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

We apply the formalism developed earlier (Kovchegov and Sievert, 2019) 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. 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 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 (2019). We analyze the sub-eikonal corrections to the quark Sivers function and construct new small-x evolution equations resumming double-logarithmic powers of α_s ln² (1/x) with α_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.