Deep Neural Network extraction of Unpolarized Transverse Momentum Distributions

TL;DR

This work develops a momentum-space, physics-informed deep-learning framework to extract unpolarized transverse momentum dependent parton distributions (TMDs) directly from fixed-target Drell–Yan data without transforming to impact-parameter space. A two-stage pipeline first learns a structure kernel $S(q_T;x_1,x_2,Q_M)$ from cross-sections and then reconstructs a normalized intrinsic profile $s(x,k_ot;Q)$ via a differentiable inverse convolution, ensuring per-$(x,Q)$ normalization and smoothness. Applied to E288 and E605 data, the method reproduces the observed $q_T$ spectra and yields flavor-separated TMDs that broaden with $Q$, displaying a consistent $x$- and flavor-dependent structure and Sudakov-like evolution in momentum space. The framework provides a minimally biased, data-driven template for TMD extraction directly in $k_ot$ space and offers a transferable foundation for polarized TMDs and related QCD inverse problems, with uncertainty propagation through a robust Monte Carlo replica approach. The approach complements traditional $b_T$-space analyses and holds promise for integrating additional datasets and extending to more complex TMD phenomena in future studies.

Abstract

Building on the first-ever application of neural networks in TMD phenomenology: "Extraction of the Sivers function with deep neural networks", we now present a momentum space, physics-informed deep learning framework for the direct extraction of unpolarized transverse momentum dependent parton distributions (TMDs) from fixed target Drell-Yan data (E288, E605). Rather than transforming to impact-parameter space, we remain in k and embed a normalized integrand s(x, k; Q) whose auto-convolution produces the observed qT spectra. The extraction proceeds in two steps. Stage I learns the structure kernel S(qT , x1, x2; QM ) by regressing the cross-section with known kinematic prefactors and charge-weighted PDF combinations factored out; experimental and PDF uncertainties are propagated with Monte Carlo replicas. Stage II reconstructs s(x, k; Q) with an end-to-end differentiable k quadrature layer. Applied to Fermilab cross-section data from experiments E288 and E605, the method reproduces the measured qT spectra across Q and yields x and Q dependent TMDs that broaden with Q, with uncertainty bands that consistently propagate experimental, PDF, algorithmic and methodological components. The approach is minimally biased (no factorized Ansatze and no bT transform) and provides a transferable template for polarized TMDs and related QCD inverse problems.

Figures (10)

• Figure 1: Cross‑section closure with pseudo‑data at E288‑like kinematics (with $\Upsilon$ veto). The dashed line (in blue) show the pseudo‑data (true cross-sections); the solid line and the error band (in red) represent the 'mean' and 1-$\sigma$ DNN prediction (with 150 replicas) respectively; reconstructed at bin centers using Eq. \ref{['eq:inv']}.
• Figure 2: Closure for $S(q_T)$ and $s(k)$ using the inverse‑integration workflow (results with 1000 replicas). Left:$S(q_T)$ comparison (learned vs. generator). Right:$s(k)$ comparison (extracted vs. generator). Where shown, the shaded band indicates the $1\sigma$ algorithmic spread across independent trainings.
• Figure 3: Representative cross-section comparison after the joint $s(x,k_\perp,Q)$ extraction. Points with error bars are E288 data (with standard cuts); the curve and band are the DNN prediction and its $1\sigma$ uncertainty obtained by propagating the replica ensemble through Eq. \ref{['eq:fit_summary']}.
• Figure 4: Representative cross-section comparison after the joint $s(x,k_\perp,Q)$ extraction. Points with error bars are E605 data (with standard cuts); the curve and band are the DNN prediction and its $1\sigma$ uncertainty obtained by propagating the replica ensemble through Eq. \ref{['eq:fit_summary']}.
• Figure 5: Unpolarized TMDs reconstructed at $x=0.1$ and $Q=4.5~\mathrm{GeV}$ using Eq. \ref{['eq:fit_summary']}. Curves show the central values for selected flavors; shaded bands indicate the propagated $1\sigma$ uncertainties from experimental replicas (Step 1) and collinear PDF replicas.