Prospects for measuring exclusive diffractive $\eta,\eta'$ at the LHC

This paper reviews the Regge approach to pomeron-pomeron fusion and investigates the feasibility of measuring exclusive central diffractive production of $\eta$ and $\eta'$ mesons at the LHC to determine the pomeron's spin structure.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle accelerator, a giant circular racetrack where protons (tiny subatomic particles) zoom around at nearly the speed of light and crash into each other. Usually, when these protons smash together, it's like a car crash: everything explodes, and a chaotic cloud of debris flies in every direction.

But sometimes, something much more graceful happens. This paper is about a specific, rare type of "graceful crash" called exclusive diffractive production.

Here is the story of the paper, broken down into simple concepts:

1. The "Ghostly" Glue: The Pomeron

In the world of particle physics, there are invisible forces holding things together. The paper talks about a theoretical "glue" called the Pomeron.

• The Analogy: Imagine two cars driving past each other on a highway. Usually, they don't touch. But in this quantum world, they can exchange an invisible "handshake" (the Pomeron) without actually crashing.
• The Mystery: Physicists know this handshake exists, but they don't know what shape it is. Is it a simple dot (scalar)? A spinning arrow (vector)? Or a complex, stretchy sheet (tensor)? Figuring out the shape of this "handshake" is a huge puzzle.

2. The Perfect Crime Scene: Exclusive Events

In a normal crash, debris flies everywhere. But in an exclusive event, the two protons barely touch. They exchange this invisible Pomeron, create a new particle in the middle, and then bounce away almost untouched.

• The Setup: Imagine two billiard balls hitting each other gently. They bounce off to the sides, and in the exact center of the table, a new, fragile object (like a meson) appears.
• The Clue: Because the protons didn't smash into pieces, there is a "quiet zone" (no debris) between the bouncing protons and the new particle. This silence is the key to identifying these rare events.

3. The Target: The Eta and Eta-Prime

The author wants to catch two specific, shy particles: the Eta ($\eta$) and the Eta-Prime ($\eta'$).

• Why them? These particles are like special test dummies. If we can see them being created by the Pomeron "handshake," it will tell us the shape of that handshake.
• The Stakes: If we see these particles, it proves the Pomeron isn't a simple "dot." It helps us understand the fundamental rules of the universe's glue.

4. The Detective Work: How to Catch Them

Catching these particles is like trying to find a specific needle in a haystack, but the needle is made of light and the haystack is full of other needles that look almost identical.

Step A: The Forward Detectors (The Proton Catchers)
Since the protons bounce off at very shallow angles (like skipping stones on a pond), we need special detectors placed far down the track to catch them.

• The Challenge: The beam is incredibly tight. The detectors need to be precise enough to spot a proton that has moved just a tiny bit from its original path. The paper calculates that with the right "optics" (lenses for the beam), we can catch them.

Step B: The Middle Detectors (The Particle ID)
Once the protons are caught, we look at the center. The Eta and Eta-Prime particles are unstable; they instantly fall apart into other things (photons and pions).

• The Eta-Prime ($\eta'$): It breaks into a smaller Eta, two pions, and then the Eta breaks into two photons. It's a "Russian Nesting Doll" of decay.
• The Eta ($\eta$): It breaks into pions and a neutral pion, which then breaks into two photons.

Step C: The Imposter Problem
The tricky part is that other particles (like the $\omega$ or $\phi$ mesons) can break apart into the exact same pieces (photons and pions). It's like two different brands of candy melting into the same puddle of sugar.

• The Solution: The author shows how to tell them apart using math and geometry.
  • The "Back-to-Back" Trick: In a true exclusive event, the pieces fly out in a very specific, balanced pattern. If the pieces are flying in a messy, unbalanced way, it's an imposter.
  • The Mass Check: By measuring the energy of the pieces very precisely, we can calculate the "weight" of the original particle. The real Eta-Prime weighs 958 units; the imposter weighs 1285 units. With the right tools, the LHC can clearly see the difference.

5. The Conclusion: What's Next?

The paper concludes that with the current setup at the LHC (specifically using detectors at the "IP2" location with special beam settings), it is feasible to catch these events.

• The Goal: We need to build or use detectors that can see the forward protons and the faint photons in the center.
• The Payoff: If we succeed, we solve the mystery of the Pomeron's shape. We move from guessing what the "glue" of the universe looks like to actually seeing it.

In a nutshell: This paper is a blueprint for a high-stakes game of "Pin the Tail on the Donkey," where the donkey is a subatomic particle, the tail is a specific decay pattern, and the blindfold is the confusion caused by look-alike particles. If the authors' plan works, we finally get to see the invisible "glue" that holds our universe together.

Technical Summary

1. Problem Statement

The primary objective of this study is to investigate the feasibility of measuring exclusive central diffractive production of pseudoscalar mesons ($\eta$ and $\eta'$) in proton-proton collisions at the Large Hadron Collider (LHC).

• Scientific Context: In the Regge approach, high-energy hadronic interactions are mediated by the exchange of trajectories. While meson trajectories (e.g., $\rho, \omega$) dominate at lower energies, the Pomeron trajectory dominates at LHC energies.
• The Core Challenge: The internal spin structure of the soft Pomeron remains a subject of intense theoretical debate. While it is established that the Pomeron carries vacuum quantum numbers ($Q=0, C=1, I=0$), its spin is disputed:
  • Scalar ($J=0$): The vacuum expectation value.
  • Vector ($J=1$): Successfully used to describe elastic scattering and high-$p_T$ jet production but implies a relative minus sign between $pp$ and $p\bar{p}$ amplitudes.
  • Tensor ($J=2$): Proposed as an effective rank-2 symmetric tensor, supported by STAR Collaboration data and analyses of central production.
• Specific Goal: The observation of exclusive $\eta$ and $\eta'$ production via Pomeron-Pomeron fusion serves as a critical test. Theoretical models suggest that the observation of this specific reaction channel would exclude the scalar Pomeron hypothesis, thereby helping to determine the Pomeron's spin structure.

2. Methodology

The study employs a phenomenological approach combining Regge theory, kinematic phase-space analysis, and detector simulation to assess experimental feasibility.

A. Event Topology and Kinematics

• Process: The reaction is modeled as a $2 \to 3$ process: $pp \to p_A p_B \eta^{(\prime)}$, where two protons scatter forward (remaining intact or excited) and a central meson is produced.
• Phase Space: The event is characterized by five independent parameters:
  1. Momentum transfers to the protons ($t_A, t_B$).
  2. Azimuthal angles of the protons ($\phi_A, \phi_B$).
  3. Rapidity of the produced meson.
• Input Data: Differential cross-sections ($d\sigma/dt$ and $d\sigma/d\phi_{pp}$) are derived from Ref. [13], utilizing parameter sets for $\eta$ and $\eta'$ production.

B. Detector Configuration and Acceptance

• Forward Proton Detection:
  • Simulations assume LHC optics with $\beta^* = 30$ m at Interaction Point 2 (IP2).
  • Forward detectors are positioned at 80 m and 112 m from the interaction point.
  • Selection Criteria: Protons must be displaced by $>14\sigma_{x,y}$ from the beam center to be detected. Position resolution is assumed to be 100 $\mu$m.
• Central Detector (Mid-rapidity):
  • Requires detection of charged pions and photons with energies down to 100 MeV.
  • Resolution: Relative momentum resolution $\Delta P/P = 3\%$ for pions; relative energy resolution $\Delta E/E = 3\%$ for photons.
  • Rapidity Coverage: Charged particles and photons are restricted to $-1.6 < y < 1.6$.

C. Decay Channels and Background Suppression

The study focuses on specific decay chains to reconstruct the mesons while distinguishing them from background resonances.

1. $\eta'$ Measurement:

   • Channel: $\eta' \to \eta \pi^+ \pi^- \to (\gamma\gamma) \pi^+ \pi^-$.
   • Final State: Two protons, two pions, two photons ($p p \gamma \gamma \pi^+ \pi^-$).
   • Background: The $f_1(1285)$ resonance decays to the same final state ($f_1 \to \eta \pi^+ \pi^-$).
   • Discrimination: Utilizes invariant mass reconstruction of the $\gamma\gamma$ system (identifying the $\eta$) and the full system ($\gamma\gamma\pi^+\pi^-$). The study demonstrates that with the specified resolution, the $\eta'(958)$ and $f_1(1285)$ peaks are clearly separable.

2. $\eta$ Measurement:

   • Channel: $\eta \to \pi^+ \pi^- \pi^0 \to \pi^+ \pi^- (\gamma\gamma)$.
   • Final State: Two protons, two pions, two photons ($p p \gamma \gamma \pi^+ \pi^-$).
   • Backgrounds: $\omega(782)$ and $\phi(1020)$ resonances also decay to $\pi^+ \pi^- \pi^0$.
   • Discrimination: Similar to the $\eta'$ case, the invariant mass of the $\pi^+\pi^-\gamma\gamma$ system allows for the clear separation of the $\eta(548)$ from $\omega$ and $\phi$ resonances.

3. Key Results

• Kinematic Distributions: The study generated distributions for momentum transfer ($t$) and the azimuthal opening angle between protons ($\phi_{pp}$), confirming that these variables are sensitive to the underlying Pomeron spin structure.
• Background Rejection:
  • Transverse Momentum Conservation: The correlation between the transverse momentum of the forward proton system ($p_A, p_B$) and the central system ($\pi^+\pi^-\gamma\gamma$) was used to identify exclusive events.
  • Azimuthal Correlation: Background events where the central system and proton system are back-to-back ($\Delta\phi \approx \pi$) were identified and distinguishable.
  • Mass Resolution: Simulations (Figures 5 and 6) show that with the assumed experimental resolutions (3% for $P$ and $E$), the signal resonances ($\eta, \eta'$) can be clearly distinguished from background resonances ($f_1, \omega, \phi$) in the invariant mass spectra.
• Feasibility: The analysis confirms that the proposed detector setup (forward protons + mid-rapidity tracking/calorimetry) is sufficient to isolate the exclusive $\eta$ and $\eta'$ signals.

4. Significance and Contributions

• Probing Pomeron Spin: This work provides a concrete experimental pathway to resolve the long-standing debate regarding the Pomeron's spin. By demonstrating that exclusive $\eta/\eta'$ production is measurable and that a scalar Pomeron would be excluded by such observations, the paper offers a critical test for QCD in the non-perturbative regime.
• Experimental Strategy: It outlines a specific, viable strategy for the LHC, detailing the necessary optics ($\beta^*=30$m), detector placement (80m/112m), and kinematic cuts required to separate signal from significant hadronic backgrounds.
• Validation of Tensor Pomeron: The results support the tensor Pomeron model (which agrees with STAR data) over the scalar model, as the observation of these pseudoscalar mesons in central exclusive production is theoretically forbidden for a scalar Pomeron.

5. Conclusion and Outlook

The paper concludes that measuring exclusive diffractive $\eta$ and $\eta'$ production at the LHC is feasible with current or near-future detector capabilities. The next steps involve calculating the achievable data statistics based on the cross-section ranges provided in Ref. [13] to determine the required integrated luminosity for a statistically significant observation. This measurement represents a crucial step toward understanding the fundamental nature of the Pomeron and the vacuum structure of QCD.