Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF
Dennis Bollweg, Xiang Gao, Jin-Chen He, Swagato Mukherjee, Yong Zhao
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: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.