Top cross section in the LHeC and FCC-he energy range

Predictions of the top-pair production cross section in electron-proton (ep) collisions at the future circular collider hadron-electron (FCC-he) and the large hadron electron collider (LHeC) in the leading order (LO) and the next-to-leading order (NLO) approximations in perturbative quantum chromodynamics (QCD) are presented. Numerical results for the top structure functions and the ratio of structure functions are computed at the renormalization scale $\mu^{2}=Q^{2}+4m^{2}_{t}$ in the collinear generalized double asymptotic scaling (DAS) approach. These results are presented in terms of the effective parameters of the parameterization of $F_{2}(x,Q^{2})$, where relying on a Froissart-bounded. These quantitative results can be important for future collider experiments at the center-of-mass energy frontier and the improvement of the phenomenological models for the development of the cosmic ray cascades in ultra-high-energy domain. We obtained the uncertainties because of the power-law behavior $x^{-\Delta}$ of the collinear parton distribution functions (PDFs), the factorization, and renormalization scales, on the top properties at high inelasticity for the FCC-he center-of-mass (CM) energy at the NLO approximation.