⚛️ phenomenology

Freeze-in at all couplings

This paper presents a comprehensive analysis of charged parent freeze-in dark matter models in low-reheating-temperature scenarios, demonstrating that Boltzmann suppression enables stronger couplings to the Standard Model while revealing critical dependencies on mass scales and the necessity of tracking non-equilibrium mediator dynamics to accurately predict relic abundance and constrain the parameter space via LHC and lepton-flavor-violation searches.

Original authors: Andreas Goudelis, Andre Lessa, Lucas Magno Dantas Ramos, Thomas Reggio

Published 2026-06-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andreas Goudelis, Andre Lessa, Lucas Magno Dantas Ramos, Thomas Reggio

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

The Big Picture: A New Way to Make Dark Matter

For decades, scientists have been trying to figure out what Dark Matter is. It's the invisible stuff that holds galaxies together, but we can't see it or touch it.

Usually, scientists think Dark Matter was made in the early universe when things were super hot and energetic. They imagine a "Freeze-out" scenario: Dark Matter particles were everywhere, bumping into each other, until the universe cooled down so much that they stopped bumping and just stayed there.

However, this paper explores a different idea called "Freeze-in." Imagine the universe was a giant party. In the "Freeze-out" story, the party was wild, and the guests (Dark Matter) were everywhere. In the "Freeze-in" story, the party was very quiet, and the guests (Dark Matter) were almost non-existent. They only showed up very slowly, trickling in from the outside, never really getting to know the other guests (the Standard Model particles).

The Twist: The "Low-Temperature" Party

The authors of this paper ask a specific question: What if the universe reheated to a very low temperature after the Big Bang?

Think of the early universe like a pot of soup.

Standard Theory: The soup is boiling hot. The ingredients (particles) are moving fast and mixing perfectly.
This Paper's Theory: The soup is barely warm. It's lukewarm.

If the soup is lukewarm, the heavy ingredients (like the "Mediator" particles) can't move around easily. They get "Boltzmann-suppressed," which is a fancy way of saying they are too heavy to be created in large numbers because there isn't enough heat energy.

The Characters in Our Story

To explain how Dark Matter is made in this lukewarm soup, the authors use three characters:

The Dark Matter (The Ghost): A invisible particle that we want to find. Let's call it "S."
The Mediator (The Messenger): A heavy, charged particle that acts as a bridge between the visible world and the invisible Dark Matter. Let's call it "F."
The Standard Model (The Crowd): The normal particles we know (electrons, muons, etc.).

The Mechanism: How the Ghost Gets In

In this model, the "Ghost" (S) is created when the "Messenger" (F) decays (breaks apart).

The Surprising Discovery:
Usually, scientists think that for "Freeze-in" to work, the connection between the Ghost and the Messenger must be extremely weak (like a whisper). If the connection is strong, the Ghost would have been created too fast and would have overwhelmed the universe.

But this paper says: Not necessarily!

If the universe is lukewarm (low temperature), the Messenger (F) is so rare because it's too heavy to be made easily. Because there are so few Messengers, even if they are very eager to break apart into Ghosts (a strong connection), they still can't make too many Ghosts. The lack of Messengers acts as a bottleneck.

The Analogy:
Imagine a factory making toys (Dark Matter).

Standard View: The factory has a huge supply of raw material (Messengers). To make the right number of toys, the workers must work very slowly (weak connection).
This Paper's View: The factory has almost no raw material because the delivery truck broke down (low temperature). Even if the workers are super fast and eager (strong connection), they can't make too many toys because they are out of raw materials.

The Complicated Dance: Tracking the Messenger

The authors found that the story gets even more interesting. They had to track the number of Messengers (F) very carefully.

The "Too Fast" Problem: If the connection between the Messenger and the Ghost is too strong, the Messenger might decay (break apart) so fast that it never gets a chance to reach a stable balance with the rest of the universe. It's like a runner who trips and falls before they even reach the starting line.
The Switch: Depending on how hot the universe was and how heavy the particles are, the way Dark Matter is made switches between two modes:
- Decay Mode: Messengers break apart to make Ghosts.
- Collision Mode: Normal particles crash into each other to make Ghosts.

The paper shows a smooth transition between these two modes, which hadn't been fully explored before.

Checking the Clues: Can We Find Them?

Since the authors suggest the connection between the Messenger and the Ghost could be strong, this changes how we look for them in experiments like the Large Hadron Collider (LHC).

Weak Connection (Old View): The Messenger lives a long time. It travels far before breaking apart. We look for "Heavy Stable Charged Particles" (HSCPs) or "Displaced Leptons" (particles that appear far from where they started).
Strong Connection (This Paper's View): If the connection is strong, the Messenger breaks apart almost instantly. It looks like a normal particle collision. We look for "Prompt" signals (immediate flashes of light and missing energy).

The Result:
The paper maps out where we might find these particles.

If the universe was very hot, we need to look for long-lived particles (HSCPs).
If the universe was lukewarm, we might need to look for instant decays (Prompt signals).
They also checked constraints from other experiments (like looking for rare decays of muons) and found that these different searches cover different parts of the map, working together like a team of detectives.

The Main Takeaway

The paper argues that we shouldn't just assume the universe was always super hot. If it was cooler, the rules change. Dark Matter could be made with much stronger connections than we thought, as long as the universe didn't have enough heat to create the "Messengers" in the first place.

This means we need to look for Dark Matter in more places than before, using different types of experiments, because the "recipe" for making Dark Matter depends heavily on the temperature of the early universe.

Technical Summary: Freeze-in at all couplings

Problem Statement
The paper addresses the production mechanism of dark matter (DM) within the "freeze-in" paradigm, specifically challenging the conventional assumption that freeze-in requires extremely weak ("feeble") couplings between the DM sector and the Standard Model (SM) thermal bath. While traditional freeze-in scenarios assume a high reheating temperature ( $T_{rh}$ ) where interactions are feeble to avoid thermalization, this work investigates configurations where the Universe reheats to a temperature comparable to or lower than the mass scales of the theory. In such low- $T_{rh}$ scenarios, DM production is Boltzmann-suppressed from the outset. The authors aim to determine if successful freeze-in can occur with significantly stronger couplings in these regimes and to understand the complex interplay between the reheating temperature, particle masses, and coupling strengths.

Methodology
The authors analyze a specific extension of the Standard Model featuring a real gauge-singlet scalar field $s$ (the DM candidate) and a vector-like fermion $F$ (a charged lepton mediator). The interaction is governed by Yukawa-like couplings $y_s^\alpha$ between $s$ , $F$ , and SM right-handed leptons.

The methodology involves:

Boltzmann Equation System: Unlike standard approximations that assume the mediator $F$ is always in thermal equilibrium or that DM production is solely from decays, the authors solve a coupled system of Boltzmann equations for both the DM yield ( $Y_s$ ) and the mediator yield ( $Y_F$ ). This allows for the tracking of "backreactions" and the non-trivial evolution of the mediator abundance.
Regime Analysis: They explore various hierarchies between the reheating temperature ( $T_{rh}$ ), the mediator mass ( $m_F$ ), and the DM mass ( $m_s$ ). This includes scenarios where $T_{rh} < m_F$ (Boltzmann suppression of the mediator) and $T_{rh} < m_s$ (Boltzmann suppression of DM production).
Numerical Implementation: The model is implemented using FeynRules and analyzed with CalcHEP/micrOMEGAs 6 to compute relic densities and particle yields.
Phenomenological Constraints: The viable parameter space is constrained by:
- Direct Detection: DM-electron scattering limits (e.g., XENON1T, SENSEI).
- Flavor Physics: Constraints from the radiative decay $\mu \to e\gamma$ .
- Collider Searches: Reinterpretation of LHC searches for Heavy Stable Charged Particles (HSCP), Displaced Leptons (DL), and prompt leptons with missing transverse energy ( $E_T^{miss}$ ), utilizing tools like SModelS, MadAnalysis5, and LLP recasting repositories.

Key Contributions

Strong Coupling Freeze-in: The paper demonstrates that freeze-in can be achieved with couplings orders of magnitude stronger than the "feeble" regime, provided $T_{rh}$ is sufficiently low to suppress the production rates via Boltzmann factors.
Mediator Non-Equilibration: A significant finding is the non-trivial behavior of the mediator $F$ . In regions where $T_{rh} \sim T_{F}^{fo}$ (the naive freeze-out temperature), increasing the coupling $y_s$ can actually prevent the mediator from ever reaching thermal equilibrium. This occurs because $F$ decays into DM faster than it can be replenished by SM annihilations, leading to a "freeze-in" of the mediator itself.
Continuous Transition: The authors identify a continuous transition between decay-dominated freeze-in (where $F$ is in equilibrium) and scattering-dominated freeze-in (where $F$ never equilibrates and DM is produced via SM annihilations). This bridges the gap between traditional freeze-in and thermal freeze-out scenarios.
Complementarity of Searches: The study highlights that the viability of the model and the relevant experimental constraints depend heavily on the assumed thermal history ( $T_{rh}$ ). Different reheating temperatures shift the required coupling strength, moving the model's predictions between the sensitivity ranges of HSCP, displaced lepton, and prompt search channels.

Results

Relic Density Contours: In the $(m_s, y_s)$ plane, the authors find two distinct families of solutions. For high $T_{rh}$ , a feeble coupling ( $y_s \lesssim 10^{-7}$ ) is required. As $T_{rh}$ decreases below $m_F$ , the required coupling increases dramatically (up to $y_s \gtrsim 10^{-3}$ ) to compensate for the Boltzmann suppression of the mediator and DM production.
SuperWIMP Contribution: For high DM masses ( $m_s \sim m_F$ ), a "superWIMP" contribution from the decay of frozen-out $F$ particles becomes dominant, often overclosing the universe unless $y_s$ is tuned to be extremely small.
Experimental Exclusions:
- HSCP Searches: Constrain the model for high $T_{rh}$ (low $y_s$ ) where $F$ is long-lived, excluding mediator masses up to $\sim 800$ GeV.
- Displaced Lepton Searches: Constrain intermediate coupling values ( $10^{-9} \lesssim y_s \lesssim 10^{-7}$ ) where $F$ decays within the tracker.
- Prompt Searches: Constrain the high-coupling, low- $T_{rh}$ regime ( $y_s \gtrsim 10^{-3}$ ), excluding mediator masses up to $\sim 650$ GeV.
- Flavor Constraints: The $\mu \to e\gamma$ limit constrains $y_s$ at the $O(10^{-2})$ level, providing complementary coverage to collider searches, particularly for higher mass scales.

Significance and Claims
The paper claims that the distinction between microscopic dark matter models, thermal histories, and production mechanisms is not clear-cut. It argues that assessing the cosmologically viable parameter space of a model requires simultaneously tracking the number density evolution of multiple particle species (both DM and mediators) rather than relying on simplified assumptions.

The authors emphasize that "freeze-in" is not a single mechanism defined solely by feeble couplings but a broad class of scenarios where sub-Hubble production rates can be achieved either by weak interactions or by low temperatures. Consequently, experimental searches for dark matter are not just probing specific particle models but are probing combinations of microscopic models and the thermal history of the early Universe. The work underscores the necessity of considering low reheating temperature scenarios to fully explore the phenomenological landscape of charged parent freeze-in models.