pith. sign in

arxiv: 2606.11564 · v1 · pith:FSURBRNTnew · submitted 2026-06-10 · ✦ hep-ph

Strong First-Order Electroweak Phase Transition and Gravitational Waves in a mathbb{Z}₄ Fermion-Scalar Dark Matter Model

J. P. Carvalho-Corr\^ea , J. P. Cunha-Melo , I. M. Pereira , B. L. S\'anchez-Vega , A. C. D. Viglioni This is my paper

Pith reviewed 2026-06-27 09:30 UTC · model grok-4.3

classification ✦ hep-ph
keywords electroweak phase transitiondark mattergravitational wavesZ4 symmetryscalar singletDirac fermionWIMP-FIMPphase transition
0
0 comments X

The pith

A Z4 fermion-scalar model permits strong first-order electroweak phase transition only in two dark matter regimes after constraints.

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

The paper checks whether a minimal Z4-symmetric extension of the Standard Model containing one real scalar singlet and one Dirac fermion can simultaneously satisfy dark matter requirements and drive a strong first-order electroweak phase transition capable of producing observable gravitational waves. After enforcing theoretical consistency, the correct electroweak vacuum, relic density, direct detection limits, and invisible Higgs decay bounds, the surviving parameter points are fed into a finite-temperature analysis. This shows that strong transitions occur only in the thermal two-component regime where the fermion mass is less than the scalar mass which is less than twice the fermion mass, and in the decay-driven WIMP-FIMP regime where the scalar mass exceeds twice the fermion mass. Other regimes, including thermal cases with the scalar lighter than the fermion and stable mixed scenarios with the scalar lighter than twice the fermion mass, concentrate at small portal couplings or near resonance and fail to produce strong transitions. Successful cases typically pass through an intermediate singlet-dominated vacuum, and selected benchmark points yield gravitational wave spectra from sound waves and turbulence that may reach future detectors.

Core claim

After current dark-matter constraints are imposed, the strong-transition criterion along the Higgs direction is satisfied only in two viable regimes: the thermal two-component case with M_ψ < M_S < 2M_ψ and the decay-driven WIMP-FIMP case with M_S > 2M_ψ. The successful transitions typically proceed through an intermediate singlet-like phase. For representative nucleating benchmark points, gravitational-wave spectra from sound waves and turbulence are computed, with some entering the projected reach of future space-based interferometers.

What carries the argument

Finite-temperature effective potential along the Higgs direction together with the nucleation criterion for strong first-order transitions, applied only to points that already satisfy all dark matter and theoretical bounds.

If this is right

  • The thermal regime with M_S < M_ψ and the stable mixed WIMP-FIMP scenario with M_S < 2M_ψ largely fail to produce strong transitions.
  • Successful points require a sufficiently active Higgs portal in combination with the scalar mass and remaining dark sector parameters.
  • Gravitational wave spectra computed for benchmark points in the viable regions arise from sound waves and turbulence.
  • Detectable signals appear only in selected dark-matter-compatible regions.
  • The model accommodates thermal two-component dark matter, mixed WIMP-FIMP histories, and an effectively fermionic relic abundance generated by scalar decays.

Where Pith is reading between the lines

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

  • The sharp selectivity after dark matter cuts implies that similar singlet-fermion extensions will face comparable restrictions when both requirements are imposed simultaneously.
  • Future gravitational wave detectors could provide an independent test that further narrows or eliminates the two surviving regimes.
  • The need for an intermediate singlet phase may recur in other models that add a real scalar to the Standard Model.
  • The correlation between an active Higgs portal and the scalar mass could be probed by combining collider measurements with relic density data.

Load-bearing premise

The finite-temperature effective potential calculation and the nucleation criterion used to identify a strong first-order transition are reliable for the scanned parameter space.

What would settle it

A computation of the effective potential for the identified mass regimes that shows either no barrier or a transition strength below the nucleation threshold would falsify the claim that these regimes support strong transitions.

Figures

Figures reproduced from arXiv: 2606.11564 by A. C. D. Viglioni, B. L. S\'anchez-Vega, I. M. Pereira, J. P. Carvalho-Corr\^ea, J. P. Cunha-Melo.

Figure 1
Figure 1. Figure 1: Theoretical constraints on the scalar sector. Left: viable parameter space in the (𝜆𝑆, 𝜆𝐻𝑆) plane. The dashed curves mark the bounded-from-below boundaries. The green region satisfies both boundedness-from-below and perturbative-unitarity requirements, whereas the yellow region remains perturbatively unitary but is excluded by boundedness from below. Right: lower bound 𝜆 min 𝑆 imposed by the possible singl… view at source ↗
Figure 2
Figure 2. Figure 2: Representative Z4 processes relevant for the relic-density calculation: portal annihilation (𝑆𝑆 → SM), conversion (𝜓𝜓¯ ↔ 𝑆𝑆), and semi-annihilation (𝜓𝜓 → 𝑆ℎ), together with its CP-conjugate channel. 4.1. Production Mechanisms and Relic Abundance We now describe the Boltzmann system used to compute the relic abundances of 𝑆 and 𝜓. The relevant cosmological histories are controlled by the 𝑆–𝜓 mass hierarchy … view at source ↗
Figure 3
Figure 3. Figure 3: Diagrams for dark matter–nucleus scattering. Left: tree-level Higgs exchange for scalar dark matter. Right: loop-induced Higgs exchange for fermionic dark matter. Solid black lines denote nucleons, while dashed lines denote mediators. For the fermion, the loop-induced SI cross section reads [20] 𝜎 SI 𝜓 = 1 𝜋 𝜇 2 𝜓𝑁 𝑚 2 𝑁 𝑓 2 𝑁 𝑀4 ℎ  𝜆𝐻𝑆 16𝜋 2𝑀𝜓 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Viable parameter-space structure for Scenario I, the thermal two-component regime with 𝑀𝜓 < 𝑀𝑆 < 2𝑀𝜓. The surviving region is typically dominated by the fermionic component, while direct-detection bounds remove a significant fraction of the relic-density-compatible parameter space. Figure 4a shows that the latest LZ spin-independent bound removes a significant part of the relic-density￾compatible region. H… view at source ↗
Figure 5
Figure 5. Figure 5: Viable parameter-space structure for Scenario II, the thermal two-component regime with 𝑀𝑆 < 𝑀𝜓. Direct-detection bounds strongly restrict the scalar component and drive the surviving points towards the Higgs￾resonance region. As shown in Fig. 5a, the surviving region is concentrated near the Higgs resonance, 𝑀𝑆 ≃ 𝑀ℎ/2. In this region, resonant annihilation can deplete the scalar abundance efficiently even… view at source ↗
Figure 6
Figure 6. Figure 6: Viable parameter space for Scenario III, the mixed two-component WIMP–FIMP regime with 𝑀𝑆 < 2𝑀𝜓. The Higgs-portal coupling 𝜆𝐻𝑆 is shown against the fermion mass 𝑀𝜓, with the fermionic relic fraction Ω𝜓/ΩDM encoded in the colour scale. The surviving points reflect the interplay between scalar freeze-out, fermion freeze-in, and the latest LZ spin-independent bound on the stable scalar component. As shown in … view at source ↗
Figure 7
Figure 7. Figure 7: Viable parameter space for Scenario IV, the effectively one-component WIMP–FIMP regime with 𝑀𝑆 > 2𝑀𝜓. The Higgs-portal coupling 𝜆𝐻𝑆 is shown against the scalar mass 𝑀𝑆, with the fermion mass 𝑀𝜓 encoded in the colour scale. Since the scalar decays before today, direct-detection bounds on scalar dark matter do not constrain the present-day relic abundance. Sizeable portal couplings therefore remain allowed, … view at source ↗
Figure 8
Figure 8. Figure 8: Electroweak phase-transition strength for Scenario I, the thermal two-component regime with 𝑀𝜓 < 𝑀𝑆 < 2𝑀𝜓. The panels show the correlations of the projected order parameter 𝜁𝑐 = 𝑣𝑐/𝑇𝑐 with the model parameters. Green points satisfy the strong-transition criterion, 𝜁𝑐 ≳ 1, at the critical temperature, whereas blue points do not. All points satisfy the theoretical and dark-matter constraints discussed in the… view at source ↗
Figure 9
Figure 9. Figure 9: Electroweak phase-transition strength in Scenario IV, the effectively one-component WIMP–FIMP regime with 𝑀𝑆 > 2𝑀𝜓. The scalar is thermally produced but unstable, and the surviving dark matter is fermionic. The panels show the correlations of the projected order parameter 𝜁𝑐 = 𝑣𝑐/𝑇𝑐 with the model parameters. Green points satisfy the strong-transition criterion, 𝜁𝑐 ≳ 1, at the critical temperature, whereas… view at source ↗
Figure 10
Figure 10. Figure 10: Predicted stochastic gravitational-wave spectra for the benchmark points in Scenario I, with 𝑀𝜓 < 𝑀𝑆 < 2𝑀𝜓. The spectra include the sound-wave and turbulence contributions computed with the benchmark choice 𝑣𝑤 = 1, and are compared with the projected sensitivity curves shown in the figure, including LISA [6], BBO [52], and the DECIGO-related configurations DECIGO, UD, DC, and UDC [7,53]. The benchmark poi… view at source ↗
Figure 11
Figure 11. Figure 11: Predicted stochastic gravitational-wave spectra for the benchmark points in Scenario IV, with 𝑀𝑆 > 2𝑀𝜓. The spectra include the sound-wave and turbulence contributions computed with the benchmark choice 𝑣𝑤 = 1, and are compared with the projected sensitivity curves shown in the figure, including LISA [6], BBO [52], and the DECIGO-related configurations DECIGO, UD, DC, and UDC [7,53]. The strongest spectra… view at source ↗
read the original abstract

We investigate whether a minimal $\mathbb{Z}_4$-symmetric fermion-scalar extension of the Standard Model can simultaneously realise viable dark matter, a strong electroweak phase transition, and a stochastic gravitational-wave signal. The model contains a real scalar singlet and a Dirac fermion, allowing thermal two-component dark matter, mixed WIMP-FIMP histories, and an effectively fermionic relic abundance generated by scalar decays. We impose theoretical consistency, the correct electroweak vacuum, and dark-matter constraints from relic density, direct detection, and invisible Higgs decays before using the surviving points as input for the finite-temperature analysis. This reveals that the compatibility between dark matter and a strong first-order electroweak phase transition is highly selective. After current dark-matter constraints are imposed, the strong-transition criterion along the Higgs direction is satisfied only in two viable regimes: the thermal two-component case with $M_\psi<M_S<2M_\psi$ and the decay-driven WIMP-FIMP case with $M_S>2M_\psi$. By contrast, the thermal regime with $M_S<M_\psi$ and the stable mixed WIMP-FIMP scenario with $M_S<2M_\psi$ are largely concentrated at small portal couplings or near the Higgs-resonance region, and do not yield a strong transition in the parameter space considered. The successful transitions typically proceed through an intermediate singlet-like phase. For representative nucleating benchmark points in the viable strong-transition regions, we compute the gravitational-wave spectra from sound waves and turbulence. Some spectra enter the projected reach of future space-based interferometers, showing that detectable signals arise only in selected dark-matter-compatible regions where a sufficiently active Higgs portal appears in correlated combination with the scalar mass and the remaining dark sector parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper studies a minimal Z_4-symmetric SM extension containing a real scalar singlet S and a Dirac fermion ψ. It imposes theoretical consistency, correct electroweak vacuum, and dark-matter constraints (relic density, direct detection, invisible Higgs decays) on the parameters (M_ψ, M_S, portal couplings), then performs a finite-temperature analysis on the surviving points. The central claim is that a strong first-order electroweak phase transition (FOPT) along the Higgs direction occurs only in two regimes: the thermal two-component case with M_ψ < M_S < 2M_ψ and the decay-driven WIMP-FIMP case with M_S > 2M_ψ. Successful transitions typically proceed via an intermediate singlet-like vacuum; gravitational-wave spectra from sound waves and turbulence are computed for benchmark points in these regimes, with some entering the projected sensitivity of future interferometers.

Significance. If the finite-temperature results hold, the work establishes a concrete selectivity: after current DM constraints, strong FOPT is viable only in two narrow regimes of this Z_4 model, while other DM-compatible regions (M_S < M_ψ thermal and stable mixed WIMP-FIMP with M_S < 2M_ψ) do not produce strong transitions. The explicit mapping of viable points to GW spectra provides falsifiable predictions for space-based detectors and illustrates how DM requirements can restrict the parameter space available for detectable gravitational waves.

major comments (3)
  1. [§4] §4 (finite-temperature effective potential): the manuscript does not specify the explicit one-loop form (tree-level + Coleman-Weinberg + thermal integrals) or the daisy-resummation scheme (Parwani vs. Arnold-Espinosa) used for the thermal masses; without this, it is impossible to verify whether the reported selectivity of the two regimes survives changes in the resummation prescription or the treatment of the multi-field potential when an intermediate singlet-like vacuum appears.
  2. [§5] §5 (nucleation criterion): the definition of 'strong' transition is not stated (e.g., v_c/T_c > 1 at the critical temperature versus S_3/T ≈ 140 at the nucleation temperature); the central claim that only the two cited regimes survive therefore rests on an unspecified numerical threshold whose variation could alter the viability map.
  3. [Table 2 / Figure 5] Table 2 / Figure 5 (benchmark points): the reported GW spectra are computed only for points already selected by the unspecified FOPT criterion; no scan-variation or error-band analysis is shown to demonstrate that the 'highly selective' conclusion is robust against reasonable changes in the thermal-potential implementation.
minor comments (3)
  1. [§2] Notation for the portal couplings λ_HS and λ_ψS is introduced without an explicit Lagrangian term in the model-definition section.
  2. [Figure 3] The caption of Figure 3 does not indicate whether the plotted points include the full thermal history or only the zero-temperature relic-density slice.
  3. [§5] A reference to the numerical package or custom code used for the multi-field minimization and bounce-action calculation is missing.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help improve the clarity and technical transparency of the manuscript. We address each major comment point-by-point below and will revise the text to incorporate the requested details.

read point-by-point responses

  1. Referee: [§4] §4 (finite-temperature effective potential): the manuscript does not specify the explicit one-loop form (tree-level + Coleman-Weinberg + thermal integrals) or the daisy-resummation scheme (Parwani vs. Arnold-Espinosa) used for the thermal masses; without this, it is impossible to verify whether the reported selectivity of the two regimes survives changes in the resummation prescription or the treatment of the multi-field potential when an intermediate singlet-like vacuum appears.

    Authors: We agree that the explicit implementation details were omitted. The effective potential follows the standard one-loop form with Coleman-Weinberg corrections and thermal integrals evaluated via the high-temperature expansion; daisy resummation is performed in the Parwani scheme for the thermal masses of the Higgs and singlet fields. The multi-field potential is minimized numerically along the Higgs direction after accounting for the intermediate singlet vacuum. We will add a dedicated subsection (or appendix) with the explicit expressions for V_eff, the thermal mass corrections, and the resummation prescription. While a full cross-check against the Arnold-Espinosa scheme lies outside the present scope, the selectivity is driven primarily by the interplay of tree-level barriers, DM relic constraints, and the requirement of a sufficiently deep electroweak minimum; we will note this limitation explicitly. revision: yes

  2. Referee: [§5] §5 (nucleation criterion): the definition of 'strong' transition is not stated (e.g., v_c/T_c > 1 at the critical temperature versus S_3/T ≈ 140 at the nucleation temperature); the central claim that only the two cited regimes survive therefore rests on an unspecified numerical threshold whose variation could alter the viability map.

    Authors: The strong-transition criterion is the standard nucleation condition S_3(T_n)/T_n ≈ 140, where T_n is determined by solving the bounce equation for the three-dimensional Euclidean action; v_c/T_c > 1 is used only as a preliminary filter at the critical temperature. We will state this definition explicitly in §5, including the numerical procedure for locating T_n and the bounce action, and will clarify that the final viability map is based on the nucleation threshold rather than the critical-temperature ratio alone. revision: yes

  3. Referee: [Table 2 / Figure 5] Table 2 / Figure 5 (benchmark points): the reported GW spectra are computed only for points already selected by the unspecified FOPT criterion; no scan-variation or error-band analysis is shown to demonstrate that the 'highly selective' conclusion is robust against reasonable changes in the thermal-potential implementation.

    Authors: The benchmark points are representative nucleating solutions drawn from the two viable regimes after all DM and theoretical constraints. We acknowledge that a systematic variation over resummation schemes or a full error-band scan would strengthen the robustness claim. We will expand the discussion around Table 2 and Figure 5 to emphasize that the GW spectra are illustrative and to note the dependence on the chosen potential implementation; a comprehensive parameter-variation study is beyond the scope of the present work but could be addressed in follow-up analyses. revision: partial

Circularity Check

0 steps flagged

No circularity; DM constraints applied first, PT checked sequentially on survivors

full rationale

The derivation applies relic-density, direct-detection and invisible-Higgs constraints first, then feeds only the surviving points into the finite-temperature analysis. No equation equates a derived quantity to its own input by construction, no parameter is fitted to a subset and then relabeled a prediction, and no load-bearing premise rests on a self-citation whose content is itself unverified. The reported selectivity (strong FOPT only for M_ψ < M_S < 2M_ψ and M_S > 2M_ψ) is therefore an output of independent external constraints rather than a definitional or self-referential step. The paper is self-contained against external benchmarks for the purpose of this circularity check.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Review performed on abstract only; specific free parameters, axioms, and invented entities cannot be enumerated in detail without the full text. The model introduces a Z4 symmetry and two new particles whose masses and couplings are constrained by external data.

free parameters (1)
  • M_ψ, M_S and portal couplings
    Masses and interaction strengths are scanned subject to dark matter relic density and direct detection constraints.
axioms (1)
  • domain assumption Z4 symmetry protects the dark matter candidates from decay
    The abstract states the model contains a real scalar singlet and Dirac fermion allowing thermal two-component or decay-driven dark matter.
invented entities (1)
  • Z4-protected scalar singlet and Dirac fermion no independent evidence
    purpose: Provide dark matter candidates and enable strong first-order electroweak phase transition
    New particles and symmetry introduced in the model extension.

pith-pipeline@v0.9.1-grok · 5898 in / 1632 out tokens · 35680 ms · 2026-06-27T09:30:04.665781+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

58 extracted references · 49 canonical work pages · 26 internal anchors

  1. [1]

    Planck 2018 results. VI. Cosmological parameters

    N. Aghanim et al. “Planck 2018 results. VI. Cosmological parameters”.Astron. Astrophys.641 (2020), A6.doi:10.1051/0004-6361/201833910. arXiv:1807.06209 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1051/0004-6361/201833910 2018

  2. [2]

    Electroweak baryogenesis

    David E. Morrissey and Michael J. Ramsey-Musolf. “Electroweak baryogenesis”.New J. Phys.14 (2012), p. 125003.doi:10.1088/1367-2630/14/12/125003. arXiv:1206.2942 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1367-2630/14/12/125003 2012

  3. [3]

    Mariano Quiros.Finite temperature field theory and phase transitions. 1999. arXiv:hep- ph/ 9901312 [hep-ph]

    1999

  4. [4]

    Review of cosmic phase transitions: their significance and signatures

    Anupam Mazumdar and Graham White. “Review of cosmic phase transitions: their significance and signatures”.Rept. Prog. Phys.82(July 2019), p. 076901.doi:10.1088/1361-6633/ab1f55. arXiv:1811.01948 [hep-ph]

    work page doi:10.1088/1361-6633/ab1f55 2019

  5. [5]

    Detecting gravitational waves from cosmological phase transitions with LISA: an update

    Chiara Caprini et al. “Detecting gravitational waves from cosmological phase transitions with LISA: an update”.JCAP03(2020), p. 024.doi:10.1088/1475-7516/2020/03/024 . arXiv: 1910.13125 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2020/03/024 2020

  6. [6]

    Pau Amaro-Seoane et al.Laser Interferometer Space Antenna. 2017. arXiv: 1702 . 00786 [astro-ph.IM]

    2017

  7. [7]

    Current status of space gravitational wave antenna DECIGO and B-DECIGO

    Seiji Kawamura et al. “Current status of space gravitational wave antenna DECIGO and B- DECIGO”.Prog. Theor. Exp. Phys.2021, 5 (2021), 05A105.doi:10.1093/ptep/ptab019 . arXiv:2006.13545 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/ptep/ptab019 2021

  8. [8]

    Beyond LISA: Exploring Future Gravitational Wave Missions

    Jeff Crowder and Neil J. Cornish. “Beyond LISA: Exploring future gravitational wave missions”. Phys.Rev.D72(2005),p.083005.doi: 10.1103/PhysRevD.72.083005.arXiv: gr-qc/0506015

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.72.083005.arxiv: 2005

  9. [9]

    New Sensitivity Curves for Gravitational-Wave Signals from Cosmological Phase Transitions

    Kai Schmitz. “New sensitivity curves for gravitational-wave signals from cosmological phase transitions”.JHEP01(2021), p. 097.doi: 10.1007/JHEP01(2021)097 . arXiv:2002.04615 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2021)097 2021

  10. [10]

    Complex Scalar Singlet Dark Matter: Vacuum Stability and Phenomenology

    Matthew Gonderinger, Hyungjun Lim, and Michael J. Ramsey-Musolf. “Complex scalar singlet dark matter: Vacuum stability and phenomenology”.Phys. Rev. D86(Aug. 2012), p. 043511.doi: 10.1103/PhysRevD.86.043511. arXiv:1202.1316 [hep-ph]. 34

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.86.043511 2012

  11. [11]

    Update on scalar singlet dark matter

    James M. Cline, Kimmo Kainulainen, Pat Scott, and Christoph Weniger. “Update on scalar singlet dark matter”.Phys. Rev. D88(2013). [Erratum: Phys. Rev. D 92, 039906 (2015)], p. 055025.doi: 10.1103/PhysRevD.88.055025. arXiv:1306.4710 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.88.055025 2013

  12. [12]

    Marciu, Dynamical description of a quintom cosmo- logical model nonminimally coupled with gravity, The 15 European Physical Journal C80, 10.1140/epjc/s10052- 020-08476-9 (2020)

    P. Athron, C. Balázs, T. Bringmann, A. Buckley, M. Chrzaszcz, J. Conrad, et al. “Status of the scalar singlet dark matter model”.Eur. Phys. J. C77(2017), p. 568.doi:10.1140/epjc/s10052- 017-5113-1. arXiv:1705.07931 [hep-ph]

    work page doi:10.1140/epjc/s10052- 2017

  13. [13]

    Revisiting electroweak phase transition in the standard model with a real singlet scalar

    Cheng-Wei Chiang, Yen-Ting Li, and Eibun Senaha. “Revisiting electroweak phase transition in the standard model with a real singlet scalar”.Phys. Lett. B789(2019), pp. 154–159.doi: 10.1016/j.physletb.2018.12.017. arXiv:1808.01098 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2018.12.017 2019

  14. [14]

    Electroweakbaryogenesisandgravitationalwavesfromarealscalarsinglet

    VilleVaskonen.“Electroweakbaryogenesisandgravitationalwavesfromarealscalarsinglet”.Phys. Rev. D95, 12 (2017), p. 123515.doi:10.1103/PhysRevD.95.123515 . arXiv:1611.02073 [hep-ph]

    work page doi:10.1103/physrevd.95.123515 2017

  15. [15]

    Electroweak phase transition in singlet extensions of the standard model with dimension-six operators

    V. K. Oikonomou and Apostolos Giovanakis. “Electroweak phase transition in singlet extensions of the standard model with dimension-six operators”.Phys. Rev. D109, 5 (2024), p. 055044.doi: 10.1103/PhysRevD.109.055044. arXiv:2403.01591 [hep-ph]

    work page doi:10.1103/physrevd.109.055044 2024

  16. [16]

    Semi-annihilation of Dark Matter

    Francesco D’Eramo and Jesse Thaler. “Semi-annihilation of Dark Matter”.JHEP06(2010), p. 109. doi:10.1007/JHEP06(2010)109. arXiv:1003.5912 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep06(2010)109 2010

  17. [17]

    Freeze-In Production of FIMP Dark Matter

    Lawrence J. Hall, Karsten Jedamzik, John March-Russell, and Stephen M. West. “Freeze-in production of FIMP dark matter”.JHEP03(2010), p. 080.doi:10.1007/JHEP03(2010)080. arXiv:0911.1120 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep03(2010)080 2010

  18. [18]

    SuperWIMP Dark Matter Signals from the Early Universe

    Jonathan L. Feng, Arvind Rajaraman, and Fumihiro Takayama. “Superweakly interacting massive particle dark matter signals from the early Universe”.Phys. Rev. D68(2003), p. 063504.doi: 10.1103/PhysRevD.68.063504. arXiv:hep-ph/0306024

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.68.063504 2003

  19. [19]

    Superweakly Interacting Massive Particles

    Jonathan L. Feng, Arvind Rajaraman, and Fumihiro Takayama. “Superweakly Interacting Massive Particles”.Phys.Rev.Lett.91(2003),p.011302.doi: 10.1103/PhysRevLett.91.011302.arXiv: hep-ph/0302215

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.91.011302.arxiv: 2003

  20. [20]

    Fermion and scalar two-component dark matter from a𝑍4 symmetry

    Carlos E. Yaguna and Óscar Zapata. “Fermion and scalar two-component dark matter from a𝑍4 symmetry”.Phys. Rev. D105(May 2022), p. 095026.doi:10.1103/PhysRevD.105.095026. arXiv:2112.07020 [hep-ph]

    work page doi:10.1103/physrevd.105.095026 2022

  21. [21]

    Minimal model of fermion FIMP dark matter

    Carlos E. Yaguna and Óscar Zapata. “Minimal model of fermion FIMP dark matter”.Phys. Rev. D109(Jan. 2024), p. 015002.doi: 10.1103/PhysRevD.109.015002 . arXiv: 2308.05249 [hep-ph]

    work page doi:10.1103/physrevd.109.015002 2024

  22. [22]

    Theoretical and experimental constraints onZ2𝑛 multi-component dark matter models

    J. P. Carvalho-Corrêa, I. M. Pereira, B. L. Sánchez-Vega, and A. C. D. Viglioni. “Theoretical and experimental constraints onZ2𝑛 multi-component dark matter models”.Eur. Phys. J. C85(2025), p. 1353.doi:10.1140/epjc/s10052-025-15042-8. arXiv:2502.19489

    work page doi:10.1140/epjc/s10052-025-15042-8 2025

  23. [23]

    Dark Matter Search Results from 4.2 Tonne-Years of Exposure of the LUX-ZEPLIN (LZ) Experiment

    LZ Collaboration. “Dark Matter Search Results from 4.2 Tonne-Years of Exposure of the LUX- ZEPLIN (LZ) Experiment”.Phys. Rev. Lett.135(2025), p. 011802.doi:10.1103/4dyc-z8zf. arXiv:2410.17036 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/4dyc-z8zf 2025

  24. [24]

    Vacuum Stability Conditions From Copositivity Criteria

    Kristjan Kannike. “Vacuum stability conditions from copositivity criteria”.Eur. Phys. J. C72 (2012), p. 2093.doi:10.1140/epjc/s10052-012-2093-z. arXiv:1205.3781 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-012-2093-z 2012

  25. [25]

    Weak interactions at very high energies: The role of the Higgs-boson mass

    Benjamin W. Lee, C. Quigg, and H. B. Thacker. “Weak interactions at very high energies: The role of the Higgs-boson mass”.Phys. Rev. D16(1977), p. 1519.doi:10.1103/PhysRevD.16.1519. 35

    work page doi:10.1103/physrevd.16.1519 1977

  26. [26]

    Strength of Weak Interactions at Very High- Energies and the Higgs Boson Mass

    Benjamin W. Lee, C. Quigg, and H. B. Thacker. “Strength of Weak Interactions at Very High- Energies and the Higgs Boson Mass”.Phys. Rev. Lett.38(1977), pp. 883–885.doi:10.1103/ PhysRevLett.38.883

    1977

  27. [27]

    Logan.Lectures on perturbative unitarity and decoupling in Higgs physics

    Heather E. Logan.Lectures on perturbative unitarity and decoupling in Higgs physics. July 2022. arXiv:2207.01064 [hep-ph]

    arXiv 2022

  28. [28]

    Unitarity constraints on general scalar couplings with SARAH

    M. D. Goodsell and F. Staub. “Unitarity constraints on general scalar couplings with SARAH”.Eur. Phys. J. C78, 8 (2018), p. 649.doi:10.1140/epjc/s10052-018-6127-z. arXiv:1805.07306

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-018-6127-z 2018

  29. [29]

    micrOMEGAs 6.0: N-component dark matter

    G. Alguero, G. Belanger, F. Boudjema, S. Chakraborti, A. Goudelis, S. Kraml, A. Mjallal, and A. Pukhov. “micrOMEGAs 6.0: N-component dark matter”.Comput. Phys. Commun.299(2024), p. 109133.doi:10.1016/j.cpc.2024.109133. arXiv:2312.14894 [hep-ph]

    work page doi:10.1016/j.cpc.2024.109133 2024

  30. [30]

    Combination of searches for invisible decays of the Higgs boson using 139 fb−1 of proton-proton collision data at√𝑠=13 TeV collected with the ATLAS experiment

    ATLAS Collaboration. “Combination of searches for invisible decays of the Higgs boson using 139 fb−1 of proton-proton collision data at√𝑠=13 TeV collected with the ATLAS experiment”.Phys. Lett. B842(2023), p. 137963.doi:10.1016/j.physletb.2023.137963. arXiv:2301.10731

    work page doi:10.1016/j.physletb.2023.137963 2023

  31. [31]

    Dark Matter

    Particle Data Group.Review of Particle Physics: Dark Matter. PDG Review. See PDG review “Dark Matter”. 2023

    2023

  32. [32]

    How arbitrary are perturbative calculations of the electroweak phase transition?

    Peter Athron, Csaba Balazs, Andrew Fowlie, Lachlan Morris, Graham White, and Yang Zhang. “How arbitrary are perturbative calculations of the electroweak phase transition?”JHEP01(2023), p. 050.doi:10.1007/JHEP01(2023)050. arXiv:2208.01319 [hep-ph]

    work page doi:10.1007/jhep01(2023)050 2023

  33. [33]

    Theo- retical uncertainties for cosmological first-order phase transitions

    Djuna Croon, Oliver Gould, Philipp Schicho, Tuomas V. I. Tenkanen, and Graham White. “Theo- retical uncertainties for cosmological first-order phase transitions”.JHEP04(2021), p. 055.doi: 10.1007/JHEP04(2021)055. arXiv:2009.10080 [hep-ph]

    work page doi:10.1007/jhep04(2021)055 2021

  34. [34]

    On the perturbative expansion at high temperature and implications for cosmological phase transitions

    Oliver Gould and Tuomas V. I. Tenkanen. “On the perturbative expansion at high temperature and implications for cosmological phase transitions”.JHEP06(2021), p. 069.doi:10.1007/ JHEP06(2021)069. arXiv:2104.04399 [hep-ph]

    arXiv 2021

  35. [35]

    Resummation in a Hot Scalar Field Theory

    Rajesh R. Parwani. “Resummation in a hot scalar field theory”.Phys. Rev. D45(1992), pp. 4695– 4705.doi:10.1103/PhysRevD.45.4695. arXiv:hep-ph/9204216 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.45.4695 1992

  36. [36]

    Symmetry Restoration and Breaking at Finite Temperature: An Introductory Review

    Eibun Senaha. “Symmetry Restoration and Breaking at Finite Temperature: An Introductory Review”.Symmetry12, 5 (2020), p. 733.doi:10.3390/sym12050733

    work page doi:10.3390/sym12050733 2020

  37. [37]

    Self consistent thermal resummation: a case studyofthephasetransitionin2HDM

    Pedro Bittar, Subhojit Roy, and Carlos E. M. Wagner. “Self consistent thermal resummation: a case studyofthephasetransitionin2HDM”.JHEP12(2025),p.021.doi: 10.1007/JHEP12(2025)021. arXiv:2504.02024 [hep-ph]

    work page doi:10.1007/jhep12(2025)021 2025

  38. [38]

    Baryon Washout, Electroweak Phase Transition, and Perturbation Theory

    Hiren H. Patel and Michael J. Ramsey-Musolf. “Baryon washout, electroweak phase transition, and perturbation theory”.JHEP07(2011), p. 029.doi: 10.1007/JHEP07(2011)029 . arXiv: 1101.4665 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep07(2011)029 2011

  39. [39]

    What is the Criterion for a Strong First Order Electroweak Phase Transition in Singlet Models?

    Amine Ahriche. “What is the criterion for a strong first order electroweak phase transition in singlet models?”Phys. Rev. D75(8 2007), p. 083522.doi:10.1103/PhysRevD.75.083522 . arXiv: hep-ph/0701192 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.75.083522 2007

  40. [40]

    Singlet Higgs Phenomenology and the Electroweak Phase Transition

    StefanoProfumo,MichaelJRamsey-Musolf,andGabeShaughnessy.“SingletHiggsphenomenology and the electroweak phase transition”.JHEP2007, 08 (2007), pp. 010–010.doi:10.1088/1126- 6708/2007/08/010. arXiv:0705.2425 [hep-ph]. 36

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1126- 2007

  41. [41]

    PhaseTracer2: from the effective potential to gravitational waves

    Peter Athron, Csaba Balázs, Andrew Fowlie, Lachlan Morris, William Searle, Yang Xiao, and Yang Zhang. “PhaseTracer2: from the effective potential to gravitational waves”.Eur. Phys. J. C85, 5 (2025), p. 559.doi: 10 . 1140 / epjc / s10052 - 025 - 14258 - y. arXiv: 2412 . 04881 [astro-ph.CO]

    2025

  42. [42]

    Fate of the false vacuum: Semiclassical theory

    Sidney R. Coleman. “Fate of the false vacuum: Semiclassical theory”.Phys. Rev. D15(1977). [Erratum: Phys. Rev. D 16, 1248 (1977)], pp. 2929–2936.doi:10.1103/PhysRevD.15.2929

    work page doi:10.1103/physrevd.15.2929 1977

  43. [43]

    Fate of the false vacuum. II. First quantum corrections

    Curtis G. Callan and Sidney R. Coleman. “Fate of the false vacuum. II. First quantum corrections”. Phys. Rev. D16(1977), pp. 1762–1768.doi:10.1103/PhysRevD.16.1762

    work page doi:10.1103/physrevd.16.1762 1977

  44. [44]

    Decay of the false vacuum at finite temperature

    Andrei D. Linde. “Decay of the false vacuum at finite temperature”.Nucl. Phys. B216(1983). [Erratum: Nucl. Phys. B 223, 544 (1983)], p. 421.doi:10.1016/0550-3213(83)90072-X

    work page doi:10.1016/0550-3213(83)90072-x 1983

  45. [45]

    Fate of the false vacuum at finite temperature: Theory and applications

    Andrei D. Linde. “Fate of the false vacuum at finite temperature: Theory and applications”.Phys. Lett. B100(1981), pp. 37–40.doi:10.1016/0370-2693(81)90281-1

    work page doi:10.1016/0370-2693(81)90281-1 1981

  46. [46]

    Cosmological phase transitions from the functional measure

    Bruno Berganholi, Gláuber C. Dorsch, Iberê Kuntz, Beatriz M. D. Sena, and Giovanna F. do Valle. “Cosmological phase transitions from the functional measure”.JHEP07(2025), p. 129.doi: 10.1007/JHEP07(2025)129. arXiv:2502.09593 [hep-ph]

    work page doi:10.1007/jhep07(2025)129 2025

  47. [47]

    How fast can the wall move? A study of the electroweak phase transition dynamics

    GuyD.MooreandTomislavProkopec.“Howfastcanthewallmove?Astudyoftheelectroweakphase transitiondynamics”.Phys.Rev.D52(1995),pp.7182–7204.doi: 10.1103/PhysRevD.52.7182. arXiv:hep-ph/9506475

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.52.7182 1995

  48. [48]

    Bubble wall dynamics at the electroweak phase transition

    Stefania De Curtis, Luigi Delle Rose, Andrea Guiggiani, Ángel Gil Muyor, and Giuliano Panico. “Bubble wall dynamics at the electroweak phase transition”.JHEP03(2022), p. 163.doi: 10.1007/JHEP03(2022)163. arXiv:2201.08220 [hep-ph]

    work page doi:10.1007/jhep03(2022)163 2022

  49. [49]

    Supercool subtleties of cosmological phase transitions

    Peter Athron, Csaba Balázs, and Lachlan Morris. “Supercool subtleties of cosmological phase transitions”.JCAP03(2023), p. 006.doi: 10 . 1088 / 1475 - 7516 / 2023 / 03 / 006. arXiv: 2212.07559 [hep-ph]

    arXiv 2023

  50. [50]

    Phase transition dynamics and gravitational wave spectra of strong first-order phase transition in supercooled universe

    Xiao Wang, Fa Peng Huang, and Xinmin Zhang. “Phase transition dynamics and gravitational wave spectra of strong first-order phase transition in supercooled universe”.JCAP05(2020), p. 045.doi: 10.1088/1475-7516/2020/05/045. arXiv:2003.08892 [hep-ph]

    work page doi:10.1088/1475-7516/2020/05/045 2020

  51. [51]

    Di-Higgs production in the 4b channel and gravitational wave complementarity

    Alexandre Alves, Dorival Gonçalves, Tathagata Ghosh, Huai-Ke Guo, and Kuver Sinha. “Di-Higgs production in the 4b channel and gravitational wave complementarity”.JHEP2020, 3 (2020), p. 53. doi:10.1007/JHEP03(2020)053. arXiv:1909.05268v2 [hep-ph]

    work page doi:10.1007/jhep03(2020)053 2020

  52. [52]

    Detector configuration of DECIGO/BBO and identification of cosmological neutron-star binaries

    Kent Yagi and Naoki Seto. “Detector configuration of DECIGO/BBO and identification of cosmological neutron-star binaries”.Phys. Rev. D83(2011), p. 044011.doi:10.1103/PhysRevD. 83.044011. arXiv:1101.3940 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd 2011

  53. [53]

    Gravitational Wave Background and Non-Gaussianity as a Probe of the Curvaton Scenario

    Kazunori Nakayama and Jun’ichi Yokoyama. “Gravitational wave background and non-Gaussianity as a probe of the curvaton scenario”.JCAP01(2010), p. 010.doi:10.1088/1475-7516/2010/ 01/010. arXiv:0910.0715 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2010/ 2010

  54. [54]

    Numerical simulations of acoustically generated gravitational waves at a first order phase transition

    Mark Hindmarsh, Stephan J Huber, Kari Rummukainen, and David J Weir. “Numerical simulations of acoustically generated gravitational waves at a first order phase transition”.Phys. Rev. D92, 12 (2015), p. 123009.doi:10.1103/PhysRevD.92.123009. arXiv:1504.03291 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.92.123009 2015

  55. [55]

    Erratum: Shape of the acoustic gravitational wave power spectrum from a first order phase transition [Phys. Rev. D 96, 103520 (2017)]

    Mark Hindmarsh, Stephan J Huber, Kari Rummukainen, and David J Weir. “Erratum: Shape of the acoustic gravitational wave power spectrum from a first order phase transition [Phys. Rev. D 96, 103520 (2017)]”.Physical Review D101, 8 (2020), p. 089902.doi:10.1103/PhysRevD.101. 089902. 37

    work page doi:10.1103/physrevd.101 2017

  56. [56]

    The stochastic gravitational wave background from turbulence and magnetic fields generated by a first-order phase transition

    ChiaraCaprini,RuthDurrer,andGeraldineServant.“Thestochasticgravitationalwavebackground from turbulence and magnetic fields generated by a first-order phase transition”.JCAP2009, 12 (2009), pp. 024–024.doi:10.1088/1475-7516/2009/12/024. arXiv:0909.0622 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2009/12/024 2009

  57. [57]

    Cosmological Backgrounds of Gravitational Waves and eLISA/NGO: Phase Transitions, Cosmic Strings and Other Sources

    Pierre Binetruy, Alejandro Bohe, Chiara Caprini, and Jean-Francois Dufaux. “Cosmological backgrounds of gravitational waves and eLISA/NGO: phase transitions, cosmic strings and other sources”.JCAP2012, 06 (2012), pp. 027–027.doi:10.1088/1475-7516/2012/06/027. arXiv: 1201.0983 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2012/06/027 2012

  58. [58]

    Energy budget of cosmological first-order phase transitions

    Jose R Espinosa, Thomas Konstandin, Jose M No, and Geraldine Servant. “Energy budget of cosmological first-order phase transitions”.JCAP2010, 06 (2010), pp. 028–028.doi:10.1088/ 1475-7516/2010/06/028. arXiv:1004.4187 [hep-ph]. 38

    Pith/arXiv arXiv 2010