⚛️ phenomenology

Factorization of elastic, single, and double diffractive $pp$ scattering

Using effective field theory, this paper factorizes elastic, single, and double diffractive $pp$ scattering in the Regge limit to demonstrate that while rapidity anomalous dimensions and Pomeron evolution are universal, the diffractive parton distribution functions differ between $ep$ and $pp$ processes, thereby establishing their non-universality in hadronic collisions.

Original authors: Philipp B. Aretz, Kyle Lee, Stella T. Schindler, Iain W. Stewart

Published 2026-07-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Philipp B. Aretz, Kyle Lee, Stella T. Schindler, Iain W. Stewart

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

Imagine two proton racers zooming toward each other at nearly the speed of light. Usually, when they crash, they shatter into a chaotic explosion of particles. But sometimes, something magical happens: they bounce off each other, staying mostly intact, leaving behind a giant, empty "quiet zone" in the middle of the track where no particles are detected. This is called diffraction. It's a bit like two cars swerving to avoid a collision, but instead of crashing, they just graze each other and keep driving, leaving a massive gap of silence between them.

This phenomenon is huge—it makes up nearly half of all the collisions happening at giant particle smashers like the LHC. For a long time, physicists tried to predict exactly how these "grazing" collisions work using a set of rules they had developed for a different type of experiment (electron-proton collisions). They thought the "blueprint" for how protons are built (called diffractive PDFs) would be the same for both electron crashes and proton crashes.

The Big Discovery: The Blueprint is Different
In this new paper, the authors used a powerful mathematical toolkit called SCET (Soft Collinear Effective Theory) to build a brand-new, first-principles map of what happens during these proton grazes. Their main finding is a bit of a buzzkill for the old way of thinking: The blueprints are not universal.

Think of it like this: If you have a recipe for a cake that works perfectly when you bake it in an electric oven (electron-proton collisions), you can't just assume that same recipe will work if you try to bake it in a wood-fired oven (proton-proton collisions). The authors proved that the "soft" ingredients—the invisible, fuzzy clouds of energy that surround the protons—behave differently depending on whether you are smashing electrons into protons or protons into protons.

Specifically, they showed that for the messy, low-energy collisions where the protons wiggle a lot (where the momentum transfer $|t|$ is around the scale of $\Lambda^2_{QCD}$ ), the functions describing these particles are non-universal. You cannot simply take the data from electron experiments and plug it into proton equations. The old models that tried to do this were missing the mark because they assumed a universality that simply doesn't exist.

The "Ghost" in the Gap
One of the most playful and surprising parts of their work involves what happens in that "quiet zone" or rapidity gap. In the simplest version of diffraction, the gap is truly empty. But the authors found that in "double diffraction" (where both protons break apart into jets), there can be a sneaky intruder.

Imagine a party where everyone is supposed to stay in their own room, leaving the hallway empty. In some cases, a "ghost" particle (ultrasoft radiation) can sneak through the hallway without being seen by the detectors, provided it's wearing a "color-nonsinglet" disguise. This means the gap isn't always truly empty; it can be filled with invisible soft radiation that we can't detect. This "quasi-diffraction" is a background noise that previous models struggled to account for properly.

What They Can and Can't Do Yet
The authors are very careful not to overpromise. They have built the mathematical skeleton of the problem. They have derived the exact formulas (factorization) that describe how these collisions break down into smaller, manageable pieces.

However, they explicitly state that they cannot yet make precise predictions for the total number of collisions or the exact shape of the data curves. Why? Because the formulas involve infinite matrices of "Glauber exchanges" (a fancy way of describing how gluons, the glue of the proton, interact). It's like having a recipe that lists "add an infinite number of spices" without telling you exactly how much of each.

They suggest that if certain simplifications exist—like if the "spices" follow a simple pattern or if the number of interactions can be limited—then we might eventually be able to predict things like the "dip-bump" region (a specific wiggly pattern in the data near $|t| \sim 0.5 \text{ GeV}^2$ ) or the interference between nuclear and Coulomb forces near $|t| \sim 5 \times 10^{-4} \text{ GeV}^2$ . But right now, those predictions are just possibilities, not facts.

The "Odderon" Mystery
The paper also tackles a mystery called the "Odderon." This is a theoretical particle made of three gluons that acts differently on matter (protons) versus antimatter (antiprotons). The authors' new framework provides a rigorous way to look at the difference between proton-proton and proton-antiproton collisions. They show that the difference comes down to how "C-parity" (a kind of symmetry charge) is mixed in the collision. While they don't solve the Odderon mystery, they provide the correct map to find it.

The Bottom Line
This paper doesn't give us the final answer key for proton collisions. Instead, it tears down the old, incorrect assumption that electron and proton diffraction are the same game. It replaces a broken map with a new, complex, and highly detailed one. It tells us that the "soft" parts of the proton are unique to the collision type, and that invisible ghosts can sneak into the gaps. While we still can't perfectly predict the outcome of every crash, we now know exactly why our old guesses were wrong and have the right tools to eventually get it right.

Technical Summary: Factorization of Elastic, Single, and Double Diffractive $pp$ Scattering

Problem Statement
Diffractive proton-proton ($pp$) scattering, encompassing elastic, single diffractive (SD), and double diffractive (DD) processes, constitutes approximately half of the total $pp$ cross section at Tevatron and LHC energies. Despite its prevalence, a rigorous QCD factorization for these processes in the Regge limit ( $|t| \ll s$ ) has remained elusive. Existing phenomenological approaches, such as the Ingelman-Schlein model utilizing diffractive parton distribution functions (dPDFs) extracted from electron-proton ($ep$) diffraction, fail to describe $pp$ data. This discrepancy is often attributed to "factorization breaking," modeled ad hoc via gap survival probabilities. Furthermore, the universality of hadronic functions between $ep$ and $pp$ diffraction is unproven, and the theoretical framework for describing the rapidity gap and non-perturbative dynamics in $pp$ collisions lacks a first-principles derivation.

Methodology
The authors employ Soft Collinear Effective Theory (SCET) extended with Glauber operators to derive factorization formulas for forward $pp$ scattering from first principles. The analysis focuses on the kinematic regime where the squared momentum transfer $t$ is small compared to the center-of-mass energy squared $s$ , and a large rapidity gap exists between the forward hadronic states.

Key methodological steps include:

Kinematic Power Counting: Defining power counting parameters $\lambda = \sqrt{-t}/\sqrt{s}$ and $\rho_i = M_i/\sqrt{-t}$ (where $M_i$ are the invariant masses of the diffractive systems). The analysis establishes that diffraction requires $\lambda \ll 1$ , while allowing for various regimes of $\rho_i$ .
Mode Decomposition: The scattering amplitude is decomposed into collinear ( $n$ and $\bar{n}$ ), soft ( $s$ ), and ultrasoft ($us$) modes. Crucially, Glauber operators mediate interactions between these sectors.
Operator Construction: The authors utilize two- and three-sector Glauber operators ( $O_{ns}$ , $O_{ns\bar{n}}$ ) involving collinear and soft Wilson lines. They apply a BPS field redefinition to decouple ultrasoft radiation from collinear sectors, allowing for a systematic treatment of radiation penetrating the rapidity gap.
Factorization Derivation: By expanding the cross section in terms of Glauber exchanges and grouping operators with their corresponding matrix elements, the authors derive a general factorization formula for double diffraction. This formula expresses the differential cross section as a convolution of beam functions ( $B$ ), soft functions ( $S$ ), and an ultrasoft function ( $U$ ), connected by transverse momentum convolutions and color projectors.

Key Contributions and Results

General Factorization Formula: The paper derives a unified factorization formula (Eq. 19) for double diffraction ( $pp \to X_1 X_2$ ) that simultaneously describes elastic, single, and double diffraction as limiting cases. The formula involves an infinite sum over the number of Glauber exchanges ( $N, M$ ) and color representations, structured as:
$\frac{d\sigma}{dM_1^2 dM_2^2 dt} \sim \sum_{N,M} \int \dots B \otimes B \otimes S \otimes S \otimes U$
where $B$ represents beam functions, $S$ soft functions, and $U$ the ultrasoft function accounting for radiation in the gap.
Non-Universality of Hadronic Functions: The authors demonstrate that the non-perturbative hadronic functions in $ep$ and $pp$ diffraction are non-universal when $|t| \sim \Lambda_{QCD}^2$ .
- In $ep$ diffraction, the soft function involves an electromagnetic current and specific Glauber operators.
- In $pp$ diffraction, the absence of an electromagnetic current and the presence of a larger set of Glauber operators ( $O_{j\bar{n}}^s, O_{jn}^s, O_s$ ) result in a distinct soft function structure.
- Consequently, dPDFs derived from $ep$ data cannot be directly applied to $pp$ scattering, explaining the failure of the Ingelman-Schlein model.
Universality of Rapidity Evolution: While the hadronic functions themselves are non-universal, the authors prove that the rapidity anomalous dimensions governing the Renormalization Group (RG) evolution are universal between $ep$ and $pp$ processes (specifically in the color-singlet channel). This allows the determination of color-singlet (Pomeron) evolution equations at the amplitude level.
Treatment of Quasi-Diffraction: The framework explicitly accounts for "quasi-diffraction" in double diffraction, where color-nonsinglet exchanges allow undetected ultrasoft radiation to penetrate the rapidity gap. This is mediated by the ultrasoft function $U$ , which leads to exponential Sudakov suppression in the virtuality scale $\mu$ .
Amplitude-Level Factorization: For elastic diffraction, the factorization simplifies to occur at the amplitude level, reducing the cross section to the square of a sum of amplitudes involving radiative jet functions ( $J$ ) and soft functions.

Significance and Claims
The paper claims to provide the first first-principles QCD factorization for elastic, single, and double $pp$ diffraction in the Regge limit. Its primary significance lies in:

Resolving the Factorization Breaking Puzzle: It explains why dPDFs from $ep$ fail in $pp$ collisions without invoking ad hoc gap survival probabilities, attributing the difference to the distinct structure of the soft function in hadron-hadron versus lepton-hadron scattering.
Rigorous Framework for Regge Dynamics: It establishes a formalism where Regge dynamics (Pomeron evolution) can be derived from SCET operators, moving beyond phenomenological Regge theory fits.
Foundation for Future Predictions: The authors note that immediate phenomenological predictions are currently limited by the complexity of the factorization formula, which involves infinite matrices in color and Glauber number space. However, the derived formulas provide the necessary groundwork for future simplifications, such as identifying universal non-perturbative objects or truncating the Glauber series.
Extension to Other Phenomena: The framework offers a rigorous basis for analyzing the difference between $pp$ and $p\bar{p}$ scattering (Odderon effects) via C-parity properties of the beam functions and provides a path to understanding Donnachie-Landshoff scaling through the universality of soft anomalous dimensions across different hadron pairs.

The authors conclude that while the full predictive power depends on future calculations to simplify the non-perturbative ingredients (specifically the soft and beam functions), this work lays the essential theoretical foundation for a deeper understanding of diffraction at the LHC.

Technical Summary: Factorization of Elastic, Single, and Double Diffractive $pp$ Scattering

More like this