Effect of sub-nucleon fluctuations on the DVCS process in proton and nuclear targets at the EIC

This paper investigates the impact of sub-nucleon fluctuations on Deeply Virtual Compton Scattering at the Electron-Ion Collider using a hot-spot model, predicting distinct energy dependencies and $t$-distribution features for coherent and incoherent cross-sections in both proton and nuclear targets.

The Gist

Imagine the proton, the tiny particle at the center of every atom, not as a smooth, solid marble, but as a bustling city made of smaller, shifting neighborhoods. This paper explores what happens when we shoot a high-energy "probe" (an electron) at these cities to see how they are built, specifically looking at a process called Deeply Virtual Compton Scattering (DVCS).

Here is a simple breakdown of what the authors did and found, using everyday analogies.

The Setup: The "Hot Spot" City

Usually, scientists might imagine a proton as a uniform ball of dough. However, this paper uses a model called the "Hot Spot" model.

• The Analogy: Think of the proton as a city where the population isn't spread out evenly. Instead, the city is made up of distinct, glowing "hot spots" (clusters of energy).
• The Twist: As the energy of the collision gets higher, the city doesn't just get brighter; it gets crowded. New hot spots appear, and they shift around randomly every time you take a snapshot. The paper argues that these shifting, fluctuating neighborhoods are crucial to understanding how the proton behaves.

The Experiment: Taking a Photo vs. Breaking the Window

The researchers looked at two ways the electron interacts with the proton (or a larger nucleus like Lead or Calcium):

1. Coherent Scattering (The Group Photo):

   • What happens: The electron hits the target, and the target stays perfectly intact, like a group photo where everyone stands still.
   • The Result: This measures the average layout of the city. The paper found that the "Hot Spot" model predicts this very well, matching existing data from older experiments (HERA).

2. Incoherent Scattering (The Broken Window):

   • What happens: The electron hits the target, and the target gets shaken up or breaks apart into a cloud of debris.
   • The Result: This measures the fluctuations—the fact that the city layout changes from moment to moment. This is where the paper's big discovery lies.

The Big Discovery: The "Energy Turn-Over"

The most exciting finding concerns the Incoherent process (the one where the target gets shaken up).

• The Prediction: The authors predict that as you increase the energy of the collision, the number of times this "shaking up" happens will rise, reach a peak (a maximum), and then suddenly drop.
• The Analogy: Imagine throwing a stone into a pond. At first, the bigger the stone (energy), the bigger the splash. But in this specific quantum world, if you throw the stone too hard, the splash actually gets smaller again.
• The Catch: The exact point where this splash peaks depends on how "virtual" (intense) the photon is. For less intense photons, the peak happens at lower energies; for more intense ones, it happens at higher energies.

The Nuclear Targets: Bigger Cities, Different Rules

The paper also looked at Nuclei (like Calcium or Lead), which are essentially clusters of many protons stuck together (like a whole neighborhood block instead of a single house).

• The Difference: For these larger targets, the "turn-over" (the peak and drop) does not happen in the energy range the new Electron-Ion Collider (EIC) will be able to test. The "splash" keeps getting bigger as energy increases.
• The Ratio: The paper predicts that as energy goes up, the "Group Photo" (coherent) becomes much more common compared to the "Broken Window" (incoherent) for protons, but this ratio changes differently for larger nuclei.

The Map: Where the Action Happens

The researchers also mapped out the "shape" of the collision (called the t-distribution).

• For Protons: The "Broken Window" events vanish if you look straight on (zero angle) and show a specific pattern elsewhere.
• For Nuclei: The "Broken Window" events create a hump (a maximum) at a specific angle. The position of this hump depends on the size of the nucleus and the intensity of the photon. It's like a shadow cast by the nucleus that changes shape based on the light source.

The Bottom Line

The authors are saying: "If we build the new Electron-Ion Collider (EIC) and run these experiments, we should see these specific patterns."

• If we see the peak and drop in the proton data, it proves the "Hot Spot" model is correct and that protons are full of shifting, fluctuating sub-structures.
• If we see the hump in the nuclear data, it confirms how these fluctuations behave in larger, heavier atoms.

Essentially, this paper is a set of instructions for what to look for in future experiments to prove that the inside of a proton is a chaotic, shifting city of "hot spots" rather than a smooth ball.

Technical Summary

1. Problem Statement

The Deeply Virtual Compton Scattering (DVCS) process ($\gamma^* h \to \gamma h$) is a primary tool for probing the 3D structure of hadrons (protons and nuclei) and understanding the gluon content at high energies. While the coherent cross-section (where the target remains intact) measures the average spatial distribution of gluons, the incoherent cross-section (where the target dissociates) is sensitive to event-by-event fluctuations in the gluon density.

Current theoretical models often treat the proton as a smooth object. However, recent evidence suggests that at high energies, the proton's internal structure is dominated by sub-nucleon fluctuations, often modeled as "hot-spots" (localized regions of high gluon density). The authors aim to investigate how these sub-nucleon fluctuations affect DVCS observables, specifically:

• The energy dependence of coherent and incoherent cross-sections.
• The differential distributions in momentum transfer ($t$).
• The behavior of these observables for both proton and nuclear targets (specifically Calcium and Lead) within the kinematic reach of the future Electron-Ion Collider (EIC).

2. Methodology

The authors employ a theoretical framework combining the Good-Walker formalism with the Color Dipole Model and a specific Hot-Spot Model for sub-nucleon structure.

• Formalism:

  • Good-Walker Approach: Used to separate the coherent and incoherent cross-sections. The coherent cross-section is proportional to the square of the average scattering amplitude, while the incoherent cross-section is proportional to the variance (fluctuations) of the amplitude.
  • Color Dipole Model: The scattering amplitude is factorized into the photon wave function (fluctuating into a $q\bar{q}$ dipole), the dipole-target interaction, and the recombination into a real photon.
  • Photon Wave Function: Calculated using QED, considering only transversely polarized overlaps since the final state is a real photon.

• The Hot-Spot Model:

  • Proton Target: The proton profile is modeled as a superposition of $N_{hs}$ hot-spots (Gaussian distributions of color charge) in the impact parameter plane.
    • The number of hot-spots ($N_{hs}$) is energy-dependent, increasing as the Bjorken-$x$ decreases (following a zero-truncated Poisson distribution).
    • The positions of these hot-spots fluctuate event-by-event.
  • Nuclear Target: The dipole-nucleus cross-section is calculated using the Glauber-Gribov model. The hot-spot model is applied to individual nucleons, which are distributed according to a Woods-Saxon distribution.

• Parameters:

  • The saturation scale $Q_s(x)$ follows the GBW (Golec-Biernat-Wusthoff) model.
  • The study covers EIC kinematic ranges, varying photon virtuality ($Q^2$), center-of-mass energy ($W$), and atomic number ($A$).

3. Key Contributions

• Extension to DVCS: This is the first study to apply the energy-dependent hot-spot model specifically to the DVCS process (previously applied mainly to vector meson photoproduction).
• Prediction of a "Turn-Over": The authors predict a non-monotonic behavior (a maximum followed by a decrease) in the energy dependence of the incoherent cross-section for proton targets, a feature driven by the increasing number of hot-spots.
• Nuclear $t$-Distribution Prediction: They predict a distinct maximum in the $t$-distribution of the nuclear incoherent cross-section, a feature absent in simpler models.
• EIC Forecasts: The paper provides specific quantitative predictions for the EIC, including the ratio of coherent to incoherent cross-sections and differential $t$-distributions for various nuclei (Ca, Pb).

4. Key Results

A. Proton Targets

• Energy Dependence:
  • Coherent Cross-section: Increases with energy ($W$) and decreases with photon virtuality ($Q^2$). The model successfully describes existing HERA data.
  • Incoherent Cross-section: Exhibits a turn-over. It rises with energy, reaches a maximum, and then decreases steeply at higher energies.
    • Crucial Finding: The position of this maximum is strongly dependent on $Q^2$. Lower $Q^2$ values result in the maximum occurring at lower energies. For small $Q^2$, this turnover is expected to be observable within the EIC energy range.
• Ratio ($\sigma_{coh}/\sigma_{incoh}$): This ratio increases with energy, atomic number, and decreases with $Q^2$. The turnover in the incoherent cross-section leads to a steeper increase in this ratio for $W \gtrsim 80$ GeV at low $Q^2$.
• $t$-Distributions:
  • Coherent: Shows a typical diffractive pattern with dips. The dip positions are largely independent of $Q^2$.
  • Incoherent: No dip is predicted; the distribution vanishes as $|t| \to 0$. The presence or absence of a dip in the coherent channel depends on $Q^2$.

B. Nuclear Targets (Ca and Pb)

• Energy Dependence:
  • Similar to protons, coherent cross-sections are larger than incoherent ones.
  • No Turn-Over: Unlike the proton case, the incoherent cross-section for nuclear targets does not show a turn-over in the energy range considered. It continues to rise or plateau.
  • Ratio Behavior: The ratio of coherent to incoherent cross-sections increases with the atomic number ($A$) and has a steeper slope for heavier nuclei (Pb vs. Ca).
• $t$-Distributions:
  • Coherent: Exhibits diffractive dips. The position of the first dip shifts to smaller $|t|$ values for heavier nuclei.
  • Incoherent: The model predicts a maximum in the $t$-distribution (a peak at non-zero $|t|$).
    • The position of this maximum depends on both $Q^2$ and $A$.
    • The peak occurs at larger $|t|$ values for lower $Q^2$ and for heavier nuclei.

5. Significance

• Validation of Sub-Nucleon Structure: The predicted "turn-over" in the proton incoherent cross-section and the specific shape of the nuclear incoherent $t$-distribution serve as unique signatures of the hot-spot model. Observing these at the EIC would provide strong evidence for the existence and energy evolution of sub-nucleon fluctuations.
• EIC Physics Program: The paper provides concrete benchmarks for the EIC physics program. It highlights that measuring the incoherent DVCS cross-section at different energies and $Q^2$ values is critical to distinguishing between smooth proton models and fluctuating hot-spot models.
• Nuclear Geometry: The results offer new insights into how gluon fluctuations behave in nuclear environments, suggesting that the interplay between nucleon fluctuations and nuclear geometry leads to distinct signatures (like the $t$-distribution peak) that differ from single nucleon behavior.

In conclusion, the paper demonstrates that sub-nucleon fluctuations are not merely a correction but a dominant factor shaping the energy and momentum transfer dependence of DVCS, offering testable predictions that will guide future experimental analyses at the EIC.