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Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF
Dennis Bollweg, Xiang Gao, Jinchen He, Swagato Mukherjee, Yong Zhao
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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 (NLL) factorization of the quasi-TMD beam function and the soft function, we obtain x-dependent pion valence-quark TMDPDF for transverse separations b_⊥ ≳1 fm. Interestingly, we find that the b_⊥ 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.
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