Double SM-like Higgs Production at future $e^{+}{e}^{-}$ colliders in the 3-Higgs Doublet Model under the $S_3$ symmetry

In this paper, we present the production of double SM-like Higgs $(h)$ at the future electron positron colliders within the context of $S_3$ model with three Higgs doublets (S3-3H) and no CP violation to describe beyond the Standard Model Higgs Physics. We focus first on the numerically allowed parameter space of the model, taking into account theoretical bounds from perturbative unitarity and vacuum stability, as well as by data at the Large Hadron Collider (LHC) and the Tevatron. The double Higgs production in the S3-3H model can deviate from the SM predictions up to a few orders of magnitude in both the $e^{+}{e}^{-}\to hhZ$ and the $e^{+}{e}^{-}\to hh{\nu }_e{\bar{\nu }}_e$ channels. Thus, our findings indicate that the S3-3H model can lead to measurable deviations from the SM predictions of Higgs production at future $e^{+}{e}^{-}$ colliders. These results highlight the importance of studies at such colliders for searching physics beyond the Standard Model.