Lensing Mechanism Meets Small-$x$ Physics: Single Transverse Spin Asymmetry in $p^{\uparrow}+p$ and $p^{\uparrow}+A$ Collisions

We calculate the single transverse spin asymmetry (STSA) in polarized proton-proton ($p^{\uparrow}+p$) and polarized proton-nucleus ($p^{\uparrow}+A$) 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. 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 $1/N_c^2$ 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 and with the preliminary data on $A_N$ reported by the STAR collaboration.