Quark Sivers Function at Small-x: Return of a Spin-Dependent Odderon

We apply the formalism developed earlier $[1,2]$ for studying the small-$x$ asymptotics of transverse momentum dependent parton distribution functions (TMDs) to construct the small-$x$ asymptotics of the quark Sivers function. We explicitly construct the transversely polarized, fundamental ``Wilson line'' operator to sub-sub-eikonal order. We then express the Sivers function in terms of dipole scattering amplitudes containing the transversely polarized ``Wilson line'', and show that the main term which contributes to the Sivers function is the spin-dependent odderon, similar to the case of the gluon Sivers function derived by Boer, Eschevarria, Mulders and Zhou [3] (see also [4]). \vskip 0.2in \noindent [1] Y. V. Kovchegov and M. D. Sievert, Phys. Rev. D99, 054032 (2019). \newline [2] Y. V. Kovchegov and M. D. Sievert, Phys. Rev. D99, 054033 (2019). \newline [3] D. Boer, M. G. Echevarria, P. Mulders, and J. Zhou, Phys. Rev. Lett. 116, 122001 (2016). \newline [4] L. Szymanowski and J. Zhou, Phys. Lett. B760, 249 (2016).