Global Bayesian Analysis of $\mathrm{J}/ψ$ Photoproduction on Proton and Lead Targets

This paper presents a global Bayesian analysis of diffractive $\mathrm{J}/\psi$ photoproduction on proton and lead targets using a color glass condensate framework, revealing that while simultaneous description of HERA and LHC data is challenging, introducing an overall $K$-factor significantly improves the model's ability to fit both datasets.

The Gist

Imagine you are trying to bake the perfect cake, but you have two very different recipes to follow: one for a small, delicate cupcake (representing a single proton) and one for a massive, dense bundt cake (representing a heavy Lead nucleus).

In the world of high-energy physics, scientists use a theoretical "recipe book" called the Color Glass Condensate (CGC) to predict how these cakes behave when hit by a beam of light (photons). This light is used to create a specific type of particle called a J/ψ (pronounced "J-psi"), which is like a tiny, heavy cherry on top of the cake.

The Problem: The Recipe Doesn't Fit Both Cakes

For a long time, physicists noticed a frustrating problem. When they used their CGC recipe to predict the results for the cupcake (proton), it worked perfectly. The predictions matched the data from particle colliders like HERA and the LHC.

However, when they used that exact same recipe to predict the results for the bundt cake (Lead nucleus), it went wrong. The recipe predicted that the bundt cake would produce way too many J/ψ particles, especially when the collision energy was high. It was as if the recipe said, "Add a cup of sugar for the cupcake," and then, without changing the amount, said, "Add a cup of sugar for the bundt cake," resulting in a cake that was far too sweet.

The scientists wanted to know: Is there a single set of ingredients (parameters) that can explain both the small cupcake and the giant bundt cake simultaneously?

The Investigation: A Bayesian "Taste Test"

To solve this, the authors performed a Global Bayesian Analysis. Think of this as a super-smart, automated taste test.

1. The Ingredients (Parameters): They had a list of variables they could tweak, such as the "size" of the proton, how "fluffy" the inside is, and how the ingredients mix together at high speeds.
2. The Simulator (The Emulator): Because baking these theoretical cakes takes a massive amount of computer power, they built a "smart guesser" (a Gaussian Process emulator). This tool learned to predict the outcome of the baking process without having to run the full, slow simulation every time.
3. The Test: They ran thousands of simulations, tweaking the ingredients to see which combination could make both the cupcake and the bundt cake taste right (match the experimental data) at the same time.

The Findings: The "Magic Scaling Factor"

Here is what they discovered:

• The Standard Recipe Failed: When they tried to fit both datasets using the standard recipe (without any extra tricks), they couldn't do it. The settings that made the cupcake perfect made the bundt cake too sweet (too many particles). The settings that made the bundt cake perfect made the cupcake too dry (too few particles). The two datasets seemed to want different "evolution speeds" for the energy.
• The "K-Factor" Solution: The breakthrough came when they introduced a K-factor. Imagine this as a universal "volume knob" or a "scaling dial" that you can turn up or down for the entire recipe.
  • When they turned this dial down to about 0.3 (meaning they reduced the predicted output by 70%), something magical happened.
  • By lowering the overall output, the model was forced to adjust the internal ingredients (specifically, increasing the density of the "glue" holding the particles together).
  • This higher density created stronger "nuclear suppression" (like a denser cake that resists being broken apart), which naturally slowed down the growth of particles in the Lead nucleus.
  • Result: Suddenly, the same recipe could perfectly describe both the small proton and the large Lead nucleus.

What Didn't Work

The scientists also tried other fancy modifications to the recipe, such as:

• Changing the shape of the proton from a smooth ball to something more jagged.
• Adding or removing "hot spots" (clumps of energy) inside the proton.
• Filtering out high-frequency noise.

They found that none of these fancy tweaks helped as much as simply turning down the volume knob (the K-factor). The data strongly preferred the simple scaling solution over these complex structural changes.

The Bottom Line

The paper concludes that while the Color Glass Condensate framework is powerful, it currently needs a "correction factor" (the K-factor) to describe both protons and heavy nuclei simultaneously.

This suggests that our current understanding of the "non-perturbative" parts of the recipe (the messy, complex parts of how particles bind together) or the higher-order effects (the subtle chemical reactions in the oven) are not yet fully understood. The K-factor acts as a placeholder for these missing pieces, allowing the theory to fit the data for now, but hinting that the underlying theory needs further refinement to explain why that knob needs to be turned down so low.

In short: The same physics rules apply to both small and large targets, but our current mathematical "recipe" needs a global volume adjustment to get the proportions right for both.

Technical Summary

Technical Summary: Global Bayesian Analysis of J/ψ Photoproduction on Proton and Lead Targets

Problem Statement
The paper addresses a persistent tension in high-energy quantum chromodynamics (QCD) regarding the description of diffractive $J/\psi$ photoproduction. While Color Glass Condensate (CGC) based models have successfully described exclusive $J/\psi$ production in $\gamma + p$ collisions at HERA and the LHC, they typically overpredict the cross sections for coherent and incoherent production in $\gamma + \text{Pb}$ collisions at high center-of-mass energies ($W$). Previous calculations constrained by proton data fail to reproduce the observed degree of nuclear suppression in lead targets, predicting a more rapid increase with $W$ than experimental data indicates. The central question is whether a single set of model parameters can simultaneously describe both $\gamma + p$ and $\gamma + \text{Pb}$ data within the standard CGC framework, or if the tension is irreconcilable without modifying the theoretical assumptions.

Methodology
The authors perform a global Bayesian analysis to constrain the probability distributions of model parameters ($\theta$) by comparing theoretical predictions with experimental measurements ($y_{\text{exp}}$).

• Theoretical Framework: The calculation utilizes the CGC framework, specifically the McLerran-Venugopalan (MV) model at an initial scale $x_P = 0.01$ evolved via the JIMWLK equation. The scattering amplitude is calculated using the Boosted Gaussian model for the non-perturbative vector meson wave function and a dipole-target amplitude constructed from Wilson lines.
• Model Extensions: Beyond the "standard" setup (fixed parameters for proton shape, hot spots, and UV damping), the authors introduce several extensions to test their flexibility:
  • A flexible proton transverse shape parameter ($\omega$).
  • A variable number of sub-nucleonic hot spots ($N_q$).
  • A UV damping factor ($v_{\text{UV}}$) to suppress high-frequency modes.
  • A global scaling factor ($K$) applied to all cross sections to absorb uncertainties in the wave function or missing higher-order effects.
• Bayesian Inference: The analysis employs the pocoMC sampler and Gaussian Process (GP) emulators (specifically the PCGP emulator from the surmise package and Scikit-learn GP) to approximate the computationally expensive forward model. The training strategy involves Latin Hypercube Designs (LHD) and active learning to ensure emulator accuracy across the 11-dimensional parameter space.
• Datasets: The analysis incorporates coherent and incoherent differential and integrated cross-section data for $\gamma + p$ (H1, ZEUS, ALICE, LHCb) and $\gamma + \text{Pb}$ (ALICE, CMS) across a wide range of $W$.

Key Contributions

1. Global Bayesian Framework: This work provides the first simultaneous Bayesian analysis of $\gamma + p$ and $\gamma + \text{Pb}$ diffractive $J/\psi$ data within a unified CGC framework, systematically propagating model uncertainties.
2. Quantification of Tension: The study rigorously demonstrates that the standard model (without a scaling factor) cannot simultaneously fit both datasets. Separate fits to $\gamma + p$ and $\gamma + \text{Pb}$ yield conflicting constraints on the energy evolution speed and saturation scales.
3. Model Comparison via Bayes Factors: The authors utilize Bayes factors to quantitatively compare the standard model against various extensions. This statistical approach determines which model extensions are genuinely supported by the data versus those that are merely over-parameterized.
4. Resolution via Scaling Factor: The analysis identifies that the introduction of a global scaling factor $K$ is the only extension strongly favored by the data, while other structural modifications (proton shape, hot spot number, UV damping) do not significantly improve the fit.

Results

• Standard Model Failure: When $K$ is fixed to 1, the model cannot describe both systems simultaneously. A fit to $\gamma + p$ data overestimates $\gamma + \text{Pb}$ cross sections at high $W$, while a fit to $\gamma + \text{Pb}$ data underestimates $\gamma + p$ cross sections. The posterior distributions for the global fit exhibit a double-peak structure in the JIMWLK evolution parameters, indicating the tension between the two datasets.
• Preferred Model: The model extended with a free scaling factor $K$ yields a significantly better fit ($\chi^2/\text{dof} \approx 1.82$) compared to the standard model ($\chi^2/\text{dof} \approx 5.94$). The Bayes factor strongly favors the inclusion of $K$ ($\ln B_{A/B} \approx -18.56$).
• Parameter Values: The preferred value for the scaling factor is $K \approx 0.3$. This reduction in the overall cross-section normalization is compensated by a preference for a larger color charge density (smaller $Q_s/(g^2\mu)$ ratio), leading to a larger effective saturation scale.
• Physical Interpretation: The larger saturation scale enhances non-linear effects, resulting in stronger nuclear suppression and a slower energy dependence for the lead target, which aligns with experimental observations.
• Other Extensions: Variations in the proton shape ($\omega$), the number of hot spots ($N_q$), and the UV damping ($v_{\text{UV}}$) were found to be either disfavored or only weakly preferred compared to the standard model. Adding these parameters to the $K$-extended model further reduced the Bayesian evidence.

Significance and Claims
The paper concludes that within the current CGC-based framework, a simultaneous description of $\gamma + p$ and $\gamma + \text{Pb}$ data is challenging and requires a significant rescaling of the cross sections ($K \sim 0.3$). The authors interpret this necessity as an indication that uncertainties in the non-perturbative vector meson wave function and/or missing higher-order (NLO) corrections are substantial and currently unaccounted for in the leading-order formalism.

The authors modestly claim that their results indicate calculations performed entirely in the linear regime (or without these specific non-linear adjustments) cannot describe both datasets simultaneously. They suggest that while NLO corrections might resolve the issue, full NLO calculations are not yet available to confirm this. The study motivates future work to improve the theoretical description of vector meson production, including NLO extensions and better descriptions of non-perturbative physics, and highlights the need for data from intermediate nuclear systems (e.g., $\gamma + \text{O}$, $\gamma + \text{Xe}$) to further explore saturation effects.