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

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:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>a</a:mi><a:mo>=</a:mo><a:mn>0.06</a:mn><a:mtext> </a:mtext><a:mtext> </a:mtext><a:mi>fm</a:mi></a:math>, 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 (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>b</c:mi><c:mo>⊥</c:mo></c:msub></c:math>) between the quarks. At larger <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>b</e:mi><e:mo>⊥</e:mo></e:msub></e:math>, 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>x</g:mi></g:math>-dependent pion valence-quark TMDPDF for transverse separations <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>b</i:mi><i:mo>⊥</i:mo></i:msub><i:mo>≳</i:mo><i:mn>1</i:mn><i:mtext> </i:mtext><i:mtext> </i:mtext><i:mi>fm</i:mi></i:math>. Interestingly, we find that the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>b</k:mi><k:mo>⊥</k:mo></k:msub></k:math> dependence of the phenomenological parametrizations of TMDPDF for moderate values of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>x</m:mi></m:math> 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.