Flavor violating signatures of neutral Higgs bosons at the LHeC

We analyze the prospect for observing the lightest and heaviest CP-even neutral Higgs bosons ($\phi=h,\;H$) in their decays to flavor violating $b\bar{s}$ (with charge conjugate) final states at the future Large Hadron electron Collider (LHeC), in the framework of the Two Higgs Doublet Model type III (2HDM-III) assuming a four-zero texture in the Yukawa matrices and a general Higgs potential. We consider the charge current production processes $e^- p \to \nu_e \phi q_f$ ($\phi= h, \; H$), with the flavor violating decays of the Higgs bosons. We consider scenarios of the model which are consistent with current experimental data from flavor and Higgs physics. We also consider the most relevant Standard Model (SM) backgrounds. We find that the lightest (SM-like) CP-even Higgs boson $h$ and the heaviest one $ H$ (with mass up to 150 GeV) would have 1--2 $\Sigma$ and 1 $\Sigma$ significances, respectively, with 100 fb$^{-1}$ of data. At the end of the LHeC running, the significances would be increased by ten times and the detection of both Higgs bosons ($h, H$) would be guaranteed.