Inclusive diffraction at the LHeC and FCC-eh

We analyse the possibilities for the study of inclusive diffraction offered by future electron-proton/nucleus colliders in the TeV regime, the Large Hadron-electron Collider as an upgrade of the HL-LHC and the Future Circular Collider in electron-hadron mode. Compared to $ep$ collisions at HERA, we find an extension of the available kinematic range in $x$ by a factor of order $20$ and of the maximum $Q^2$ by a factor of order $100$ for LHeC, while the FCC version would extend the coverage by a further order of magnitude both in $x$ and $Q^2$. This translates into a range of available momentum fraction of the diffractive exchange with respect to the hadron ($\xi$), down to $10^{-4}-10^{-5}$ for a wide range of the momentum fraction of the parton with respect to the diffractive exchange ($\beta$). Using the same framework and methodology employed in previous studies at HERA and under very conservative assumptions for the luminosities and systematic errors, we find an improvement in the extraction of diffractive parton densities from fits to reduced cross sections for inclusive coherent diffraction in $ep$ by about an order of magnitude. We analyse the sensitivity to kinematic cuts and variations of the fit framework. We also note sensitivity to the shape of the gluon distribution, and to physics beyond linear twist-2 DGLAP evolution at moderate $Q^2$. For $eA$, we find that an extraction of the currently unmeasured nuclear diffractive parton densities is possible with similar accuracy to that in $ep$.