Braking protons at the EIC: from invisible meson decay to new physics searches

This paper proposes that the Electron-Ion Collider (EIC) can significantly enhance sensitivity to invisible decays of pseudoscalar mesons and axion-like particles by detecting forward protons with reduced energy in coherent exclusive electroproduction, potentially improving existing bounds on meson branching ratios by up to four orders of magnitude and probing new physics couplings in the 0.1–2 GeV mass range.

The Gist

Imagine you are at a massive, high-speed racetrack called the Electron-Ion Collider (EIC). This isn't a track for cars, but for subatomic particles: tiny electrons zooming one way and protons (the building blocks of atoms) zooming the other. When they crash, they create a spectacular fireworks display of new particles.

Usually, scientists want to see everything that comes out of the crash. They have giant cameras (detectors) all around the track to catch every spark, every flash of light, and every flying piece of debris.

But this paper proposes a very different kind of detective work. It's like playing a game of "Where's Waldo?" but with invisible objects.

The Mystery: The "Ghost" Proton

In this experiment, the scientists are looking for a specific type of crash where:

1. An electron hits a proton.
2. The electron bounces off and is caught by a camera.
3. The proton bounces off, but it's slower than it started.
4. Crucially: There is nothing else visible in the entire detector. No flashes, no sparks, no debris.

It's as if the proton hit a wall, lost some of its energy, and that missing energy simply vanished into thin air.

The Analogy: Imagine you are playing billiards. You hit the cue ball, and it strikes another ball. You see the cue ball slow down significantly. But when you look at the table, the other ball isn't there. It didn't roll away; it didn't break. It just... disappeared.

In the world of physics, energy cannot just vanish. If the proton lost energy, that energy had to go somewhere. The paper suggests that "somewhere" might be a hidden world (often called the "Dark Sector") that our current cameras can't see.

The Two Suspects

The scientists are hunting for two types of "invisible ghosts" that could be stealing the proton's energy:

1. The "Shy" Standard Particles:
   Normally, particles like the neutral pion ($\pi^0$) or the eta meson ($\eta$) are very chatty. They decay (break apart) instantly into photons (light) or other particles that our cameras can see.

   • The Theory: What if, very rarely, these particles decide to be shy and decay into invisible particles instead?
   • The Goal: The EIC is so sensitive it might catch these rare moments of silence, proving that these "chatty" particles can sometimes whisper into the dark.

2. The "Axion-Like" Intruders (ALPs):
   These are hypothetical new particles that physicists have been dreaming about for decades. They are like "ghosts" that interact very weakly with normal matter.

   • The Theory: Maybe the proton crash creates one of these ALPs, which then flies off and decays into invisible dark matter particles.
   • The Goal: Finding an ALP would be a Nobel Prize-level discovery, potentially explaining what Dark Matter is made of.

How They Catch the Ghosts

Since they can't see the invisible particle, they have to be clever detectives. They use a technique called "Missing Proton Energy" (MPE).

Think of it like a perfectly balanced scale:

• Before the crash: You know exactly how much energy the proton and electron had.
• After the crash: You measure the energy of the electron that bounced back and the proton that slowed down.
• The Calculation: If you add up the energy of the things you can see, and it's less than what you started with, the difference is the "Missing Energy."

The paper argues that the EIC has special "forward-facing cameras" (detectors) that can catch the proton even if it's moving at a very shallow angle. This allows them to measure the proton's speed with incredible precision. If the math doesn't add up, they know a ghost is present.

Why This is a Big Deal

The authors show that the EIC is a "super-powered" version of previous experiments.

• The Sensitivity: They claim the EIC could be 10,000 times more sensitive than current experiments at finding these invisible decays.
• The Analogy: Imagine previous experiments were like trying to hear a whisper in a noisy room with a regular microphone. The EIC is like having a super-microphone in a soundproof room that can hear a leaf dropping from a tree.

The "Invisible" Safety Net

One of the biggest challenges is making sure the "missing energy" isn't just a mistake. Maybe a particle flew into a gap in the detector, or a camera glitched.

• The paper details how they will use multiple layers of safety checks. They will look at the angles, the timing, and the energy to ensure that no visible particle "escaped" through a crack in the net.
• They estimate that with their new strategy, they can filter out almost all the "fake" ghosts (background noise) and focus on the real ones.

The Bottom Line

This paper is a blueprint for a new treasure hunt. It tells the builders of the Electron-Ion Collider exactly how to set up their "cameras" and "scales" to look for the most elusive things in the universe: particles that don't leave a trace.

If they succeed, they might:

1. Prove that known particles have secret, invisible lives.
2. Discover a whole new family of particles (Axions) that could explain the mysterious Dark Matter holding our galaxy together.

It's a search for the invisible, using the most powerful microscope humanity has ever built.

Technical Summary

1. Problem Statement

The paper addresses the challenge of detecting invisible final states in high-energy physics, specifically in the context of the upcoming Electron-Ion Collider (EIC). While the EIC is primarily designed to study the partonic structure of nucleons, its unique kinematic reach and detector capabilities offer a powerful platform for searching for Beyond the Standard Model (BSM) physics.

The specific problem involves identifying processes where a proton scatters and loses energy, but the recoiling system ($X$) is invisible to the detector. This signature is relevant for two main categories:

1. Standard Model (SM) Mesons: Neutral pseudoscalar mesons ($\pi^0, \eta, \eta'$) which have extremely suppressed invisible decay modes in the SM but could decay into hidden sectors (e.g., sterile neutrinos or dark matter) in BSM scenarios.
2. New Physics Particles: Axion-Like Particles (ALPs) or other BSM states that couple to hadrons and decay invisibly into a dark sector.

Current experimental bounds on the invisible branching ratios of these mesons leave significant room for improvement (e.g., $BR(\eta \to \text{inv}) \sim 10^{-4}$). The paper proposes a strategy to probe these decays and ALP parameters with unprecedented sensitivity.

2. Methodology: Missing Proton Energy (MPE) Search

The authors propose a "Missing Proton Energy" (MPE) search strategy, distinct from traditional missing transverse energy searches. The core process is coherent exclusive electroproduction:
$$p(p_p) + e^-(p_e) \to p(p_{p'}) + e^-(p_{e'}) + X(p_X)$$
where $X$ is an invisible state.

Key Experimental Strategy:

• Forward Proton Tagging: The signal is characterized by a forward-going proton ($p'$) with reduced energy ($E_{p'} < E_p$) and a scattered electron ($e'$). The EIC's Far-Forward Detector (FFD) system is crucial for reconstructing the forward proton with high efficiency ($\eta_{p'} > 4.5$).
• Kinematic Reconstruction: By precisely measuring the momenta of the outgoing proton ($p'$) and electron ($e'$), the kinematics of the invisible system $X$ can be indirectly reconstructed. Specifically, the pseudorapidity of the missing system, $\eta_{X, \text{reco}}$, is calculated.
• Veto Strategy: The analysis focuses on events where the reconstructed $\eta_X$ falls within the fully instrumented region of the detector ($|\eta_X| < 4$). In this region, any visible decay products of $X$ (e.g., photons from $\pi^0 \to \gamma\gamma$) would be detected. Therefore, if no additional activity is observed in the central and forward detectors, the event is classified as "invisible."
• Background Suppression:
  • Irreducible Backgrounds: Processes involving neutrinos (e.g., $ep \to ep\nu\bar{\nu}$) are estimated to yield $<1$ event over the full EIC run and are negligible.
  • Reducible Backgrounds: The main concern is "leakage" where visible particles escape detection (e.g., due to detector gaps or reconstruction failures). The authors simulate dominant SM processes ($\pi^0 \to \gamma\gamma$, $\rho^0 \to \pi^+\pi^-$, etc.) and apply a global veto on additional activity. They estimate a conservative background rejection efficiency, aiming for a background-free environment or a very low background yield ($N_{bg} \sim 2 \times 10^5$ before veto, reduced significantly after).

Detector Configuration:
The study utilizes the ePIC detector layout. Two beam energy configurations are analyzed:

• Baseline: $E_p = 100$ GeV, $E_e = 10$ GeV.
• Optimal: $E_p = 41$ GeV, $E_e = 5$ GeV. The lower energy configuration is preferred because it directs more scattered electrons into the central detector (improving reconstruction) and keeps the missing system $X$ within the fully instrumented $|\eta| < 4$ region.

3. Key Contributions

• Novel Search Channel: The paper introduces the MPE search as a unique probe for particles with hadronic interactions, complementing existing missing-electron-energy searches.
• Comprehensive Background Analysis: A detailed simulation of reducible backgrounds (particle leakage) and instrumental effects, establishing that the EIC can achieve a background-free regime with optimized selection cuts.
• Data-Driven Validation: The authors propose that the signal cross-section can be validated using visible meson decays (e.g., $\pi^0 \to \gamma\gamma$) collected in the same kinematic region, minimizing theoretical uncertainties.
• ALP Phenomenology: The study connects the search to a specific model of gluon-coupled Axion-Like Particles (ALPs), analyzing both on-shell ALP production ($a \to \text{inv}$) and off-shell contributions via meson-ALP mixing ($P \to a^* \to \text{inv}$).

4. Results and Sensitivity Projections

The authors present projected sensitivities for invisible decays of pseudoscalar mesons and ALPs, assuming an integrated luminosity of $L_{\text{int}} = 100 \text{ fb}^{-1}$.

A. Invisible Meson Decays:
The EIC is projected to improve existing bounds on invisible branching ratios by up to four orders of magnitude:

• $\pi^0 \to \text{inv}$: Sensitivity down to $BR \sim 4 \times 10^{-9}$ (Baseline) and $2 \times 10^{-10}$ (Optimal). This is roughly 2 times better than current limits.
• $\eta \to \text{inv}$: Sensitivity down to $BR \sim 1 \times 10^{-9}$ (Optimal), improving current limits ($1.1 \times 10^{-4}$) by $\sim 10^5$.
• $\eta' \to \text{inv}$: Sensitivity down to $BR \sim 4 \times 10^{-9}$ (Optimal), improving current limits ($2.1 \times 10^{-4}$) by $\sim 10^5$.

B. Axion-Like Particles (ALPs):
The search probes the parameter space of ALPs coupled to gluons ($\mathcal{L} \supset \frac{\alpha_s}{4\pi} \frac{a}{f_a} G\tilde{G}$) and dark fermions ($\chi$).

• Mass Range: $0.1 \text{ GeV} < m_a < 2 \text{ GeV}$.
• Coupling Reach: The EIC can probe ALP decay constants up to $f_a \sim 10^5$ GeV.
• Complementarity: The search is sensitive to both on-shell ALP production (dominant for small couplings $g_{a\chi}$) and off-shell meson decays mediated by ALPs (dominant for larger couplings). The combined sensitivity covers a broad region of the $\{m_a, f_a^{-1}\}$ plane, surpassing current bounds from NA62, NA64, and B-factories.

5. Significance

• New Physics Frontier: The MPE search offers a unique window into "dark sector" physics and hidden sectors that couple to quarks and gluons, which are difficult to probe in purely leptonic colliders.
• Precision Hadron Physics: The ability to normalize the signal against visible meson decays allows for precision measurements of rare SM processes and reduces theoretical uncertainties.
• Detector Optimization: The study highlights the critical importance of the Far-Forward Detector (FFD) and suggests potential upgrades (e.g., a second detector with dedicated forward muon/photon systems) to further enhance sensitivity.
• Broader Applicability: The methodology is not limited to ALPs; it can be applied to search for other new states with invisible decays, such as milli-charged particles or new spin-1 bosons.

In conclusion, the paper demonstrates that the EIC, through the Missing Proton Energy search, can transform the study of invisible decays, pushing the boundaries of sensitivity by several orders of magnitude and providing a powerful tool to explore the interface between the Standard Model and the dark sector.