⚛️ phenomenology

A Precise Determination of the Shape of the Proton's Gluon Cloud from HERA data

By fitting a leading-twist hotspot model to exclusive $J/\psi$ production data from HERA, this study precisely determines the proton's gluon cloud shape as a perturbative Gaussian core of 0.105 fm surrounded by a non-perturbative exponential halo of 0.220 fm, achieving an excellent fit with $\chi^2/{\rm ndf}=0.77$ .

Original authors: Tobias Toll, Nahid Vasim

Published 2026-07-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tobias Toll, Nahid Vasim

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 the proton not as a solid, smooth marble, but as a bustling, chaotic city made of invisible energy. Inside this city, tiny particles called quarks are the mayors, but they are surrounded by a swirling, foggy cloud of even smaller particles called gluons. These gluons are the glue that holds the city together, zipping back and forth at nearly the speed of light. Scientists have long wondered: what does this foggy cloud actually look like? Is it a uniform mist, or does it have distinct neighborhoods? Understanding this shape is crucial because the proton is the building block of almost everything we see. If we know how its internal "city" is laid out, we can better understand how the universe behaves in extreme conditions, like the fiery collisions that happen in giant particle smashers or the moments just after the Big Bang.

To get a peek inside this invisible city, scientists use a clever trick called "exclusive diffraction." Imagine throwing a pebble at a foggy window. If the window stays intact but the pebble bounces off at a specific angle, you can learn about the fog's density just by watching the bounce. In the world of particle physics, they fire electrons at protons and watch for a specific type of bounce where a particle called a $J/\psi$ meson is created, but the proton itself remains whole. By measuring how these particles scatter, scientists can map the proton's internal structure, much like using sound waves to create an ultrasound image of a baby.

Now, let's look at what Tobias Toll and Nahid Vasim discovered in their recent study. They took a fresh look at data collected from the HERA laboratory, where electrons and protons collided at incredible speeds. Previous models tried to describe the proton's gluon cloud as a simple, smooth Gaussian shape (like a perfect bell curve), but these models struggled to explain the data, especially when the scattering happened at sharp angles. The authors proposed a new, more detailed map of the proton's "hotspots"—the dense clusters where gluons gather.

Their findings suggest that these hotspots are not just simple blobs. Instead, they have a very specific, two-layered structure. Think of a hotspot like a fluffy marshmallow with a hard, crunchy center. The authors found that each hotspot has a "perturbative Gaussian core" that is incredibly small, measuring about 0.105(1) femtometers (a femtometer is one quadrillionth of a meter). Surrounding this hard core is a "non-perturbative exponential halo" that is softer and more spread out, stretching out to about 0.220(2) femtometers. This shape—a hard center wrapped in a soft, fuzzy halo—was consistent across the data, regardless of how much energy was used or how many hotspots were present.

The researchers tested their model against 104 different measurements from the H1 and ZEUS experiments. The result was a remarkably good match, with a statistical score ( $\chi^2$ /ndf) of 0.77, indicating their "marshmallow" model fits the data much better than the old, smooth-blob models. They also looked at how the size of the proton's cloud changes with energy. They found that the cloud's shape is surprisingly stable; it doesn't seem to change its fundamental geometry as the energy of the collision changes. The "halo" size they measured matches predictions from other theories about how energy flows through empty space, giving them confidence that they are seeing the real structure of the proton's gluon cloud.

One interesting detail they uncovered involves the number of these hotspots. The data suggested that the proton might have around 4 to 6 hotspots. While the "incoherent" data (where the proton gets excited and breaks apart slightly) preferred a lower number (around 3 or 4), and the "coherent" data (where the proton stays perfectly intact) preferred a higher number (around 9 or 10), the authors settled on a model with 4 hotspots. This number makes physical sense: it could represent the three valence quarks (the "mayors") plus one extra high-energy gluon acting as a source for the rest.

The paper also addressed some alternative ideas. They tested models where the size of the hotspots or the number of hotspots would change depending on the energy of the collision. However, the data showed no significant improvement when these variables were allowed to change. This suggests that the geometry of the proton's gluon cloud is "frozen" and does not shift its shape based on the energy level, at least within the range of what HERA could measure.

In summary, Toll and Vasim have provided a precise, analytical picture of the proton's gluon cloud. They found it is composed of distinct hotspots with a hard, tiny core and a soft, fuzzy halo. This shape is stable, independent of the collision energy, and fits the experimental data with high precision. While they couldn't determine the exact number of hotspots with absolute certainty, their model of 4 hotspots offers a compelling and consistent explanation for how the proton's internal city is built.

Technical Summary: A Precise Determination of the Shape of the Proton's Gluon Cloud from HERA data

Problem Statement
The geometric structure of hadronic matter, specifically the spatial distribution of the proton's gluon cloud at small momentum fractions ( $x$ ), remains a fundamental question in Quantum Chromodynamics (QCD). While exclusive diffractive events at high-energy colliders like HERA provide a sensitive probe of this structure via the Mandelstam variable $t$ (where $\Delta = \sqrt{-t}$ is the Fourier conjugate to the transverse spatial coordinate), existing theoretical models have struggled to provide a consistent, precise description of the data. Previous constituent quark models, which assume the gluon cloud is concentrated in three "hotspots" around valence quarks, successfully described coherent scattering but failed to simultaneously fit the incoherent spectrum, particularly at large $|t| > 2 \text{ GeV}^2$ . Furthermore, attempts to resolve these discrepancies using Bose-Einstein corrections or specific evolution schemes often lacked a firm perturbative QCD motivation or failed to describe the full dataset consistently. Additionally, lattice QCD studies suggest a specific non-perturbative structure for flux tubes—a perturbative Gaussian core surrounded by a non-perturbative exponential halo—which had not been rigorously tested against exclusive HERA data.

Methodology
The authors present an analytical calculation of coherent and incoherent cross-sections for exclusive $J/\psi$ photoproduction within the dipole model framework, utilizing the "hotspot" model of the proton.

Theoretical Framework: The study employs the Good-Walker formalism, where the coherent cross-section is proportional to the square of the average scattering amplitude, and the incoherent cross-section is proportional to the variance of the amplitude. The dipole amplitude is calculated using the IPsat model (including saturation effects) at leading twist, which is sufficient for describing HERA data.
Hotspot Geometry: The proton's gluon content is modeled as $N_{hs}$ hotspots. Crucially, the authors model the thickness function of a single hotspot ( $T_{hs}$ ) as a convolution of a perturbative Gaussian core and a non-perturbative Bessel function ( $K_0$ ) halo. This functional form is inspired by lattice QCD results regarding flux tube entanglement entropy.
Analytical Derivation: Unlike previous approaches requiring numerical generation of thousands of hotspot configurations, the authors derive an analytical expression for the cross-sections. They account for the center-of-mass (CM) shift of the proton to ensure the variance vanishes for a single hotspot and for $|t| \to 0$ . They also incorporate event-by-event fluctuations in the saturation scale via a log-normal distribution factor ( $\xi_i$ ).
Data Fitting: The model is fitted to 104 exclusive $J/\psi$ photoproduction data points from the H1 and ZEUS experiments at HERA, covering center-of-mass energies $40 \le W \le 251 \text{ GeV}$ and momentum transfers $0 < |t| \le 30 \text{ GeV}^2$ . The fit includes 67 coherent and 37 incoherent data points. The free parameters include the width of the hotspot distribution ( $B_{qc}$ ), the hotspot core size ( $B_{hs}$ ), the halo penetration depth ( $\lambda$ ), the saturation fluctuation parameter ( $\sigma_S$ ), and the energy dependence slope ( $\alpha'$ ).

Key Contributions

Analytical Precision: The paper provides a fully analytical calculation of the coherent and incoherent cross-sections in the hotspot model, avoiding the computational cost of Monte Carlo averaging while maintaining precision.
Hybrid Hotspot Profile: The authors introduce and validate a specific hotspot profile consisting of a perturbative Gaussian core surrounded by a non-perturbative exponential halo (modeled via a $K_0$ Bessel function). This directly connects the phenomenological fit to lattice QCD predictions regarding flux tube structures.
Simultaneous Fit: The model successfully describes both the coherent and incoherent $t$ -spectra simultaneously, resolving previous shortcomings where models could only fit one or the other, or failed at high $|t|$ .

Results

Fit Quality: The model achieves an excellent fit to the data with a reduced chi-squared of $\chi^2/\text{ndf} = 0.77$ (specifically $75.4/98$ for the best-fit configuration).
Hotspot Geometry: The extracted shape of the gluon cloud is found to be independent of the momentum fraction $x_{IP}$ and the number of hotspots ( $N_{hs}$ ) within the best-fit range. The hotspots possess a perturbative Gaussian core with a size of $\sqrt{2B_{hs}} = 0.105(1) \text{ fm}$ , surrounded by a non-perturbative exponential halo with a penetration depth of $\lambda = 0.220(2) \text{ fm}$ . This halo size is consistent with lattice predictions for flux tubes in a vacuum.
Number of Hotspots: While the incoherent data prefers $N_{hs} \approx 3-4$ and coherent data prefers $N_{hs} \approx 9-10$ , the authors settle on $N_{hs}=4$ (representing three valence quarks plus one large- $x$ gluon source) as a physically motivated choice that yields a stable fit ( $\chi^2/\text{ndf} = 0.78$ ).
Energy Dependence: The effective Pomeron slope is determined to be $\alpha'_{\text{eff}} = 0.046(19) \text{ GeV}^{-2}$ . This value is stable across the fit and is consistent with running coupling BFKL predictions, though lower than values extracted from previous Regge fits to HERA data.
Stability Tests: Variations allowing the hotspot size or the average number of hotspots to evolve with $x_{IP}$ yielded no significant improvement in fit quality ( $\Delta \chi^2 \approx 0.5 - 1.0$ ), indicating that HERA data cannot resolve such geometric variations and that the hotspot geometry is effectively frozen.

Significance
The paper claims to provide a precise determination of the proton's gluon cloud shape, establishing that the gluon distribution is not merely a simple Gaussian but possesses a distinct two-component structure: a hard perturbative core and a soft non-perturbative halo. This result bridges the gap between phenomenological descriptions of exclusive diffraction and non-perturbative QCD predictions from lattice simulations. By demonstrating that a single, stable geometric model can describe the full range of HERA exclusive $J/\psi$ data (both coherent and incoherent), the study offers a robust baseline for understanding the initial state of proton-proton and heavy-ion collisions, where the proton structure serves as a critical input for subsequent QCD flow dynamics.

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