Scotoseesaw mechanism from a $$Z_3$$ symmetry of matter

Abstract We show that the neutrino mass generation and the dark matter stability can be governed by the center of the QCD group, which is a $$Z_3$$ Z 3 group. Three right-handed neutrinos $$N_{1,2,3R}$$ N 1 , 2 , 3 R transform under $$Z_3$$ Z 3 as $$1,w,w^2$$ 1 , w , w 2 , where $$w=e^{i2\pi /3}$$ w = e i 2 π / 3 is the cube root of unity, and they couple to usual lepton doublets via the usual Higgs doublet H and two new scalar doublets $$\eta ,\chi $$ η , χ , which transform under $$Z_3$$ Z 3 as $$1,w^2,w$$ 1 , w 2 , w , respectively. This leads to a scotoseesaw mechanism in which the seesaw and scotogenic neutrino mass generations are induced by the Majorana $$N_{1R}$$ N 1 R mass and the Dirac $$N_{2,3R}$$ N 2 , 3 R mass, respectively. Although the lightest of the $$Z_3$$ Z 3 fields is stabilized, responsible for dark matter, the model lacks an explanation for relic density and/or direct detection. The issue can be solved in a $$U(1)_{B-L}$$ U ( 1 ) B - L gauge completion of the model, for which the center of the QCD group is isomorphic to $$Z_3=\{1,T,T^2\}$$ Z 3 = { 1 , T , T 2 } for $$T=w^{3(B-L)}$$ T = w 3 ( B - L ) .