The eikonal spin-dependent Odderon and gluon Sivers function of a proton, and its small-$x$ evolution

The matrix element in the proton of the eikonal Odderon operator, with a helicity flip, has been shown to correspond to the dipole gluon Sivers function. We employ a three quark light-front model of the proton to determine the Sivers function at moderately small $x_0\sim 0.1$ and transverse momentum $k_{\perp }\lesssim 1$~GeV. The model light-cone (LC) wave function predicts the properties of $x{f}_{1T}^{\perp g}(x,k_{\perp })$ such as its overall magnitude, the position of its peak in $k_{\perp }$, and its behavior at small $k_{\perp }$. We then compute numerically the BFKL anomalous dimension characterizing the power-law tail at $k_{\perp }\gtrsim 1.5$~GeV of the gluon Sivers function at small LC momentum fractions, ${\alpha }_s\log x_0\mathrm{/}x=1$: $x{f}_{1T}^{\perp g}(x,k_{\perp })\sim k_{\perp }^{-3.3}$.