Probing the $h c\bar{c} $ coupling at a Future Circular Collider in the electron-hadron mode

We study the production of a neutral Higgs boson at a Future Circular Collider in the electron-hadron mode (FCC-eh) through the leading process $e^- p \to \nu_e h q $ assuming the decay channel $h \to c \bar{c} $, where $h$ is the Standard Model (SM)-like state discovered at the Large Hadron Collider (LHC). This process is studied in the context of a 2-Higgs Doublet Model Type III (2HDM-III) embedding a four-zero texture in the Yukawa matrices and a general Higgs potential, where both Higgs doublets are coupled with up- and down-type fermions. Flavour Changing Neutral Currents (FCNCs) are well controlled by this approach through the adoption of a suitable texture once flavour physics constraints are taken in account. Considering the parameter space where the signal is enhanced and in agreement with both experimental data and theoretical conditions, we analyse the aforementioned signal by taking into account the most important SM backgrounds, separating $c$-jets from light-flavour and gluon ones as well as $b$-jets by means of efficient flavour tagging. We find that the $h c \bar{c} $ coupling strength can be accessed with good significance after a luminosity of $1$ ab$^{-1}$ for a 50 TeV proton beam and a 60 GeV electron one, the latter with a 80% (longitudinal) polarisation.