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We apply the formalism developed earlier for studying transverse momentumdependent parton distribution functions (TMDs) at small Bjorken $x$ toconstruct the small-$x$ asymptotics of the quark Sivers function. First, weexplicitly construct the complete fundamental "polarized Wilson line" operatorto sub-sub-eikonal order: this object can be used to study a variety of quarkTMDs at small-$x$. We then express the quark Sivers function in terms of dipolescattering amplitudes containing various components of the "polarized Wilsonline" and show that the dominant (eikonal) term which contributes to the quarkSivers function at small $x$ is the spin-dependent odderon, confirming therecent results of Dong, Zheng and Zhou. Our conclusion is also similar to thecase 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-eikonalcorrections to the quark Sivers function using the constructed "polarizedWilson line" operator. We derive new small-$x$ evolution equations re-summingdouble-logarithmic powers of $\alpha_s \, \ln^2 (1/x)$ with $\alpha_s$ thestrong coupling constant. We solve the corresponding novel evolution equationsin the large-$N_c$ limit, obtaining a sub-eikonal correction to thespin-dependent odderon contribution. We conclude that the quark Sivers functionat 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 ellipsisdenoting the sub-asymptotic and sub-sub-eikonal (order-$x$) corrections.

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