosα=0.98, κS=-300 GeV, vS=280 GeV
    Chosen to maximise ξn in the high-vS region (Eq. 39).
  • SNR observability threshold = 10
    Chosen as detection criterion at LISA.
assumptions (6)
  • domain assumption Perturbative finite-temperature effective potential at one loop with Arnold-Espinosa daisy resummation is sufficiently accurate for EWPT dynamics.
    Used in Sec. 3.1; two-loop/3D lattice corrections are not included (see Ref. [159] for possible shifts).
  • domain assumption SFOEWPT criterion ξn = vn/Tn ≥ 1.
    Standard sphaleron-suppression condition (Eq. 24).
  • ad hoc to paper Bubble wall velocity vw can be treated as an external parameter; assumed 0.95 or 0.6.
    vw is not computed; Fig. 9 shows strong SNR dependence.
  • ad hoc to paper Benchmark plane relations (Eq. 37 and Eq. 39) are representative of the two SFOEWPT regions.
    Chosen to maximize ξn; the paper itself calls the study a proof of concept.
  • domain assumption Non-runaway bubble regime (α < 1): GW spectrum dominated by sound waves and turbulence; bubble collisions neglected.
    Used in Sec. 3.3.2; valid for the scanned points with α<1.
  • domain assumption OS-like and full OS renormalisation schemes yield compatible trilinear couplings.
    Claimed in Sec. 5 without showing the comparison; affects consistency between EWPT and collider analyses.

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Cite this review

Pith. "Pith review of Complementarity of gravitational wave analyses and di-Higgs production in the exploration of the Electroweak Phase Transition dynamics in the RxSM." pith.science (2026). https://pith.science/paper/OEACCBCR

@misc{pith2026251012569,
  author       = {Pith},
  title        = {Pith review of: Complementarity of gravitational wave analyses and di-Higgs production in the exploration of the Electroweak Phase Transition dynamics in the RxSM},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OEACCBCR}},
  note         = {Machine review of arXiv:2510.12569}
}
abstract

The real singlet extension of the Standard Model (SM), RxSM, is one of the simplest Beyond-the-Standard Model (BSM) theories that can accommodate a strong first-order electroweak phase transition (SFOEWPT). We survey the possible thermal histories of the early Universe in the RxSM, and find that a SFOEWPT can occur in this model as single- or two-step phase transitions. We investigate complementary approaches to probe such scenarios experimentally: either via searches for a stochastic background of gravitational waves (GWs) or via searches for di-Higgs production processes at future collider experiments: the HL-LHC, or a possible high-energy $e^+e^-$ collider. For these analyses we consistently include one-loop corrections to the trilinear Higgs couplings. We find that entirely different phenomenological signals are possible, depending on how the SFOEWPT occurs. In scenarios where such a transition is driven by the Higgs doublet direction in field space, BSM deviations in properties of the detected Higgs boson, particularly in the trilinear scalar coupling, typically lead to observable signals at colliders, while the regions of parameter space with detectable GW signals are very narrow. On the other hand, if the SFOEWPT is triggered by the singlet field direction, the detected Higgs boson is very SM-like and no signs of BSM physics would appear in di-Higgs production processes. However, strong GW signals could be produced for significant parts of the RxSM parameter space with singlet-driven SFOEWPT. This work highlights the crucial importance of exploiting complementary experimental directions to determine the dynamics of the electroweak phase transition and access the shape of the Higgs potential realised in Nature.

Figures

Figures reproduced from arXiv: 2510.12569 by the authors.

Figure 1
Figure 1. The tracing of the different minima of the potential with respect to the temperature [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Results for ξn ≡ vn/Tn for the points from our RxSM parameter scan. Top left: {cos α, mH} plane; top right: {cos α, vS} plane; bottom left: {κS, κSH} plane; bottom right: {mH, vS} plane. ξn in this region κS = −900 GeV , κSH = (5662.9 cos α − 5688.4) GeV , vS = (4239.5 cos α − 4067.6) GeV . (37) We are then left with only two free parameters, over which we scan with the ranges1 mH ∈ [260, 1000] GeV , cos α ∈ [0.975,… view at source ↗
Figure 3
Figure 3. Parameter scan results in the RxSM for benchmark plane 1. The colour coding [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: Parameter scan results in the RxSM for the points with a SFOEWPT in the [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: Temperature-dependent one-loop effective potential at the nucleation temperature, [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: Parameter scan results in the RxSM for benchmark plane 2. The colour coding [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: Parameter scan results for the points with a SFOEWPT in benchmark plane 2. [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: Temperature-dependent one-loop effective potential at the nucleation temperature, [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: SNR at LISA after three years of data taking as a function of [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]
Figure 10
Figure 10. Figure 10: SNR at LISA for a bubble wall velocity of [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]
Figure 11
Figure 11. Figure 11: One-loop corrected trilinear scalar couplings in the benchmark plane 2. ()() [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: One-loop corrections to the trilinear scalar couplings in the benchmark plane 2. () () () () [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: Results for ξn in benchmark plane 2. Left: projected in the plane {mH, κ (0) λ }; right: projected in the plane {mH, κ (1) λ }. To investigate the correlation between a BSM deviation in κλ and the strength of the phase transition, we show ξn in the colour coding of […
Figure 14
Figure 14. Figure 14: Results for the di-Higgs cross-section at the HL-LHC in the benchmark plane 2. () [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]
Figure 15
Figure 15. Figure 15: Left: Differential di-Higgs production cross-section distributions w.r.t. mhh. Right: Distribution of the number of di-Higgs events, taking into account the BR(h → b ¯b), as well as smearing and binning effects, with respect to the invariant mass of the four re￾constr…
Figure 16
Figure 16. Figure 16: Plots and line styles as in Fig. 15 [PITH_FULL_IMAGE:figures/full_fig_p032_16.png]
Figure 17
Figure 17. Figure 17: Results of the di-Higgs cross section computation for a 1 TeV ()() [PITH_FULL_IMAGE:figures/full_fig_p034_17.png]
Figure 18
Figure 18. Figure 18: Results of the di-Higgs cross section computation for a 1 TeV ()() [PITH_FULL_IMAGE:figures/full_fig_p035_18.png]
Figure 19
Figure 19. Figure 19: Differential polarised di-Higgs production cross-section distributions (for the po [PITH_FULL_IMAGE:figures/full_fig_p037_19.png]
Figure 20
Figure 20. Figure 20: Plots and line styles as in Fig. 19 [PITH_FULL_IMAGE:figures/full_fig_p038_20.png]
Figure 21
Figure 21. Figure 21: Plots and line styles as in Fig. 19, but for the process [PITH_FULL_IMAGE:figures/full_fig_p040_21.png]
Figure 22
Figure 22. Figure 22: Plots and line styles as in Fig. 20, but for the process [PITH_FULL_IMAGE:figures/full_fig_p041_22.png]

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Forward citations

Cited by 7 Pith papers

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