Quark sivers function at small x: spin-dependent odderon and the sub-eikonal evolution

We apply the formalism developed earlier [1, 2] for studying transverse momentum dependent parton distribution functions (TMDs) at small Bjorken x to construct the small- x asymptotics of the quark Sivers function. First, we explicitly construct the complete fundamental polarized Wilson line operator to sub-sub-eikonal order: this object can be used to study a variety of quark TMDs at small x. We then express the quark Sivers function in terms of dipole scattering amplitudes containing various components of the polarized Wilson line and show that the dominant (eikonal) term which contributes to the quark Sivers function at small x is the spin-dependent odderon, confirming the recent results of Dong, Zheng and Zhou [3]. Our conclusion is also similar to the case of the gluon Sivers function derived by Boer, Echevarria, Mulders and Zhou [4] (see also [5]). We also analyze the sub-eikonal corrections to the quark Sivers function using the constructed polarized Wilson line operator. We derive new small- x evolution equations re-summing double-logarithmic powers of alpha(s) ln(2)(1/x) with alpha(s) the strong coupling constant. We solve the corresponding novel evolution equations in the large-N-c limit, obtaining a sub-eikonal correction to the spin-dependent odderon contribution. We conclude that the quark Sivers function at small x receives contributions from two terms and is given by f(1)(T)(perpendicular to)(q)(x, k(T)(2)) = C-O(x, k(T)(2))1/x + C-1(k(T)(2)) (1/x)(0) + ... with the function C-O (x, k(T)(2)) varying slowly with x and the ellipsis denoting the subasymptotic and sub-sub-eikonal (order-x) corrections.

Automated summary

This study investigates the small-x behavior of the quark Sivers function using a specific formalism, revealing that it is mainly influenced by the spin-dependent odderon contribution at small x. Additionally, new evolution equations are derived to provide insights into the structure of the quark Sivers function at small x.