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Spontaneous Peccei-Quinn symmetry breaking renders sterile neutrino, axion and $χ$boson to be candidates for dark matter particles

We study the Peccei-Quinn (PQ) symmetry of the sterile right-handed neutrino sector and the gauge symmetries of the Standard Model. Due to four-fermion interactions, spontaneous breaking of these symmetries at the electroweak scale generates top-quark Dirac mass and sterile-neutrino Majorana mass. The top quark channel yields massive Higgs, $W^\pm$ and $Z^0$ bosons. The sterile neutrino channel yields the heaviest sterile neutrino Majorana mass, sterile Nambu-Goldstone axion (or majoron) and massive scalar $χ$boson. Four-fermion operators effectively induce their tiny couplings to SM particles. We show that a sterile QCD axion is the PQ solution to the strong CP problem. The lightest and heaviest sterile neutrinos ($m_N^e\sim 10^2$ keV and $m_N^τ\sim 10^2$ GeV), a sterile QCD axion ($m_a< 10^{-8}$ eV, $g_{aγ}< 10^{-13} {\rm GeV}^{-1}$) and a Higgs-like $χ$boson ($m_χ\sim 10^2$ GeV) can be dark matter particle candidates, for the constraints of their tiny couplings and long lifetimes inferred from the $W$-boson decay width, Xenon1T and precision fine-structure-constant experiments. The axion and $χ$boson couplings to SM particles are below the values reached by current laboratory experiments and astrophysical observations for directly or indirectly detecting dark matter particles.