An $O(3)$ spinor, Φ, as a doublet denoted by ${\mathbf{2}}_D$ consists of an $SO(3)$ spinor, ϕ, and its complex conjugate, ${\varphi }^{⁎}$, which form $\mathrm{\Phi }={(\varphi ,{\varphi }^{⁎})}^T$ to be identified with a Majorana-type spinor of $O(4)$. The four gamma matrices ${\mathrm{\Gamma }}_{\mu }$ ($\mu =1\sim 4$) are given by ${\mathrm{\Gamma }}_i=\text{diag.}({\tau }_i,{\tau }_i^{⁎})$ ($i=1,2,3$) and ${\mathrm{\Gamma }}_4=-{\tau }_2\otimes {\tau }_2$, where ${\tau }_i$ denote the Pauli matrices. The rotations and axis-reflections of $O(3)$ are, respectively, generated by ${\mathrm{\Sigma }}_{ij}$ and ${\mathrm{\Sigma }}_{i4}$, where ${\mathrm{\Sigma }}_{\mu \nu }=[{\mathrm{\Gamma }}_{\mu },{\mathrm{\Gamma }}_{\nu }]/2i$. While Φ 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 $SO(3)\times {\mathit{Z}}_2$ equivalent to $O(3)$, where the ${\mathit{Z}}_2$ 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.

一个 $Ø（3）$ 旋子Φ ${\mathbf{2}}_d$ 由一个 $\mathrm{小号}Ø（3）$旋子ϕ及其复共轭，${\varphi }^{⁎}$，哪种形式 $\mathrm{\Phi }={（\varphi ，{\varphi }^{⁎}）}^{Ť}$ 被识别为马约拉那型的 $Ø（4）$。四种伽玛矩阵${\mathrm{\Gamma }}_{\mu }$ （$\mu =\mathrm{1个}〜4$）由 ${\mathrm{\Gamma }}_{\mathrm{一世}}=\text{诊断。}（{\tau }_{\mathrm{一世}}，{\tau }_{\mathrm{一世}}^{⁎}）$ （$\mathrm{一世}=\mathrm{1个}，2，3$）和 ${\mathrm{\Gamma }}_4=-{\tau }_2\otimes {\tau }_2$，在哪里 ${\tau }_{\mathrm{一世}}$表示保利矩阵。的旋转和轴反射$Ø（3）$ 分别是由 ${\mathrm{\Sigma }}_{\mathrm{一世}Ĵ}$ 和 ${\mathrm{\Sigma }}_{\mathrm{一世}4}$，在哪里 ${\mathrm{\Sigma }}_{\mu \nu }=[{\mathrm{\Gamma }}_{\mu }，{\mathrm{\Gamma }}_{\nu }]/2\mathrm{一世}$。Φ被视为标量，而费米离子$Ø（3）$ 旋转器是由一个 $\mathrm{小号}Ø（3）$双重Dirac旋子及其电荷共轭物。这些$Ø（3）$旋转子被限制为中性的，并且不能携带标准模型量子数，因为它们包含粒子和反粒子。我们的$Ø（3）$棘是暗物质的候选者。的$Ø（3）$通过明确包括相互作用的共轭来考虑相互作用的不变性，就可以看到粒子物理学中的对称性。可以引入基于$\mathrm{小号}Ø（3）\times {\mathit{ž}}_2$ 相当于 $Ø（3）$，其中 ${\mathit{ž}}_2$ 奇偶性由 $ü（\mathrm{1个}）$电荷使粒子为1，反粒子为-1。的$\mathrm{小号}Ø（3）$ 和 $ü（\mathrm{1个}）$ 规范玻色子最终转变为的轴向矢量 $Ø（3）$ 和的伪标量 $Ø（3）$， 分别。此属性与以下项的一致定义有关：$Ø（3）$或O（3）转换的自旋子的U（1）电荷。为了了解我们暗物质模型的可行性，我们讨论基于暗物质的标量暗物质现象学$ü（\mathrm{1个}）$ 量规模型。