⚛️ phenomenology

Analysis of freeze-in scenario with a scalar Leptoquark and a scalar Dark Matter

This paper investigates a freeze-in Dark Matter scenario mediated by a heavy scalar leptoquark interacting with both the Standard Model and Dark Matter sectors, numerically exploring the resulting relic density constraints across a parameter space defined by two masses and three dimensionless couplings.

Original authors: Joydeep Roy

Published 2026-01-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Joydeep Roy

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: Two Ways to Fill the Universe's "Dark" Bucket

Imagine the early Universe as a giant, boiling pot of soup. Inside this pot, there are two types of ingredients:

The Visible Soup: This is the "Standard Model" stuff we know (atoms, light, stars).
The Invisible Secret Ingredient: This is Dark Matter. We know it's there because of gravity, but we can't see it or touch it.

For decades, scientists thought Dark Matter was like a WIMP (Weakly Interacting Massive Particle). Think of a WIMP as a social butterfly at a party. It mingles with the visible soup, shakes hands, and eventually settles down to a specific amount as the party cools. This is called the "Freeze-out" scenario.

However, this paper explores a different idea: Freeze-in.
In this scenario, Dark Matter is like a ghost at the party. It never actually mingles with the visible soup. It is so shy and weakly connected that it never becomes part of the main crowd. Instead, it slowly "leaks" into existence from the visible soup, but so slowly that it never reaches a point where it can interact with itself. It just accumulates quietly until the Universe cools down enough that no more can be made. This is the "Freeze-in" scenario.

The New Characters: The Leptoquark and the Scalar

The authors of this paper introduce two new characters to their story:

Scalar Dark Matter (The Ghost): A simple, invisible particle that makes up the Dark Matter.
Scalar Leptoquark (The Heavy Bridge): A hypothetical, very heavy particle that acts as a bridge between the visible world (quarks and leptons) and the invisible Dark Matter world.

Think of the Leptoquark as a massive, heavy construction crane in the kitchen. It's too heavy to be part of the daily cooking (it's over 1.5 TeV, which is incredibly heavy for a particle), but it can help move ingredients around.

The Experiment: How the Ghost is Made

The paper asks: If we have this heavy crane (Leptoquark) and a ghost (Dark Matter) in our kitchen, how does the ghost get created?

The authors found that the ghost is created in two different ways, depending on the "temperature" of the Universe:

Before the Kitchen Cools Down (High Energy):
Imagine the kitchen is super hot. The heavy crane (Leptoquark) and the Higgs boson (a standard particle) are bumping into each other. Occasionally, they smash together and create a pair of ghosts.
- The Catch: The connection between the crane and the ghost must be extremely weak. If they were too friendly, the ghosts would start interacting with each other and ruin the "Freeze-in" recipe. The paper calculates that this connection (a coupling called $\lambda_{\chi\Delta}$ ) must be tiny—about one-millionth of a standard interaction.
After the Kitchen Cools Down (Low Energy):
Once the Universe cools, the Higgs boson changes its nature (it gets a "Vacuum Expectation Value," or a steady state). Now, the Higgs particle itself acts like a factory machine that occasionally decays (breaks apart) into two ghosts.
- The Catch: This connection (called $\lambda_{\chi H}$ ) must be even weaker—about one ten-billionth of a standard interaction.

The Results: What the Numbers Say

The authors ran the numbers to see if this scenario works and what the rules are:

The Heavy Crane is Allowed: They found that having this super-heavy Leptoquark (1.5 TeV) in the mix doesn't break the recipe. It fits perfectly with current experimental limits. It's like having a giant crane in a small kitchen; it doesn't get in the way as long as it doesn't touch the food too much.
The "Shyness" Rule: The most important finding is how weak the interactions must be.
- The Leptoquark and Dark Matter must barely know each other ( $\lambda_{\chi\Delta} \lesssim 10^{-6}$ ).
- The Higgs and Dark Matter must barely know each other ( $\lambda_{\chi H} \lesssim 10^{-10.5}$ ).
- Analogy: If the Standard Model particles are shouting, the Dark Matter is whispering so softly it's almost silent. This explains why we haven't found it yet in experiments; it's just too quiet to hear.
The Size of the Ghost: To get exactly the right amount of Dark Matter in the Universe today (the "relic density" that matches our observations), the Dark Matter particle itself must be incredibly light.
- The paper concludes the Dark Matter mass must be less than 10 electron-volts (eV).
- Analogy: If a proton is a bowling ball, this Dark Matter particle is lighter than a single grain of sand. It's a "featherweight" ghost.

The Conclusion

This paper proposes a specific recipe for how the Universe's Dark Matter could have been made. Instead of being a heavy, social particle that froze out of the crowd, it suggests Dark Matter is a featherweight ghost that was slowly "leaked" into existence by a heavy, shy bridge (the Leptoquark) and the Higgs boson.

The key takeaway is that for this to work, the connections between the visible world and the dark world must be incredibly weak, and the Dark Matter itself must be very light. This scenario is consistent with all current experiments because the particles are so shy and light that they have easily evaded detection so far.

Technical Summary: Analysis of Freeze-in Scenario with a Scalar Leptoquark and a Scalar Dark Matter

Problem Statement
While the Weakly Interacting Massive Particle (WIMP) paradigm has dominated Dark Matter (DM) research, the lack of experimental evidence from direct and indirect searches motivates the exploration of alternative mechanisms. Specifically, the "freeze-in" scenario, where DM particles (Feebly Interacting Massive Particles or FIMPs) are never in thermal equilibrium with the Standard Model (SM) bath, offers a viable alternative. This paper addresses a gap in the literature regarding the freeze-in mechanism within a framework involving a scalar Leptoquark (LQ) and a scalar Dark Matter (SDM) candidate. While previous studies have explored LQs as portals for WIMPs (freeze-out), similar investigations for FIMPs are scarce. The authors aim to analyze the parameter space of a model where a real singlet scalar DM interacts with a scalar LQ and the SM Higgs via renormalizable quartic couplings.

Methodology
The analysis extends the Standard Model by introducing a real singlet scalar $\chi$ (DM) and a scalar Leptoquark ( $S_1$ , an $SU(2)_L$ singlet with quantum numbers $(\bar{3}, 1, 1/3)$ ). Stability of the DM is ensured by a $Z_2$ symmetry ( $\chi \to -\chi$ ). The Lagrangian includes kinetic terms, mass terms, and quartic interaction terms:

Higgs-DM coupling ( $\lambda_{\chi H}$ )
LQ-DM coupling ( $\lambda_{\chi \Delta}$ )
Higgs-LQ coupling ( $\lambda_{H \Delta}$ )
Self-couplings for stability.

The production of DM is modeled using Boltzmann equations in a radiation-dominated universe. The process is divided into two epochs relative to Electroweak Symmetry Breaking (EWSB):

Before EWSB ( $T > T_{EW}$ ): DM is produced primarily via $2 \to 2$ scattering processes ( $HH \to \chi\chi$ and $\Delta\Delta \to \chi\chi$ ).
After EWSB ( $T < T_{EW}$ ): DM production is dominated by the decay of the Higgs boson ( $h \to \chi\chi$ ) and Higgs-mediated annihilation of LQs.

The authors numerically solve the Boltzmann equation for the yield $Y = n_\chi/s$ to determine the relic density $\Omega_\chi h^2$ . They enforce the freeze-in condition, ensuring that the interaction rate of bath particles to DM remains below the Hubble expansion rate ( $\Gamma_{int} < H$ ) to prevent thermalization.

Key Contributions and Results
The study explores a parameter space defined by two masses ( $m_\chi, m_\Delta$ ) and three dimensionless couplings ( $\lambda_{\chi H}, \lambda_{\chi \Delta}, \lambda_{H \Delta}$ ). The primary findings are:

Coupling Constraints:
- $\lambda_{H \Delta}$ : Set to unity ( $\sim 1$ ) as it is not responsible for DM production before EWSB and has no known experimental bounds from LHC or other sources.
- $\lambda_{\chi \Delta}$ (LQ-DM coupling): To satisfy the freeze-in condition (interaction rate < Hubble rate), this coupling must be extremely small. For a DM mass of 10 GeV and an LQ mass of 1.5 TeV, the analysis finds $\lambda_{\chi \Delta} \lesssim 10^{-6}$ . This constraint holds across a wide range of DM masses.
- $\lambda_{\chi H}$ (Higgs-DM coupling): This coupling is constrained differently before and after EWSB. Before EWSB, $\lambda_{\chi H} \lesssim 10^{-6}$ ; after EWSB, the decay channel $h \to \chi\chi$ requires a much stricter bound, $\lambda_{\chi H} \lesssim 10^{-10.5}$ , to satisfy the freeze-in condition. The authors adopt $\lambda_{\chi H} \lesssim 10^{-10.5}$ as the safe upper limit.
Mass Constraints and Relic Density:
- LQ Mass ( $m_\Delta$ ): The study assumes an LQ mass $\ge 1.5$ TeV, consistent with current collider limits. The results indicate that heavier LQ masses in the thermal bath do not significantly alter the DM yield in the freeze-in scenario.
- DM Mass ( $m_\chi$ ): Unlike the freeze-out scenario, the freeze-in yield is largely independent of the DM mass. However, to match the observed relic density ( $\Omega h^2 \approx 0.12$ ) given the extremely small couplings derived above, the DM mass is constrained to be very light. The analysis concludes that $m_\chi \lesssim 10$ eV is required to reproduce the correct relic abundance.
Experimental Consistency: The authors note that the tiny couplings required for the freeze-in mechanism ( $\sim 10^{-11}$ ) naturally evade constraints from direct detection experiments (e.g., XENON1T, LUX) and Higgs invisible decay limits, as these bounds typically apply to the WIMP regime or require larger couplings.

Significance
The paper claims to fill a void in the literature by providing a detailed phenomenological analysis of the freeze-in mechanism specifically involving a scalar Leptoquark and scalar Dark Matter. The significance of the work lies in demonstrating that:

Heavy particles like Leptoquarks can exist in the thermal bath without disrupting the freeze-in mechanism, provided the couplings to the dark sector are sufficiently feeble.
The stability of the scalar potential and perturbativity do not impose significant additional constraints beyond the standard freeze-in requirements.
In this specific framework, satisfying the observed relic density necessitates an ultra-light Dark Matter candidate ( $m_\chi \lesssim 10$ eV) due to the suppression of production rates by the required small couplings.

The study concludes that the FIMP scenario remains a predictive and viable alternative to the WIMP paradigm, particularly in models where DM communicates with the SM via scalar portals like Leptoquarks.

The Big Picture: Two Ways to Fill the Universe's "Dark" Bucket

The New Characters: The Leptoquark and the Scalar

The Experiment: How the Ghost is Made

The Results: What the Numbers Say

The Conclusion

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