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

published in Nuclear Physics B 956 (2020) 115031 (May 16, 2020) with Teruyuki Kitabayashi

arXiv:1903.04439

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

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