Spinorial Structure of $O(3)$ and Application to Dark Matter

An $O(3)$ spinor, $\Phi$, as a doublet denoted by ${\bf 2}_D$ consists of an $SO(3)$ spinor, $\phi$, and its complex conjugate, $\phi^\ast$, which form $\Phi=\left(\phi,\phi^\ast\right)^T$ to be identified with a Majorana-type spinor of $O(4)$. The four gamma matrices $\Gamma_\mu$ ($\mu=1\sim 4$) are given by $\Gamma_i=\text{diag.}\left(\tau_i,\tau^\ast_i\right)$ ($i=1,2,3$) and $\Gamma_4=-\tau_2\otimes\tau_2$, where $\tau_i$ denote the Pauli matrices. The rotations and axis-reflections of $O(3)$ are, respectively, generated by $\Sigma_{ij}$ and $\Sigma_{i4}$, where $\Sigma_{\mu\nu}=[\Gamma_\mu,\Gamma_\nu]/2i$. While $\Phi$ is regarded as a scalar, a fermionic $O(3)$ spinor is constructed out of an $SO(3)$ doublet Dirac spinor and its charge conjugate. These $O(3)$ spinors are restricted to be neutral and cannot carry the standard model quantum numbers because they contain particles and antiparticles. Our $O(3)$ spinors serve as candidates of dark matter. The $O(3)$ symmetry in particle physics is visible when the invariance of interactions is considered by explicitly including their complex conjugates. It is possible to introduce a dark gauge symmetry based on $SO(3)\times\boldsymbol{Z}_2$ equivalent to $O(3)$, where the $\boldsymbol{Z}_2$ parity is described by a $U(1)$ charge giving 1 for a particle and $-1$ for an antiparticle. The $SO(3)$ and $U(1)$ gauge bosons turn out to transform as the axial vector of $O(3)$ and the pseudoscalar of $O(3)$, respectively. This property is related to the consistent definition of the nonabelian field strength tensor of $O(3)$ or of the U(1) charge of the O(3)-transformed spinor. To see the feasibility of our dark matter models, we discuss scalar dark matter phenomenology based on the dark $U(1)$ gauge model.