Minimal Dirac seesaw dark matter

We propose a minimal Type-I Dirac seesaw which accommodates a thermal scalar dark matter (DM) candidate protected by a charge conjugation symmetry in dark sector $C_{\rm dark}$, without introducing any additional field beyond the ones taking part in the seesaw. A $Z_4$ symmetry is introduced to realise the tree level Dirac seesaw while the Majorana mass terms are prevented by an unbroken global lepton number symmetry. While the spontaneous $Z_4$ breaking together with electroweak symmetry breaking lead to the generation of light Dirac neutrino mass, it also results in the formation of domain walls. These cosmologically catastrophic walls can be made to annihilate away by introducing bias terms while also generating stochastic gravitational waves (GW) within reach of near future experiments like \texttt{LISA}, \texttt{BBO}, $\mu$-\texttt{ARES} etc. The scalar DM parameter space can be probed at direct and indirect search experiments. Light Dirac neutrinos also enhance the relativistic degrees of freedom $N_{\rm eff}$ within reach of future cosmic microwave background (CMB) experiments. The model can also explain the observed baryon asymmetry via Dirac leptogenesis.