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 re- cent 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 powers of αs ln2(1/x) with αs the strong coupling constant. We solve the corresponding novel evolution equations in the large-Nc 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$$ {f}_{1T}^{\perp q}\left(x,{k}_T^2\right)={C}_O\left(x,{k}_T^2\right)\frac{1}{x}+{C}_1\left({k}_T^2\right){\left(\frac{1}{x}\right)}^0+\cdots $$ f 1 T ⊥ q x k T 2 = C O x k T 2 1 x + C 1 k T 2 1 x 0 + ⋯ with the function CO(x,$$ {k}_T^2 $$ k T 2 ) varying slowly with x and the ellipsis denoting the subasymptotic and sub-sub-eikonal (order-x) corrections.

We calculate the single transverse spin asymmetry in polarized proton-proton (p ↑ þ p) and polarized proton-nucleus (p ↑ þ A) collisions (A N) generated by a partonic lensing mechanism. The polarized proton is considered in the quark-diquark model while its interaction with the unpolarized target is calculated using the small-x/saturation approach, which includes multiple rescatterings and small-x evolution. The phase required for the asymmetry is caused by a final-state gluon exchange between the quark and diquark, as is standard in the lensing mechanism of Brodsky, Hwang, and Schmidt [Phys. Lett. B 530, 99 (2002)]. Our calculation combines the lensing mechanism with small-x physics in the saturation framework. The expression we obtain for the asymmetry A N of the produced quarks has the following properties: (i) The asymmetry is generated by the dominant elastic scattering contribution and the 1=N 2 c suppressed inelastic contribution (with N c the number of quark colors). (ii) The asymmetry grows or oscillates with the produced quark's transverse momentum p T until the momentum reaches the saturation scale Q s , and then only falls off as 1=p T for larger momenta. (iii) The asymmetry decreases with increasing atomic number A of the target for p T below or near Q s , but is independent of A for p T significantly above Q s. We discuss how these properties may be qualitatively consistent with the data on A N published by the PHENIX Collaboration [

We present the first global analysis of generalized parton distributions (GPDs) combing lattice quantum chromodynamics (QCD) calculations and experiment measurements including global parton distribution functions (PDFs), form factors (FFs) and deeply virtual Compton scattering (DVCS) measurements. Following the previous work where we parameterize GPDs in terms of their moments, we extend the framework to allow for the global analysis at non-zero skewness. Together with the constraints at zero skewness, we fit GPDs to global DVCS measurements from both the recent JLab and the earlier Hadron-Electron Ring Accelerator (HERA) experiments with two active quark flavors and leading order QCD evolution. With certain choices of empirical constraints, both sea and valence quark distributions are extracted with the combined inputs, and we present the quark distributions in the proton correspondingly. We also discuss how to extend the framework to accommodate more off-forward constraints beyond the small ξ expansion, especially the lattice calculated GPDs.

We present the calculation of the leading contribution to the quark Sivers function at small-Bjorken x as in [4]. This calculation uses the high energy scattering approximation and operator formalism developed in [3, 15] to obtain a dominant contribution to the quark Sivers function coming from the spin-dependent odderon, in agreement with the results of [7]. We then calculate this dominant contribution in the diquark model of the proton to obtain a small-x estimate for the Sivers function.