⚛️ phenomenology

Minimal Scale-Invariant Dark Matter

This paper proposes a minimal classically scale-invariant extension of the Standard Model where a real scalar singlet dark matter particle acquires its mass through radiative electroweak symmetry breaking and is produced via the freeze-in mechanism, predicting a mass of approximately 2 MeV that evades current direct-detection constraints.

Original authors: Alexander Belyaev, Roman Pasechnik, Rishav Roshan, Alfonso Zerwekh

Published 2026-08-05

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CC BY 4.0

Original authors: Alexander Belyaev, Roman Pasechnik, Rishav Roshan, Alfonso Zerwekh

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

Imagine the universe as a giant, invisible stage where particles act out a cosmic drama. For decades, physicists have been trying to figure out why the actors have the weights they do. In our current script, the Standard Model, the "weight" of the Higgs boson (which gives other particles their mass) is just a number we plug in by hand, like a director shouting, "Let's say this actor weighs 125 kilograms!" It feels a bit unsatisfying, like a story with a plot hole. Scientists love the idea of Classical Scale Invariance, a rule that says the universe shouldn't have any built-in weights at all; everything should be weightless until the story itself creates the mass through the interactions of the characters.

Then there's the mystery of Dark Matter. We know it's there because it holds galaxies together with its invisible gravity, but we've never seen it. Most theories assume Dark Matter is a heavy, shy particle that was once bumping into regular matter in the hot, early universe before freezing out, like a party guest who leaves once the music stops. But what if the guest never actually arrived at the party? What if they were just slowly trickling in from the outside, never mixing with the crowd? This paper explores a version of the universe where the rules of scale invariance are strictly followed, and Dark Matter is a tiny, ghostly particle that never really "thawed out" with the rest of the universe.

The Ghost in the Machine: A Tiny, Invisible Particle

In this paper, a team of physicists builds a new, stricter version of the Standard Model. They imagine a universe where no particle has a "bare" mass written in the laws of physics. Instead, masses are generated dynamically, like a snowball rolling down a hill, picking up size only because of how it interacts with the snow around it. To make this work, they add just one new character: a single, real scalar particle called a "singlet." This particle is the Dark Matter candidate. It's protected by a special rule (a $Z_2$ symmetry) that prevents it from decaying, making it stable enough to last from the Big Bang until today.

The authors set out to see if this minimalist model could actually explain the amount of Dark Matter we see in the universe today. They had to do some heavy lifting with math, constructing a complex "effective potential"—think of this as a topographical map of energy landscapes. They needed to make sure that the "valleys" in this map (where particles settle) were stable and that the height of the hills (the masses) matched what we observe in our labs, specifically the mass of the Higgs boson. They developed a new, modified way of doing the calculations (a "renormalisation scheme") to ensure their predictions didn't change just because they tweaked a mathematical dial.

The "Freeze-In" Discovery

When the team ran their simulations, they found that the usual way Dark Matter is thought to be created—thermal freeze-out—is impossible for their model. In the freeze-out scenario, Dark Matter particles are hot and active in the early universe, colliding with regular matter until the universe cools down and they stop interacting. However, the rules of their scale-invariant model mean that if Dark Matter were this active, it would interact too strongly with regular matter, and we would have already detected it in experiments looking for Dark Matter hitting atomic nuclei. Since those experiments have come up empty, the "freeze-out" idea is ruled out for this specific model.

Instead, the paper points to a different, much more subtle mechanism called freeze-in. Imagine a leaky faucet dripping water into a bucket. The water (Dark Matter) enters the bucket so slowly that it never mixes with the air or splashes around; it just quietly accumulates. In this model, the Dark Matter particle is so weakly connected to the rest of the universe that it never reaches thermal equilibrium. It is slowly "cooked" into existence from the hot soup of the early universe, but it never gets hot enough to interact with the other particles.

The Sweet Spot: A 2.6 MeV Ghost

The most exciting part of the paper is that this "freeze-in" idea doesn't just work; it is incredibly specific. The math forces the Dark Matter particle to have a very precise mass. The authors found that for the model to work, the Dark Matter must weigh approximately 2.59 MeV to 2.68 MeV. To put that in perspective, an electron weighs about 0.511 MeV, so this Dark Matter particle is only about five times heavier than an electron. It is incredibly light compared to the heavy WIMP particles (Weakly Interacting Massive Particles) that most other experiments are hunting for.

The paper also calculated how likely this particle is to hit an electron in a detector. They found a reference cross-section (a measure of how likely a collision is) of roughly $5.5 \times 10^{-65}$ cm $^2$ . This number is mind-bogglingly small. The authors compare this to the sensitivity of the DAMIC-M experiment, a cutting-edge detector designed to catch light Dark Matter. Their calculation shows that the predicted signal is about 28 orders of magnitude (that's a 1 followed by 28 zeros) smaller than what DAMIC-M can currently see.

Why This Matters

The paper concludes that while this model is mathematically beautiful and consistent with the rules of scale invariance, it is essentially invisible to our current technology. The "heavy" solution (where the Dark Matter is hundreds of GeV) was ruled out because it would require unnatural mathematical cancellations and would likely lead to the particle thermalizing, which contradicts the model's strict rules.

So, the paper leaves us with a highly predictive, albeit elusive, picture: Dark Matter exists as a tiny, MeV-scale particle that was born through a "freeze-in" process. It is so faintly connected to our world that it has never left a trace in our detectors. The authors suggest that the persistent lack of signals in Dark Matter searches might not be a failure of our experiments, but rather a confirmation of this specific mechanism: if Dark Matter is this kind of "freeze-in" ghost, it is supposed to be undetectable with current tools. The absence of a signal is actually the prediction of the theory.

Technical Summary: Minimal Scale-Invariant Dark Matter

Problem Statement
The origin of the electroweak and dark matter (DM) mass scales remains a central open question in particle physics. In the Standard Model (SM), the electroweak scale is introduced via an explicit dimensionful parameter in the Higgs potential, and most DM models introduce the DM mass as an independent input scale. Classically scale-invariant extensions of the SM offer a framework where explicit dimensionful parameters are absent, and physical scales arise radiatively through dimensional transmutation (the Coleman–Weinberg mechanism).

Previous studies established that a minimal real scalar singlet coupled to the Higgs can trigger radiative electroweak symmetry breaking (EWSB). However, reproducing the observed relic density in this minimal setup typically requires a large Higgs portal coupling. This places the DM in thermal equilibrium with the SM plasma, leading to a thermal freeze-out production mechanism. Such large couplings are currently excluded by direct-detection limits. The challenge addressed in this work is to determine if the same minimal extension can accommodate a viable DM candidate produced via the freeze-in mechanism, which avoids thermalization and direct-detection constraints, while maintaining a consistent renormalized quantum framework.

Methodology
The authors construct a minimal classically scale-invariant extension of the SM containing a single real scalar singlet $S$ , stabilized by an exact $Z_2$ symmetry ( $S \to -S$ ). The scalar potential contains no bare mass terms:
$V_0(H, S) = \lambda_H(H^\dagger H)^2 + \frac{\lambda_{HS}}{2}(H^\dagger H)S^2 + \frac{\lambda_S}{4}S^4$

The core methodological contribution is the construction of a renormalized one-loop effective potential that ensures physical predictions are stable under variations of the renormalization scale $\mu$ .

Two-Field Effective Potential: The authors calculate the full two-field one-loop Coleman–Weinberg potential in the $\overline{\text{MS}}$ scheme and Landau gauge.
Modified On-Shell Renormalization: Unlike the strict Gildener–Weinberg treatment where the scalar mass along the flat direction is a pure one-loop prediction, this work introduces a modified on-shell scheme. Finite counterterms are fixed by physical conditions at the electroweak vacuum $(v, 0)$ $(v, 0)$ :
- The vacuum position is fixed at $v \simeq 246$ GeV.
- The Higgs mass is fixed to the physical value $m_h$ .
- Higgs–singlet mixing is forced to vanish.
- The singlet curvature (mass) is preserved at its tree-level value determined by the portal coupling.
- These conditions are imposed independently at each chosen $\mu$ , ensuring that the physical vacuum and curvatures remain invariant under scale variation, while residual scale dependence is confined to off-shell quantities as an estimate of higher-order uncertainties.
Effective Couplings: The authors distinguish between the physical singlet mass (determined by the curvature of the potential) and the effective portal coupling $\lambda_{HS}^{\text{eff}}$ (extracted from the renormalized $hSS$ vertex). This distinction is crucial because freeze-in production and scattering rates depend on $\lambda_{HS}^{\text{eff}}$ , while vacuum stability depends on the singlet self-coupling $\lambda_S^{\text{eff}}$ .

Key Contributions and Results
The study identifies two theoretically viable regions in the parameter space defined by perturbativity, vacuum stability, and the observed relic abundance ( $\Omega_S h^2 \approx 0.12$ ). However, only one region survives as a consistent freeze-in solution.

Exclusion of Thermal Freeze-Out: The conventional thermal freeze-out scenario, requiring large portal couplings to match the relic density, is confirmed to be excluded by current direct-detection bounds.
The Light Mass Solution (MeV Scale):
- A narrow window exists where the effective singlet self-coupling $\lambda_S^{\text{eff}} > 0$ and the relic density condition is met via freeze-in.
- This solution is confined to a very specific mass range: $m_S \simeq 2.59$ MeV to $2.68$ MeV.
- In this region, the effective portal coupling is extremely small, $\lambda_{HS}^{\text{eff}} \sim 2.4 \times 10^{-10}$ , ensuring the singlet never thermalizes with the SM plasma.
- The solution is robust: the required tree-level couplings are perturbative, and the vacuum is globally stable.
The Heavy Mass Region (Excluded):
- A second region appears for $m_S \in [436, 632]$ GeV where $\lambda_S^{\text{eff}} > 0$ .
- However, this region is theoretically disfavored. To achieve the necessary tiny effective coupling ( $\lambda_{HS}^{\text{eff}} \sim 10^{-11}$ ) from tree-level inputs ( $\lambda_{HS} \approx 6.3–13.2$ ), an extreme and unnatural cancellation between tree-level and one-loop terms is required.
- Furthermore, the required tree-level couplings approach the non-perturbative regime ( $\lambda_{HS} \gtrsim 4\pi$ ) near the upper bound, and the singlet self-coupling $\lambda_S$ becomes negative in parts of this interval, violating stability conditions.
Direct Detection Constraints:
- For the MeV-scale benchmark ( $m_S \approx 2.6$ MeV), the predicted electron-scattering cross-section is $\bar{\sigma}_e \simeq 5.5 \times 10^{-65} \text{ cm}^2$ .
- This value is approximately 28 orders of magnitude below the current sensitivity of the DAMIC-M experiment.
- Current direct-detection experiments (DAMIC-M, SENSEI, SuperCDMS) do not constrain this scenario. The absorption limits for hidden photons are noted as inapplicable to this scalar Higgs-portal model.

Significance and Claims
The paper claims to provide a robust and highly predictive framework that connects radiative electroweak symmetry breaking with freeze-in dark matter genesis.

Predictivity: By enforcing classical scale invariance, perturbativity, vacuum stability, and the observed relic abundance simultaneously, the model reduces the allowed parameter space to a single, isolated solution in the MeV mass range.
Natural Explanation for Null Results: The authors argue that the persistent null results of dark matter direct-detection searches find a natural explanation within this scenario. If minimal conformal dark matter is realized through freeze-in, its coupling to the visible sector is structurally forced to be so feeble that the predicted signals lie many orders of magnitude below current and foreseeable experimental sensitivities. The absence of a signal is presented as a structural prediction of the mechanism rather than an accident of parameter choice.
Theoretical Consistency: The work demonstrates that a consistent renormalized one-loop framework can be constructed where physical observables are stable against renormalization scale variations, resolving potential ambiguities in the effective potential approach for scale-invariant models.

The Ghost in the Machine: A Tiny, Invisible Particle

The "Freeze-In" Discovery

The Sweet Spot: A 2.6 MeV Ghost

Why This Matters

Technical Summary: Minimal Scale-Invariant Dark Matter

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