⚛️ phenomenology

Central exclusive production of $\eta$ and $\eta'$ mesons in diffractive proton-proton collisions at the LHC

This paper utilizes the tensor-pomeron approach, constrained by WA102 data, to predict the central exclusive production cross sections and differential distributions of $\eta$ and $\eta'$ mesons in proton-proton collisions at $\sqrt{s} = 13$ TeV, demonstrating the feasibility of using these processes to probe the nature of the pomeron at the LHC.

Original authors: Piotr Lebiedowicz, Otto Nachtmann, Antoni Szczurek

Published 2026-10-01

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CC BY 4.0

Original authors: Piotr Lebiedowicz, Otto Nachtmann, Antoni Szczurek

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, particles do not merely bounce off one another like billiard balls; they interact through invisible forces that can be thought of as exchanges of other, fleeting particles. When two protons collide at extremely high speeds, they can sometimes pass each other so closely that they exchange these force carriers without shattering, leaving the protons intact but creating new particles in the empty space between them. This process is known as central exclusive production. The force carrier responsible for this interaction at high energies is called the pomeron. For decades, physicists have debated the true nature of the pomeron: is it a simple, scalar object, or does it possess a more complex, tensor structure? Understanding this distinction is crucial because the pomeron governs how matter behaves at the highest energy levels found in the universe, yet its internal character remains one of the great mysteries of particle physics.

A team of researchers has now taken a significant step toward solving this mystery by simulating the creation of two specific particles, the eta and eta-prime mesons, during proton collisions. Using a theoretical framework that treats the pomeron as a tensor object, they calculated what should happen when protons collide at the energies achieved by the Large Hadron Collider. Their work builds on previous measurements taken at lower energies, where the team successfully matched their calculations to experimental data. By applying this same model to the much higher energies of the Large Hadron Collider, they generated precise predictions for how often these mesons should appear and how they should be distributed in space. The results demonstrate the feasibility of studying the production of these mesons to probe the nature of the pomeron, suggesting that if the pomeron were a simple scalar object, these particles would not be produced in the way the model predicts.

The researchers focused on a scenario where two protons collide and emerge unchanged, while a new meson is created in the center of the collision zone. To ensure their model was reliable, they first tuned its parameters using data from an earlier experiment conducted at a much lower energy. Once the model could accurately reproduce the results of that past experiment, they projected its behavior forward to the current energy levels of the Large Hadron Collider. They found that the production of eta and eta-prime mesons is highly sensitive to the internal structure of the pomeron. Because the eta and eta-prime have specific quantum properties that a simple scalar pomeron cannot generate, the successful observation of these particles would confirm that the pomeron must have a tensor nature. The team calculated that at the Large Hadron Collider, the production of eta mesons should occur with a cross section, a measure of probability, reaching up to 2.5 microbarns in the central region and up to 5.6 microbarns in the forward regions. For the heavier eta-prime meson, the predicted rates are lower, ranging between 0.3 and 0.7 microbarns in the center and 0.9 to 2.1 microbarns in the forward regions.

A critical part of their analysis involved accounting for the fact that protons are not just single points but complex clouds of particles that can interact with each other even before the main collision occurs. These secondary interactions, known as absorption effects, significantly reduce the number of particles that are actually produced. The researchers included these effects in their calculations, finding that they play a major role in lowering the final numbers. They also examined the influence of other, less dominant force carriers that might contribute to the process. At the lower energies of the past experiment, these other carriers were important, but at the high energies of the Large Hadron Collider, their contribution becomes negligible, leaving the pomeron interaction as the clear dominant force. This shift allows for a much cleaner test of the pomeron's properties than was possible before.

The study concludes that the experimental observation of these mesons at the Large Hadron Collider would provide a definitive test of the pomeron's nature. If the particles are found at the rates predicted by the tensor model, it would confirm that the pomeron is indeed a tensor object and not a scalar one. This would also settle a long-standing debate regarding whether the eta meson, which has a complex internal structure involving gluons, can be produced by the pomeron. Some had argued that because the pomeron is a singlet and the eta is part of a different group, the specific reaction producing the eta meson should be forbidden. However, the researchers show that the breaking of symmetry in the subatomic world allows this interaction to happen. The findings offer a clear roadmap for experimentalists: by measuring the production rates and distributions of these specific mesons, they can finally pin down the true character of the pomeron, a fundamental piece of the puzzle in understanding how the universe holds itself together at its most energetic levels.

Technical Summary: Central Exclusive Production of $\eta$ and $\eta'$ Mesons in Diffractive Proton-Proton Collisions

Problem and Motivation
This work addresses the central exclusive production (CEP) of pseudoscalar mesons ( $J^{PC} = 0^{-+}$ ), specifically the $\eta$ and $\eta'(958)$ , in high-energy proton-proton collisions ( $p + p \to p + M + p$ ). The primary theoretical challenge lies in determining the nature of the pomeron, the exchange object responsible for diffractive processes at high energies. While scalar pomeron models have been proposed, the paper notes that a scalar pomeron is not compatible with STAR measurements regarding the helicity structure of elastic $pp$ scattering [8, 9]. Furthermore, scalar pomeron models cannot naturally produce the required quantum numbers for pseudoscalar mesons via pomeron-pomeron ( $P-P$ ) fusion. Conversely, vector pomeron models violate charge-conjugation ( $C=+1$ ) symmetry and crossing relations. The authors argue that the tensor-pomeron approach is the only consistent framework to describe these diffractive vertices. The study aims to constrain model parameters using existing data from the WA102 experiment ( $\sqrt{s} = 29.1$ GeV) and provide predictions for LHC energies ( $\sqrt{s} = 13$ TeV) to test the pomeron's tensor character.

Methodology
The authors employ the tensor-pomeron formalism to calculate the scattering amplitude for the reaction $pp \to ppM$ . The total amplitude is constructed as the sum of a Born term (via $P-P$ fusion) and a rescattering term (absorption corrections due to $pp$ interactions):
$\mathcal{M}_{pp \to ppM} = \mathcal{M}^{\text{Born}}_{pp \to ppM} + \mathcal{M}^{\text{rescattering}}_{pp \to ppM}$

Key components of the formalism include:

Tensor-Pomeron Exchange: The pomeron is treated as a rank-2 tensor. The effective pomeron-proton vertex and propagator are defined according to the tensor-pomeron approach [6], utilizing a standard pomeron trajectory $\alpha_P(t) = 1.0808 + 0.25 t$ .
$PPM$ Coupling: The central vertex coupling two pomerons to a pseudoscalar meson ( $PPM$ ) includes two distinct tensor structures corresponding to orbital angular momentum and spin configurations $(l, S) = (1, 1)$ and $(3, 3)$ . These are weighted by dimensionless coupling constants $g'_{PPM}$ and $g''_{PPM}$ , which are fitted to experimental data. A form factor $F(t_1, t_2)$ is applied to the central vertex to regulate high-momentum transfers.
Secondary Reggeons: To describe lower-energy data (WA102), the model incorporates subleading exchanges involving the $f_2$ Reggeon ( $Pf_2$ , $f_2P$ , and $f_2f_2$ ). At LHC energies, these contributions are expected to be negligible compared to the dominant $P-P$ fusion.
Absorption Corrections: The model includes $pp$-rescattering effects at the amplitude level, calculated via an integral over the transverse momentum loop, which significantly reduces the predicted cross sections at high energies.

Key Contributions and Results
The paper presents a comprehensive analysis of CEP for $\eta$ and $\eta'$ mesons, constrained by WA102 data and extrapolated to the LHC.

WA102 Data Description ( $\sqrt{s} = 29.1$ GeV):
- The model successfully reproduces the total cross sections measured by WA102: $\sigma_{\eta} \approx 3859$ nb and $\sigma_{\eta'} \approx 1717$ nb.
- Differential distributions, specifically the angle $\phi_{pp}$ between the transverse momenta of the outgoing protons and the momentum transfer $t$ , are well-described. The $\phi_{pp}$ distribution, peaking at $\pi/2$ , requires the inclusion of both $(l, S) = (1, 1)$ and $(3, 3)$ coupling terms.
- Multiple fit scenarios (Sets A–D for $\eta$ ; Sets 1–6 for $\eta'$ ) were explored. These fits vary the relative contributions of $P-P$ , $P-f_2$ , and $f_2-f_2$ exchanges. At WA102 energies, the $P-P$ contribution ranges from 25% to 40%, with significant interference from secondary reggeons, making it difficult to isolate the pure pomeron contribution at this energy.
LHC Predictions ( $\sqrt{s} = 13$ TeV):
- Extrapolating to LHC energies, the subleading $f_2$ -reggeon contributions become negligible, isolating the $P-P$ fusion mechanism.
- Absorption effects play a dominant role, significantly suppressing the cross sections compared to the Born approximation.
- Predicted Cross Sections:
  - For $pp \to pp\eta$ : The upper limit for the total cross section is 2.5 $\mu$ b for pseudorapidity $|\eta_M| < 1$ and 5.6 $\mu$ b for $2 < \eta_M < 5$ .
  - For $pp \to pp\eta'$ : The predicted cross sections are 0.3–0.7 $\mu$ b for $|\eta_M| < 1$ and 0.9–2.1 $\mu$ b for $2 < \eta_M < 5$ .
- The paper notes that different parameter sets, which were indistinguishable at WA102 energies, yield significantly different predictions at LHC energies, with Sets 4–6 (where $P-P$ is less dominant at low energy) predicting cross sections roughly half the size of Sets A–3.

Significance and Claims
The authors assert that the observation of CEP for $\eta$ , $\eta'(958)$ , or $f_1(1285)$ mesons at the LHC would provide a critical test of the pomeron's nature:

Tensor vs. Scalar: Since the fusion of two scalar pomerons cannot produce $J^{PC} = 0^{-+}$ states, the successful description of these processes via the tensor-pomeron model implies that the pomeron must have tensor character. Experimental observation of these CEP events would be strictly incompatible with a scalar pomeron.
Flavor Singlet Production: The paper addresses the theoretical objection that $\eta$ production should be forbidden because the $\eta$ is an $SU(3)_F$ octet while the pomeron is a singlet. The authors argue this is invalidated by significant flavor symmetry breaking and the presence of a non-negligible gluon-gluon component in the $\eta$ wave function, allowing direct coupling to pomerons.
Future Prospects: The study concludes that comparing CEP data from WA102 and the LHC will allow for the disentangling of leading $P-P$ fusion from non-leading $f_2$ -reggeon exchanges. Furthermore, the feasibility of studying these processes at the LHC is established, offering a pathway to probe the nature of the pomeron and diffractive flavor-singlet meson production.

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