⚛️ phenomenology

Cosmological Signatures of the Conformal and Non-Conformal Next-to-Minimal Two-Higgs-Doublet Model

This paper investigates cosmological signatures of the Next-to-Minimal Two-Higgs-Doublet Model in both conformal and non-conformal realizations, demonstrating that while both scenarios can generate observable gravitational waves from a first-order electroweak phase transition, they occupy distinct regions of the scalar mass spectrum and exhibit different patterns in Higgs-pair production rates, thereby offering a pathway to discriminate between the two models through combined cosmological and collider data.

Original authors: Filip Gustavsson, Roman Pasechnik, Johan Rathsman

Published 2026-08-18

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CC BY 4.0

Original authors: Filip Gustavsson, Roman Pasechnik, Johan Rathsman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the earliest moments of the universe, a fundamental shift occurred that shaped everything we see today. Just as water freezes into ice, the universe cooled and underwent a phase transition, changing the way particles interacted and acquiring the properties that allow atoms to exist. Physicists call this the electroweak phase transition. In the standard model of particle physics, our best current description of nature, this change happened smoothly, like water slowly turning to slush. However, many scientists suspect the transition might have been violent and abrupt, like a sudden freeze that shatters a glass. If it was violent, the collision of bubbles of the new state would have sent ripples through the fabric of space-time itself. These ripples are gravitational waves, and if they were strong enough, they would still be echoing through the cosmos today, waiting to be detected by future instruments.

The search for these ancient ripples has led researchers to look beyond the standard model, exploring theories that add new particles to the known family of forces and matter. One such theory is the Next-to-Minimal Two-Higgs-Doublet Model, or N2HDM. This model suggests that the universe contains not just one type of Higgs particle, which gives other particles their mass, but several. Specifically, it proposes two pairs of these particles and an additional, invisible scalar field. Within this framework, researchers investigated two distinct possibilities: a version where the laws of physics look the same at all energy scales, known as the conformal model, and a version where they do not, the non-conformal model. The goal was to see if either of these scenarios could produce a violent enough phase transition to create gravitational waves that future space-based detectors could hear, and whether the specific pattern of those waves could tell us which version of reality is correct.

To answer this, the researchers built a detailed computer simulation of the early universe, calculating how the energy of the vacuum changes as the temperature drops. They mapped out the behavior of the Higgs particles in both the conformal and non-conformal versions of their model, checking every step against the strict rules of particle physics and the known limits from particle colliders like the Large Hadron Collider. They ensured that the models they tested were stable and that the particles they predicted would not have been seen already by existing experiments. Once they had a set of viable scenarios, they calculated the strength of the phase transition and the resulting gravitational waves. They found that both the conformal and non-conformal models could indeed produce a violent transition, creating a stochastic background of gravitational waves—essentially a constant hum of ripples from the early universe—that would be strong enough to be detected by the upcoming Laser Interferometer Space Antenna, or LISA, a mission planned for the next decade.

However, the discovery that both models work is only half the story. The researchers found that while both can produce detectable waves, they do so in very different ways. The strength of the gravitational wave signal is closely tied to the masses of the extra Higgs particles predicted by the models. In the conformal version, the particles that would generate a strong signal tend to be heavier, with charged Higgs masses generally above ∼350 GeV for observable signals. In the non-conformal version, the particles that produce a similar signal are generally smaller, typically less than ∼350 GeV. This means that if LISA detects these ancient waves, and if future colliders can measure the masses of these new Higgs particles, scientists will be able to distinguish between the two theories. The combination of a gravitational wave detection and a measurement of particle masses would act as a fingerprint, revealing whether the universe follows the conformal or non-conformal path.

The study also looked at how these new particles might interact with each other, specifically how often they could be produced in pairs during high-energy collisions. They found that in the conformal model, there is a higher chance of seeing unusual rates of these particle pairs compared to what the standard model predicts. In the non-conformal model, such deviations are much less common. This suggests a complementary relationship between different types of experiments: while gravitational wave detectors listen for the echo of the universe's birth, particle colliders look for the specific particles that caused the echo. If a future experiment sees a strong signal of gravitational waves but finds no strange behavior in Higgs pair production, it would point toward the non-conformal model. Conversely, if both the waves and the strange particle interactions are found, it would strongly favor the conformal model.

The researchers were careful to note that their results are based on simulations and specific assumptions about how the phase transition unfolds. They found that in most cases, the transition happened quickly once it started, but in the most extreme scenarios, the universe cooled significantly before the transition completed, a phenomenon known as supercooling. These extreme cases produced the strongest gravitational waves, but they also introduced more uncertainty into the predictions. Despite these uncertainties, the core finding remains robust: both versions of the model are capable of producing observable signals, but they leave distinct signatures in the mass spectrum of the new particles. This offers a clear path forward for the next generation of experiments, turning the search for the universe's origins into a process of cross-referencing data from the sky and the laboratory to uncover the true nature of the Higgs sector.

Technical Summary: Cosmological Signatures of the Conformal and Non-Conformal Next-to-Minimal Two-Higgs-Doublet Model

Problem Statement
The Standard Model (SM) predicts a crossover electroweak phase transition (EWPT), which fails to satisfy the out-of-equilibrium condition required for electroweak baryogenesis. Beyond the Standard Model (BSM) scenarios, such as the Next-to-Minimal Two-Higgs-Doublet Model (N2HDM), are investigated for their ability to generate a strong first-order EWPT. This paper addresses a specific gap in the literature: a systematic comparison between the ordinary, non-conformal N2HDM and its classically scale-invariant (SI), or conformal, realization. The central question is whether a stochastic gravitational wave (GW) signal from a first-order EWPT, combined with collider data on the scalar spectrum and Higgs-pair production, can discriminate between these two theoretical realizations.

Methodology
The authors perform a comprehensive parameter scan of the N2HDM, analyzing both the non-conformal case (with explicit mass terms in the potential) and the conformal case (where all dimensionful parameters are set to zero at tree level, with symmetry breaking induced radiatively).

Theoretical Framework: The model extends the SM with two Higgs doublets ( $\Phi_1, \Phi_2$ ) and a real singlet ( $S$ ), constrained by $Z_2$ and $Z'_2$ symmetries to suppress tree-level flavor-changing neutral currents. The analysis utilizes a one-loop finite-temperature effective potential ( $V_{\text{eff}}$ ) including the Coleman-Weinberg potential, thermal corrections, and daisy resummation (implemented via the BSMPT package).
Renormalization: A critical methodological distinction is the treatment of counterterms (CTs). For the non-conformal model, a standard on-shell (OS) scheme is used. For the conformal model, the OS scheme is modified to account for the tree-level flat direction (the "scalon"). The modification ensures that the radiatively generated mass of the scalon is correctly incorporated while preserving the vacuum position, as detailed in Appendix A.
Constraints: The parameter space is constrained by:
- Theoretical: Boundedness from below (BFB), perturbative unitarity, and the requirement of a stable electroweak minimum.
- Phenomenological: Collider limits (LHC, Tevatron, LEP) and flavor constraints (e.g., $b \to s\gamma$ ) are enforced using ScannerS, HiggsBounds, and HiggsSignals. Oblique parameters are also calculated.
Observables: The study calculates EWPT observables (nucleation temperature $T_n$ , percolation temperature $T_p$ , transition strength $\alpha$ , and duration $\beta/H$ ) and the resulting GW power spectrum. The GW signal is estimated using the sound-wave contribution as the baseline, assuming a bubble wall velocity $v_w = 0.95$ . Signal-to-noise ratios (SNR) are calculated for future space-based interferometers (LISA, BBO, DECIGO).
Complementarity: The paper also evaluates partonic Higgs-pair production ( $gg \to h_i h_j$ ) in the heavy-top approximation at $\sqrt{\hat{s}} = 400$ GeV to assess correlations between GW signals and di-Higgs rates.

Key Results

Existence of Observable Signals: Both the conformal and non-conformal N2HDM scenarios contain viable parameter regions that support a strong first-order EWPT with potentially observable stochastic GW signals. Several points in both scenarios yield an SNR $> 10$ for LISA, assuming a 3-year observation time.
Transition Dynamics: In the viable regions, the transitions complete promptly, with nucleation and percolation temperatures being nearly identical ( $T_n \simeq T_p$ ). Pronounced supercooling (where $T_p \ll T_c$ ) is confined to the strongest transitions ( $\alpha \gtrsim 1$ ), which also correspond to the highest GW signal strengths.
Discrimination via Scalar Spectrum: The most significant finding is that conformal and non-conformal scenarios populate distinct regions of the scalar mass spectrum when the GW signal is observable:
- Conformal Model: Observable GWs are associated with heavier charged Higgs masses ( $m_{H^\pm} \gtrsim 350$ GeV) and a broader range of CP-even masses.
- Non-Conformal Model: Observable GWs are found for lighter charged Higgs masses ( $m_{H^\pm} \lesssim 350$ GeV) and specific correlations with the heavier CP-even Higgs mass ( $m_H$ ) and $\tan\beta$ .
- This suggests that a measurement of the stochastic GW background, combined with mass measurements of additional Higgs states at colliders, could distinguish between the two realizations.
Higgs-Pair Production Complementarity:
- Large deviations in the SM-like di-Higgs production rate tend to occur in parameter regions away from those with the strongest GW signals.
- Enhanced BSM Higgs-pair production rates are primarily associated with the conformal model, driven by larger quartic couplings required to generate physical masses without tree-level mass terms.
- In the non-conformal model, the di-Higgs rate remains closer to the SM prediction in the GW-observable regions.

Significance and Claims
The paper claims that its primary contribution is not merely demonstrating that the N2HDM can support a first-order EWPT (a known result), but rather establishing that conformal and non-conformal realizations can be distinguished through a combination of cosmological and collider observables.

The authors emphasize that while the N2HDM can produce observable GWs in both scenarios, the "fingerprint" of the underlying theory (scale-invariant vs. non-scale-invariant) is encoded in the correlation between the GW signal strength and the masses/couplings of the extended Higgs sector. Specifically, the distinct mass ranges for charged and CP-odd scalars in the GW-observable regions provide a "scan-level discriminator."

The study also highlights the complementary nature of GW searches and Higgs-pair probes: while GWs signal the strength of the phase transition, Higgs-pair production probes the shape of the scalar potential (trilinear and quartic couplings). The authors note that their Higgs-pair analysis is a partonic diagnostic rather than a full collider reach projection, and their GW forecasts rely on the sound-wave approximation, which may require refinement for very strong transitions ( $\alpha \gtrsim 1$ ). Nevertheless, the qualitative conclusion remains robust: the two realizations offer testable, distinguishable signatures for future multi-messenger observations.

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