A limit on top quark pair production at future electron-proton colliders

The ratio of the structure functions for deep inelastic scattering in top pair production at future electron-proton colliders is analyzed at a fixed $\sqrt{s}$ and $Q^2$ relative to the minimum value of $x$ given by $Q^2/s$ using collinear factorization. This compact formula for the ratio $F_{L2}(Q^2/s,Q^2,m_{t})$ is useful for extracting a bound on the top structure function. The reduced cross-section for top production at this limit is determined, establishing a bound value for $t\overline{t}$ production at the LHeC and FCC-eh center-of-mass energies based on renormalization scales. The modification of the Bjorken scaling is applied to this bound of the reduced top cross-section at the renormalization scale $μ^2=Q^2+4m_{t}^{2}$, which improves the scaling quantity at $Q^2{