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

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 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.


I. INTRODUCTION
The cosmological observation of dark matter [1] has inspired theoretical interest in seeking possible physics of dark matter [2]. However, theoretical description of dark matter to date remains unclear and a candidate of dark matter is not provided by the standard model. A simple candidate exhibits the property that dark matter does not couple to the standard model particles and is present in the so-called dark sector [3]. The dark sector (or hidden sector) may contain new particles that couple only indirectly to ordinary matter. These new particles are expected to have masses well below the weak-scale. The dark sector particles communicate with the standard model particles via "dark matter portals". The typical dark matter portal includes the coupling of Higgs particle to dark scalars (as a Higgs portal), of flavor neutrinos to sterile neutrinos (as a neutrino portal) and of photons to dark photons (as a vector portal) [4].
The key issue to discuss is how to realize the constraint on the dark sector particles that they do not have direct couplings to ordinary matter. The appropriate invariance of the dark sector based on certain symmetries may forbid the dark sector particles to couple to ordinary matter. Discrete symmetries such as Z 2 and a U (1) symmetry are the simpler candidates that also ensure stability of dark matter although the origin of the symmetries is not * Electronic address: yasue@keyaki.cc.u-tokai.ac.jp naturally understood except for the need to constrain the dark sector.
The constraint on the dark sector is naturally satisfied if dark matter transforms as the recently advocated spinorial doublet of the O(3) symmetry [5]. We have presented an irreducible representation 2 D as the spinorial doublet of O(3), which is applied to models of quarks and leptons possessing the discrete S 4 symmetry that has been considered as a promising flavor symmetry of quarks and leptons [6][7][8][9]. Since 2 D ⊗ 2 D = 1 ⊕ 3, the O(3) spinor provides S 4 -invariant couplings to the standard model particles. Since S 4 is known as a subgroup of O(3), the S 4 spinor can be based on a four component O(3) spinor, which is composed of the two component SO(3) spinor and its complex conjugate. A remarkable feature is that the O(3) spinor as an elementary particle contains a particle as the SO(3) spinor and an antiparticle as its conjugate in the same multiplet of 2 D so that our O(3) spinors cannot have no quantum numbers of the standard model. It is clear that the definition of the O(3) spinor itself forbids dark sector particles to directly couple to ordinary matter if dark matter consists of the O(3) spinor.
In this paper, we would like to enlarge our previous argument [5] on the possible existence of an O(3) spinor on the basis of the four component spinor of the O(4) symmetry and to include the physical aspect of the O(3) spinor. Since particle physics involves fermionic degrees of freedom, our O(3) spinors must include a fermionic O(3) spinor. To introduce the fermionic O(3) spinor needs careful examination because it is simultaneously the Dirac spinor. As expected, it is understood that the charge conjugate is employed instead of the complex conjugate to be consistent with the Dirac spinor. We have to confirm that such a fermionic spinor correctly transforms under the O(3) transformation so that the internal O(3) symmetry becomes orthogonal to the Lorentz symmetry.
It is further demonstrated that a gauged O(3) symmetry is based on the mathematical equivalence of O(3) to SO(3) × Z 2 , where Z 2 is generated by the parity inversion. For the O(3) spinor, it is found that Z 2 is described by a spinorial Z 2 parity operator, whose eigenvalues are 1 for the particle as the SO(3) spinor and −1 for the antiparticle as its conjugate that does imply the appearance of a U (1) symmetry. The typical feature of the gauged O(3) symmetry is that the gauge bosons transform as the axial vector of O(3) (or the pseudoscalar of O (3)). This property is related to the consistent definition of the nonabelian field strength tensor (or of the U (1) charge of the O(3)-transformed spinor).
The present article is organized as follows: In Sec. II, we clarify spinorial structure of O(3) based on the O(4) symmetry. In Sec.III, we discuss application of the O(3) spinor to particle physics. For a scalar and a fermion taken as the O(3) spinor, we construct O(3)-invariant lagrangians. The consistency with the Dirac spinor is clarified. Also discussed is the possible inclusion of the gauged O(3) symmetry based on SO(3) × Z 2 . Section IV deals with discussions on our candidates of dark matter and dark gauge bosons. A variety of implementation of dark gauge symmetries based on O(3) is discussed. The final section Sec.V is devoted to summary and discussions.

A. Manifestation of Spinorial Structure
We have advocated the use of the O(3) spinor doublet as an S 4 spinor representation, which is composed of an SO(3) spinor φ: from which Φ as the O(3) spinor is defined to be: as a four component spinor, transforming as 2 D [5]. Since O(3) is contained in the O(4) symmetry, which enables us to discuss the parity inversion, we first introduce the standard form of the gamma matrices γ µ (µ = 1 ∼ 4): where τ i (i = 1, 2, 3) stand for the three Pauli matrices. The action of generators γ ij = [γ i , γ j ]/2i on Φ must be compatible with the definition of Φ = (φ, φ * ) T . It is known that this kind of Φ takes the form of (φ, iτ 2 φ * ) T as the proper spinor of O(4), which can be regarded as a Majorana representation of the O(4) spinor. For (φ, φ * ), it is straightforward to find Γ µ from γ µ by W = diag.(I, iτ 2 ): , which are explicitly written as Altogether, we obtain the following rotation matrices of O(3) denoted by D ij (σ, θ) for σ = 1 and σ = −1, respectively, taking care of the SO(3) rotations and those including the parity inversion: In terms of the SO(3) rotations on φ denoted by S ij (1, θ) = exp(−iτ ij θ/2), where τ ij = [τ i , τ j ]/2i, D ij (σ, θ) are expressed as follows [5]: and where The definition of Φ given by Eq.(2) can be generalized to include a parameter η, which commutes with S ij (1, θ). The generalized form of the O(3) spinor takes the form of for η * η = I, or equivalently, for a = 1, 2, as the consistency condition on Φ, which will be used to find the fermionic O(3) spinor. The action of The consistency condition ensures that the complex conjugate of Eq.(12) coincides with Eq. (13). For D ij (1, θ), the condition is obviously satisfied. The simplest choice of η = I giving corresponds to Eq.(2).

B. Parity inversion
Although Σ i4 is the axis-inversion operator, it is instructive to cast Σ i4 into the following form: where Σ i = ijk Σ jk given by To see the rôles of Σ i andP , we discuss the case of Σ 14 for D 12 (−1, θ). Noticing that Σ 1 = iD 23 (1, π), we obtain that D 12 (−1, θ) = D 12 (1, θ)D 23 (1, π)iP , which suggests that 23 (1, π) describes the reflection in the vector space; •P describes the reflection in the spinor space as the parity operator, leading to the interchange of φ and φ * ; and similarly for others. As a result, we obtain that This decomposition of Eq.(17) is equivalent to the one for T ij (σ, θ) (=−T ji (σ, θ)) describing the corresponding transformation matrices acting on the O(3) vector space: where P (3) denotes the parity operator given by P (3) = diag.(−1, −1, −1) and The operatorP corresponding to P (3) plays a rôle of the spinorial parity operator, whose eigenvalues are (1, 1, −1, −1). As a result,P is equivalent to the following operator P : which is responsible for the Z 2 parity in the spinorial space. As a result, φ and φ * are distinguished by the parity operator, implying the appearance of a U (1) symmetry associated with P . Note that there is a relation of

C. G-Parity
We introduce a G-conjugate state, Φ G , which corresponds to a complex conjugate transforming as 2 D . The G-conjugate state can be defined by where Instead of Eq. (22), we also find that This spinor doublet is transformed by D G ij (σ, θ): which should be equal to D ij (σ, θ) up to phases so that Φ G transforms as 2 D . For Eq.

D. Axial vector and pseudoscalar of O(3)
Since Γ µ is the vector of O(4), Γ i is the O(3) vector. By the action ofP as the spinorial version of P (3) , we have the following results, whereΣ i = Σ i4 andΓ i = PΣ i . These relations correctly describe the property of the parity inversion that the vector changes its sign but the axial vector does not change its sign and that the pseudoscalar changes its sign. Therefore, we obtain that • Γ and Γ are the vectors of O(3); • Σ and Σ are the axial vectors of O(3); and similarly for Σ, Γ and Σ . The rotations generated by D ij (σ, θ) provide the correct description of the spinorrotations because D ij (σ, θ) turn out to give These transformation properties are consistent with those indicated by Eq. (26). For the remaining SO(3) vectors, Γ and Σ , since there are relations of Γ Φ = −i ΓΦ G and Σ Φ = −i ΣΦ G , the action on Γ and Σ can be understood from that on Γ and Σ. We note that the property of P being the pseudoscalar operator provides us an important observation on the transformed state of Φ given by Φ =D ij (−1, θ)Φ. The eigenvalues of P for Φ are calculated by from which we find that −P Φ is transformed into P Φ . The eigenvalues of P for Φ are opposite in signs to Φ.
In this pseudoscalar case, φ = S ij (−1, θ)φ * and its complex conjugate are consistent with P Φ =D ij (−1, θ)(−P Φ). The parity operator P being the pseudoscalar is a key ingredient to later introduce a U (1) symmetry into the O(3) spinor.

III. PARTICLE PHYSICS
To see the feasibility of the O(3) spinors in particle physics, we construct O(3)-invariant lagrangians for a bosonic and fermionic O(3) spinor. It is found that the appearance of the O(3) symmetry in particle physic can be rephrased as "the O(3) symmetry gets visible when we consider the invariance of interactions by including their complex conjugates". For example, the SU (2) gauge interaction with an SU (2)-doublet scalar (as the SO(3) spinor), φ, contains igτ i with g as the gauge coupling of the SU (2) gauge boson, which is equivalent to

A. Scalar
Let us start by utilizing Φ as a scalar. As already demonstrated, Φ † Φ and Φ † Γ i Φ behave as 1 and 3, respectively.
The computations are straightforward to obtain that are satisfied. Therefore, interactions of scalars consist of Φ † Φ as the scalar, Φ † Γ i Φ and Φ G † Γ i Φ as the vector and Φ G † Σ i Φ as the axial vector. The mass and kinetic terms are described by the form of Φ † Φ. Quartic couplings consist of Φ † Φ 2 and of the appropriate products out A Fierz identity relates these quartic terms. In fact, is satisfied. Similarly, we find that As quartic couplings, it is sufficient to use Φ † Φ 2 .
where µ Φ and λ Φ stand for a mass and a quartic coupling, respectively. L Φ is also invariant under a phase transformation induced by φ = e −iq φ as a global U (1) symmetry, where q is real. In terms of Φ, the invariance of L Φ is associated with the transformation of where P changes its sign for σ = −1 because of Eq. (28), reflecting the fact that P for φ * is equal to P for φ . This charge given by P can be interpreted as a Φ-number, which is referred to the "dark number". It should be noted that the transformation of Φ = e −iq Φ such as in U (1) of the standard model is incompatible with the definition of Φ because the transformed state Φ cannot satisfy the consistency condition of Eq. (14).
Although L Φ expressed in terms of Φ is nothing but a lagrangian expressed in terms of φ 1,2 , the O(3) structure can be recognized if its complex conjugate is treated on the same footing. For example, the identity of 2φ In this simple case, it is invariant under the interchange of φ ↔ φ * , which is caused by the parity operator iP of Eq.(15).

Consistency with the Dirac fermion
For the Dirac fermion denoted by ψ, the fermionic O(3) spinor, which is consistent with the Dirac spinor, should be composed of ψ C instead of ψ * to form (ψ, ψ C ) T , where ψ C is a charge conjugation of ψ defined by ψ = Cψ T . The charge conjugation operator, C, satisfies that where γ µ (µ = 0, 1, 2, 3) represents the Dirac gamma matrices. The definition of the O(3) spinor demands that ψ C = ψ. Since the O(3) spinor is required to be neutral, the possible candidate is a Majorana fermion, which is, however, excluded because of ψ C = ψ. Our neutral O(3) spinor carries the different Z 2 charge; therefore, ψ and ψ C are distinguished. Let us start with a pair of Dirac fields, ψ 1 and ψ 2 , which form as the SO(3) doublet spinor and ξ C = (ψ C 1 , ψ C 2 ) T as its charge conjugate. In terms of ξ, the O(3) doublet spinor is expressed to be Ψ: The fermionic spinor behaves as the spinor compatible with the transformation must satisfy Eq. (11). For the fermionic O(3) spinor, since Ψ a+2 = ξ C a =Cγ 0T Ψ * a holds, the parameter η in Eq.(11) is given by which turns out satisfy the necessary condition of η * η = I. As a result, we confirm that Ψ is the O(3) spinor, which is subject to the correct O(3) transformation: The use of ψ C a instead of ψ * a to form Ψ assures the orthogonality of the O(3) symmetry to the Lorentz symmetry. It is obvious that the fermionic O(3) spinor is vectorlike. If it is not vectorlike, ξ and ξ C have different chiralities so that ξ C = S * ij (−1, θ)ξ given by Eq.(40) cannot be satisfied.

Lagrangian of the fermionic O(3) spinor
Treating charge conjugates on the same footing, we use the familiar relations given by ψ a i ¡ ∂ψ a = ψ C a i ¡ ∂ψ C a up to the total derivative and ψ a ψ a = ψ C a ψ C a . The lagrangian of ψ a can be defined by where m Ψ is a mass of ψ a . The invariance under the O(3) transformation of D ij (σ.θ) acting on Ψ is satisfied by L Ψ because ofΨ=Ψ † γ 0 , where γ µ commutes with D ij (σ, θ). The possible bilinears includingΨΨ are expressed in terms of ξ as follows: and those containing γ 5 arē Ψγ 5 Ψ = 2ξγ 5 ξ,ΨP γ 5 Ψ = 0, Ψγ 5 Γ i Ψ = 2ξγ 5 τ i ξ,Ψγ 5 Σ i Ψ = 0, Ψγ 5 γ µ Ψ = 2ξγ 5 γ µ ξ,Ψγ 5 γ µ P Ψ = 0, Ψγ 5 γ µ Γ i Ψ = 2ξγ 5 γ µ τ i ξ,Ψγ 5 γ µ Σ i Ψ = 0. (43) As for Yukawa coupling of Ψ, Φ accompanied by an O(3)-singlet Majorana fermion N satisfying that N C = N can yield where f is a coupling constant. Considering relations of N φ † a ψ a = ψ C a φ * a N and ψ a φ a N =N φ T a ψ C a to form Φ and Ψ, L Y turns out to be The invariance under the O(3) symmetry is satisfied because Φ † Ψ is a singlet of O(3). In L Y , Φ can be replaced by its G-conjugate Φ G . If both couplings of Φ and Φ G are present, the U (1) symmetry is not respected. The lagrangian for N can be expressed in terms of N R and Ψ L . Since the resulting lagrangian is In a practical sense, N R can be taken as a right-handed neutrino [5] that couples to flavor neutrinos (to be denoted by ν). The Majorana mass of N R generates very tiny masses of ν due to the seesaw mechanism [11][12][13][14][15][16].

C. Gauge boson
Let us start with a lagrangian for an SO(3)-triplet vector, V iµ , and an SO(3)-singlet vector U µ accompanied by φ and ξ: for g and g as coupling constants and Y U stands for a U (1) charge, where mass terms are omitted for simplicity and To introduce Φ and Ψ, the relations to the complex conjugate of L φ,ξ : are explicitly used. Now, after V T µ = V * µ is used for ξ C , τ i for φ and ξ can be replaced by diag.(τ i , −τ * i ) for Φ and Ψ, which is nothing but Σ i . As a result, we find that where Hereafter, Ψ is assumed to possess the same Y U as Φ so as to yield the vanishing respectively, transform as the axial vector and the pseudoscalar as indicated by Eqs. (26) and (28). It is concluded that • V iµ is the axial vector transformed by V kµ = σT ij (σ, θ) k V µ ; • U µ is the pseudoscalar transformed by U µ = σU µ ,  Our gauge symmetry may serve as a dark gauge symmetry [2,10], which includes a dark photon as a dark gauge boson [17,18]. Our gauged O(3) model provides a vector portal dark matter [19].
The dark photon sometimes remains massless [17]. As the worst case, Φ and Ψ, provided that they are dark charged, are annihilated into the massless dark photon if the dominance of dark matter over dark antimatter is not effective. In the rest of discussions, we assume that this dominance is based on the universal mechanism over the universe [20,21] so that dark matter remains in the dark sector after the annihilation is completed. For instance, since our interactions may contain N R as a right-handed neutrino in L Y , the dominance of matter over antimatter can be based on the leptogenesis utilizing the lepton number violation due to N R [22], which in turn produces a dark matter particle-antiparticle asymmetry in the dark sector [23,24]. Reviews on the dark matter and baryon abundances are presented in Ref. [25,26], where further references can be found.
Dark matter scenarios based on the gauged O(3) symmetry as a maximal gauge symmetry provide the similar scenarios to the existing ones [3]. However, the essential difference from the existing scenarios lies in the fact that our dark matter and dark gauge bosons are theoretically forced to be all neutral once the spinorial structure of O(3) is built in the dark sector. From this feature, the kinetic mixing with the photon [27] is not a primary mixing with the standard model gauge bosons. Instead, the mixing with the Z boson becomes important and arises as an induced loop effect that allows Z to couple to intermediate flavor neutrinos. Another difference is that our gauge bosons do not transform as the triplet vector for V iµ and the singlet scalar for U µ but as the axial vector and the pseudoscalar, respectively.

A. Unbroken U (1) symmetry
The simplest realization of the dark photon denoted byγ can be based on the U (1) gauge model. When the U (1) symmetry is not spontaneously broken, the dark photon stays massless. The dark photon interacts with the standard model particles via H. The relevant part of the lagrangian of Φ including H is where µ H,Φ and λ H,Φ , respectively, stand for mass parameters and quartic couplings. The physical Higgs scalar h is parameterized in H = (0, Since Φ couples to the U (1) gauge boson but does not couple to the standard model gauge bosons, the decay of h proceeds via The next example is based on the O(3) gauge symmetry without the U (1) symmetry, which we refer as an "O(3)/U (1)"symmetry. The relevant part of the lagrangian of Φ and Ψ is In addition to L Y , another L Y as L Y given by can be included because of the absence of U (1). 1 To generate spontaneous breakdown, we assume that φ 2 in φ acquires a VEV. 2 For µ 2 Φ < 0 in Eq.(54), φ 2 develops a VEV that spontaneously breaks O(3)/U (1). The physical scalar denoted by ϕ is parameterized in φ = (0, (v Φ + ϕ)/ √ 2) for v Φ as a VEV of Φ. Three of the four real components in Φ are absorbed into three dark gauge bosons, which become massive. The remaining scalar is a massive ϕ. Interactions of scalars provide the following results: • h and ϕ are mixed via the mass term of M mix calculated to be: where m 2 h = 2λ H v 2 and m 2 ϕ = 2λ Φ v 2 Φ ; • The massive dark gauge bosons have a degenerated mass of gv Φ /2 and play a rôle ofγ; • h →γγ via the one-loop by V (FIG.1-(B) and by Ψ and N R (FIG.1-(C)) if the decay is kinematically allowed.

C. Spontaneously broken O(3) symmetry
The relevant part of the lagrangian of Φ and Ψ is In the case that the O(3) symmetry is spontaneously broken, if the appropriately defined U (1) charge is not carried by φ 2 , there remains a massless gauge boson chosen to beγ as a mixed state of V 3µ and U µ as in the standard model [34]. Since the U (1) invariance is assumed in the O(3) symmetry, L Y cannot be included. To be specific, without the loss of the generality, Y U = 1 for Φ can be chosen. Since everything is the same as in the standard model, we use the tilde to denote dark gauge bosons with the obvious notations. The massless dark photon identified withÃ can couple to dark charged particles. For Ψ, ψ 1 is a dark charged particle while ψ 2 is a dark neutral particle be- • h →γγ via the one loop byW ± , becauseW ± interact with both ϕ andγ as shown in FIG.1-(B).
Our dark matter consists of ϕ and ψ 1,2 . For other candidates ofW ± andZ,W ± disappear becauseW +W − are annihilated intoγγ andZ decays into νν owing to theZ mixing with the Z boson (see Sec.IV D). There arises an interaction between ϕ and ψ 2 , which isνψ 2 ϕ induced by the collaboration of N R ϕψ 2 andνhN R . Due to this interaction, the heavier O(3) spinor will decay according to the decay mode of either ϕ → ψ 2ν /ψ 2 ν or ψ 2 → ϕν/φ 2ν . For ψ 1 , these decay modes yielding ν are absent because L Y is absent. Since ψ 1,2 are degenerated, ψ 1 is stable. The dark chargedW ± are annihilated into dark photons whileZ may decay into + − , νν and so on. We conclude that • our dark matter consists of ψ 1 and the lighter spinor, which is either ϕ or ψ 2 ;  •γ has a direct coupling to the dark charged ψ 1 while ϕ has a direct coupling to the dark neutral ψ 2 .
The recent discussions on the phenomenology of the massless dark photon of this type can be found in Ref. [35,36].

D. Dark gauge bosons
The dark gauge bosons imported into these three subsections consist of • the massless U boson identified withγ in Sec.IV A; • three degenerated massive V bosons identified with γ in Sec.IV B; • the masslessÃ identified withγ and three massive bosons ofW ± andZ in Sec.IV C.
It is a general feature thatγ cannot possess the hypercharge associated with the U (1) Y symmetry of the standard model, which enablesγ to kinematically mix with the photon. Instead, there appears the mixing betweeñ γ and Z. The mixing ofγ with Z arises as the following loop effects due to • two loops by Ψ, N R and ν in the external loop involving the internal exchange of Φ for Sec.IV A or one loop with the same external loop as in Sec.IV A but involving the nonvanishing Φ for Sec.IV B; • two loops byW + , ψ 2 , N R and ν in the external loop involving the internal exchange of ψ 1 for Sec.IV C, whereγ only couples to ψ 1 , which cannot couple to N R because of the absence of L Y .
These induced couplings are depicted in FIG.2. 3 The similar mixing with Z is applied toZ. Due to the same one-loop coupling as theγ-Z mixing but one of H G 's is replaced with h, the h decay proceeds via • h → Zγ, which is necessarily induced for the masslessγ.
The particle content of each realization of the dark gauge symmetry is summarized in TABLE.I. For the masslessγ, the mixing betweenγ and Z is restricted to the kinetic mixing. For the massiveγ, the induced decay ofγ:γ → νν is possible to occur via the intermediate Z boson decaying into νν. Considering higher loop effects involving the inducedγ-Z mixing, we have the kineticγ-γ mixing, which is generated by the Z-boson exchanged between the loop(s) giving theγ-Z mixing and the one loop by quarks and leptons, which finally couples to the photon (FIG.3). Our kineticγγ mixing turns out to be further suppressed by the Z boson mass.
for (φ, φ * ), which describes the U (1) charge of Φ. We have also noted that P as the pseudoscalar of O(3) ensures that the transformed O(3) spinor with the reflections included has the opposite U (1) charge to the original O(3) spinor In particle physics, the O(3) spinor consisting of (φ, φ * ) serve as the bosonic O(3) spinor. The fermionic spinor is composed of the Dirac spinor and its charge conjugate instead of the complex conjugate so that the internal O(3) symmetry gets orthogonal to the Lorentz symmetry. The O(3) symmetry gets visible when the invariance of interactions is extended to explicitly include contributions from their complex conjugates. Owing to the inherent property that both the particle and antiparticle are contained in the O(3) spinor, which is constrained to be neutral, the application to particle physics is limited. One of the viable possibilities is to regard the O(3) spinors as dark matter. To see the interactions of the dark sector particles, we have constructed the lagrangians of the bosonic spinor, the fermionic spinor and the gauge bosons. The results indicate that the fermionic spinor is vectorlike and that the gauge bosons, V iµ and U µ , associated with SO(3) × Z 2 behave as the axial vector and pseudoscalar, respectively.
We have described the following scenarios of dark gauge bosons: 1. The massless dark photon is associated with the unbroken U (1) symmetry generated by the spinorial Z 2 parity, and h →γγ proceeds via the one loop by Φ; 2. Three degenerated massive dark photons are associated with the spontaneously broken O(3)/U (1) symmetry and h →γγ proceeds via the one loop by V with the gauge coupling toγ as well as by Ψ and N R with the Ψ coupling toγ if the decay is kinematically allowed; 3. The massless dark photon is associated with the spontaneously broken O(3) symmetry and h →γγ proceeds via the one loop byW ± with the gauge coupling toγ.
We have further obtained that 1. theγ-Z mixing is generated as loop corrections to give the decay of h as h → Zγ and the kineticγ-γ mixing further suppressed by the Z boson mass; 2. massive dark gauge bosons decay into νν; In case of the nonvanishing Φ , the O(3)-doublet vectorlike fermion supplies four species of Majorana fermions of ψ 1,2 and ψ C 1,2 , which all couple to N R as in Eq.(46), and will mix with flavor neutrinos via the seesaw mechanism. These Majorana fermions can be sterile neutrinos as dark matter [37]. To clarify detailed dark matter physics influenced by the O(3)-doublet vectorlike fermion as sterile neutrinos as well as to estimate their effects on neutrino oscillations is left for our future study. Finally, since the CP transformation exchanges particles and antiparticles, which are also exchanged by D 23 (−1, 0), there might be a chance to implement the CP violation in the dark sector associated with the breaking of O(3).

ACKNOWLEGMENTS
The author would like to thank T. Kitabayashi for collaboration in the early stages of this work and for valuable comments and advices.