Exploring Leptogenesis, WIMP Dark Matter, and Gravitational Waves in an extended Scalar Framework

We explore extensions of type I seesaw framework with a scalar mediator ($\Phi$) connecting to a complex scalar dark field ($S$), and right handed neutrinos ($N_i$), with an aim to correlate neutrino mass generation, leptogenesis, and dark matter. $\mathcal{Z}_4\times CP$ turns out to be a phenomenologically viable choice of the extended symmetry, which can accommodate a dimension five effective interaction $\bar{l}_L^\alpha \tilde{H}\Phi N_i$, involving the SM lepton isodoublet ${l}_L$, and Higgs $H$; prohibiting the canonical Yukawa term $\bar{l}_L^\alpha \tilde{H} N_i$. The $\mathcal{Z}_{4}$ symmetry is spontaneously broken via the vacuum expectation value (VEV) of the $\Phi$ filed, which directly affects neutrino mass generation and leptogenesis; while the $CP$ symmetry stabilises one component of $S$, making it a viable dark matter candidate. The discrete symmetry breaking creates domain wall, which needs to be annihilated before the over-closure of the Universe. This paves the way for gravitational wave signal associated with the model set up, which probes the symmetry breaking scale, and indirectly connects to the other phenomena.