Spin-flip gluon generalized transverse-momentum-dependent parton distribution F1 ,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) Re (F_{1 ,2}) as Re (F_{1 ,2})∼(1/x^{) α_{¯ s}(4 ln 2 -8 /3 )}(cos 3 ϕ_{k Δ}+cos ϕ_{k Δ}) . This is done by analytically solving the integro-differential evolution equation for 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 ϕ_{k Δ}+cos ϕ_{k Δ} azimuthal dependence. As the imaginary part of F_{1 ,2}, is related to the spin-dependent odderon or gluon Siver function and scales as Im (F_{1 ,2})∼x⁰ , the positive intercept for 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 unpolarized GTMDs and associated spin-flip processes.