⚛️ phenomenology

Exclusive production of $\pi^+ \pi^-$ pairs in diffractive $\gamma p$ and in $pp$ collisions within the tensor-pomeron approach

This paper employs the tensor-pomeron approach to model the exclusive production of $\pi^+ \pi^-$ pairs in diffractive $\gamma p$ and $pp$ collisions, successfully describing H1 data for the $\gamma p$ reaction while predicting differential cross sections for $pp$ collisions at 13 TeV to guide future LHC and electron-ion collider measurements.

Original authors: Piotr Lebiedowicz, Otto Nachtmann, Antoni Szczurek

Published 2026-10-01

📖 4 min read🧠 Deep dive

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

High-energy physics often feels like trying to understand a complex machine by smashing its parts together and watching what flies out. In the world of particle accelerators, scientists collide protons or fire beams of light at protons to see how nature behaves under extreme conditions. One specific area of interest involves "diffractive" collisions, a process where particles graze past each other without shattering completely, instead exchanging energy to create new pairs of particles while the originals survive the encounter. A key player in these interactions is the pomeron, a theoretical concept that acts like a carrier of the strong nuclear force at high speeds, allowing particles to interact without exchanging electric charge. Another important idea is the "resonance," which is a short-lived particle that forms briefly before decaying into lighter pieces, much like a bubble that pops almost instantly. Understanding how these resonances form and how they mix with other, non-resonant processes is crucial for mapping out the fundamental forces that hold matter together.

A team of researchers from Poland and Germany has refined the mathematical framework used to describe how pairs of positively and negatively charged pions are created when a photon hits a proton or when two protons collide. Their work focuses on a specific energy range where a well-known particle called the rho meson usually appears. In simpler experiments, such as when electrons and positrons annihilate, this rho meson produces a clean, symmetric peak in the data. However, in the messy environment of proton collisions, this peak looks skewed or tilted. The researchers set out to explain why this distortion happens and to create a more accurate model that accounts for the subtle interference between the formation of the rho meson and the direct creation of pion pairs, a process known as the Drell-Söding mechanism.

The team employed a theoretical approach called the tensor-pomeron model, which treats the force-carrying particles involved in these collisions as specific types of mathematical objects to ensure the calculations respect the fundamental laws of physics, particularly the requirement that electric charge and energy are conserved. They improved upon previous calculations by paying closer attention to the behavior of pions that exist only for a fleeting moment inside the collision, known as off-shell effects. By adjusting how these temporary particles interact with the force carriers, the researchers were able to reproduce the distorted shape of the rho meson peak seen in real-world data collected by the H1 experiment at the HERA accelerator. Their model successfully described the data for pion pairs with a combined mass up to 1.2 GeV, capturing the subtle tilt that earlier theories missed.

Beyond explaining past data, the researchers used their refined model to predict what will happen in future collisions at the Large Hadron Collider, where protons collide at an energy of 13 TeV. They calculated the likelihood of producing these pion pairs in a specific way where the protons remain intact, a scenario known as central exclusive production. Their simulations suggest that at these high energies, the production of the rho meson will dominate the scene, while the creation of a heavier particle called the f2(1270) will be more than a thousand times less frequent. They also found that the interference between the resonant production and the direct creation of pions will create a distinct, skewed pattern in the data, similar to what was seen in the lower-energy experiments. This prediction is vital for experiments at the LHC, such as those conducted by the ATLAS, CMS, and ALICE detectors, which are looking for these rare events using specific criteria to filter out background noise.

The implications of this work extend beyond just proton-proton collisions. The same theoretical tools can be applied to understand how pion pairs are created when a proton passes very close to a heavy atomic nucleus without hitting it, a process called an ultraperipheral collision. Furthermore, as new electron-proton colliders are planned for the future, this model provides a reliable foundation for interpreting how light interacts with matter to produce these particle pairs. By grounding their calculations in the rigorous rules of quantum field theory and incorporating the subtle details of how particles behave when they are not in their stable state, the researchers have provided a robust tool for experimentalists. This ensures that when new data arrives from the world's most powerful machines, scientists will have a clear and accurate map to interpret the complex dance of particles being created and destroyed.

Technical Summary: Exclusive Production of $\pi^+\pi^-$ Pairs in Diffractive $\gamma p$ and $pp$ Collisions within the Tensor-Pomeron Approach

Problem Statement
This work addresses the exclusive production of $\pi^+\pi^-$ pairs in high-energy diffractive photon-proton ( $\gamma p \to \pi^+\pi^- p$ ) and proton-proton ( $pp \to pp\pi^+\pi^-$ ) collisions. A central challenge in this domain is the accurate description of the $\rho^0(770)$ resonance line shape. Unlike the symmetric shape observed in $e^+e^-$ annihilation, hadronic reactions exhibit a distinct skewing of the $\rho^0$ peak. This asymmetry arises from the interference between the resonant decay $\rho^0 \to \pi^+\pi^-$ and the non-resonant production of pion pairs, a phenomenon known as the Drell-Söding (DS) mechanism. Theoretical calculation of the DS term is complicated by the strict requirement of gauge invariance. Previous applications of the tensor-pomeron model to this problem achieved some skewing but failed to fully reproduce experimental data, particularly regarding the magnitude of the effect. Furthermore, the treatment of pion off-shell effects in the interaction vertices had not been fully integrated into the DS amplitude calculations in earlier iterations.

Methodology
The authors employ the tensor-pomeron approach, where the pomeron ( $\mathbb{P}$ ) and $C=+1$ reggeons ( $f_{2R}, a_{2R}$ ) are modeled as effective rank-two symmetric tensor exchanges, while the odderon ( $\mathbb{O}$ ) and $C=-1$ reggeons ( $\rho_R, \omega_R$ ) are treated as effective vector exchanges.

The formalism is developed for two primary reactions:

$\gamma^{(*)}p \to \pi^+\pi^- p$ : The total scattering amplitude is decomposed into three components:
- The non-resonant Drell-Söding term ( $M^{(DS)}$ ), involving exchanges of the pomeron, $f_{2R}$ , $\rho_R$ , and photon.
- Resonant vector-meson production ( $M^{(res)}$ ), driven by $C$ -even exchanges ( $\mathbb{P}, f_{2R}, a_{2R}$ ).
- Tensor $f_2(1270)$ meson production ( $M^{(f_2)}$ ), mediated by $C$ -odd exchanges.
A critical methodological advancement presented here is the explicit inclusion of pion off-shell effects within the DS term. The authors introduce off-shell pion form factors, $F_\pi(t_{\pi,i}) = \exp((t_{\pi,i} - m_\pi^2)/\Lambda_\pi^2)$ , at the $P\pi\pi$ , $f_{2R}\pi\pi$ , and $\rho_R\pi\pi$ vertices. To maintain gauge invariance ( $q_\mu M^\mu = 0$ ), the amplitudes for the "crossed" diagrams are updated to satisfy the generalized Ward identity, ensuring the total amplitude remains consistent with Quantum Field Theory (QFT) principles.
$pp \to pp\pi^+\pi^-$ : The authors extend the formalism to central exclusive production (CEP) in proton-proton collisions. The amplitude is constructed as a sum of contributions where at least one proton emits a virtual photon:
- $\gamma h$ and $h \gamma$ terms: One proton emits a photon, the other interacts hadronically (via pomeron/reggeon exchange).
- $\gamma \gamma$ fusion: Both protons emit photons.
  The calculation utilizes high-energy approximations for the photon-proton vertices and includes $\rho-\omega$ interference effects via propagator mixing.

Key Contributions

Improved Drell-Söding Amplitude: The paper presents a QFT-consistent calculation of the DS term that incorporates pion off-shell effects and updated vertex functions. This corrects previous prescriptions where a common energy variable was assumed in Regge factors, which yielded insufficient skewing.
Unified Framework: The work provides a unified description of both real and virtual photon photoproduction ( $\gamma^{(*)}p$ ) and central exclusive production in $pp$ collisions within the same tensor-pomeron framework.
Gauge Invariance: The derivation explicitly ensures gauge invariance for the non-resonant terms by updating the amplitude structure to satisfy the Ward identity, a necessary condition for physical consistency.

Results

$\gamma p \to \pi^+\pi^- p$ Comparison: The improved model, including off-shell effects, provides a satisfactory description of H1 experimental data for invariant masses $M_{\pi\pi} \lesssim 1.2$ GeV. The model successfully reproduces the observed skewing of the $\rho^0$ line shape. Discrepancies at higher masses ( $M_{\pi\pi} > 1.2$ GeV) are attributed to higher-mass resonances ( $\rho(1450), \rho(1700)$ ) not included in the current analysis.
$pp \to pp\pi^+\pi^-$ Predictions: Differential cross-section predictions for $\sqrt{s} = 13$ $s = 13$ TeV (LHC energy) are presented.
- The production of the $f_2(1270)$ resonance is found to be more than three orders of magnitude smaller than the $\rho(770)$ production in the specified kinematic regime ( $|\eta_\pi| < 2$ ).
- The $\gamma\gamma \to \pi^+\pi^-$ fusion contribution is negligible in this regime.
- The interference between the resonant ( $\rho, \omega$ ) and non-resonant (DS) contributions is clearly visible, reproducing the characteristic skewing and the $\rho-\omega$ interference pattern.

Significance and Claims
The authors claim that their improved treatment of the Drell-Söding mechanism is essential for describing experimental data, particularly the spectral shape of the $\rho(770)$ . The results are positioned as directly relevant for:

LHC Experiments: Interpreting central exclusive production measurements by ALICE, ATLAS, CMS, and LHCb, especially under rapidity-gap selection criteria.
Ultraperipheral Collisions: Applying the formalism to $\pi^+\pi^-$ production in ultraperipheral $pA$ and $AA$ collisions at the LHC.
Future Colliders: Providing a robust theoretical baseline for future electron-proton/ion colliders (EIC, LHeC) to study photo- and electroproduction of pion pairs.

The paper emphasizes that the framework is firmly grounded in QFT, offering a reliable tool for constraining model parameters (primarily form factor shapes) against current HERA data and future measurements.

More like this