Unbiased Data-Driven Determination of the Nuclear Dipole Amplitude in the Color Glass Condensate

Technical Summary: Unbiased Data-Driven Determination of the Nuclear Dipole Amplitude in the Color Glass Condensate

Problem Statement
In the high-energy limit of Quantum Chromodynamics (QCD), the growth of parton densities at small Bjorken-$x$ is tamed by gluon saturation, leading to the Color Glass Condensate (CGC) regime. While the saturation scale $Q_s(x)$ grows with nuclear mass number $A$ ($Q_s^2 \propto A^{1/3}$), quantitative extractions of the nuclear gluon dipole amplitude have historically relied on parametrized initial conditions. These assumptions introduce uncontrolled model dependence, particularly regarding the nuclear profile and the functional form of the initial amplitude. Consequently, extracted values for the nuclear saturation scale vary widely across the literature, obscuring genuine nuclear effects and hindering the use of nuclear data as a precision probe of saturation physics. Furthermore, existing extractions for protons have struggled with violations of momentum-space positivity when using standard parametrizations.

Methodology
The authors introduce a physics-informed neural network (PINN) framework to determine the impact-parameter-averaged nuclear dipole amplitude $N(r, x)$ for $^{208}\text{Pb}$ without assuming a functional form for its initial condition.

• Physics-Informed Architecture: The dipole amplitude is represented by a neural network $N_\theta(r, Y)$, where $Y = \ln(1/x)$. The network is trained to satisfy the collinearly improved Balitsky-Kovchegov (ciBK) evolution equation directly within the loss function, rather than solving the equation iteratively during a fitting loop. This ensures the extracted amplitude strictly obeys QCD evolution dynamics.
• Training Constraints: The training objective minimizes a total loss function $L_{tot}$ comprising:
  • ciBK Evolution Loss: Penalizes residuals of the ciBK equation using automatic differentiation and differentiable quadrature.
  • Observable Losses: Constrains the network against experimental data, specifically the nuclear modification factor $R_{p\text{Pb}}$ for forward hadron production and the coherent $J/\psi$ photoproduction cross section in ultraperipheral Pb-Pb collisions. The framework utilizes a heteroscedastic Gaussian loss to account for experimental uncertainties and learned data-dependent variances.
  • Physical Regularization: Enforces the black-disk limit at large dipole sizes, Fourier positivity in momentum space, and bounds on the anomalous dimension at small $r$.
• Optimization: The network employs a gradient-calibration procedure to balance the weights of distinct loss components (evolution, data, and physical constraints). Uncertainty quantification is achieved via deep ensembles, separating data-dependent and model-dependent uncertainties.
• Data Inputs: The analysis utilizes LHCb measurements of $R_{p\text{Pb}}$ at $\sqrt{s_{NN}} = 5.02$ TeV and coherent $J/\psi$ photoproduction data from ALICE, CMS, ATLAS, and LHCb.

Key Results

• Extraction of the Pb Amplitude: The framework successfully extracts the $^{208}\text{Pb}$ dipole amplitude at the starting scale $x_0 = 0.01$. The evolved amplitude reproduces the measured $R_{p\text{Pb}}$ and coherent $J/\psi$ cross sections across the available kinematic range.
• Saturation Scale Ratio: The extracted impact-parameter-averaged initial saturation scales yield a ratio $Q_{s0,\text{Pb}}^2 / Q_{s0,p}^2 = 3.17^{+0.17}_{-0.10}$. This value is consistent with, though approximately 20% higher than, a simple geometric scaling estimate ($A S_{\perp,p} / S_{\perp,A}$), reflecting the expected higher color-charge density in heavy nuclei.
• Initial Condition Form: Unlike the proton case, where an MV-type (McLerran-Venugopalan) parametrization was found insufficient, the extracted Pb initial condition is well-described by an MV-type functional form. This emerges naturally from the data-driven fit, validating the Gaussian-source approximation for large nuclei.
• Momentum-Space Positivity: The extracted amplitude maintains positivity in momentum space throughout the evolution, resolving a known issue in previous parametrized fits.
• Predictive Power: Using the same extracted amplitude, the authors predict the rapidity dependence of the transverse-momentum ratio $\langle p_T \rangle / \langle p_T \rangle(\eta_0)$ for $pp$, $p\text{Pb}$, and $\text{Pb}p$ collisions. The predictions agree with recent LHCb measurements in the low-multiplicity regime without introducing system-dependent parameters. Discrepancies at higher multiplicities are attributed to final-state effects outside the scope of the initial-state dipole framework.
• Generalization: The authors provide a physically motivated interpolation scheme to construct initial conditions for intermediate nuclei (e.g., $^{12}\text{C}$, $^{16}\text{O}$, $^{197}\text{Au}$) based on the extracted proton and Pb opacities, offering benchmark inputs for future small-$x$ phenomenology.

Significance and Claims
The paper claims to provide the first unbiased, data-driven determination of the nuclear dipole amplitude in the saturation regime. By embedding the nonlinear ciBK evolution equation directly into the machine learning training objective, the work eliminates the persistent source of bias associated with pre-defined functional forms for nuclear initial conditions.

The authors assert that this strategy establishes a general methodology for extracting dynamically constrained observables in QCD. The results offer a model-independent quantitative test of geometric scaling arguments and provide a robust foundation for interpreting forward particle production at the LHC and RHIC, as well as future measurements at the Electron-Ion Collider (EIC). The work demonstrates that the interplay between QCD evolution and experimental data can uniquely determine the nuclear structure in the saturation regime, removing historical ambiguities in comparing saturation predictions across different nuclear species.