⚛️ phenomenology

The Pomeron loop as a perturbative correction to single Pomeron exchange revisited

This paper revisits the magnitude of Pomeron loop corrections to single Pomeron exchange by numerically evaluating the expression using Korchemsky's triple Pomeron vertex representation, finding that while the loops do not exceed the single Pomeron contribution as previously claimed, they still provide a significant 24%–39% correction at the lowest accessible $x$ values and hard scales near the non-perturbative boundary.

Original authors: Martin Hentschinski, Karina Mendoza-Ramírez, Miguel A. Ocaña-Bribiesca

Published 2026-08-31

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CC BY 4.0

Original authors: Martin Hentschinski, Karina Mendoza-Ramírez, Miguel A. Ocaña-Bribiesca

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, where particles collide at speeds approaching that of light, nature follows rules that are both incredibly precise and surprisingly chaotic. When physicists study these high-energy crashes, they often look at how particles scatter off one another, a process governed by the strong force that binds the building blocks of matter together. At the heart of this interaction lies a concept known as the Pomeron, which acts like a carrier of force, mediating the collision between particles. In the simplest view, this force is carried by a single Pomeron, much like a single messenger delivering a message. However, as the energy of the collision increases, this simple picture breaks down. The messengers can split, multiply, and interact with one another, creating complex loops and networks of force that are difficult to calculate. Understanding these loops is crucial because they determine how likely a collision is to happen and how much energy is released, which is essential for interpreting data from the world's most powerful particle accelerators.

For some time, a debate has simmered among theorists regarding the importance of these complex loops. One camp argued that the loops were negligible, mere tiny ripples in a vast ocean of simpler interactions. A study published in 2013, however, suggested a dramatic shift in this perspective, claiming that these loops were so powerful that they would overwhelm the simple single-messenger interactions even at the energies currently achievable in modern laboratories. This claim, if true, would mean that our current understanding of high-energy collisions was missing a massive piece of the puzzle. The authors of the new study, Martin Hentschinski and his colleagues, set out to revisit this specific claim with a fresh, rigorous eye. They wanted to see if the 2013 conclusion held up under a more careful mathematical microscope, specifically by re-deriving the equations from scratch and checking every step of the logic.

The team focused their investigation on the scattering of two virtual photons, which are fleeting particles that act as probes in high-energy experiments. They constructed a detailed model of how a single Pomeron could split into two, form a loop, and then recombine, a process that adds a layer of complexity to the standard calculation. By carefully tracing the flow of energy and momentum through these loops, and by using a specific mathematical technique to handle the triple interaction point where the particles meet, they were able to calculate the size of this correction. Their work involved a painstaking re-evaluation of the mathematical tools used in previous studies, ensuring that the normalization factors and symmetry rules were applied exactly as the fundamental laws of physics require.

The results of this careful re-examination did not support the dramatic conclusion of the 2013 study. The authors found that the Pomeron loop does not dominate the interaction at the energies currently accessible to the Large Hadron Collider. Instead of taking over the process, the loop provides a significant but manageable correction. In the most extreme scenarios they tested, where the energy scales were pushed to the very edge of what can be described by current theories, the loop contributed an additional effect ranging from 24 percent to 39 percent of the main interaction. This is a substantial amount, certainly large enough to matter for precise measurements, but it is far from the overwhelming dominance predicted by the earlier work. The study suggests that while these loops are important and cannot be ignored, the simpler picture of a single Pomeron exchange remains the primary driver of the collision at current energies.

The researchers also discovered that the size of this correction is highly sensitive to the strength of the strong force, which changes depending on the energy scale of the interaction. When they applied more sophisticated methods to account for how this force changes with energy, the contribution from the loops became even smaller, further reinforcing the idea that the single Pomeron exchange is the dominant effect. However, the study leaves the door open for future exploration. The authors note that at even higher energies, perhaps those found in cosmic rays or planned future colliders, the balance might shift. For now, their work provides a clearer, more reliable map of the subatomic landscape, confirming that while the Pomeron loops are real and significant, they are not the giants they were once thought to be. This refined understanding allows physicists to continue their work with greater confidence, knowing exactly how much weight to give to these complex, looping interactions in their models of the universe.

Technical Summary: The Pomeron Loop as a Perturbative Correction to Single Pomeron Exchange Revisited

Problem Statement
In the high-energy limit of Quantum Chromodynamics (QCD), scattering amplitudes are traditionally described by the Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution equation, which resums leading logarithmic terms $(\alpha_s \ln 1/x)^n$ . This framework effectively describes the exchange of a single Pomeron (an interacting two-reggeized-gluon state). However, the BFKL approach neglects fluctuations in the number of exchanged Pomerons, known as Pomeron loops. While the emergence of these loops is a natural consequence of including the triple-Pomeron vertex (splitting of one Pomeron into two), their quantitative impact remains debated. A 2013 study [49] suggested that complete QCD Pomeron loops provide large corrections, potentially exceeding the single Pomeron contribution at Large Hadron Collider (LHC) energies. Conversely, other arguments suggest these effects are strongly suppressed. This paper aims to re-investigate the derivation and numerical magnitude of the single Pomeron loop correction to single Pomeron exchange, specifically addressing the discrepancies with the 2013 findings.

Methodology
The authors employ a rigorous perturbative approach based on the high-energy effective action of reggeized gluons. The methodology proceeds through the following steps:

Formalism and Notation: The scattering of two color dipoles is formulated using reggeized gluon fields within the effective action framework [27, 56]. The authors establish a precise matching between dipole correlators and physical scattering amplitudes, utilizing the conformal solution to the non-forward BFKL equation by Lipatov [50].
Derivation of the Loop: Unlike previous studies that extracted the triple-Pomeron vertex via multiple discontinuities, the authors perform an independent derivation starting from the JIMWLK-BK evolution equation. By treating the non-linear term as a perturbation, they isolate the BFKL Pomeron and the triple-Pomeron vertex with uncontracted reggeized gluon fields. The Pomeron loop configuration is constructed by contracting solutions from the "plus" and "minus" rapidity sides at central rapidities.
Analytical Calculation: The authors derive the explicit expression for the Pomeron loop amplitude ( $M_2$ ) by integrating over rapidities and transverse coordinates. A key component is the evaluation of the triple-Pomeron vertex projection onto conformal eigenfunctions, utilizing the representation by Korchemsky [51]. The authors focus on the dominant configuration where the conformal spin $n=0$ , as contributions with finite $n$ are suppressed at high energies.
Regularization and Integration: The resulting coordinate integrals are divergent; the authors employ dimensional regularization ( $d = 2 + 2\epsilon$ ) to handle these divergences, showing that the result is proportional to a delta function in coordinate space.
Numerical Evaluation: The final expressions for the single Pomeron exchange ( $A_1$ ) and the Pomeron loop correction ( $A_2$ ) are evaluated numerically. The study considers the scattering of two virtual photons with different virtualities ( $Q_A^2, Q_B^2$ ). The authors utilize a Mellin-Barnes representation for the triple-Pomeron vertex function $\Omega$ , ensuring convergence up to $|\nu| = 8$ . They also explore the impact of resummation effects (kinematic constraints and running coupling) inspired by Next-to-Leading Order (NLO) BFKL studies [68, 69].

Key Contributions

Re-derivation and Correction: The paper provides a complete re-derivation of the single Pomeron loop result, identifying specific discrepancies with the 2013 study [49]. The authors identify a relative suppression factor of $1/32$ in their result compared to [49], attributing this to differences in the normalization of the BFKL Green's function and the treatment of the triple-Pomeron vertex.
Explicit Expressions: The authors provide explicit, self-contained expressions for the triple-Pomeron vertex function $\Omega$ in terms of Meijer G-functions and double Mellin-Barnes integrals, facilitating future numerical evaluations.
Numerical Precision: A detailed numerical analysis of the convergence of the Mellin-Barnes integrals and the stability of the vertex function is presented, establishing a cutoff of $\nu_{max} = 4$ for reliable results.

Results
The numerical evaluation yields the following findings regarding the ratio of the Pomeron loop correction ( $\sigma_2$ ) to the single Pomeron exchange ( $\sigma_1$ ):

Discrepancy with Previous Claims: The authors do not confirm the findings of [49] that the Pomeron loop dominates single Pomeron exchange at LHC energies.
Dependence on Coupling and Scale: The magnitude of the correction is highly sensitive to the strong coupling constant $\alpha_s$ (scaling as $\alpha_s^4$ ) and the hard scales involved.
Standard Perturbative Regime: For typical perturbative scales (e.g., $Q_A^2 = 16 \text{ GeV}^2, Q_B^2 = 4 \text{ GeV}^2$ ) and using a resummed BFKL eigenvalue, the Pomeron loop correction is modest: approximately 8% at $x = 10^{-6}$ and 5% at $x = 10^{-5}$ .
Non-Perturbative Boundary: For lower hard scales near the non-perturbative boundary (e.g., $Q_A^2 = Q_B^2 = 2 \text{ GeV}^2$ , relevant for $J/\psi$ production), the correction is significantly enhanced. In this scenario, the loop provides a 39% correction at $x = 10^{-6}$ and a 24% correction at $x = 10^{-5}$ .
Energy Dependence: While the ratio grows with energy ( $Y = \ln 1/x$ ), it only exceeds unity (dominating the single Pomeron) at extremely high energies ( $Y \approx 15-20$ ), corresponding to center-of-mass energies beyond current collider capabilities (e.g., Future Circular Collider or cosmic rays).

Significance and Claims
The paper concludes that while the Pomeron loop configuration does not dominate single Pomeron exchange at current LHC energies (contrary to [49]), it cannot be entirely discarded for phenomenological studies. The authors assert that for processes probing hard scales near the non-perturbative boundary (such as exclusive vector meson production at the LHC), the Pomeron loop provides a sizeable correction (24%–39%) that must be considered for precision phenomenology.

The work emphasizes that the size of the correction is intrinsically linked to the value of the strong coupling constant. If resummed prescriptions for the running coupling are applied, the loop effects are naturally suppressed. However, if the coupling is determined directly by external hard scales, the corrections remain significant. The authors suggest that future research should extend these calculations to Deep Inelastic Scattering (DIS) and exclusive photoproduction, and investigate the necessity of all-order resummation of Pomeron loops in scenarios where the correction is large.

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