⚛️ phenomenology

Low- $x$ parton densities accounting for absorptive effects and exclusive $J/\psi$ and $\Upsilon$ data from the LHC

This paper utilizes combined HERA DIS and LHC exclusive $J/\psi$ and $\Upsilon$ data within the $\texttt{xFitter}$ framework to investigate the impact of absorptive corrections and non-linear evolution on low- $x$ parton distribution functions, ultimately determining the effective size of proton hot spots and discussing implications for future global PDF analyses.

Original authors: V. Bertone, C. A. Flett, A. D. Martin, M. G. Ryskin

Published 2026-08-27

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CC BY 4.0

Original authors: V. Bertone, C. A. Flett, A. D. Martin, M. G. Ryskin

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

To understand the structure of the proton, physicists often look at it not as a solid sphere, but as a seething cloud of smaller particles called quarks and gluons. These components zip around at nearly the speed of light, carrying the energy that holds the proton together. A key challenge in mapping this internal landscape is understanding what happens when we look at the proton from a very specific angle: when we focus on the particles that carry only a tiny fraction of the proton's total momentum. In this low-momentum region, the density of particles becomes so high that they begin to interact with one another in complex ways, crowding each other out. For decades, standard models of particle physics have struggled to describe this crowded environment accurately, often leading to predictions that clash with experimental data. This uncertainty matters because the proton is the building block of visible matter; if our map of its interior is wrong, our understanding of how the universe works at its most fundamental level is incomplete.

A team of researchers has now taken a significant step toward fixing this map by combining two different types of experimental data. They brought together measurements from deep-inelastic scattering, where electrons are fired at protons to probe their inner structure, and data from the Large Hadron Collider, where protons are smashed together to produce heavy particles like the J/ψ and Υ mesons. By merging these datasets, the scientists were able to test a specific idea: that when the proton's interior becomes too crowded, the particles effectively shield one another, a phenomenon known as absorptive corrections. In their analysis, they found that ignoring this shielding effect leads to a distorted picture of the proton's interior, particularly at very low momentum fractions. When they included these corrections in their calculations, the resulting model fit the experimental data much better, resolving inconsistencies that had plagued previous attempts to describe this region.

The researchers used a sophisticated computer framework to simulate how the proton's internal particles evolve as the energy scale changes. They discovered that without accounting for the shielding effect, the model predicted a gluon density that was too low to explain the heavy particle production seen at the collider. To make the model work without the shielding, they would have to artificially tweak the starting conditions of the simulation, introducing a mathematical term that had no clear physical justification. However, when they allowed the particles to screen one another, the model naturally produced the correct results without needing these artificial adjustments. This suggests that the shielding effect is a real physical phenomenon that must be included to get an accurate description of the proton's structure.

One of the most tangible outcomes of this work is a new measurement of the size of "hot spots" within the proton. The team found that the data is best explained if the proton's interior is not a uniform cloud, but rather contains concentrated regions of high particle density. They calculated that these hot spots have a radius of approximately 2.5 to 2.6 inverse GeV, which is significantly smaller than the overall size of the proton itself. This finding provides the first quantitative estimate of these structures based on a combined analysis of deep-inelastic scattering and exclusive heavy meson production. While the effect of particle crowding is present, the researchers noted that the hot spots are still large enough that the extreme saturation of particles is not yet fully visible in the current data, leaving room for future experiments to probe even deeper.

The study also explored how these findings hold up when applying more advanced levels of theoretical calculation. By using a method that approximates higher-order corrections, they confirmed that the inclusion of absorptive effects remains crucial, even as the calculations become more precise. In these advanced scenarios, the need for the artificial mathematical adjustments disappeared entirely, reinforcing the conclusion that the shielding effect is the correct physical explanation for the observed data. The team has made their new calculation tools and the resulting maps of the proton's interior publicly available, allowing other scientists to use these improved models in their own work. This ensures that future studies of high-energy physics will start with a more accurate and reliable foundation, bringing us closer to a complete understanding of the matter that makes up our world.

Technical Summary: Low-x Parton Densities Accounting for Absorptive Effects and Exclusive $J/\psi$ and $\Upsilon$ Data from the LHC

Problem Statement
Parton distribution functions (PDFs) at low longitudinal momentum fractions ( $x$ ) are critical for understanding high-energy hadronic interactions. While inclusive Deep-Inelastic Scattering (DIS) data from HERA provide primary constraints, uncertainties grow rapidly at very small $x$ and low factorization scales ( $\mu_F$ ). Exclusive production of heavy vector mesons ( $J/\psi$ and $\Upsilon$ ) at the LHC offers a complementary probe, extending knowledge of the gluon PDF down to $x \simeq 3 \times 10^{-6}$ at scales $\mu_F \simeq m_q$ . However, at these low scales and small $x$ , linear DGLAP evolution is insufficient; absorptive (screening) effects, which asymptotically lead to parton saturation, become significant. Previous analyses incorporating exclusive LHC data into PDF fits using linear evolution required ad-hoc modifications to the input gluon parametrization (specifically a "negative gluon component") to accommodate the data, potentially masking the underlying physics of non-linear evolution.

Methodology
The authors perform a combined global analysis of inclusive HERA DIS data and exclusive heavy-vector-meson production data from the LHC (specifically LHCb $J/\psi$ and $\Upsilon$ data). The core methodological innovation is the implementation of non-linear DGLAP evolution equations that include absorptive corrections directly into the evolution machinery.

Non-Linear Evolution Framework:
- The standard DGLAP equations are modified by introducing absorptive screening factors, $F_g$ and $F_q$ , which depend on the gluon density and the transverse spatial extent of the parton distribution ( $R$ ).
- These factors are derived from eikonalized approximations of multi-gluon screening, summing contributions from gluon ladders. The screening factor for gluons is given by $F_g = \exp\left(-\frac{C_A 9 \alpha_s \pi}{16 R^2 \mu_F^2} x'g(x', \mu_F^2)\right)$ .
- The coupled evolution equations for gluons and singlet quarks are solved directly within an extended version of APFEL++, which is interfaced with xFitter. Unlike linear evolution, where pre-computed operators can be used, the non-linear terms require solving the system step-by-step in the evolution variable $t = \ln \mu_F^2$ because the screening factors depend on the running scale.
Data and Parametrization:
- The fit utilizes HERA Run I+II neutral and charged-current data and effective gluon PDF points extracted from exclusive $J/\psi$ and $\Upsilon$ production at 13 TeV (LHCb).
- The input PDFs at the scale $Q_0 = 1.54$ GeV are parametrized using a standard functional form ( $x^B(1-x)^C$ ) without the "negative gluon component" term ( $A'_g$ ) previously used in HERAPDF2.0 to mimic low- $x$ behavior.
- The parameter $R$ , governing the strength of absorptive corrections, is treated as a free parameter determined by minimizing $\chi^2$ .
NNLO Approach:*
- Since full NNLO coefficient functions for exclusive vector meson production are not yet available, the authors employ the NNLO* method.
- This involves using NNLO accuracy for inclusive DIS data while approximating the exclusive process via a constant NNLO/NLO $K$ -factor ( $K \approx 1.2 \pm 0.1$ ). This $K$ -factor is determined by comparing NNLO-based calculations with NLO coefficient functions against HERA exclusive photoproduction data. The effective gluon points are rescaled by this factor, with the uncertainty treated as a correlated systematic error.

Key Results

Impact of Absorptive Corrections: Including absorptive corrections leads to a larger input gluon distribution at low $x$ ( $10^{-5} < x < 0.1$ ) compared to linear evolution fits. This is necessary because the non-linear evolution suppresses the growth of the gluon density; thus, a larger initial density is required to reproduce the same data.
Fit Quality and Stability:
- NLO: The non-linear fit combined with exclusive data yields a global $\chi^2/\text{d.o.f.} \approx 1.18$ , significantly better than the linear fit ( $\approx 1.25$ ). The linear fit struggles to describe both DIS and exclusive data simultaneously without artificially inflating the low- $x$ power ( $\lambda$ ) or relying on the negative gluon component.
- NNLO:* Similar improvements are observed. The non-linear fit achieves $\chi^2 = 1416$ (1166 d.o.f.), compared to $\chi^2 = 1495$ for the linear case.
Redundancy of the Negative Gluon Component: When absorptive corrections are included, the "negative gluon component" term ( $A'_g$ ) becomes redundant. Fits including non-linear evolution show that the $A'_g$ term provides only marginal improvement to $\chi^2$ (e.g., reducing $\chi^2$ from 1416 to 1412 in NNLO*), whereas in linear fits, this term was essential to describe the exclusive data. The non-linear evolution naturally accounts for the suppression effects that the $A'_g$ term was previously compensating for.
Hot-Spot Size Determination: By fitting the parameter $R$ $R$ , the authors determine the effective size of "hot spots" in the proton.
- At NLO, the best-fit value is $R \approx 3.6 \text{ GeV}^{-1}$ .
- At NNLO*, the best-fit value is $R \approx 3.3 \text{ GeV}^{-1}$ .
- Assuming a proton wave function with $n=3$ hot spots, this corresponds to a hot-spot radius $R_{h.s.} \approx 2.57 \text{ GeV}^{-1}$ (NLO). This is the first evaluation of hot-spot size derived directly from a global PDF fit including exclusive data.
Uncertainties: The inclusion of exclusive data significantly reduces the uncertainties of the gluon PDF at very low $x$ . The uncertainty associated with the parameter $R$ is found to be small, with a clear minimum in the $\chi^2$ profile.

Significance and Claims
The paper claims that absorptive corrections are essential for a consistent description of combined inclusive DIS and exclusive heavy-vector-meson data. The primary significance lies in:

Physical Consistency: Demonstrating that non-linear evolution naturally resolves the tension between inclusive and exclusive data without requiring ad-hoc parametrizations like the negative gluon component.
Methodological Advancement: Successfully implementing non-linear DGLAP evolution within the standard APFEL++/xFitter framework, making it available for future global analyses.
Proton Structure: Providing the first quantitative estimate of the "hot-spot" size in the proton wave function ( $R_{h.s.} \approx 2.5-2.6 \text{ GeV}^{-1}$ ) based on global PDF fits.
Future Implications: The authors suggest that future global PDF determinations, particularly those aiming to constrain the gluon density in the small- $x$ , low- $Q^2$ region, should incorporate these absorptive effects to avoid unphysical local minima and improve accuracy.

The authors remain modest regarding the observation of saturation, noting that while the hot-spot size is determined, the effect of saturation is "hardly seen" in the present data due to the relatively large size of the hot spots compared to the saturation scale.

Technical Summary: Low-x Parton Densities Accounting for Absorptive Effects and Exclusive J/ψJ/\psiJ/ψ and Υ\UpsilonΥ Data from the LHC

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Technical Summary: Low-x Parton Densities Accounting for Absorptive Effects and Exclusive $J/\psi$ and $\Upsilon$ Data from the LHC