Kaon Portal to Freeze-in Dark Matter

This article proposes a low-reheating cosmological scenario in which light dark matter particles are produced via kaon decays and scatterings mediated by a flavor-changing operator, thereby establishing a testable connection between the dark matter relic abundance and the search for rare kaon decays in experiments such as NA62 and KOTO.

The Gist

The Great Mystery: What is Dark Matter?

Imagine the universe as a huge, dark room. We can see the furniture (stars, planets, ourselves), but we know that much invisible material fills the room and holds it together. We call this Dark Matter.

For a long time, scientists thought this invisible material was like heavy, slow-moving ghosts that occasionally collide with things. But after years of searching with huge detectors, we haven't found any. Therefore, scientists are now looking for another kind of ghost: Light Dark Matter. These are tiny, fast-moving particles that hardly interact with anything.

The Problem: Too Weak to Be Seen

The leading theory for how these light particles got here is called "Freeze-in".

Imagine the early universe as a crowded, hot party.

• Standard particles (like electrons and quarks) are the loud, dancing guests everyone knows.
• Dark Matter is a shy guest who never steps onto the dance floor. They only slowly seep in, one by one, from the outside because they are too shy to interact with the crowd.

The problem with this theory is that this "shyness" (the coupling strength) must be incredibly tiny. If it is too tiny, we can possibly never detect it in the lab. It is like trying to hear a whisper in a hurricane.

The Paper's Idea: The "Cooler" Party

This paper suggests a twist in the party story. Normally, we assume the party started super-hot. But what if the party started cooler?

The authors suggest that in the very early universe, the temperature dropped below a critical point before Dark Matter arrived (the so-called QCD crossover).

• The hot party: If the party is hot, the guests are energetic quarks and gluons (the fundamental building blocks).
• The cool party: If the party is cooler (below 150 MeV), the guests are hadrons (particles made of quarks, like protons and pions).

In this "cool party" scenario, the main guests are Kaons (a specific type of unstable particle) and pions.

The "Kaon Portal"

The paper suggests that Dark Matter enters the universe through a specific door: the Kaon.

Imagine Kaons like delivery trucks.

1. The Decay (The delivery truck drops a package): A Kaon can spontaneously decay into a pion and a pair of Dark Matter particles ($K \to \pi + \chi + \bar{\chi}$).
2. The Scattering (The delivery truck crashes into a car): A Kaon can collide with a pion, and during this collision, Dark Matter is created ($K + \pi \to \chi + \bar{\chi}$).

Because the universe is "cool" (low temperature), there are not many Kaons. They are rare, like finding a specific rare coin in a pile of pennies. To get enough Dark Matter to fill today's universe, the "shyness" of the Dark Matter must not be too great. It only needs to be loud enough to be produced through these rare Kaon events.

The key insight: The colder the universe was, the fewer Kaons there were. To compensate for this, the Dark Matter must interact somewhat more strongly with the Kaons. This makes the interaction strong enough that we might actually be able to see it in experiments!

The Detective Work: NA62 and KOTO

The paper connects this cosmic story with real experiments in Japan and Europe (NA62 and KOTO).

These experiments are searching for "rare Kaon decays."

• The standard story: A Kaon sometimes decays into a pion and a pair of invisible neutrinos ($K \to \pi + \nu + \bar{\nu}$). This is rare, but it happens.
• The new story: What if the Kaon instead decays into a pion and a pair of Dark Matter particles?

Since the mathematics for producing Dark Matter and producing neutrinos in this model are almost identical, the experiments looking for the neutrino signal are also looking for the Dark Matter signal.

What the Numbers Say

The authors have crunched the numbers (solved complex equations, so-called Boltzmann equations) to see if this works.

• The result: If the universe was heated to a low temperature (between 60 and 100 MeV), the amount of Dark Matter produced by Kaons matches exactly what we observe in the universe today.
• The catch: For this to work, the interaction strength must be just right.
  • If the temperature was very low (60 MeV), the Kaons were very rare, so the Dark Matter had to interact more strongly. This places the signal exactly in the range that current experiments (NA62) can already see.
  • If the temperature was somewhat higher (100 MeV), the signal is weaker, but future experiments (KOTO II) should be able to find it.

The "Fingerprint"**

A cool point the paper makes is that Dark Matter leaves a different "fingerprint" than neutrinos.

• Neutrinos are massless (or very light), so they carry away a specific amount of energy.
• Dark Matter has mass. If the Dark Matter is heavy, it requires more energy to be produced.
• This changes the spectrum of "missing energy." If you look closely at the data from KOTO, you might see a bump or a distortion in the energy distribution pattern that doesn't fit the neutrino story. This would be the smoking gun (the definitive proof) for Dark Matter.

Summary

This paper says:

1. Dark Matter could be light and shy, produced by Kaons in a cool early universe.
2. Because the universe was cool, the Dark Matter does not have to be too shy to exist; it only needs to be strong enough to be produced by rare Kaon events.
3. This makes the theory testable. The same experiments searching for rare Kaon decays (NA62 and KOTO) are searching for this Dark Matter.
4. If the experiments find a signal that looks like a heavy invisible particle, this could be the "Kaon Portal" to Dark Matter.

It is a bridge between the very small (particle physics in the lab) and the very large (the history of the universe), with the Kaon serving as the messenger.

Technical Summary

Technical Summary: Kaon Portal to Freeze-in Dark Matter

Problem and Motivation
The origin of Dark Matter (DM) remains a central open question. While the paradigm of Weakly Interacting Massive Particles (WIMPs) faces challenges due to the lack of direct detection signals, the Freeze-in mechanism offers an alternative where DM never reaches thermal equilibrium with the Standard Model (SM) bath. However, a fundamental challenge for Freeze-in scenarios is testability: the couplings required to reproduce the observed relic density are typically so weak that they evade laboratory investigations.

A potential solution arises in cosmological histories with low reheating temperatures ($T_{RH}$). If $T_{RH}$ is low compared to the mass scales of DM production, the initial state particles are suppressed by Boltzmann factors. To compensate for this suppression and achieve the correct relic density, the coupling strength must be significantly larger than in high $T_{RH}$ scenarios, potentially bringing the interaction within the experimental detection range. This article investigates a specific realization of this scenario, where DM is produced via quark-flavor-changing interactions in a hadronic bath ($T_{RH} < T_{QCD} \sim 150\text{--}160$ MeV).

Methodology
The authors propose a light Dirac fermion DM candidate ($\chi$) coupled to the SM via a dimension-six quark-flavor-changing operator:
$$\mathcal{L}_{\text{int}} = \frac{C_V}{\Lambda^2} (\bar{s}\gamma^\mu d)(\bar{\chi}\gamma_\mu \chi) + \text{h.c.}$$
The analysis focuses on the regime where $T_{RH}$ is below the QCD crossover transition but above the primordial nucleosynthesis (BBN) limit ($T_{RH} \gtrsim 6$ MeV). In this epoch, the relevant degrees of freedom are hadrons (Kaons and Pions) rather than quarks and gluons.

The methodology comprises three main components:

1. Derivation of Decay Rates: The authors calculate the differential decay rates for rare Kaon decays $K^+ \to \pi^+ \chi \bar{\chi}$ and $K_L \to \pi^0 \chi \bar{\chi}$ (and their charge conjugates) using form factors from chiral perturbation theory. They account for isospin relations and the vector form factor $f_+(q^2)$, which is linearly parametrized for the relevant kinematic range.
2. Derivation of Scattering Cross Sections: The article calculates the cross sections for scattering processes $K \pi \to \chi \bar{\chi}$. Crucially, the scattering amplitude is linked to the decay form factors via crossing symmetry. The authors model the timelike form factor $F_+(s)$ using a Breit-Wigner approximation dominated by the $K^*(892)$ resonance, which is significant near the $K\pi$ threshold.
3. Solution of the Boltzmann Equation: The evolution of the DM yield ($Y_\chi$) is determined by solving the Boltzmann equation. The collision terms include contributions from both Kaon decays ($K \to \pi \chi \bar{\chi}$) and Kaon-Pion scattering ($K \pi \to \chi \bar{\chi}$). The authors treat neutral Kaons in the thermal bath as flavor eigenstates ($K^0, \bar{K}^0$) rather than mass eigenstates ($K_L, K_S$), due to rapid elastic scattering with pions at $T \sim 100$ MeV.

Main Contributions and Results
The article solves the Boltzmann equation numerically to identify the parameter space ($m_\chi$, $\Lambda/\sqrt{|C_V|}$) that reproduces the observed relic density $\Omega h^2 \approx 0.12$ for reheating temperatures $T_{RH} = 60, 80, 100$ MeV.

• Production Mechanisms: The analysis shows that at higher temperatures (closer to $T_{RH}$), scattering processes ($K\pi \to \chi\bar{\chi}$) dominate DM production, driven by the thermal population of pions and resonant enhancement via the $K^*(892)$. At lower temperatures, decay processes ($K \to \pi \chi \bar{\chi}$) become relatively more important as pions are suppressed by Boltzmann factors.
• Coupling Strength vs. $T_{RH}$: A central result is the inverse relationship between $T_{RH}$ and the required coupling strength. Lower reheating temperatures lead to stronger Boltzmann suppression of Kaons, necessitating larger couplings (smaller $\Lambda/\sqrt{|C_V|}$) to generate the correct relic density.
• Experimental Testability: The same operator governing DM production induces rare Kaon decays with missing energy. The authors compare the Freeze-in target region with current experimental limits:
  • NA62: Constraints on $BR(K^+ \to \pi^+ \nu \bar{\nu})$ are used as a proxy for $K^+ \to \pi^+ \chi \bar{\chi}$.
  • KOTO: Constraints on $BR(K_L \to \pi^0 \nu \bar{\nu})$ are used for the neutral mode.
  • Results: For $T_{RH} = 60$ MeV, the required coupling places the model within the detection range of current NA62 and KOTO data. For $T_{RH} = 80$ and $100$ MeV, the parameter space is accessible to the projected sensitivity of the KOTO-II experiment (target $BR \sim 10^{-12}$).
• Spectral Signatures: The article analyzes the differential missing mass spectrum ($q^2$) for $K_L \to \pi^0 + \text{invisible}$. The spectrum exhibits a kinematic threshold at $q^2 = 4m_\chi^2$. The shape and magnitude of the excess over the SM background are correlated with both the DM mass $m_\chi$ and the reheating temperature $T_{RH}$.

Significance and Claims
The article claims to close a gap in the literature by directly linking quark-flavor-changing interactions to both the cosmological Freeze-in relic density and rare meson observables. In contrast to previous studies focusing on light mediators or specific UV completions, this work demonstrates that "Kaon-driven" Freeze-in in a low-reheating cosmology mitigates the "bottleneck" problem of weak couplings.

The authors claim that this scenario offers a concrete, testable framework in which the cosmological origin of light, weakly interacting DM can be probed via flavor experiments. They emphasize that the correlation between the relic density and the branching ratios of rare decays enables laboratory investigations (NA62, KOTO, KOTO II) to directly test the cosmological history of the universe. The work is presented as a benchmark study using a vector current operator, noting that while other operator structures (e.g., scalar) could alter the specific correlations, they would likely maintain the general connection between low $T_{RH}$ cosmology and the search for rare decays.