Illuminating sequential freeze-in dark matter with dark photon signal at the CERN SHiP experiment

This paper demonstrates that the CERN SHiP experiment can probe sequential freeze-in dark matter mediated by dark photons with masses between $10^{-2}$ and $10$ GeV, predicting a specific dark charge and a narrow mixing parameter range ($\sim 10^{-11}$) that survives current constraints while remaining testable via proton bremsstrahlung signals.

The Gist

The Big Picture: Hunting for "Invisible" Ghosts

Imagine the universe is a giant, dark house. We know there is furniture in it (Dark Matter), but we can't see it, touch it, or hear it. For decades, scientists have been trying to find this furniture by looking for "footprints" left behind when it bumps into normal matter.

However, in many theories, this "furniture" is so shy and interacts so weakly with the rest of the house that it leaves no footprints at all. It's like a ghost that walks through walls without making a sound. This makes it nearly impossible to detect with current experiments.

This paper proposes a new way to catch this ghost. Instead of looking for the ghost directly, the authors suggest looking for a messenger that the ghost might be holding hands with.

The Cast of Characters

1. Dark Matter (The Ghost): The invisible stuff making up most of the universe's mass.
2. The Dark Photon (The Messenger): A new, heavy particle that acts like a bridge between our visible world and the dark world. It's not a real photon (light), but a "dark" cousin.
3. The SHiP Experiment (The Net): A proposed experiment at CERN (the European particle physics lab) designed to act like a giant net to catch these messengers.

The Story: How the Ghosts Were Born (Sequential Freeze-In)

Usually, scientists think Dark Matter was created when the universe was hot and crowded, like people at a mosh pit bumping into each other until they settled down. This is called "Freeze-Out."

But this paper suggests a different story: Sequential Freeze-In.

Imagine a factory assembly line:

1. Step 1: Normal particles (like protons) crash together in the early universe.
2. Step 2: These crashes create the Dark Photon (the messenger).
3. Step 3: The Dark Photon is unstable. It quickly decays (breaks apart) into a pair of Dark Matter particles (the ghosts).

The key here is that the Dark Matter is so weakly connected to the messenger that it never really "mixes" with the rest of the universe. It just gets produced and then quietly disappears into the dark. Because it's so weakly connected, it's invisible to standard detectors.

The Twist: The "Goldilocks" Zone

The authors did some math to figure out exactly how strong the connection between the Dark Matter and the Dark Photon needs to be for this to work.

• The Connection (Dark Charge): They found that the Dark Matter must have a very specific, tiny "charge" (a measure of how much it interacts). It's fixed at a value of roughly 1.3 × 10⁻¹². That is an incredibly small number, like finding one specific grain of sand on all the beaches on Earth.
• The Mixing (The Bridge): The connection between the Dark Photon and our normal world (called the "mixing parameter," $\epsilon$) has to be just right.
  • If it's too strong, the Dark Photons would have mixed with normal light too much, and we would have seen them already.
  • If it's too weak, the factory line stops, and no Dark Matter is made.

The paper calculates that this "Goldilocks" zone for the mixing parameter is between $10^{-11}$ and $10^{-8}$.

The Hunt: The SHiP Experiment

Now, how do we find this? The authors looked at the SHiP experiment at CERN.

• The Setup: SHiP will fire a massive beam of protons (like a giant cannon) at a solid target.
• The Process: When the protons hit the target, they might create Dark Photons (the messengers).
• The Signal: These Dark Photons will fly into a detector and decay into pairs of electrons, muons, or other particles we can see.

The authors ran simulations to see if SHiP could catch these messengers.

The Results: Closing the Door

The findings are a bit of a "good news, bad news" situation:

1. The Good News: The SHiP experiment is powerful enough to test almost the entire range of possibilities where this theory could be true.
2. The Bad News: If the experiment runs for 5 to 15 years, it will likely rule out most of the possible values for the mixing parameter.
   • It will likely prove that the mixing parameter is not between $10^{-8.5}$ and $10^{-7.5}$.
   • This leaves only a tiny, narrow sliver of possibility left (around $10^{-11}$) for this specific theory to be true.

The Analogy: The Silent Alarm

Imagine you are trying to find a thief (Dark Matter) who never leaves the house. You suspect the thief uses a specific type of walkie-talkie (Dark Photon) to talk to a friend outside.

• Old Theory: You thought the thief was loud and left footprints everywhere. You didn't find any, so you thought the thief didn't exist.
• New Theory: You realize the thief is silent, but the walkie-talkie might make a tiny "click" sound when it turns on.
• The Experiment: You build a super-sensitive microphone (SHiP) to listen for that "click."
• The Conclusion: The paper says, "If we listen for 15 years, we will almost certainly hear that click if the thief is using a walkie-talkie with a volume setting between 1 and 10. If we don't hear it, we know the volume setting must be lower than 0.1."

Why This Matters

Even if SHiP doesn't find the Dark Photon, this paper is a success. It tells us exactly where not to look and narrows down the search. It forces scientists to either:

1. Look for the Dark Matter in that tiny remaining sliver of possibilities.
2. Come up with a completely new theory about how the universe works.

It's like a detective saying, "The killer isn't in the kitchen or the living room. If they are in this house, they are hiding in this one tiny closet." That is a huge step forward in solving the mystery of the universe.

Technical Summary

1. Problem Statement

The nature of Dark Matter (DM) remains unknown. While the standard "freeze-out" mechanism (where DM is in thermal equilibrium with the Standard Model (SM) bath) is well-studied, it has faced null results in direct detection and collider experiments. Consequently, the "freeze-in" mechanism, where DM is produced via tiny couplings from the SM bath and never reaches thermal equilibrium, has gained attention.

However, single-field freeze-in models (e.g., millicharged particles, sterile neutrinos) typically involve couplings so small that they leave no observable footprints in current or near-future experiments. The authors address the question: Can a two-field freeze-in scenario, mediated by a massive dark photon, be probed by the proposed CERN SHiP (Search for Hidden Particles) experiment?

2. Methodology

A. Theoretical Model

The authors propose a sequential freeze-in model involving a massive dark photon ($A'$) acting as a mediator between the SM sector and a fermionic DM candidate ($\chi$).

• Lagrangian: The model includes kinetic mixing ($\epsilon$) between the dark photon and the SM hypercharge, and a dark gauge coupling ($e'$) between the dark photon and the DM.
• Sequential Mechanism: Unlike standard freeze-in where DM is produced directly from SM particles, this model involves a two-step process:
  1. Production: SM particles ($\psi$) annihilate to produce off-shell or on-shell dark photons ($\psi \bar{\psi} \to A'$). This occurs when the mixing parameter $\epsilon$ is below the thermal equilibrium threshold ($\epsilon < \epsilon_{th} \sim 10^{-8}$).
  2. Decay: The non-thermal dark photons subsequently decay into DM pairs ($A' \to \chi \bar{\chi}$).
• Boltzmann Equations: The authors derive and solve coupled Boltzmann equations for the number densities of the dark photon ($n_{A'}$) and the DM ($n_\chi$). They utilize yield variables ($Y = n/s$) to track the evolution of the system from the early universe to the present day.

B. Experimental Simulation (SHiP)

The study evaluates the detectability of the dark photon mediator at the SHiP experiment, a proposed proton beam-dump experiment at CERN.

• Production Mechanism at SHiP: The primary production channel considered is proton bremsstrahlung ($pp \to ppA'$). The authors explicitly calculate and discard the contribution from DM bremsstrahlung ($pp \to \chi \bar{\chi} A'$) due to severe suppression by the small couplings.
• Decay Channels: The dark photon decays into visible SM particles (leptons $\ell^+\ell^-$ and hadrons) and invisible DM pairs ($\chi \bar{\chi}$).
  • The branching ratio depends on the ratio $e'/\epsilon$.
  • In the sequential freeze-in regime, $e' \ll \epsilon$ is not required; rather, the model allows for $e' \sim 1.3 \times 10^{-12}$ while $\epsilon$ is larger, making the invisible decay ($A' \to \chi \bar{\chi}$) potentially dominant or significant.
• Sensitivity Calculation: The expected number of events is calculated using the integrated luminosity ($N_{POT}$) for 5-year ($2 \times 10^{20}$ protons) and 15-year ($6 \times 10^{20}$ protons) operations. The analysis accounts for detector acceptance and efficiency, comparing the predicted signal against existing bounds from experiments like NA64, LHCb, and BaBar.

3. Key Contributions

1. Parameter Space Definition: The paper rigorously defines the viable parameter space for sequential freeze-in DM mediated by a massive dark photon. It establishes that for the observed relic abundance ($\Omega_{DM}h^2 \approx 0.12$), the dark charge is fixed to $e' \sim 1.3 \times 10^{-12}$, largely independent of the mass ratios, provided the mixing parameter $\epsilon$ is within a specific range.
2. SHiP Projections: The authors provide the first detailed projection of the SHiP experiment's sensitivity to this specific sequential freeze-in scenario. They distinguish between Dipole Dominance and Vector Meson Dominance (VMD) models for the hadronic production cross-section.
3. Exclusion Limits: The study demonstrates that SHiP can probe a significant portion of the theoretically allowed parameter space, specifically testing the mixing parameter $\epsilon$ in the range of $10^{-11}$ to $10^{-7.5}$.

4. Key Results

• Fixed Dark Charge: The requirement to reproduce the observed DM relic abundance fixes the dark coupling to $e' \approx 1.3 \times 10^{-12}$.
• Allowed Mixing Range: The mixing parameter $\epsilon$ is constrained to the range:
  $$10^{-11} \le \epsilon < 10^{-8} - 10^{-7.5}$$
  The upper bound is set by the thermal equilibrium condition ($\epsilon_{th}$), and the lower bound is determined by the relic density requirement.
• SHiP Exclusion Power:
  • Assuming Vector Meson Dominance (VMD), SHiP's 5-year data excludes $\epsilon \ge 10^{-8.5}$.
  • Assuming Dipole Dominance, the 5-year data excludes $\epsilon \ge 10^{-7.9}$.
  • Extending to 15 years of operation further tightens these bounds.
• Remaining Parameter Space: After applying SHiP's projected limits, only a narrow region of $\epsilon$ close to $\sim 10^{-11}$ remains untested. This implies that the sequential freeze-in scenario is highly testable and will either be discovered or severely constrained by SHiP.
• Mass Range: The analysis covers dark photon masses ($m_{A'}$) from $10^{-2}$ GeV to 10 GeV, with DM masses ($m_\chi$) ranging from $10^{-2}$ GeV to 10 GeV.

5. Significance

• Bridging the Gap: This work connects the theoretical concept of sequential freeze-in (often considered "dark" and unobservable) with concrete experimental signatures. It shows that two-field models can be accessible even when single-field models are not.
• Target for SHiP: The paper provides specific targets for the SHiP collaboration, identifying a narrow but critical window of parameters ($\epsilon \sim 10^{-11}$) that SHiP is uniquely positioned to explore.
• Model Discrimination: By analyzing the interplay between the visible decay (SM particles) and invisible decay (DM), the study highlights how the ratio $e'/\epsilon$ dictates the experimental signature, offering a way to distinguish this model from other dark photon scenarios (like millicharged particles or DM-as-dark-photon).
• Future Physics: If SHiP finds no signal in the predicted region, it would effectively rule out the sequential freeze-in mechanism mediated by a massive dark photon for the specified mass ranges, forcing a re-evaluation of DM production mechanisms. Conversely, a discovery would provide the first evidence of a multi-step freeze-in process and a massive dark photon mediator.