An $S_4$ model inspired from self-complementary neutrino mixing

We build an $S_4$ model for neutrino masses and mixings based on the self-complementary (SC) neutrino mixing pattern. The SC mixing is constructed from the self-complementarity relation plus $\delta_{\rm CP}=-\frac{\pi}{2}$. We elaborately construct the model at a percent level of accuracy to reproduce the structure given by the SC mixing. After performing a numerical study on the model's parameter space, we find that in the case of normal ordering, the model can give predictions on the observables that are compatible with their $3\sigma$ ranges, and give predictions for the not-yet observed quantities like the lightest neutrino mass $m_1\in [0.003,0.010]$ eV and the Dirac CP violating phase $\delta_{\rm CP}\in[256.72^\circ,283.33^\circ]$.