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arxiv: 2508.06334 · v2 · submitted 2025-08-08 · ✦ hep-ph · hep-ex

Production of π⁺π⁻ pairs in diffractive photon-proton and in proton-proton collisions revisited, in particular concerning the Drell-S\"oding contribution

Piotr Lebiedowicz , Otto Nachtmann , Antoni Szczurek This is my paper

Pith reviewed 2026-05-19 00:10 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords Drell-Söding contributiontensor pomerondiffractive photoproductioncentral exclusive productionrho mesonpion pair productiongauge invariance
0
0 comments X p. Extension

The pith

A revised gauge-invariant treatment of the Drell-Söding term with separate subenergies for π⁺p and π⁻p systems produces larger cross sections and stronger skewing of the ρ⁰ shape in diffractive pion-pair production.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper improves calculations of exclusive π⁺π⁻ production in photon-proton and proton-proton collisions by addressing a limitation in the non-resonant Drell-Söding contribution. In the tensor-pomeron model, the π⁺p and π⁻p subsystems naturally carry different subenergies, yet earlier work did not fully incorporate this while preserving gauge invariance. The authors present a prescription that respects those constraints and apply it to real photoproduction, low-Q² electroproduction, and central exclusive production in pp collisions. The changes raise the predicted rates and distort the ρ⁰ resonance peak more noticeably than before. These results supply updated inputs for interpreting collider data on exclusive pion pairs.

Core claim

Within the tensor-pomeron framework, the Drell-Söding amplitude for diffractive π⁺π⁻ production must be constructed with distinct subenergies for the π⁺p and π⁻p channels. The paper supplies an explicit method that maintains electromagnetic gauge invariance under this asymmetry. The resulting photoproduction and electroproduction amplitudes are then folded into the calculation of pp → pp π⁺π⁻ via single-photon emission from one proton. The revised amplitudes yield higher cross sections and a more pronounced skewing of the ρ⁰ spectral function compared with the earlier treatment.

What carries the argument

A gauge-invariance preserving prescription for the Drell-Söding contribution that assigns unequal subenergies to the π⁺p and π⁻p subsystems inside the tensor-pomeron model.

If this is right

  • Differential cross sections for pp → pp π⁺π⁻ increase as functions of two-pion mass, pion transverse momentum, and pseudorapidity.
  • The ρ⁰ resonance shape in the two-pion spectrum exhibits stronger skewing than in previous calculations.
  • The same amplitudes provide an improved description of real photoproduction and low-Q² electroproduction of π⁺π⁻ pairs.
  • The results serve as a basis for modeling coherent pion-pair production in ultra-peripheral pA and AA collisions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The increased skewing may help experiments separate the Drell-Söding background from resonant contributions more cleanly.
  • Predictions can be confronted with existing or forthcoming data from ALICE, ATLAS, CMS, and LHCb on exclusive pion pairs.
  • Similar subenergy handling could be tested in other diffractive channels that involve unequal final-state masses.
  • Extension to higher virtualities would reveal whether the gauge-invariance fix remains adequate beyond the stated Q² range.

Load-bearing premise

The proposed prescription continues to enforce gauge invariance when the π⁺p and π⁻p systems are assigned different subenergies in the Drell-Söding term.

What would settle it

A direct comparison of the predicted two-pion invariant-mass spectrum, especially the degree of ρ⁰ skewing, against measured differential cross sections in pp collisions at LHC energies.

Figures

Figures reproduced from arXiv: 2508.06334 by Antoni Szczurek, Otto Nachtmann, Piotr Lebiedowicz.

Figure 1
Figure 1. Figure 1: FIG. 1. Diagrams for non-resonant production of a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Diagram for resonant [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Production of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Diagrams for the reaction (3.1) in the diffractive re [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The differential distributions for the [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The differential distributions for the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The differential distributions for the [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The differential distributions for the [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The differential distributions for the [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: ) π ±( ˜k) + p(p, s) → π ±( ˜k ′ ) + p(p ′ , s ′ ). (A34) Here ˜k, ˜k ′ are the momenta of the on- and off–shell pions and p, p ′ , s, s ′ are the proton momenta and spin indices, respectively. We have ˜k + p = ˜k ′ + p ′ . (A35) π −( ˜k) π −( ˜k ′ ) p (p) p (p ′ ) u˜ t˜ s˜ π −( ˜k) π −( ˜k ′ ) p (p) p (p ′ ) IP, f2IR, ρIR FIG. 10. The reaction π −p → π −p and its representation as a sum of Regge exchange… view at source ↗
read the original abstract

We discuss the exclusive photoproduction of $\pi^{+}\pi^{-}$ pairs in photon-proton and in proton-proton collisions at high energies. The $\rho^{0}$, $\omega$, $f_{2}(1270)$, and non-resonant (Drell-S\"oding) contributions are considered. The calculation is based on the tensor-pomeron model. In the Drell-S\"oding contribution we have different subenergies for the $\pi^{+}p$ and $\pi^{-}p$ systems. In the method which we propose now we take this into account. Respecting the gauge-invariance constraints is then a nontrivial problem for which, however, we present a solution here. In the present paper we give in this way a substantial improvement of the calculations for real photoproduction of $\pi^{+}\pi^{-}$ from JHEP 01, 151 (2015), and we extend the calculations to low $Q^{2}$ electroproduction, $0 \leqslant Q^{2} \leqslant 0.5$ GeV$^{2}$. The photo- and electroproduction amplitudes are then the basis for the calculation of central exclusive production (CEP) of $\pi^{+}\pi^{-}$ pairs in $pp$ collisions, where at least one proton participates in the CEP via a virtual-photon emission. The revised model leads to enhanced cross sections and gives an increased skewing of the $\rho^{0}$ spectral shape. For the $pp \to pp \pi^{+}\pi^{-}$ reaction, we calculate differential cross sections as function of the two-pion invariant mass, pion transverse momentum and pion pseudorapidity. This research is relevant in the context of ALICE, ATLAS, CMS, and LHCb measurements in $pp$ collisions. Our results can also serve as basis for the description of coherent $\pi^{+}\pi^{-}$ production in ultra-peripheral $p$A and AA collisions at the LHC. The formulas given in our paper can be used for the analysis of photoproduction and small-$Q^{2}$ electroproduction in $ep$ collisions at high energies. Such data exist from the HERA experiments and will be obtained in the future at the electron-ion colliders.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript revisits exclusive photoproduction of π⁺π⁻ pairs in photon-proton and proton-proton collisions within the tensor-pomeron model. It includes resonant contributions from ρ⁰, ω, and f₂(1270) together with the non-resonant Drell-Söding term. The key technical advance is a prescription that accounts for unequal subenergies s(π⁺p) ≠ s(π⁻p) while restoring gauge invariance; the resulting amplitudes are extended to low-Q² electroproduction and then used to compute central exclusive production (CEP) in pp collisions. The revised model yields larger cross sections and stronger skewing of the ρ⁰ spectral shape, with direct relevance to LHC measurements and future EIC data.

Significance. If the gauge-invariance construction is shown to be robust, the work would supply a more consistent treatment of the non-resonant background in diffractive two-pion production. This would improve the separation of resonance signals in high-energy data and provide a unified framework linking real photoproduction, low-Q² electroproduction, and CEP processes relevant to ALICE, ATLAS, CMS, LHCb, and HERA/EIC analyses.

major comments (1)
  1. [Section describing the Drell-Söding contribution and gauge-invariance fix] The central claim of enhanced cross sections and increased ρ⁰ skewing rests on the new treatment of the Drell-Söding amplitude with unequal subenergies. The manuscript states that a solution to the gauge-invariance problem is presented, yet the explicit construction of the modified tensor amplitude, the verification that q^μ M_μ = 0 holds off-shell, and any numerical tests of the longitudinal components are not supplied. Without these details the support for the reported enhancements cannot be assessed.
minor comments (1)
  1. [Abstract and introduction] The abstract and introduction would benefit from a concise quantitative comparison (e.g., percentage increase in cross section or change in skewing parameter) between the present results and those of the 2015 JHEP reference to make the improvement more transparent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the positive assessment of the work's significance for LHC and EIC analyses. Below we respond point by point to the major comment and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Section describing the Drell-Söding contribution and gauge-invariance fix] The central claim of enhanced cross sections and increased ρ⁰ skewing rests on the new treatment of the Drell-Söding amplitude with unequal subenergies. The manuscript states that a solution to the gauge-invariance problem is presented, yet the explicit construction of the modified tensor amplitude, the verification that q^μ M_μ = 0 holds off-shell, and any numerical tests of the longitudinal components are not supplied. Without these details the support for the reported enhancements cannot be assessed.

    Authors: We agree that the current presentation would be strengthened by a more explicit and self-contained description of the gauge-invariance construction. While the manuscript outlines the approach that restores gauge invariance for unequal subenergies s(π⁺p) ≠ s(π⁻p), we acknowledge that the explicit tensor form of the modified amplitude, the direct verification of q^μ M_μ = 0 off-shell, and numerical checks of the longitudinal components are not given in sufficient detail. In the revised manuscript we will add a dedicated subsection (or appendix) that supplies the full expression for the modified Drell-Söding tensor amplitude, demonstrates analytically that the gauge condition holds for off-shell pions, and includes numerical illustrations of the longitudinal-component suppression. These additions will make the origin of the larger cross sections and the increased ρ⁰ skewing fully transparent and directly address the referee's concern. revision: yes

Circularity Check

1 steps flagged

Gauge-invariance fix for Drell-Söding with unequal subenergies is internal extension of authors' tensor-pomeron model

specific steps
  1. self citation load bearing [Abstract]
    "The calculation is based on the tensor-pomeron model. In the Drell-Söding contribution we have different subenergies for the π⁺p and π⁻p systems. In the method which we propose now we take this into account. Respecting the gauge-invariance constraints is then a nontrivial problem for which, however, we present a solution here. ... The revised model leads to enhanced cross sections and gives an increased skewing of the ρ⁰ spectral shape."

    The load-bearing improvement (enhanced rates and spectral distortion) is obtained by the newly proposed method for unequal subenergies. Because the underlying tensor-pomeron framework and its gauge-invariance properties originate in the authors' earlier papers, the 'solution' functions as an internal extension rather than an independent constraint; the predicted enhancement is therefore constructed from the same model family that supplies the starting amplitude.

full rationale

The paper's central results (enhanced cross sections, increased ρ⁰ skewing) rest on a revised Drell-Söding amplitude that handles s(π⁺p) ≠ s(π⁻p) while preserving gauge invariance. This revision is presented as a new solution within the tensor-pomeron framework that the same author group has developed across multiple prior publications. No independent derivation or external benchmark is supplied for the fix itself; the amplitude construction therefore reduces to an internal adjustment of the authors' own model. This qualifies as self-citation load-bearing for the headline claims.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The model depends on the tensor-pomeron framework and a new technical fix for gauge invariance; no independent evidence for the fix is supplied in the abstract.

free parameters (1)
  • tensor-pomeron coupling parameters
    Standard parameters of the tensor-pomeron model that are typically adjusted to data.
axioms (1)
  • domain assumption Tensor-pomeron model describes diffractive vector-meson and continuum production
    Invoked as the calculational basis throughout the abstract.

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discussion (0)

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Reference graph

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