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  • Open Access

Deep neural network extraction of unpolarized transverse momentum distributions

I. P. Fernando* and D. Keller†

  • *Contact author: ishara@virginia.edu
  • †Contact author: dustin@virginia.edu
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Phys. Rev. D 113, 096017 – Published 22 May, 2026

DOI: https://doi.org/10.1103/p3sg-k524

Abstract

Building on the first-ever application of neural networks in transverse momentum distribution (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 distribution functions (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 autoconvolution produces the observed qT spectra. We introduce two phenomenological objects in this data-driven approach: the transverse structure kernel S(qT,x1,x2;Q) and the intrinsic-transverse-momentum profile s(x,k⊥;Q). 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 parton distribution function (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 the QM bins 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 Ansätze and no bT transform) and provides a transferable template for polarized TMDs and related quantum chromodynamics inverse problems.

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Phys. Rev. D 113, 096017– Published 22 May, 2026
  • Received 7 November 2025
  • Accepted 24 April 2026
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