Quark Sivers Function at Small $x$: Spin-Dependent Odderon and the Sub-Eikonal Evolution

TL;DR

This paper develops a small-$x$ theoretical framework for the quark Sivers function, revealing the dominant role of the spin-dependent odderon and deriving new evolution equations for sub-eikonal corrections.

Contribution

It constructs the polarized Wilson line operator at sub-sub-eikonal order and derives novel small-$x$ evolution equations for the quark Sivers function.

Findings

The spin-dependent odderon dominates the small-$x$ quark Sivers function.

New evolution equations resum double-logarithmic powers of $ ext{ln}(1/x)$.

The quark Sivers function has a leading $1/x$ behavior with slowly varying coefficient.

Abstract

We apply the formalism developed earlier 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. Our conclusion is also similar to the case of the gluon Sivers function derived by Boer, Echevarria, Mulders and Zhou (see also the work by Szymanowski and Zhou). 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 \begin{align} f_{1 \: T}^{\perp \: q} (x, k_T^2) = C_O (x, k_T^2) \, \frac{1}{x} + C_1 (k_T^2) \, \left( \frac{1}{x} \right)^0 + \ldots \end{align} with the function $C_O (x, k_T^2)$ varying slowly with $x$ and the ellipsis denoting the sub-asymptotic and sub-sub-eikonal (order-$x$) corrections.