Probing non-standard $HVV (V=W, Z)$ couplings in single Higgs production at future electron-proton collider

The couplings of the Higgs boson ($H$) with massive gauge bosons of weak interaction ($V= W, Z$), can be probed in single Higgs boson production at the proposed future Large Hadron-Electron Collider (LHeC). In the collision of an electron with a proton, single Higgs production takes place via so-called charged-current ($e^-p \to \nu_e H j$) and neutral-current ($e^-p \to e^-H j$) processes. We explore the potential of the azimuthal angle correlation between the forward jet and scattered neutrino or electron in probing the non-standard $HVV$ couplings at the collider center-of-mass energy of $\sqrt{s} \approx 1.3$~TeV. We choose the most general modifications (of $CP$-even and $CP$-odd nature) to these couplings due to new physics effects beyond the standard model. We derive exclusion limits on new physics parameters of $HVV$ couplings as a function of integrated luminosity at $95\%$ C.L. using the azimuthal angular correlations in charged- and neutral-current processes. We find that using 1000 $fb^{-1}$ data, the standard model-like new physics parameters in $HWW$ and $HZZ$ couplings can be constrained with accuracies of $4\%$ and $15\%$, respectively. The least constrained $CP$-even parameters of $HWW$ coupling can be as large as 0.04, while those of $HZZ$ coupling can have values around 0.31. Allowed values of $CP$-odd parameters in $HWW$ and $HZZ$ couplings are found to be around 0.14 and 0.34, respectively. We also study changes in the allowed values of non-trivial new physics parameters in the presence of other parameters.