T-odd leading-twist quark TMDs at small x

Abstract

A bstract 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- N c limit, with N c the number of quark colors. For both TMDs, the evolution equations resum all powers of the double-logarithmic parameter α s ln 2 (1 /x ), where α s 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 N c : $$ {\displaystyle \begin{array}{l}{f}_{1T}^{\perp NS}\left(x\ll 1,{k}_T^2\right)={C}_O\left(x,{k}_T^2\right)\frac{1}{x}+{C}_1\left(x,{k}_T^2\right){\left(\frac{1}{x}\right)}^{3.4\sqrt{\frac{\alpha_s{N}_c}{4\pi }}}\\ {}{h}_1^{\perp \textrm{NS}}\left(x\ll 1,{k}_T^2\right)=C\left(x,{k}_T^2\right){\left(\frac{1}{x}\right)}^{-1}.\end{array}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtable> <mml:mtr> <mml:mtd> <mml:msubsup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⊥</mml:mo> <mml:mi>NS</mml:mi> </mml:mrow> </mml:msubsup> <mml:mfenced> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mfenced> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>O</mml:mi> </mml:msub> <mml:mfenced> <mml:mi>x</mml:mi> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mfenced> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>x</mml:mi> </mml:mfrac> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mfenced> <mml:mi>x</mml:mi> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mfenced> <mml:msup> <mml:mfenced> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>x</mml:mi> </mml:mfrac> </mml:mfenced> <mml:mrow> <mml:mn>3.4</mml:mn> <mml:msqrt> <mml:mfrac> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:mfrac> </mml:msqrt> </mml:mrow> </mml:msup> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:msubsup> <mml:mi>h</mml:mi> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>⊥</mml:mo> <mml:mi>NS</mml:mi> </mml:mrow> </mml:msubsup> <mml:mfenced> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:mfenced> <mml:mi>x</mml:mi> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mfenced> <mml:msup> <mml:mfenced> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>x</mml:mi> </mml:mfrac> </mml:mfenced> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>.</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:math> The functions C O ( x, $$ {k}_T^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> ), C 1 ( x, $$ {k}_T^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> ), and C ( x, $$ {k}_T^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> ) 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 C O , along with the 1 /x factor, arises from the odderon exchange. For the sub-eikonal contribution to the quark Sivers function (the term with C 1 ), our result shown above supersedes the one obtained in [3] due to the new contributions identified recently in [4].

Topics & Concepts