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Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF

Dennis Bollweg1, Xiang Gao2, Jinchen He3,4,*, Swagato Mukherjee2, and Yong Zhao4

  • *Contact author: jinchen@umd.edu
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Phys. Rev. D 112, 034501 – Published 4 August, 2025

DOI: https://doi.org/10.1103/j3n6-8kxy

Abstract

We present the first lattice quantum chromodynamics (QCD) calculation of the pion valence-quark transverse-momentum-dependent parton distribution function (TMDPDF) within the framework of large-momentum effective theory (LaMET). Using correlators fixed in the Coulomb gauge (CG), we computed the quasi-TMD beam function for a pion with a mass of 300 MeV, a fine lattice spacing of a=0.06  fm, and multiple large momenta up to 3 GeV. The intrinsic soft functions in the CG approach are extracted from form factors with large momentum transfer, and as a byproduct, we also obtain the corresponding Collins-Soper (CS) kernel. Our determinations of both the soft function and the CS kernel agree with perturbation theory at small transverse separations (b⊥) between the quarks. At larger b⊥, the CS kernel remains consistent with recent results obtained using both CG and gauge-invariant TMD correlators in the literature. By combining next-to-leading logarithmic factorization of the quasi-TMD beam function and the soft function, we obtain an x-dependent pion valence-quark TMDPDF for transverse separations b⊥≳1  fm. Interestingly, we find that the b⊥ dependence of the phenomenological parametrizations of TMDPDF for moderate values of x are in reasonable agreement with our QCD determinations. In addition, we present results for the transverse-momentum-dependent wave function for a heavier pion with 670 MeV mass.

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Phys. Rev. D 112, 034501– Published 4 August, 2025
  • Received 28 April 2025
  • Accepted 9 July 2025
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