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

Abstract

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

Quark and scalar propagators at next-to-eikonal accuracy in the CGC through a dynamical background gluon field

“…The helicity-dependent generalization of the nonlinear rapidity evolution was studied in [40,41] at NEik order. In a recent study [42], the quark Sivers function was studied with subeikonal evolution. In [43,44], NEik corrections to both quark and gluon propagators were derived and used to study the high-energy operator product expansion (OPE) of two electromagnetic currents.…”