Missing energy signatures of inelastic magnetic dipole DM at NA64e

This paper calculates the production rate and projected sensitivity of the NA64e experiment at CERN to inelastic magnetic dipole dark matter pairs, demonstrating that including heavy vector meson decays alongside bremsstrahlung-like emission enables the probe of previously unexplored parameter spaces for light masses and modest mass splittings.

The Gist

The Big Picture: Hunting for the "Ghost" in the Machine

Imagine the universe is a giant, bustling city. We know about the people we can see, touch, and talk to (these are the Standard Model particles like electrons and protons). But astronomers have realized that about 85% of the city's population is invisible. They don't shine, they don't talk, and they don't interact with us normally. We only know they are there because their gravity is pulling on the visible buildings. This invisible crowd is Dark Matter.

For decades, scientists have been trying to catch a glimpse of these "ghosts." This paper is a proposal for a new way to catch them using a giant particle accelerator at CERN (the European Organization for Nuclear Research).

The Theory: The "Inelastic" Dark Matter

Usually, scientists imagine Dark Matter as a single, boring particle that just sits there. But this paper explores a more exciting idea: Inelastic Dark Matter (iDM).

Think of this like a two-stage rocket or a video game character with two forms:

1. The Light Form ($\chi_0$): This is the calm, everyday version of the particle. It's the "ground state."
2. The Heavy Form ($\chi_1$): This is the excited, energetic version. It's heavier and unstable.

The key rule of this theory is that the Heavy Form must eventually turn back into the Light Form. When it does, it spits out a tiny bit of energy (a photon). However, the paper argues that if the "jump" between the two forms is very small, that energy burst is so tiny it's like a whisper in a hurricane—it's invisible to our detectors.

The Experiment: The "Missing Energy" Game

The authors are looking at an experiment called NA64e at CERN. Here is how they plan to catch these particles, using a simple analogy:

The Setup:
Imagine you have a powerful cannon (the electron beam) shooting 100 GeV electrons (tiny, fast bullets) at a thick, heavy wall (a lead target).

The Normal Scenario:
When the cannonball hits the wall, it usually bounces off or breaks apart, but you can see where all the energy went. If you shoot a 100-unit energy ball, you expect to see roughly 100 units of energy coming out the other side (scattered electrons, heat, light, etc.).

The "Ghost" Scenario:
Now, imagine that occasionally, the cannonball hits the wall and creates a pair of these "Inelastic Dark Matter" particles.

1. They are created in the collision.
2. They are electrically neutral and invisible.
3. They zoom right through the wall and the detectors without stopping.
4. They carry a huge chunk of the energy away with them.

The Clue:
The scientists look at the energy that stays in the detector.

• Expected: 100 units.
• Observed: Maybe only 40 units.
• Missing: 60 units.

If they see a lot of events where energy just "disappears" (and they have ruled out all other reasons for it to vanish), they have found the Dark Matter. It's like playing a game of billiards where the cue ball hits the rack, and suddenly, half the balls vanish into thin air. You know they must be there, even if you can't see them.

The New Twist: Two Ways to Make the Ghosts

The paper calculates how often these ghosts are made. They found two main "factories" inside the wall:

1. The "Bremsstrahlung" Factory (The Spray):
   When the electron hits the nucleus, it slows down and sprays out a virtual photon (a flash of light that doesn't quite exist yet). This flash instantly turns into a pair of Dark Matter particles. This is the most common way they are made.

2. The "Heavy Meson" Factory (The Delivery Truck):
   This is the paper's big new insight. Sometimes, the collision creates a heavy, short-lived particle called a Vector Meson (like a J/psi or a Phi meson). Think of this as a delivery truck.

   • The truck drives a short distance.
   • It crashes and explodes.
   • Instead of debris, it releases a pair of Dark Matter particles.
   • Why this matters: The authors show that for lighter Dark Matter particles (under 100 MeV), this "delivery truck" method is actually very efficient. It opens up a new area of the map that other experiments haven't looked at yet.

The Goal: Mapping the Unknown

The paper does the math to predict:

• How many of these events NA64e should see if Dark Matter exists.
• How sensitive the experiment needs to be.

They conclude that if the NA64e experiment upgrades its equipment to handle more electrons (about 10 trillion hits on the target) and reduces background noise (false alarms), it will be able to find these particles if they exist in a specific "sweet spot" of mass and energy splitting.

Summary in a Nutshell

• The Problem: We know Dark Matter exists, but we can't see it.
• The Idea: Maybe Dark Matter comes in two flavors (light and heavy), and the heavy one decays into the light one so quietly we can't hear it.
• The Method: Shoot electrons at a wall and look for energy that vanishes.
• The Innovation: They realized that heavy, short-lived particles (mesons) act like delivery trucks that can drop off Dark Matter pairs, making them easier to find in certain mass ranges.
• The Hope: The NA64e experiment at CERN might be the first to catch these "invisible" particles, finally giving us a clue about what 85% of the universe is made of.

Technical Summary

1. Problem Statement

The paper addresses the search for Inelastic Dark Matter (iDM), a theoretical candidate for sub-GeV thermal dark matter. Unlike standard WIMP models, iDM consists of two fermion states ($\chi_0$ and $\chi_1$) with a small mass splitting ($\Delta = (m_{\chi_1} - m_{\chi_0})/m_{\chi_0}$). The interaction between these states and the Standard Model (SM) photon is mediated by a magnetic dipole moment (MDM) operator.

The specific challenge addressed is determining the sensitivity of the NA64e experiment at CERN's SPS to this specific iDM model. Previous studies often focused on vector mediators or scalar portals. This paper investigates the production of iDM pairs via MDM interactions, specifically analyzing two production channels:

1. Bremsstrahlung-like emission: $e^- N \to e^- N \gamma^* (\to \chi_1 \bar{\chi}_0)$.
2. Vector meson photoproduction and decay: $e^- N \to e^- N V (\to \chi_1 \bar{\chi}_0)$, where $V$ represents heavy vector mesons ($\rho, \omega, \phi, J/\psi$).

The goal is to define the projected exclusion limits on the MDM coupling scale ($\Lambda_M$) and identify unexplored regions of the iDM parameter space, particularly for small mass splittings ($\Delta \sim 10^{-3} - 5 \times 10^{-2}$) and light masses ($m_{\chi_0} \lesssim 100$ MeV).

2. Methodology

Experimental Setup (NA64e)

The analysis utilizes the NA64e fixed-target experiment at CERN, which employs a 100 GeV electron beam incident on an active lead-scintillator electromagnetic calorimeter (ECAL).

• Signal Signature: "Missing Energy." If iDM is produced, the $\chi_1 \bar{\chi}_0$ pair carries away a significant fraction of the beam energy ($E_{miss} \gtrsim 50$ GeV). The $\chi_1$ subsequently decays via $\chi_1 \to \chi_0 \gamma$. However, due to the small mass splitting and kinematics, the decay photon is soft ($E_\gamma \ll 1$ GeV) and undetectable by the calorimeter. Thus, the event appears as a single electromagnetic shower (from the recoil electron) with missing energy, indistinguishable from a background-free signal.
• Statistics: The study assumes a projected dataset of $N_{EOT} \simeq 10^{13}$ electrons on target, with a background level suppressed to $b < 1$ event.

Theoretical Framework

• Lagrangian: The interaction is defined by the effective dimension-5 operator:
  $$\mathcal{L} \supset \frac{1}{\Lambda_M} \bar{\chi}_1 \sigma_{\mu\nu} \chi_0 F^{\mu\nu} + \text{h.c.}$$
  where $\Lambda_M$ is the new physics energy scale.
• Production Mechanisms:
  1. Bremsstrahlung: Calculated using the CalcHEP software package. The differential cross-section $d\sigma/dE_{miss}$ was computed for the process $e^- N \to e^- N \chi_1 \bar{\chi}_0$.
  2. Vector Meson Decays: The production of vector mesons ($V$) via virtual photon exchange ($\gamma^* N \to NV$) followed by invisible decay $V \to \chi_1 \bar{\chi}_0$. The branching ratios were calculated based on the MDM coupling.
• Kinematics: The decay length of $\chi_1$ was estimated to ensure it decays within the detector volume (or produces a photon too soft to trigger), validating the "missing energy" assumption. Two benchmark mass splittings were used: $\Delta = 10^{-3}$ and $\Delta = 5 \times 10^{-2}$.

3. Key Contributions

1. Inclusion of Vector Meson Decays: A primary contribution is the rigorous inclusion of heavy vector meson decays ($J/\psi, \phi, \omega, \rho$) as a production channel. The authors demonstrate that for light iDM masses ($m_{\chi_0} \lesssim 100$ MeV), the contribution from coherent $J/\psi$ photoproduction is comparable to, and in some regimes dominant over, the standard bremsstrahlung channel.
2. Projected Sensitivity Maps: The paper provides the first projected sensitivity contours for NA64e specifically for magnetic dipole iDM. It maps the exclusion limits on the coupling $1/\Lambda_M$ as a function of the lightest DM mass $m_{\chi_0}$.
3. Parameter Space Exploration: The study identifies a previously unexplored region of parameter space where NA64e can probe:
   • Small mass splittings ($\Delta \approx 5 \times 10^{-2}$).
   • Relatively light masses ($m_{\chi_0} \lesssim 100$ MeV).
4. Thermal Target Comparison: The results are compared against the "thermal target" curve (the coupling required for iDM to constitute the observed relic density via thermal freeze-out). The analysis shows that while NA64e cannot cover the entire thermal target for large splittings, it can probe a significant fraction of the parameter space for $\Delta = 5 \times 10^{-2}$ and $m_{\chi_0} \lesssim 100$ MeV.

4. Results

• Signal Yield: The total number of signal events ($N_{sign.}$) is the sum of bremsstrahlung-like production and vector meson decays.
  • For $m_{\chi_0} \lesssim 100$ MeV, the $J/\psi$ contribution becomes significant, creating a "kink" in the signal yield curve near the kinematic threshold ($m_{\chi_0} \approx m_{J/\psi}/2$).
  • The signal yield is relatively flat across the sub-GeV mass range.
• Exclusion Limits (90% CL):
  • Current NA64e ($10^{12}$ EOT): Current bounds are already ruled out by LEP and CHARM II for the benchmark splittings.
  • Projected NA64e ($5 \times 10^{12}$ EOT): Can probe couplings down to $1/\Lambda_M \sim (1.0 - 1.5) \times 10^{-3} \text{ GeV}^{-1}$.
  • Projected NA64e ($10^{13}$ EOT): With background-free operation ($b=0$), the experiment can reach $1/\Lambda_M \sim 9 \times 10^{-4} \text{ GeV}^{-1}$ for $\Delta = 5 \times 10^{-2}$ and $m_{\chi_0} \lesssim 100$ MeV.
• Comparison with Other Experiments: The projected sensitivity of NA64e with $10^{13}$ EOT surpasses current limits from BaBar, NuCal, and LEP in the specific region of light masses and moderate splittings, offering a unique probe for thermal iDM scenarios that other facilities miss.

5. Significance

This paper is significant because it expands the discovery potential of the NA64e experiment beyond simple dark photon searches. By incorporating inelastic magnetic dipole interactions and vector meson decays, the authors demonstrate that NA64e is uniquely positioned to test the thermal origin of sub-GeV dark matter.

The findings suggest that:

1. Vector meson decays are crucial: Ignoring the $J/\psi$ and other meson contributions would significantly underestimate the production rate of light iDM.
2. Feasibility of Thermal DM Search: With the planned upgrade to $10^{13}$ electrons on target and improved background suppression, NA64e can probe the specific coupling strengths required for iDM to be a thermal relic, a region largely inaccessible to current collider experiments (like LEP/BaBar) for these specific mass/splitting configurations.
3. Complementarity: The results complement searches at high-energy colliders (FASER, LHC) and other beam-dump experiments, filling a gap in the sub-GeV dark matter landscape.

In conclusion, the paper provides a robust theoretical and phenomenological basis for using the NA64e missing-energy signature to discover or constrain inelastic magnetic dipole dark matter, highlighting the experiment's potential to solve the sub-GeV dark matter puzzle.