Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF
ORAL
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 ($b_\perp$) between the quarks. At larger $b_\perp$, the CS kernel remains consistent with recent results obtained using both CG and GI TMD correlators in the literature. By combining next-to-leading logarithmic (NLL) factorization of the quasi-TMD beam function and the soft function, we obtain $x$-dependent pion valence-quark TMDPDF for transverse separations $b_\perp \gtrsim 1$~fm. Interestingly, we find that the $b_\perp$ dependence of the phenomenological parameterizations 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 (TMDWF) for a heavier pion with 670~MeV mass.
*This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics through Contract No.~DE-SC0012704, No.~DE-AC02-06CH11357, within the framework of Scientific Discovery through Advanced Computing (SciDAC) award Fundamental Nuclear Physics at the Exascale and Beyond, and under the umbrella of the Quark-Gluon Tomography (QGT) Topical Collaboration with Award DE-SC0023646. The work of YZ is also partially supported by the Laboratory Directed Research and Development (LDRD) funding from Argonne National Laboratory, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No.~DE-AC02-06CH11357.
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Publication:D.~Bollweg, X.~Gao, J.~He, S.~Mukherjee and Y.~Zhao, %``Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF,'' Phys. Rev. D \textbf{112}, no.3, 034501 (2025) doi:10.1103/j3n6-8kxy [arXiv:2504.04625 [hep-lat]].
