Spin-flip gluon GTMD $F_{1,2}$ at small-$x$

Until recently the spin-flip processes in the deep inelastic scatterings are thought to be suppressed in the high energy. We found a positive intercept for the spin-flip generalized transverse momentum-dependent parton distribution (GTMDs) ${\rm Re}(F_{1,2})$ as, \begin{eqnarray} {\rm Re}(F_{1,2}) \sim \left(\frac{1}{x}\right)^{{\bar \alpha}_s\left(4\ln2-8/3\right)} \left(\cos 3\phi_{k\Delta} +\cos \phi_{k\Delta}\right). \nonumber \end{eqnarray} This is done by analytically solving the integro-differential evolution equation for ${\rm Re}(F_{1,2})$, recently proposed by Hatta and Zhou, in the dilute regime. Interestingly, the surviving solution corresponds to conformal spin $n=2$ and carries an explicit $\cos 3\phi_{k\Delta} + \cos \phi_{k\Delta}$ azimuthal dependence. As the imaginary part of $F_{1,2}$, is related to the spin-dependent odderon or gluon Siver function and scales as ${\rm Im}(F_{1,2}) \sim x^{0}$, the positive intercept for ${\rm Re}(F_{1,2})$, implies that it is expected to dominate over the gluon Siver function in the small-$x$ limit - and may directly impact the modeling of unpolarised GTMDs and associated spin-flip processes.