Production of $\pi^{+}\pi^{-}$ pairs in diffractive photon-proton and in proton-proton collisions revisited, in particular concerning the Drell-Söding contribution

This paper presents a revised tensor-pomeron model for the exclusive photoproduction and electroproduction of $\pi^{+}\pi^{-}$ pairs that incorporates a new method to handle Drell-Söding subenergies while satisfying gauge invariance, leading to enhanced cross sections and improved descriptions of central exclusive production in proton-proton collisions relevant to LHC experiments.

The Gist

Imagine the subatomic world as a giant, high-speed dance floor where particles collide, spin, and sometimes stick together to form new pairs. This paper is about a specific dance move: when a photon (a particle of light) or a proton (a building block of atoms) smashes into another proton, it can create a pair of "pions" (lightweight particles) that spin around each other like a couple.

The authors, a team of physicists, are revisiting an old problem with how they calculate the music and steps for this dance, specifically focusing on a tricky part of the choreography called the Drell-Söding contribution.

Here is the breakdown of their work in everyday terms:

1. The Main Character: The "Pomeron"

In the world of high-energy physics, when particles bounce off each other without breaking apart, they exchange invisible messengers. The most famous of these is the Pomeron.

• The Analogy: Think of the Pomeron not as a simple ball thrown back and forth, but as a complex, flexible rubber band (specifically, a "tensor" rubber band, which is a fancy math way of saying it has a specific shape and spin).
• The Old View: In previous calculations, the authors treated this rubber band exchange as if the energy of the dance was the same everywhere.
• The New View: The authors realized that in the specific "Drell-Söding" part of the dance, the energy isn't the same for all steps. One pion might be dancing with more energy than the other. Their new model accounts for these different energy levels, making the rubber band calculation much more accurate.

2. The "Drell-Söding" Puzzle: The Interference

The paper focuses on a phenomenon where two things happen at once:

1. A short-lived "resonance" (like a $\rho^0$ meson) forms and then breaks apart into the pion pair. This is like a dancer spinning so fast they blur into a single shape before separating.
2. A "non-resonant" background happens, where the pions just appear without that specific spinning shape. This is the Drell-Söding effect.

The Problem: When these two things happen together, they interfere with each other, like two sound waves clashing. This causes the "shape" of the resonance to look lopsided or skewed.

• The Old Calculation: The previous math tried to fix this skew, but it was like trying to tune a guitar with a broken tuner. It worked okay, but the skew wasn't strong enough to match what scientists actually see in experiments.
• The New Solution: The authors developed a new method to handle the "gauge invariance" (a strict rule of physics that says the laws must stay consistent no matter how you look at them). They found a way to calculate the interference that respects this rule while correctly handling the different energies of the pions.

3. The Results: A Bigger, Skewer Dance

When they applied this new, more careful math:

• The Cross-Section Jumped: The predicted number of these pion pairs being created increased by a factor of 3.5. That's a huge jump, like realizing a concert hall can hold three and a half times more people than you thought.
• The Skew Improved: The "lopsidedness" of the resonance shape became much more pronounced. This matches the real-world data from the H1 experiment (a past experiment at HERA) much better than the old model did.

4. Why This Matters (According to the Paper)

The authors aren't just doing math for fun; they are providing a better "instruction manual" for experiments happening right now and in the future:

• LHC Experiments: They mention that this improved model is relevant for the ALICE, ATLAS, CMS, and LHCb collaborations at the Large Hadron Collider (LHC). Even if the detectors don't catch the outgoing protons, they can look for "rapidity gaps" (empty spaces in the detector) to find these pion pairs.
• Future Colliders: They say their formulas can be used to analyze data from the HERA experiments (past) and future electron-ion colliders (like the EIC or LHeC).
• Heavy Ion Collisions: They note this helps describe "ultra-peripheral" collisions, where heavy ions (like lead or gold) pass each other so closely that their electromagnetic fields interact, creating these pion pairs without the nuclei actually crashing.

Summary

Think of this paper as a team of choreographers realizing they were using the wrong tempo for a specific part of a complex dance routine. By fixing the tempo (the energy variables) and ensuring the dancers followed the strict rules of the dance hall (gauge invariance), they found that the dance is actually much more energetic and has a more dramatic, lopsided style than previously thought. They are now handing this new, improved choreography to the experimentalists at the world's biggest particle accelerators so they can see if the real dancers match the new script.

Technical Summary

1. Problem Statement

The paper addresses the theoretical description of exclusive $\pi^+\pi^-$ pair production in high-energy photon-proton ($\gamma p \to \pi^+\pi^- p$) and proton-proton ($pp \to pp\pi^+\pi^-$) collisions. Specifically, it focuses on the Drell-Söding contribution, which describes the non-resonant continuum production of pion pairs that interferes with resonant production (primarily the $\rho^0(770)$ meson).

Key Issues Identified:

• Skewing of the $\rho^0$ Line Shape: Experimental data (e.g., from HERA and LHC) show an asymmetric skewing of the $\rho^0$ resonance peak compared to the symmetric shape seen in $e^+e^-$ annihilation. This is attributed to the interference between the resonant $\rho^0 \to \pi^+\pi^-$ decay and the non-resonant Drell-Söding continuum.
• Gauge Invariance Violation in Previous Models: Previous calculations using the tensor-pomeron model (e.g., in JHEP 01, 151 (2015) and Phys. Rev. D 91, 074023 (2015)) treated the energy variables in the Regge propagators of the Drell-Söding diagrams as identical ($s$). However, the diagrams involve different sub-energies ($s_1$ and $s_2$) for the $\pi^+p$ and $\pi^-p$ systems. Using a common energy variable violated gauge invariance constraints unless arbitrary ad-hoc corrections (like $s_{eff} = s/2$) were applied.
• Need for Precision: Accurate modeling of this interference is crucial for interpreting data from LHC experiments (ALICE, ATLAS, CMS, LHCb) and future Electron-Ion Colliders (EIC, LHeC), particularly for Central Exclusive Production (CEP) and ultra-peripheral collisions.

2. Methodology

The authors employ the Tensor-Pomeron Model, a framework where the pomeron ($\mathbb{P}$) and $C=+1$ reggeons are treated as effective rank-2 symmetric tensor exchanges, while the odderon ($\mathbb{O}$) and $C=-1$ reggeons are vector exchanges. This approach ensures consistency with Quantum Field Theory (QFT) principles, including crossing symmetry and charge conjugation.

Key Methodological Improvements:

• Correct Regge Kinematics: Instead of using a single center-of-mass energy squared ($s$) for all diagrams, the authors use the correct Regge variables ($2\nu_1$ and $2\nu_2$) corresponding to the specific sub-energies of the $\pi^+p$ and $\pi^-p$ subsystems in the Drell-Söding diagrams.
• Gauge-Invariant Solution: The authors derive the amplitude for the "contact" diagram (diagram (c) in their notation) strictly from the Ward identity (gauge invariance condition $q_\mu \mathcal{M}^\mu = 0$). This provides a theoretically rigorous solution that naturally incorporates the different sub-energies without ad-hoc parameter adjustments.
• Virtual Photon Extension: The formalism is extended to include slightly virtual photons ($0 \le Q^2 \le 0.5 \text{ GeV}^2$), making it applicable to electroproduction and low-$Q^2$ deep inelastic scattering regimes.
• Comprehensive Amplitude Construction: The total amplitude includes:
  • Resonant terms: $\rho^0$, $\omega$, and $f_2(1270)$ production via pomeron, reggeon, and odderon exchanges.
  • Non-resonant (Drell-Söding) terms: Calculated with the new gauge-invariant formalism for pomeron, $f_2$-reggeon, $\rho$-reggeon, and photon exchanges.
  • $\gamma\gamma$ Fusion: Included for the $pp \to pp\pi^+\pi^-$ reaction.

3. Key Contributions

1. Rigorous Drell-Söding Calculation: The paper presents the first calculation of the Drell-Söding term within the tensor-pomeron model that strictly respects gauge invariance while accounting for the distinct kinematics of the intermediate $\pi p$ states.
2. Resolution of the "Skewing" Discrepancy: The new method naturally produces a larger skewing of the $\rho^0$ spectral shape, bringing theoretical predictions into better agreement with experimental observations (specifically H1 data) without relying on arbitrary energy scaling factors.
3. Quantitative Impact on Cross Sections: The revised calculation leads to a significant enhancement of the non-resonant continuum cross section.
4. Extension to Electroproduction: The derivation of amplitudes for virtual photons allows for the analysis of low-$Q^2$ electroproduction data, bridging the gap between photoproduction and DIS.

4. Results

The authors present numerical results for the $pp \to pp\pi^+\pi^-$ reaction at $\sqrt{s} = 13$ TeV and compare them with previous models and experimental data.

• Cross Section Enhancement: The new Drell-Söding contribution is approximately 3.5 times larger than the previous calculation (based on the model in [47, 50]). This increase is attributed to the correct treatment of Regge kinematics (using $2\nu$ instead of $s$).
• Spectral Shape: The interference between the resonant $\rho^0$ and the enhanced non-resonant continuum results in a significantly more pronounced skewing of the $\rho^0$ mass distribution. This matches the asymmetric shapes observed in H1 data much better than the previous model.
• Differential Distributions:
  • Invariant Mass ($M_{\pi\pi}$): The distribution shows a clear $\rho^0$ peak with the correct asymmetry and a visible $\rho$-$\omega$ interference pattern.
  • Transverse Momentum ($p_T$) and Pseudorapidity ($\eta$): The distributions are peaked at low $|t|$ (momentum transfer) due to the photon propagator ($1/t$), indicating that the photoproduction mechanism dominates at very small $|t|$.
  • Total Cross Section: For $|\eta_\pi| < 2$, the predicted total cross section is $\sigma_{tot} \approx 3.09 \, \mu\text{b}$ (neglecting absorption corrections). The pure continuum (Drell-Söding) contribution is $\sigma_{DS} \approx 2.51 \, \mu\text{b}$.
• Comparison with H1 Data: The new model reproduces the $M_{\pi\pi}$ distribution measured by the H1 collaboration at HERA significantly better than the original model, validating the theoretical approach.
• Uncertainty Analysis: The model uncertainties regarding the Drell-Söding term are estimated to be less than 15%, which is negligible compared to the factor of 3.5 enhancement over the previous model.

5. Significance

• LHC Physics: The results are critical for interpreting Central Exclusive Production (CEP) data at the LHC. Many experiments (CMS, ATLAS, ALICE, LHCb) study $\pi^+\pi^-$ production, often with rapidity gap conditions. The enhanced continuum and correct interference pattern are essential for background estimation and signal extraction, particularly for searches involving the odderon or new physics.
• Odderon Detection: The model provides a robust baseline for detecting the odderon via $f_2(1270)$ production in $\gamma\mathbb{O}$ fusion, as the non-resonant background is now more accurately defined.
• Future Colliders: The formalism is directly applicable to future Electron-Ion Colliders (EIC, LHeC) for analyzing photoproduction and small-$Q^2$ electroproduction, offering a QFT-consistent tool for data analysis.
• Theoretical Validation: The paper demonstrates that the tensor-pomeron model, when applied with strict adherence to QFT constraints (gauge invariance and correct Regge variables), provides a superior description of diffractive processes compared to phenomenological fixes.

In summary, this work resolves long-standing theoretical inconsistencies in modeling the Drell-Söding effect, significantly improves the agreement with experimental data, and provides a refined theoretical tool for high-energy hadronic physics at current and future colliders.