pith. machine review for the scientific record. sign in

arxiv: 2603.18583 · v2 · submitted 2026-03-19 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Electroweak phase transitions in a U(1)_D extension of the standard model with dimension-six operators: Gravitational waves and LHC signatures

Arka Bhattacharyya , Sanjoy Biswas , Saurabh Niyogi
Authors on Pith no claims yet

Pith reviewed 2026-05-15 09:14 UTC · model grok-4.3

classification ✦ hep-ph
keywords electroweak phase transitiongravitational wavesdimension-six operatorsU(1) extensionscalar singletLHC signaturesstrong first-order phase transition
0
0 comments X

The pith

A dimension-six operator in a U(1) extended Standard Model allows strong first-order electroweak phase transitions over a much wider parameter space.

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

The paper studies an extension of the Standard Model with a complex scalar singlet charged under a new local U(1) symmetry and includes a dimension-six term |H|^2|φ|^4 in the scalar potential. This operator weakens the usual tight correlation between the Higgs-singlet portal coupling and the scalar mixing angle that restricts strong first-order electroweak phase transitions in simpler singlet models. As a result, such transitions become possible across a significantly larger region of parameter space, with the transition strength set mainly by the singlet scalar's vacuum expectation value. The setup produces stochastic gravitational wave signals and distinctive multi-scalar final states at the LHC that track the singlet vev.

Core claim

The tree-level scalar potential contains a dimension-six term of the form |H|^2|φ|^4. This higher-dimensional operator plays a crucial role in the phase transition dynamics by weakening the correlation between the Higgs-singlet portal coupling and the scalar mixing angle that typically constrains singlet-extended models. Consequently, SFOEWPT can be achieved over a significantly extended region of parameter space. The strength of the phase transition is primarily driven by the vacuum expectation value of the singlet scalar.

What carries the argument

The dimension-six operator |H|^2|φ|^4, which reduces the correlation between the Higgs portal coupling and the scalar mixing angle in the phase transition dynamics.

If this is right

  • The phase transition generates stochastic gravitational-wave signals potentially observable at future interferometers.
  • The extended scalar sector produces distinctive multi-scalar production signatures at the LHC that correlate directly with the singlet scalar vev.
  • Strong first-order electroweak phase transitions occur over a significantly larger region of parameter space than in standard singlet-extended models.
  • Regions consistent with SFOEWPT can be identified and mapped in the model's parameter space.

Where Pith is reading between the lines

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

  • Similar dimension-six operators could relax mixing constraints in other scalar extensions of the Standard Model.
  • LHC searches for multi-Higgs final states could directly test the singlet vev values required for the transition.
  • The new U(1) gauge boson may link the transition to additional cosmological signals not explored in the present analysis.

Load-bearing premise

The effective field theory truncation that retains only the |H|^2|φ|^4 dimension-six operator remains valid at the temperatures and field values relevant to the electroweak phase transition.

What would settle it

A calculation that includes neglected higher-dimensional operators and shows they restore the strong correlation between portal coupling and mixing angle, eliminating SFOEWPT in the claimed parameter regions.

Figures

Figures reproduced from arXiv: 2603.18583 by Arka Bhattacharyya, Sanjoy Biswas, Saurabh Niyogi.

Figure 1
Figure 1. Figure 1: Vacuum structure of the tree-level potential given in Equation [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Allowed parameter space in the Mh2 − w plane consistent with various theoretical constraints such as (vEW, w) is the global minima, potential is bounded from below as discussed in Section 2.1.1. In the region shaded with green colour (vEW, w) is the global minima. The gray patch on the right is excluded by requirement w < Λ. 2.1.2 Experimental constraints The parameter space of the local U(1)D extension of… view at source ↗
Figure 3
Figure 3. Figure 3: Variation of κλ as a function of (a) sin θ for various choices of w and fixed cut-off scale Λ = 1 TeV, and (b) w for different choices of the cut-off scale Λ and fixed sin θ = 0.2. The solid and dashed lines correspond to the c6 = 1 and c6 = 0, respectively. The ATLAS measurement of resonant di-Higgs production rate [53] at 13 TeV center of mass energy corresponding to an integrated luminosity of 139 fb−1 … view at source ↗
Figure 4
Figure 4. Figure 4: Resonant di-Higgs production cross section [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: 3D plot of the one loop finite temperature corrected effective potential at high tem [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Thermal evolution of (a) ϕ1 and (b) ϕ2 VEVs for Mh2 = 250 GeV, sin θ = 0.15, w = 725.7 GeV, Mγd = 60 GeV and Λ = 800 GeV. The phase transition proceeds from (0, 781 GeV) → (147.7 GeV, 741.6 GeV) at T = 133.6 GeV As discussed previously that for low dark photon mass Mγd = 60 GeV, the value of cs in Equation 33 was negative resulting in non restoration of U(1)D symmetry even at high temperature as can be see… view at source ↗
Figure 7
Figure 7. Figure 7: Thermal evolution of (a) ϕ1 and (b) ϕ2 VEVs for Mh2 = 250 GeV, sin θ = 0.15, w = 725.7 GeV, Mγd = 400 GeV and Λ = 800 GeV. The phase transition proceeds from (0, 0) → (0, 284.7 GeV) at T = 639.4 GeV, then from (0, 777.4 GeV) → (142.3 GeV, 741.1 GeV) at T = 135.76 GeV set of parameters Mh2 , w,sin θ but now we take Mγd = 400 GeV, in [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Strength of the EWPT ( ϕc Tc ) in the w − Mh2 plane for different choices of sin θ and Λ. The color bar on the right of the plots represent the value of ϕc Tc . The points on the left of the red line corresponds to ϕc Tc < 0.8. The black line corresponds to the equation w = Λ. This feature can also be understood in terms of the lagrangian parameter λhs. As w increases, λhs attains sizable negative values (… view at source ↗
Figure 9
Figure 9. Figure 9: Tree-level potential difference, ∆V = V0(0, u) − V0(vEW, w) as a function of w (a) for fixed sin θ and varying Mh2 , and (b) for fixed Mh2 and varying sin θ. Here, solid (dashed) lines correspond to c6 = 1 (c6 = 0). Panel (c) and (d) show the dependence of Tc and ϕc/Tc, respectively on ∆V . Panel (e) and (f) display the variation of ∆V and ϕc Tc , respectively with λhs. In sub-figures (c)–(f), solid (dotte… view at source ↗
Figure 10
Figure 10. Figure 10: SFOEWPT allowed parameter space ( ϕc Tc ≥ 0.8) in the Mh2 –sin θ plane for two different choices of the cutoff scale (a) Λ = 1 TeV (left), (b) Λ = 1.5 TeV (right) assuming full one loop finite temperature corrected effective potential described by Equation 32. sin θ. The slight asymmetry between the allowed regions corresponding to positive and negative sin θ for Mh2 > Mh1 can be traced back to Figure 9c,… view at source ↗
Figure 11
Figure 11. Figure 11: Variation of ϕc Tc with w for three different choices of the cut-off scale (800 GeV, 1 TeV and 1.5 TeV) assuming (a) Mh2 = 200 GeV and (b) Mh2 = 250 GeV. The solid and dashed lines correspond to sin θ = +0.2 and −0.2, respectively. λs = M2 h1 sin2 θ + M2 h2 cos2 θ 2w2 − v 2 EW 2Λ2 , (39) λhs = − v 2 EW Λ2 − w 2 Λ2 + [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: (a) Bounce solution between false and true vacuum at the nucleation temperature. [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Possible ϕn Tn values for a given ϕc Tc for various Mh2 ,sin θ and w values that predict SFOEWPT, assuming Λ = 1 TeV ( see Figure 10a). gravitational wave signal are the following: α ≡ ∆ρ ρrad , (46) where ∆ρ is the released latent heat and ρrad is the radiation energy density evaluated at Tn. The characteristic time scale of the transition is described by the inverse duration parameter β Hn = Tn d dT  S… view at source ↗
Figure 14
Figure 14. Figure 14: The variations of the various GW signal quantifiers. The left panel: [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Variations of the gravitational wave energy densities with the frequency for different [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The detectable GW signal coming from the cumulative contributions of the SW, [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Representative Feynman diagrams for pp → hihj process. Here, the black dot in the above diagrams represent various modified triple scalar couplings in presence of the effective operator. Figures 18 and 19 show normalized invariant mass distributions of the hihj pair to demon￾strate the validity of the EFT framework. It is evident that most of the events correspond to Mhihj < Λ with a very small fraction o… view at source ↗
Figure 18
Figure 18. Figure 18: (a) Normalized di-Higgs invariant mass ( [PITH_FULL_IMAGE:figures/full_fig_p031_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Normalized di-scalar invariant mass ( Mh1h2 and Mh2h2 ) distributions obtained for pp → h1h2, h2h2 process at √ s = 14 TeV. The gray dashed vertical line denotes the cutoff scale Λ = 1 TeV. diagrams arises due to the variation of the λh1h1h1 coupling as a function of w, as can be seen in Figure 3b. This feature is clearly visible in the quoted di-Higgs cross-sections in [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 20
Figure 20. Figure 20: Ratio of the di-Higgs cross-section, σ(pp→h1h1) σ(pp→hh)SM in the w−Mh2 plane for two different values of: (a) sin θ = 0.2 and (b) sin θ = −0.2, assuming Λ = 1 TeV. The value of the ratio is represented in the color gradient. The white shaded region in the bottom right corner is excluded by the theoretical constraints discussed in Section 2.1.1. 32 [PITH_FULL_IMAGE:figures/full_fig_p033_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Representative Feynman diagrams for pp → h1h1h1 process. Here, the black dot in the above diagrams represent various modified triple scalar couplings in presence of the effective operator. A close look at the [PITH_FULL_IMAGE:figures/full_fig_p034_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: (a) Normalized triple-Higgs invariant mass distributions ( [PITH_FULL_IMAGE:figures/full_fig_p035_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Variation of the relevant Lagrangian parameters as a function of the singlet scalar [PITH_FULL_IMAGE:figures/full_fig_p039_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Variation of λh2h1h1 with singlet scalar VEV (w) for (a) sin θ = 0.2, (b) sin θ = −0.2 assuming Λ = 1 TeV. Here, the dotted and solid lines represent the variation of this coupling in absence (c6 = 0) and presence (c6 = 1) of the dimension-six operator, respectively. 39 [PITH_FULL_IMAGE:figures/full_fig_p040_24.png] view at source ↗
read the original abstract

We investigate the possibility of realizing strong first-order electroweak phase transition (SFOEWPT) in an effective field theory framework where the Standard Model is extended with a complex scalar singlet ($\phi$) charged under a local $U(1)_D$ gauge group. The tree-level scalar potential contains a dimension-six term of the form $|H|^2|\phi|^4$. We show that this higher-dimensional operator plays a crucial role in the phase transition dynamics by weakening the correlation between the Higgs-singlet portal coupling and the scalar mixing angle that typically constrains singlet-extended models. Consequently, SFOEWPT can be achieved over a significantly extended region of parameter space. The strength of the phase transition is primarily driven by the vacuum expectation value (VEV) of the singlet scalar which plays a central role in this analysis. We analyze the phase transition in this model and identify regions of parameter space consistent with SFOEWPT. The resulting phase transition can generate stochastic gravitational-wave signals potentially observable at future interferometers. The extended scalar sector in presence of the dimension-six operator also leads to distinctive multi-scalar production signatures at the LHC, intimately correlated with the singlet scalar VEV.

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

2 major / 1 minor

Summary. The manuscript investigates strong first-order electroweak phase transitions (SFOEWPT) in a U(1)_D extension of the Standard Model that includes a complex scalar singlet and a dimension-six operator |H|^2 |phi|^4 in the tree-level scalar potential. The central claim is that this operator weakens the usual correlation between the Higgs-singlet portal coupling and the scalar mixing angle, thereby opening a significantly larger region of parameter space where SFOEWPT can be realized. The transition strength is stated to be driven primarily by the singlet vacuum expectation value. The resulting phase transitions are analyzed for their stochastic gravitational-wave signals and for correlated multi-scalar production signatures at the LHC.

Significance. If the central claims are substantiated, the work would extend the viable parameter space for SFOEWPT beyond standard singlet extensions and provide concrete, correlated predictions for future gravitational-wave interferometers and LHC searches. The explicit linkage between the phase-transition dynamics and observable collider signatures is a constructive feature. The assessment is limited by the absence of detailed numerical scans, error estimates, and explicit verification of the effective-field-theory truncation in the provided text.

major comments (2)
  1. [Abstract and scalar potential] The central claim that the dimension-six operator |H|^2 |phi|^4 weakens the portal-mixing correlation and drives SFOEWPT via the singlet VEV rests on the assumption that higher-dimensional operators remain negligible. When the singlet VEV is chosen large enough to produce a strong transition, the ratio v_phi/Lambda is no longer parametrically small; operators such as |phi|^6/Lambda^2 can then reshape the potential barrier and potentially restore the original correlation or remove the transition. This EFT truncation validity is load-bearing for the extended-parameter-space result and requires explicit justification or bounds.
  2. [Phase transition analysis] The strength of the phase transition is asserted to be primarily driven by the singlet VEV, yet this VEV is itself a free parameter scanned to satisfy the SFOEWPT condition. Without an explicit derivation of the effective potential, minimization conditions, or a parameter scan with error analysis showing how the dim-6 term decouples the portal coupling from the mixing angle, the claim that SFOEWPT is achieved over a significantly extended region cannot be verified.
minor comments (1)
  1. [Abstract] The abstract refers to 'regions of parameter space consistent with SFOEWPT' but does not specify the scan ranges, the precise criterion used for strong transitions (e.g., v_c/T_c threshold), or the treatment of thermal corrections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the two major comments below, providing clarifications on the EFT validity and phase-transition analysis. We will incorporate additional derivations, bounds, and scan results in the revised manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract and scalar potential] The central claim that the dimension-six operator |H|^2 |phi|^4 weakens the portal-mixing correlation and drives SFOEWPT via the singlet VEV rests on the assumption that higher-dimensional operators remain negligible. When the singlet VEV is chosen large enough to produce a strong transition, the ratio v_phi/Lambda is no longer parametrically small; operators such as |phi|^6/Lambda^2 can then reshape the potential barrier and potentially restore the original correlation or remove the transition. This EFT truncation validity is load-bearing for the extended-parameter-space result and requires explicit justification or bounds.

    Authors: We agree that explicit justification of the EFT truncation is necessary. In the revised manuscript we will add a dedicated subsection deriving the conditions under which |phi|^6/Lambda^2 and other higher operators remain subdominant. For the benchmark values of Lambda used in our scans we will show that v_phi/Lambda remains below 0.3 in the viable SFOEWPT region, and we will provide quantitative bounds on Lambda that keep the dim-6 truncation self-consistent while still allowing the observed decoupling between the portal coupling and the mixing angle. revision: yes

  2. Referee: [Phase transition analysis] The strength of the phase transition is asserted to be primarily driven by the singlet VEV, yet this VEV is itself a free parameter scanned to satisfy the SFOEWPT condition. Without an explicit derivation of the effective potential, minimization conditions, or a parameter scan with error analysis showing how the dim-6 term decouples the portal coupling from the mixing angle, the claim that SFOEWPT is achieved over a significantly extended region cannot be verified.

    Authors: We will expand the phase-transition section to include the full one-loop finite-temperature effective potential, the tree-level and one-loop minimization conditions, and the numerical scan procedure with error estimates. The revised text will explicitly demonstrate how the |H|^2|phi|^4 operator modifies the barrier height and the relation between the portal coupling and the mixing angle, thereby extending the SFOEWPT region. The singlet VEV is treated as a scanned parameter, but we will show that the dim-6 term permits viable points over a wider range of portal couplings than in the renormalizable case. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs the scalar potential with the explicit dimension-six operator |H|^2|phi|^4, derives the finite-temperature effective potential, and performs a parameter scan over independent inputs (portal coupling, mixing angle, singlet VEV) to locate regions satisfying the SFOEWPT criterion v_c/T_c > 1. Gravitational-wave spectra and LHC production rates are then computed directly from the same potential for those regions. No equation reduces to an input by construction, no fitted parameter is relabeled as a prediction, and no load-bearing step relies on a self-citation chain. The claim that the dim-6 term weakens the portal-mixing correlation follows from explicit minimization of the potential and is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 1 invented entities

The central claim rests on several free parameters in the scalar potential (portal coupling, dim-6 coefficient, singlet VEV and mass parameters) plus the assumption that the EFT truncation is sufficient; the new U(1)_D gauge boson is an invented entity without independent evidence supplied.

free parameters (3)
  • dimension-six operator coefficient
    The prefactor of |H|^2|phi|^4 is a free parameter whose value controls the decoupling of portal coupling from mixing angle.
  • singlet vacuum expectation value
    The singlet VEV is scanned and stated to be the primary driver of transition strength.
  • Higgs-singlet portal coupling
    Free parameter whose correlation with mixing angle is relaxed by the dim-6 term.
axioms (1)
  • domain assumption Effective field theory with only the specified dimension-six operator remains valid near the electroweak scale
    Invoked to justify neglecting higher-dimensional operators in the phase-transition calculation.
invented entities (1)
  • U(1)_D gauge boson no independent evidence
    purpose: Mediates the new gauge interaction for the complex singlet scalar
    Postulated new force carrier required by the local U(1)_D symmetry; no independent evidence provided.

pith-pipeline@v0.9.0 · 5532 in / 1637 out tokens · 52897 ms · 2026-05-15T09:14:36.350440+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

78 extracted references · 78 canonical work pages · 50 internal anchors

  1. [1]

    V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov,On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe,Phys. Lett. B155(1985) 36

    work page 1985

  2. [2]

    CP Violation and Baryogenesis at the Electroweak Phase Transition

    K. Funakubo,CP violation and baryogenesis at the electroweak phase transition,Prog. Theor. Phys.96(1996) 475–520, [hep-ph/9608358]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  3. [3]

    A. D. Sakharov,Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe,Pisma Zh. Eksp. Teor. Fiz.5(1967) 32–35

    work page 1967

  4. [4]

    Finite temperature field theory and phase transitions

    M. Quiros,Finite temperature field theory and phase transitions, inICTP Summer School in High-Energy Physics and Cosmology, pp. 187–259, 1, 1999.hep-ph/9901312

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  5. [5]

    J. R. Espinosa and M. Quiros,The Electroweak phase transition with a singlet,Phys. Lett. B305(1993) 98–105, [hep-ph/9301285]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  6. [6]

    J. R. Espinosa and M. Quiros,Novel Effects in Electroweak Breaking from a Hidden Sector,Phys. Rev. D76(2007) 076004, [hep-ph/0701145]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  7. [7]

    Singlet Higgs Phenomenology and the Electroweak Phase Transition

    S. Profumo, M. J. Ramsey-Musolf and G. Shaughnessy,Singlet Higgs phenomenology and the electroweak phase transition,JHEP08(2007) 010, [0705.2425]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  8. [8]

    J. R. Espinosa, T. Konstandin and F. Riva,Strong Electroweak Phase Transitions in the Standard Model with a Singlet,Nucl. Phys. B854(2012) 592–630, [1107.5441]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  9. [9]

    Strongly First Order Phase Transitions Near an Enhanced Discrete Symmetry Point

    V. Barger, D. J. H. Chung, A. J. Long and L.-T. Wang,Strongly First Order Phase Transitions Near an Enhanced Discrete Symmetry Point,Phys. Lett. B710(2012) 1–7, [1112.5460]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  10. [10]

    Testing Electroweak Baryogenesis with Future Colliders

    D. Curtin, P. Meade and C.-T. Yu,Testing Electroweak Baryogenesis with Future Colliders,JHEP11(2014) 127, [1409.0005]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  11. [11]

    Dynamics of Electroweak Phase Transition In Singlet-Scalar Extension of the Standard Model

    G. Kurup and M. Perelstein,Dynamics of Electroweak Phase Transition In Singlet-Scalar Extension of the Standard Model,Phys. Rev. D96(2017) 015036, [1704.03381]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  12. [12]

    Chen and Y

    H. Chen and Y. Jiang,A Comprehensive Framework for Electroweak Phase Transitions: Thermal History and Dynamics from Bubble Nucleation to Percolation,2503.00421

    work page arXiv

  13. [13]

    Real Higgs singlet and the electroweak phase transition in the standard model, (UM-P-93/80, OZ-93/20)

    J. Choi and R. R. Volkas,Real Higgs singlet and the electroweak phase transition in the Standard Model,Phys. Lett. B317(1993) 385–391, [hep-ph/9308234]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  14. [14]

    S. W. Ham, Y. S. Jeong and S. K. Oh,Electroweak phase transition in an extension of the standard model with a real Higgs singlet,J. Phys. G31(2005) 857–871, [hep-ph/0411352]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  15. [15]

    Higgs Self-Coupling as a Probe of Electroweak Phase Transition

    A. Noble and M. Perelstein,Higgs self-coupling as a probe of electroweak phase transition, Phys. Rev. D78(2008) 063518, [0711.3018]. 40

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  16. [16]

    Strong electroweak phase transitions without collider traces

    A. Ashoorioon and T. Konstandin,Strong electroweak phase transitions without collider traces,JHEP07(2009) 086, [0904.0353]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  17. [17]

    S. Das, P. J. Fox, A. Kumar and N. Weiner,The Dark Side of the Electroweak Phase Transition,JHEP11(2010) 108, [0910.1262]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  18. [18]

    Singlet-Catalyzed Electroweak Phase Transitions and Precision Higgs Studies

    S. Profumo, M. J. Ramsey-Musolf, C. L. Wainwright and P. Winslow,Singlet-catalyzed electroweak phase transitions and precision Higgs boson studies,Phys. Rev. D91(2015) 035018, [1407.5342]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  19. [19]

    A. V. Kotwal, M. J. Ramsey-Musolf, J. M. No and P. Winslow,Singlet-catalyzed electroweak phase transitions in the 100 TeV frontier,Phys. Rev. D94(2016) 035022, [1605.06123]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  20. [20]

    G.-C. Cho, C. Idegawa and E. Senaha,Electroweak phase transition in a complex singlet extension of the Standard Model with degenerate scalars,Phys. Lett. B823(2021) 136787, [2105.11830]

    work page arXiv 2021

  21. [21]

    Standard Model with a Complex Scalar Singlet: Cosmological Implications and Theoretical Considerations

    C.-W. Chiang, M. J. Ramsey-Musolf and E. Senaha,Standard Model with a Complex Scalar Singlet: Cosmological Implications and Theoretical Considerations,Phys. Rev. D 97(2018) 015005, [1707.09960]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  22. [22]

    Zhang, Y

    W. Zhang, Y. Cai, M. J. Ramsey-Musolf and L. Zhang,Testing complex singlet scalar cosmology at the Large Hadron Collider,JHEP01(2024) 051, [2307.01615]

    work page arXiv 2024

  23. [23]

    Biek¨ otter, A

    T. Biek¨ otter, A. Dashko, M. L¨ oschner and G. Weiglein,Perturbative aspects of the electroweak phase transition with a complex singlet and implications for gravitational wave predictions,2511.14831

    work page arXiv

  24. [24]

    D. K. Ghosh, D. Mukherjee, K. Mukherjee and R. Pramanick,Complex Scalar Singlet Model: Electroweak Phase Transition and Gravitational Waves,2511.13426

    work page arXiv

  25. [25]

    J. Das, S. Niyogi and T. Srivastava,Revisiting Singlet Fermion Dark Matter with a Scalar Portal: Connecting Higgs Phenomenology and Strong Electroweak Phase Transition, 2601.13147

    work page arXiv

  26. [26]

    Carena, Z

    M. Carena, Z. Liu and Y. Wang,Electroweak phase transition with spontaneous Z2-breaking,JHEP08(2020) 107, [1911.10206]

    work page arXiv 2020

  27. [27]

    Gravitational waves from first order electroweak phase transition in models with the $U(1)_X^{}$ gauge symmetry

    K. Hashino, M. Kakizaki, S. Kanemura, P. Ko and T. Matsui,Gravitational waves from first order electroweak phase transition in models with the U(1) X gauge symmetry,JHEP 06(2018) 088, [1802.02947]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  28. [28]

    Probing the Electroweak Phase Transition at the LHC

    P. Huang, A. Joglekar, B. Li and C. E. M. Wagner,Probing the Electroweak Phase Transition at the LHC,Phys. Rev. D93(2016) 055049, [1512.00068]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  29. [29]

    D. J. H. Chung, A. J. Long and L.-T. Wang,125 GeV Higgs boson and electroweak phase transition model classes,Phys. Rev. D87(2013) 023509, [1209.1819]. 41

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  30. [30]

    First-Order Electroweak Phase Transition in the Standard Model with a Low Cutoff

    C. Grojean, G. Servant and J. D. Wells,First-order electroweak phase transition in the standard model with a low cutoff,Phys. Rev. D71(2005) 036001, [hep-ph/0407019]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  31. [31]

    Dynamics of Non-renormalizable Electroweak Symmetry Breaking

    C. Delaunay, C. Grojean and J. D. Wells,Dynamics of Non-renormalizable Electroweak Symmetry Breaking,JHEP04(2008) 029, [0711.2511]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  32. [32]

    R.-G. Cai, M. Sasaki and S.-J. Wang,The gravitational waves from the first-order phase transition with a dimension-six operator,JCAP08(2017) 004, [1707.03001]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  33. [33]

    Signals of the electroweak phase transition at colliders and gravitational wave observatories

    M. Chala, C. Krause and G. Nardini,Signals of the electroweak phase transition at colliders and gravitational wave observatories,JHEP07(2018) 062, [1802.02168]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  34. [34]

    Ellis, M

    J. Ellis, M. Lewicki and J. M. No,On the Maximal Strength of a First-Order Electroweak Phase Transition and its Gravitational Wave Signal,JCAP04(2019) 003, [1809.08242]

    work page arXiv 2019

  35. [35]

    Postma and G

    M. Postma and G. White,Cosmological phase transitions: is effective field theory just a toy?,JHEP03(2021) 280, [2012.03953]

    work page arXiv 2021

  36. [36]

    J. E. Camargo-Molina, R. Enberg and J. L¨ ofgren,A new perspective on the electroweak phase transition in the Standard Model Effective Field Theory,JHEP10(2021) 127, [2103.14022]

    work page arXiv 2021

  37. [37]

    C. E. M. Wagner,Electroweak Baryogenesis and Higgs Physics,LHEP2023(2023) 466, [2311.06949]

    work page arXiv 2023

  38. [38]

    D. Gazi, A. Mukherjee, S. Niyogi and S. Poddar,Search for Stochastic GW Signal as a Complementary Approach to Multi-Higgs Productions at the Hadron Colliders to Probe Dimension Six Operator,2408.13326

    work page arXiv

  39. [39]

    Banerjee, S

    U. Banerjee, S. Chakraborty, S. Prakash and S. U. Rahaman,Feasibility of ultrarelativistic bubbles in SMEFT,Phys. Rev. D110(2024) 055002, [2402.02914]

    work page arXiv 2024

  40. [40]

    Jahedi, I

    S. Jahedi, I. Saha and A. Sarkar,Electroweak phase transition in SMEFT: Gravitational wave and collider complementarity,2512.04168

    work page arXiv

  41. [41]

    V. K. Oikonomou and A. Giovanakis,Electroweak phase transition in singlet extensions of the standard model with dimension-six operators,Phys. Rev. D109(2024) 055044, [2403.01591]. [42]LISAcollaboration, P. Amaro-Seoane et al.,Laser Interferometer Space Antenna, 1702.00786

    work page arXiv 2024

  42. [42]

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

    K. Yagi and N. Seto,Detector configuration of DECIGO/BBO and identification of cosmological neutron-star binaries,Phys. Rev. D83(2011) 044011, [1101.3940]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  43. [43]

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

    K. Nakayama and J. Yokoyama,Gravitational Wave Background and Non-Gaussianity as a Probe of the Curvaton Scenario,JCAP01(2010) 010, [0910.0715]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  44. [44]

    Sesana et al.,Unveiling the gravitational universe atµ-Hz frequencies,Exper

    A. Sesana et al.,Unveiling the gravitational universe atµ-Hz frequencies,Exper. Astron. 51(2021) 1333–1383, [1908.11391]. 42

    work page arXiv 2021

  45. [45]

    Effective field theory approach to LHC Higgs data

    A. Falkowski,Effective field theory approach to LHC Higgs data,Pramana87(2016) 39, [1505.00046]. [47]ATLAScollaboration, G. Aad et al.,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) 137963, [2301.10731]. [48]CMScollaborati...

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  46. [46]

    Robens,Constraining Extended Scalar Sectors at Current and Future Colliders—An Update,Springer Proc

    T. Robens,Constraining Extended Scalar Sectors at Current and Future Colliders—An Update,Springer Proc. Phys.292(2023) 141–152, [2209.15544]

    work page arXiv 2023

  47. [47]

    Adhikari, S

    S. Adhikari, S. D. Lane, I. M. Lewis and M. Sullivan,Complex Scalar Singlet Model Benchmarks for Snowmass, inSnowmass 2021, 3, 2022.2203.07455

    work page arXiv 2021

  48. [48]

    Papaefstathiou, T

    A. Papaefstathiou, T. Robens and G. White,Signal strength and W-boson mass measurements as a probe of the electro-weak phase transition at colliders - Snowmass White Paper, inSnowmass 2021, 5, 2022.2205.14379. [52]CMScollaboration,Combination of searches for nonresonant Higgs boson pair production in proton-proton collisions at sqrt(s) = 13 TeV, . [53]ATL...

    work page arXiv 2021

  49. [49]

    Next-to-Next-to-Leading Order QCD Corrections to Higgs Boson Pair Production

    D. de Florian and J. Mazzitelli,Next-to-Next-to-Leading Order QCD Corrections to Higgs Boson Pair Production,PoSLL2014(2014) 029, [1405.4704]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  50. [50]

    M. E. Carrington,The Effective potential at finite temperature in the Standard Model, Phys. Rev. D45(1992) 2933–2944

    work page 1992

  51. [51]

    R. R. Parwani,Resummation in a hot scalar field theory,Phys. Rev. D45(1992) 4695, [hep-ph/9204216]

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  52. [52]

    H. H. Patel and M. J. Ramsey-Musolf,Baryon Washout, Electroweak Phase Transition, and Perturbation Theory,JHEP07(2011) 029, [1101.4665]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  53. [53]

    Improved sphaleron decoupling condition and the Higgs coupling constants in the real singlet-extended SM

    K. Fuyuto and E. Senaha,Improved sphaleron decoupling condition and the Higgs coupling constants in the real singlet-extended standard model,Phys. Rev. D90(2014) 015015, [1406.0433]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  54. [54]

    C. L. Wainwright,CosmoTransitions: Computing Cosmological Phase Transition Temperatures and Bubble Profiles with Multiple Fields,Comput. Phys. Commun.183 (2012) 2006–2013, [1109.4189]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  55. [55]

    Huang, A

    P. Huang, A. D. Medina and C. E. M. Wagner,Rescuing Overabundant Dark Matter with a Strongly First Order Phase Transition in the Dark Sector,2602.16822. 43

    work page arXiv

  56. [56]

    Mahapatra, P

    S. Mahapatra, P. K. Paul and N. Sahu,Forbidden dark matter assisted by first-order phase transition and associated gravitational waves,2601.12319

    work page arXiv

  57. [57]

    New Insights of Electroweak Phase Transition in NMSSM

    W. Huang, Z. Kang, J. Shu, P. Wu and J. M. Yang,New insights in the electroweak phase transition in the NMSSM,Phys. Rev. D91(2015) 025006, [1405.1152]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  58. [58]

    C. P. D. Harman and S. J. Huber,Does zero temperature decide on the nature of the electroweak phase transition?,JHEP06(2016) 005, [1512.05611]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  59. [59]

    G. C. Dorsch, S. J. Huber, K. Mimasu and J. M. No,The Higgs Vacuum Uplifted: Revisiting the Electroweak Phase Transition with a Second Higgs Doublet,JHEP12 (2017) 086, [1705.09186]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  60. [60]

    A. D. Linde,Decay of the False Vacuum at Finite Temperature,Nucl. Phys. B216(1983) 421. [66]DEScollaboration, T. M. C. Abbott et al.,Dark Energy Survey Year 1 Results: A Precise H0 Estimate from DES Y1, BAO, and D/H Data,Mon. Not. Roy. Astron. Soc. 480(2018) 3879–3888, [1711.00403]

    work page internal anchor Pith review Pith/arXiv arXiv 1983

  61. [61]

    Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions

    C. Caprini et al.,Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions,JCAP04(2016) 001, [1512.06239]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  62. [62]

    Gravitational waves from bubble collisions: analytic derivation

    R. Jinno and M. Takimoto,Gravitational waves from bubble collisions: An analytic derivation,Phys. Rev. D95(2017) 024009, [1605.01403]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  63. [63]

    Gravitational Radiation from First-Order Phase Transitions

    M. Kamionkowski, A. Kosowsky and M. S. Turner,Gravitational radiation from first order phase transitions,Phys. Rev. D49(1994) 2837–2851, [astro-ph/9310044]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  64. [64]

    J. R. Espinosa, T. Konstandin, J. M. No and G. Servant,Energy Budget of Cosmological First-order Phase Transitions,JCAP06(2010) 028, [1004.4187]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  65. [65]

    Gravitational waves from the sound of a first order phase transition

    M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir,Gravitational waves from the sound of a first order phase transition,Phys. Rev. Lett.112(2014) 041301, [1304.2433]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  66. [66]

    Sound shell model for acoustic gravitational wave production at a first-order phase transition in the early Universe

    M. Hindmarsh,Sound shell model for acoustic gravitational wave production at a first-order phase transition in the early Universe,Phys. Rev. Lett.120(2018) 071301, [1608.04735]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  67. [67]

    Hindmarsh, S

    M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir,Shape of the acoustic gravitational wave power spectrum from a first order phase transition,Phys. Rev. D96 (2017) 103520, [1704.05871]

    work page arXiv 2017

  68. [68]

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

    C. Caprini, R. Durrer and G. Servant,The stochastic gravitational wave background from turbulence and magnetic fields generated by a first-order phase transition,JCAP12 (2009) 024, [0909.0622]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  69. [69]

    R. Zhou, L. Bian and Y. Du,Electroweak phase transition and gravitational waves in the type-II seesaw model,JHEP08(2022) 205, [2203.01561]. 44

    work page arXiv 2022

  70. [70]

    Punturo et al.,The Einstein Telescope: A third-generation gravitational wave observatory,Class

    M. Punturo et al.,The Einstein Telescope: A third-generation gravitational wave observatory,Class. Quant. Grav.27(2010) 194002

    work page 2010

  71. [71]

    FeynRules 2.0 - A complete toolbox for tree-level phenomenology

    A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks,FeynRules 2.0 - A complete toolbox for tree-level phenomenology,Comput. Phys. Commun.185(2014) 2250–2300, [1310.1921]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  72. [72]

    N. D. Christensen and C. Duhr,FeynRules - Feynman rules made easy,Comput. Phys. Commun.180(2009) 1614–1641, [0806.4194]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  73. [73]

    Automatic evaluation of UV and R2 terms for beyond the Standard Model Lagrangians: a proof-of-principle

    C. Degrande,Automatic evaluation of UV and R2 terms for beyond the Standard Model Lagrangians: a proof-of-principle,Comput. Phys. Commun.197(2015) 239–262, [1406.3030]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  74. [74]

    Generating Feynman Diagrams and Amplitudes with FeynArts 3

    T. Hahn,Generating Feynman diagrams and amplitudes with FeynArts 3,Comput. Phys. Commun.140(2001) 418–431, [hep-ph/0012260]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  75. [75]

    UFO - The Universal FeynRules Output

    C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter,UFO - The Universal FeynRules Output,Comput. Phys. Commun.183(2012) 1201–1214, [1108.2040]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  76. [76]

    ALOHA: Automatic Libraries Of Helicity Amplitudes for Feynman diagram computations

    P. de Aquino, W. Link, F. Maltoni, O. Mattelaer and T. Stelzer,ALOHA: Automatic Libraries Of Helicity Amplitudes for Feynman Diagram Computations,Comput. Phys. Commun.183(2012) 2254–2263, [1108.2041]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  77. [77]

    The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations

    J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al.,The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations,JHEP07(2014) 079, [1405.0301]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  78. [78]

    Abouabid et al.,HHH whitepaper,Eur

    H. Abouabid et al.,HHH whitepaper,Eur. Phys. J. C84(2024) 1183, [2407.03015]. [85]ATLAScollaboration, G. Aad et al.,Search for triple Higgs boson production in the 6b final state using pp collisions at s=13 TeV with the ATLAS detector,Phys. Rev. D111 (2025) 032006, [2411.02040]. 45

    work page arXiv 2024