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

Spin-flip processes in the deep inelastic scatterings are thought to be suppressed in the high energy. Recent studies by Hatta and Zhou, however, show that gluon generalized parton distribution (GPD) $E_g$, which is associated with spin-flip processes, exhibits the Regge behavior identical to the BFKL Pomeron. This was done by deriving the small-$x$ evolution equation for the real part of $F$-type spin-flip gluon GTMDs $F_{1,2}$. In this article, we have shown that though the evolution equation for ${\rm Re}(F_{1,2})$ has IR poles - they all mutually cancel - making the equation IR finite and self-consistent. We also have analytically solved the equations in the dilute regime and find small-$x$ asymptotics of the GTMDs ${\rm Re}(F_{1,2})$ as \begin{eqnarray} {\rm Re}(F_{1,2}) \sim \left(\frac{1}{x}\right)^{\alpha_s\left(4\ln2-8/3\right)} \left(\cos 3\phi_{k\Delta} +\cos \phi_{k\Delta}\right). \nonumber \end{eqnarray} 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.