T-odd leading-twist quark TMDs at small x

Abstract

We study the small-x asymptotics of the flavor non-singlet T-odd leading-twist quark transverse momentum dependent parton distributions (TMDs), the Sivers and Boer-Mulders functions. While the leading eikonal small-x asymptotics of the quark Sivers function is given by the spin-dependent odderon [1, 2], we are interested in revisiting the sub-eikonal correction considered by us earlier in [3]. We first simplify the expressions for both TMDs at small Bjorken x and then construct small-x evolution equations for the resulting operators in the large-Nc limit, with Nc the number of quark colors. For both TMDs, the evolution equations resum all powers of the double-logarithmic parameter αs ln2(1/x), where αs is the strong coupling constant, which is assumed to be small. Solving these evolution equations numerically (for the Sivers function) and analytically (for the Boer-Mulders function) we arrive at the following leading small-x asymptotics of these TMDs at large Nc:f1T⊥NSx≪1kT2=COxkT21x+C1xkT21x3.4αsNc4πh1⊥NSx≪1kT2=CxkT21x−1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\displaystyle \\begin{array}{l}{f}_{1T}^{\\perp NS}\\left(x\\ll 1,{k}_T^2\\right)={C}_O\\left(x,{k}_T^2\\right)\\frac{1}{x}+{C}_1\\left(x,{k}_T^2\\right){\\left(\\frac{1}{x}\\right)}^{3.4\\sqrt{\\frac{\\alpha_s{N}_c}{4\\pi }}}\\\\ {}{h}_1^{\\perp \ extrm{NS}}\\left(x\\ll 1,{k}_T^2\\right)=C\\left(x,{k}_T^2\\right){\\left(\\frac{1}{x}\\right)}^{-1}.\\end{array}} $$\\end{document}The functions CO(x, {k}_T^2 ), C1(x, {k}_T^2 ), and C(x, {k}_T^2 ) can be readily obtained in our formalism: they are mildly x-dependent and do not strongly affect the power-of-x asymptotics shown above. The function CO, along with the 1/x factor, arises from the odderon exchange. For the sub-eikonal contribution to the quark Sivers function (the term with C1), our result shown above supersedes the one obtained in [3] due to the new contributions identified recently in [4].