Collinear approach for top quark production at ep colliders

An approximately analysis study of the collinear approach for $t\overline{t}$ production at the Future Circular Collider hadron-electron (FCC-he) and the Large Hadron electron Collider (LHeC) is performed. To study heavy quark production processes, the collinear generalized double asymptotic scaling (DAS) approach in a wide rang of $Q^{2}$ is used. The reduced cross sections for top quark pair production at the FCC-he (for $Q^{2}{\geq}m^{2}_{t}$) and LHeC (for $Q^{2}{\leq}m^{2}_{t}$) center-of-mass energies at the high inelasticity are derived. These results may be considering for both the future collider experiments at center-of-mass energy frontier and the improvement of the phenomenological models on kinematics of low values of the Bjorken variable $x$. The evolution of the gluon density for heavy-quark production in DIS is obtained at low $x$ to include effects of heavy-quark masses. A nonlinear modification of these results shown that the nonlinear corrections, with respect to the top quark mass, are very small for $Q^{2}{\propto}m^{2}_{t}$.