Dark Photon mediated Inelastic Dark Matter in Cosmology, Astrophysics and Colliders

This paper presents a systematic phenomenological analysis of Dark Photon-mediated Inelastic Dark Matter, demonstrating that while current direct and indirect searches cannot access the parameter space yielding the correct relic abundance, future Long-Lived Particle searches at the LHC (specifically FASER and FASER 2) and neutron star observations offer significant potential to probe or rule out this model.

The Gist

The Big Picture: The "Ghostly" Upgrade

Imagine the universe is filled with invisible "dark matter" that holds galaxies together. Scientists have been trying to catch a glimpse of these particles for decades, but they are like ghosts: they don't reflect light, and they barely bump into normal matter.

This paper proposes a specific type of ghost called Inelastic Dark Matter (iDM). Think of this ghost not as a single, static object, but as a chameleon with two forms:

1. The Light Form (χ₁): This is the "sleeping" version. It's light, stable, and makes up the dark matter we see everywhere.
2. The Heavy Form (χ₂): This is the "awake" version. It's slightly heavier and unstable.

The Catch: The light form can only turn into the heavy form if it gets a specific "kick" of energy. If it doesn't get that kick, it stays light and invisible. This is the "inelastic" part—it needs a boost to change.

The Messenger: The Dark Photon

How does this dark matter talk to our world? It uses a messenger called the Dark Photon (A').

• Imagine the Standard Model (our known universe) is a house with a locked door.
• The Dark Sector (dark matter) is a neighboring house.
• The Dark Photon is a secret tunnel connecting the two. It's a bit like a "Wi-Fi signal" that leaks between the houses, allowing them to interact, but very weakly.

Why We Haven't Found It Yet (The Speed Bump)

You might ask, "If they interact, why haven't detectors on Earth found them?"

The paper explains that our dark matter ghosts are moving too slowly in our galaxy.

• The Analogy: Imagine trying to jump over a 3-foot wall. If you are walking slowly, you can't make it. You need a running start.
• In the early universe, things were moving fast (hot), so the dark matter could easily "jump the wall" (turn into the heavy form) and interact.
• Today, in our cold galaxy, they are walking too slowly to jump the wall. They just bounce off Earth's detectors without changing form, making them invisible to our current experiments.

The Three Ways to Catch Them

Since we can't catch them easily on Earth, the authors suggest three creative ways to find them:

1. The "Cosmic Accelerator" (Neutron Stars)

Neutron stars are the densest, most massive objects in the universe (like a sugar-cube-sized piece of a star weighing a billion tons).

• The Analogy: Imagine a dark matter ghost falling toward a giant magnet (the neutron star). The gravity pulls it in, accelerating it to near the speed of light.
• The Result: Now that the ghost is moving super-fast, it has enough energy to jump the wall and turn into the heavy form (χ₂) when it hits the star's surface.
• The Signal: This collision dumps a massive amount of energy into the star, heating it up. The authors predict this could heat a nearby neutron star to about 2,000 Kelvin (roughly 3,000°F).
• How to see it: If we point infrared telescopes at old, cold neutron stars near Earth, we might see one that is mysteriously glowing hot. That heat would be the fingerprint of dark matter.

2. The "Long-Lived Particle" Hunt (FASER at the LHC)

The Large Hadron Collider (LHC) smashes protons together at incredible speeds.

• The Analogy: Think of the LHC as a giant slingshot. If we smash protons hard enough, we might create the heavy dark matter ghost (χ₂) directly.
• The Twist: Because the "wall" (mass difference) is small, this heavy ghost doesn't decay immediately. It's like a slow-motion firework. It travels a long distance before it explodes (decays) back into the light ghost and some normal particles.
• The Detector: The FASER detector is a small, specialized camera placed 480 meters down the track from the collision point. It's designed to catch these "slow-motion fireworks" that travel far before exploding.
• The Upgrade (FASER 2): The authors show that a future, bigger version of this detector (FASER 2) could catch these particles even more easily, potentially ruling out or finding this specific dark matter model for masses up to 7 GeV (for FASER) and higher for FASER 2.

3. Why Earth Detectors Fail

The paper concludes that traditional underground detectors (like those using giant tanks of xenon) are likely useless for this specific model.

• The Analogy: It's like trying to catch a butterfly with a net made of steel bars. The butterfly (dark matter) is too slow to trigger the net, and the net is too heavy to feel the tiny bump. The "kinetic threshold" is too high for Earth's slow-moving dark matter.

The Main Takeaway

The authors did a massive "sweep" of all possible numbers for this model. They found:

1. It's still possible: The model isn't ruled out by current data.
2. Earth is too slow: We can't find it with current underground detectors because the dark matter isn't moving fast enough.
3. The Universe is the lab: We have two great chances to find it:
   • Look for hot neutron stars using infrared telescopes.
   • Look for long-lived particles decaying far away from the collision point at the LHC (using FASER).

In short, if this "chameleon" dark matter exists, we won't find it by waiting for it to bump into us on Earth. We have to either accelerate it with a neutron star's gravity or create it in a particle collider and wait for it to travel a long way before it reveals itself.

Technical Summary

1. Problem Statement

The search for Weakly Interacting Massive Particles (WIMPs) has yielded no definitive signals, placing increasing tension on the standard thermal decoupling scenario. Inelastic Dark Matter (iDM) offers a solution by proposing that Dark Matter (DM) consists of two nearly degenerate Majorana states, $\chi_1$ (the stable DM candidate) and $\chi_2$ (a heavier excited state), separated by a mass splitting $\delta = M_{\chi_2} - M_{\chi_1}$.

The specific model analyzed is the Dark Photon mediated iDM ($A'$iDM) scenario. In this framework:

• The Standard Model (SM) is extended by a dark $U(1)_D$ gauge symmetry.
• A dark photon ($A'$) mediates interactions between the dark sector and the SM via kinetic mixing ($\epsilon$).
• The DM candidate $\chi_1$ interacts with SM nuclei only by upscattering to $\chi_2$.

The Gap: While the $A'$iDM model is theoretically simple, previous literature has only discussed specific benchmark points. There is a lack of a systematic phenomenological scan covering the full parameter space, particularly regarding the interplay between cosmological relic abundance, direct/indirect detection limits, astrophysical heating, and collider signatures.

2. Methodology

The authors performed a systematic numerical scan of the $A'$iDM parameter space to identify regions consistent with observational constraints.

Model Parameters

The phenomenology is determined by five parameters:

1. $M_{\chi_1}$: Mass of the light DM state.
2. $\delta$: Mass splitting between $\chi_1$ and $\chi_2$.
3. $M_{A'}$: Mass of the dark photon.
4. $\epsilon$: Kinetic mixing parameter.
5. $\alpha_D$: Dark sector coupling strength.

Fixed Assumptions:

• Coupling: $\alpha_D$ is fixed to the electromagnetic coupling ($\alpha_{EM}$).
• Mixing: $\epsilon$ is fixed to its experimental upper bound ($\epsilon_{max}$) for a given $M_{A'}$ (derived from LEP and BaBar constraints).
• Mass Range: $M_{A'} \in [10, 60]$ GeV (avoiding the $Z$-pole region and low-mass constraints).
• Scan Ranges: $M_{\chi_1} \in [1, 30]$ GeV, $\delta \in [10^{-4}, 20]$ GeV.

Constraints Applied

1. Collider Constraints:
   • LEP electroweak precision limits on $\epsilon$.
   • BaBar monophoton searches ($e^+e^- \to \gamma A' \to \gamma \chi_1 \chi_2$).
   • Constraints on the invisible $Z$ decay width ($\Gamma_{inv}^Z$).
   • Monojet and displaced muon searches (noted as less sensitive for the specific long-lived particle (LLP) regime of interest).
2. Cosmological Constraints:
   • Relic Abundance: Solved the Boltzmann equation using micrOMEGAs to ensure the relic density matches the Planck observation ($\Omega_{\chi_1} h^2 = 0.12 \pm 0.0012$). This requires efficient co-annihilation ($\chi_1 \chi_2 \to A', Z \to f\bar{f}$).
   • Big Bang Nucleosynthesis (BBN): Ensured the lifetime of $\chi_2$ ($\tau_{\chi_2}$) is $< 1.0$ second to avoid injecting relativistic degrees of freedom that would alter light element abundances.
3. Direct/Indirect Detection:
   • Evaluated kinematic thresholds for upscattering in terrestrial detectors.
   • Checked for velocity-suppressed annihilation in the galactic halo.

Probes Analyzed

• Neutron Star (NS) Capture: Calculated the capture rate of $\chi_1$ in a neutron star ($M=1.5 M_\odot, R=10$ km) driven by gravitational acceleration. Assessed the resulting kinetic heating temperature.
• Collider Searches (FASER/FASER 2): Simulated the production of $\chi_1 \chi_2$ pairs via Drell-Yan processes at the LHC ($\sqrt{s}=13.6$ TeV). Used MadGraph5_aMC@NLO to generate events and calculated the probability of $\chi_2$ decaying inside the FASER and FASER 2 detectors (Long Lived Particle signature).

3. Key Contributions

• Systematic Scan: First comprehensive exploration of the $A'$iDM model with $\alpha_D = \alpha_{EM}$, challenging previous assumptions that this coupling is phenomenologically disfavored.
• Re-evaluation of Direct Detection: Demonstrated that for the parameter space satisfying the relic density and BBN constraints, Direct Detection (DD) is kinematically inaccessible. The mass splitting $\delta$ required for the correct relic abundance is too large for terrestrial WIMP velocities to overcome the upscattering threshold.
• Indirect Detection Null Result: Confirmed that indirect searches are ineffective because $\chi_2$ decays away in the current epoch (preventing $\chi_1 \chi_2$ annihilation) and $\chi_1 \chi_1$ annihilation is velocity-suppressed (p-wave).
• Astrophysical Heating Prediction: Quantified the heating of nearby neutron stars due to DM capture, predicting a surface temperature of $\approx 2000$ K for specific parameter ranges.
• Collider Reach: Defined the discovery potential of FASER and the proposed FASER 2 upgrade for LLPs in this specific model.

4. Key Results

Cosmological and BBN Constraints

• The allowed parameter space consistent with Planck data and BBN is:
  • $2 \text{ GeV} \lesssim M_{\chi_1} \lesssim 25 \text{ GeV}$
  • $2 \text{ MeV} \lesssim \delta \lesssim 12 \text{ GeV}$
  • $10 \text{ GeV} \lesssim M_{A'} \lesssim 60 \text{ GeV}$
• BBN Limit: For $\delta \lesssim 10^{-2}$ GeV, the $\chi_2$ lifetime exceeds 1 second, ruling out this region.
• Direct/Indirect Detection: In the allowed region, DD signals are suppressed because the required $\delta$ exceeds the kinematic threshold of Earth-based detectors ($\delta \gtrsim 200$ keV). Indirect detection is null due to the absence of $\chi_2$ in the halo.

Neutron Star Heating

• The inelastic scattering cross-section $\sigma_{inel}$ exceeds the geometric cross-section of a neutron star for all allowed configurations.
• Result: Capture is "optically thick," meaning all passing DM is captured.
• Heating: This process heats the neutron star to approximately 2000 K (assuming $\delta \lesssim 300$ MeV, the kinematic limit for NS gravitational acceleration).
• Significance: This offers a potential detection channel via future infrared telescope observations, overlapping partially with the parameter space accessible to FASER.

Collider Prospects (FASER & FASER 2)

• FASER (Current Run 3):
  • Sensitivity: $M_{\chi_1} \lesssim 7$ GeV, $100 \text{ MeV} \lesssim \delta \lesssim 300$ MeV, $M_{A'} \lesssim 25$ GeV.
  • Can probe or rule out a significant portion of the low-mass parameter space.
• FASER 2 (HL-LHC Upgrade):
  • Sensitivity: Extends to $M_{\chi_1} \lesssim 25$ GeV, $50 \text{ MeV} \lesssim \delta \lesssim 700$ MeV, $M_{A'} \lesssim 60$ GeV.
  • The larger decay volume ($R=1$ m, $L=10$ m) and higher luminosity ($3000 \text{ fb}^{-1}$) allow it to cover the entire cosmologically allowed parameter space for this model.

5. Significance

This paper establishes that the Dark Photon iDM scenario with $\alpha_D = \alpha_{EM}$ is a viable and testable model that evades current direct and indirect detection limits.

• Paradigm Shift: It moves the search for this specific DM candidate away from traditional underground detectors (which are kinematically blind) toward accelerator-based LLP searches and astrophysical heating observations.
• FASER 2 Potential: The study highlights that the FASER 2 upgrade at the HL-LHC is uniquely positioned to either discover or completely exclude this specific realization of Inelastic Dark Matter.
• Multi-Messenger Approach: It underscores the necessity of combining collider data with astrophysical observations (neutron star thermodynamics) to fully constrain the parameter space of light, inelastic dark matter models.