Vector dark matter from split SU(2) gauge bosons

We propose a vector dark matter model with an exotic dark SU(2) gauge group. Two Higgs triplets are introduced to spontaneously break the symmetry. All of the dark gauge bosons become massive, and the lightest one is a viable vector DM candidate. Its stability is guaranteed by a remaining Z_2 symmetry. We study the parameter space constrained by the Higgs measurement data, the dark matter relic density, and direct and indirect detection experiments. We find numerous parameter points satisfying all the constraints, and they could be further tested in future experiments. Similar methodology can be used to construct vector dark matter models from an arbitrary SO(N) gauge group.


I. INTRODUCTION
Dark matter (DM) occupies approximately 26% of the energy density of the present Universe. In the thermal paradigm, DM particles annihilate into standard model (SM) products in the early Universe before they freeze out from the thermal plasma, leaving a proper relic density [1][2][3]. Particle candidates for dark matter can be classified according to their spins. Thus, DM particles could be scalar bosons (spin-0), spin-1/2 fermions, vector bosons (spin-1), or spin-3/2 fermions. Various scalar and fermionic DM models have been adequately studied in the literature. On the other hand, vector DM models draw much less attentions, with many issues unstudied.
When the related mediators are sufficiently heavy, interactions of vector DM can be appropriately described by effective operators in a model-independent way [4,5]. Otherwise, renormalizable interactions should be considered. If extra dimensions exist, the first Kaluza-Klein mode of the U(1) Y gauge boson could be a well motivated vector DM candidate [6,7]. In the four-dimensional spacetime, a natural approach for constructing renormalizable vector DM models utilizes the gauge theories, in which at least one gauge boson acts as the DM particle. A mechanism is required to generate the mass for the vector DM particle. For U(1) gauge theories, this mechanism can be either the Stueckelberg mechanism [8,9] or the Brout-Englert-Higgs mechanism [10][11][12]. For non-abelian gauge theories, the former is inapplicable, and the latter is commonly considered. Furthermore, the DM particle could be a confined spin-1 bound state based on non-abelian gauge interactions, and thus its mass is linked to a confinement scale [13].
If the dark gauge group is SU (2), it can be spontaneously broken completely through one dark Higgs doublet, with a remaining custodial global SU(2) symmetry ensuring the stability of the three degenerate gauge bosons as vector DM particles [30]. If, instead, one real Higgs triplet is introduced to break the SU(2) gauge symmetry, a U(1) gauge symmetry would remain, leading to a massless gauge boson acting as the dark radiation [31,43]. The other two gauge bosons are massive and degenerate, forming a pair of vector DM particle and antiparticle. This scenario can also be cast to a dark SU(3) case, where one Higgs triplet partially breaks the gauge group, leaving us five DM vector bosons and three massless dark radiation particles [50]. On the other hand, two dark Higgs triplets would be able to generate masses for all the eight SU(3) gauge bosons [21,51,52]. For a general dark SU(N ) gauge group, all the gauge bosons can be made massive if N − 1 Higgs fields in the fundamental representation are introduced [21].
In this paper, we study a vector DM model with a dark SU(2) D gauge group broken by two real Higgs triplets, which develop a generic configuration of vacuum expectation values (VEVs). As a result, the three dark gauge bosons can obtain different masses. The two lighter gauge bosons are odd under a remaining Z 2 symmetry, and the lightest one is stable, playing the role of the DM candidate. Compared with Ref. [30,31,43], no mass degeneracy appears among the three dark gauge bosons. After spontaneous symmetry breaking, the dark Higgs bosons mix with the SU(2) L Higgs boson, and there are four mass eigenstates of neutral Higgs bosons. The dark sector communicates with the SM sectors only through the Higgs quartic couplings. Thus, this model belongs to a kind of Higgs-portal DM models. We will perform a random scan in the parameter space to investigate phenomenological constraints from collider measurements of the 125 GeV Higgs boson, direct collider searches for exotic Higgs bosons, electroweak precision measurements, the DM relic density, and the direct/indirect DM detection experiments. Finally, we will discuss the possibility of the generalization to dark SO(N) gauge groups for N > 2 with N −1 real Higgs multiplets in the N -dimensional fundamental representation. We will prove that an accidental Z 2 symmetry also arises to preserve the stability of a dark gauge boson, acting as a DM candidate.
In Sec. II, we describe the model setup and discuss the general formalism of the remaining Z 2 symmetry. In Sec. III, the constraints from the relic density and the direct and indirect detection experiments are displayed. In Sec. IV, we generalize the methodology to a general dark SO(N ) model. In Sec. V, we summarize the paper.

II. MODEL
Now we discuss our vector DM model based on the dark SU(2) D gauge symmetry. The corresponding gauge fields are denoted asÃ a µ (a = 1, 2, 3). Two real SU(2) D Higgs triplets, Φ a and X a (a = 1, 2, 3), are introduced to break the SU(2) D gauge symmetry. We call them dark Higgs fields, as they are assumed to be SM gauge singlets. In this section, we construct the generic Lagrangian, and study the spontaneous symmetry breaking in detail.

A. Lagrangian and symmetry analysis
Under a SU(2) D gauge transformation, the dark Higgs fields Φ a and X a transform as where U = exp[iθ a (x)T a ] with (T a ) bc being the generators in the adjoint representation of SU(2) D , and some representation indices have been omitted. Below, we adopt (T a ) bc = −iε abc , where ε abc is the three-dimensional Levi-Civita symbol. The corresponding transformation of theÃ a µ isÃ where g D is the SU(2) D gauge coupling. Φ a and X a can be treated as vectors in a three-dimensional Euclidean representation space. From three vectors E, F, and G in such a space, nonzero invariant scalars can only be constructed via either dot products, like E · F = E a F a and G · G = G a G a , or triple products, like (E × F) · G = ε abc E a F b G c . Since we only have two triplet Higgs fields Φ a and X a , the related scalar potential terms must be formed by bilinear dot products Φ a Φ a , X a X a , and Φ a X a . Therefore, the model respects an accidental Z 2 symmetry, under which the Higgs triplets transform as where P D = diag(−1, −1, −1) serves as a "dark parity", analogous to the parity P in the 3-dimensional physical space. Thus, Φ a and X a are P D -odd, while the rest fields are P Deven. The The generic renormalizable scalar potential reads where the SM part for the SU(2) L Higgs doublet H is the dark sector part is and the portal part is It is easy to verify that V is invariant under the global O(3) D transformations.
The Lagrangian of the model can be divided into where L SM is the SM Lagrangian involving −V SM , and is the Lagrangian for the dark sector withÃ a µν ≡ ∂ µÃ a ν −∂ νÃ a µ +g D ε abcÃb,µÃc,ν . The covariant derivatives of the dark Higgs triplets are L D involves gauge interaction terms in a triple-product form, e.g., g D ε abcÃa µ Φ b ∂ µ Φ c . Since the Levi-Civita symbol satisfies ε abc = det(R)R ad R be R cf ε def for R ∈ O(3) D , the O(3) D invariance of the Lagrangian requires thatÃ a µ acts as an O(3) D axial vector, whose transformation is Thus, the P D transformation ofÃ a µ is i.e.,Ã a µ is P D -even.

B. Spontaneous symmetry breaking
In order to generate masses for the dark gauge bosons, the two real dark Higgs triplets Φ a and X a should obtain VEVs Φ a and X a , respectively. Thus, the SU(2) D gauge symmetry is spontaneously broken, and so is the global O(3) D symmetry. In general, the vectors Φ a and X a are not parallel to each other, so they determine a plane in the threedimensional representation space. We can always rotate the axes to a configuration that the z-axis is along the Φ a direction and the y-axis lies inside the plane. Therefore, without loss of generality, the configuration of the VEVs can be expressed as Φ a = (0, 0, v 1 ) and X a = (0, v 2 , v 3 ). Expanding the fields around the VEVs, we have In the SM sector, the SU(2) L Higgs doublet H has a form of in the unitary gauge. Note that all the four VEVs v 0 , v 1 , v 2 , and v 3 are real constants.
Expanding the covariant kinetic terms of the dark Higgs triplets, we have where the mass-squared matrix for the dark gauge bosons is For convenience, we define by an orthogonal matrix The relation betweenÃ a µ and the mass eigenstates A a µ is Note that A 1 µ =Ã 1 µ , while A 2 µ and A 3 µ are linear combinations ofÃ 2 µ andÃ 3 µ . The physical masses squared are given by Thus, we find m A 2 ≤ m A 3 ≤ m A 1 . For the generic nonzero v 1 , v 2 , and v 3 , there is no degeneracy in the mass eigenstates.
We further identify the Goldstone bosons which are eaten by the gauge bosons A 1 , A 2 , and A 3 , respectively. The shorthand notations s θ ≡ sin θ and c θ ≡ cos θ have been used. The Higgs bosons orthogonal to these Goldstone bosons can be chosen ash Utilizing the minimization conditions for the potential V , we obtain We then derive the mass-squared matrix M 2 h for the Higgs bosons (h 0 ,h 1 ,h 2 ,h 3 ), whose elements are given by The basis (h 0 ,h 1 ,h 2 ,h 3 ) can be rotated to the mass eigenstate basis The elements of O h will be numerically calculated. We define h 0 as the SM-like Higgs boson. In the following parameter scan, we require that the h 0 mass is around 125 GeV andh 0 makes the most contribution to h 0 . For the other Higgs bosons, we adopt a mass hierarchy convention of Finally, we discuss a remaining Z 2 symmetry after the spontaneous symmetry breaking of O(3) D . This is the reflection symmetry with respect to the Φ a -X a plane, i.e., the y-z plane according to our convention (15), because the VEV configuration Φ a = (0, 0, v 1 ) and is preserved under such a reflection. The corresponding reflection matrix where (T 1 ) bc = −iε 1bc is the rotation generator about the x-axis. According to (4) and (13), Φ a , X a , andÃ a µ transform under the reflection as Thus, we obtain These are the P D -odd components. All the other components are P D -even. Therefore, the gauge bosons A 2 and A 3 together with their corresponding Goldstone bosons G 2 and G 3 are P D -odd, since A 2 and A 3 (G 2 and G 3 ) are linear combinations ofÃ 2 andÃ 3 (φ 1 and χ 1 ). The rest physical states are all P D -even. Consequently, the lightest gauge boson A 2 cannot decay, serving as a DM candidate. Generally when

III. PHENOMENOLOGY
In this section, we scan the parameter space, taking into account the constraints from the observed DM relic density and the collider measurements of the 125 GeV Higgs boson. DM scattering and annihilation cross sections for direct and indirect detection predicted by this model is further calculated. We describe the details of the parameter scan and discuss the numerical results in the following subsections.

A. Parameter scan and DM relic density
We adopt the following 14 real parameters, as the independent parameters. A random scan is carried out for the parameters in logarithmic scales within the following range: 10 −3 < λ 0 , λ 1 , λ 2 , |λ 3 |, |λ 4 |, |λ 5 |, |λ 6 |, |λ 10 |, |λ 20 |, |λ 30 | < 1. (50) Note that the positivity of the m 2 h i demands the positive λ 0 , λ 1 , and λ 2 . We numerically calculate the eigenvalues and eigenvectors of M 2 h , and construct O h . We require that the SM-like Higgs boson h 0 has the most contribution fromh 0 , and that m h 0 lies within the 3σ range of the measured value 125.10 ± 0.14 GeV [53]. Other eigenvalues of M 2 h are sorted in an ascending order and accordingly appointed to h 1 , h 2 , and h 3 .
We implement the Lagrangian in FeynRules 2.3.36 [54] to generate the CalcHEP input files, and feed them to microOMEGAs 5.4 [4,55,56] for phenomenological calculations. Based on current Higgs measurements at the Large Hadron Collider (LHC), the SM-like Higgs boson is tested by Lilith 2.0 [57,58] implemented in the micrOMEGAs, and the corresponding p-values are derived. Parameter points with p-values less than 0.05 are discarded, corresponding to the exclusion at 95% confidence level (C.L.).
The exotic neutral Higgs bosons h 1 , h 2 , and h 3 in this model might be directly produced at the LEP and the LHC. Thus, the parameter points are also constrained by the direct searches at these colliders, which are utilized to constrain the mixing angle sin θ of a second neutral Higgs boson in Fig. 3 of Ref. [59]. As a good approximation, we reinterpret these constraints on | sin θ| as the constraints on is the h i decay branching ratio to the SM final states including h 0 , and Br(h i → SM) acts as a rescaling factor of the signal strengths for comparison with the collider data.
In addition, these exotic Higgs bosons couple to the W and Z bosons through their components of the SU(2) L Higgs doublet, inducing one-loop corrections to the propagators of the electroweak gauge bosons. The corresponding shifts of the g µν coefficients of the vacuum polarization amplitudes with respect to the SM are given by where m h SM is the SM Higgs boson mass and the loop function F is defined by [59] F (p 2 , m 2 1 , m 2 with Following the discussions in Ref. [59], we calculate the shifts of the precisely measured electroweak quantities Γ Z , R , R b , sin 2 θ eff , and m W , and compare the shifted values with the 2σ ranges of the experimental values adopted from Ref. [53] to filter all the parameter points. Now we discuss the DM relic density predicted in this model, where the DM particles are A 2 gauge bosons. Since the P D -odd A 3 must decay into A 2 , its abundance can also contribute to the DM relic density. If m A 3 − m A 2 0.1m A 2 , the A 2,3 coannihilation processes at the freeze-out epoch could significantly affect the relic density [60]. Therefore, we consider all possible annihilation and coannihilation channels with two-body final states, as listed below.   We utilize micrOMEGAs to evaluate the relic density, including the coannihilation effect. In micrOMEGAs, the freeze-out temperature T f indicating the departure from thermal equilibrium is defined by Y (T f ) = 2.5Y eq (T f ), where Y = n DM /s is the DM number density n DM divided by the entropy density s, and Y eq is its equilibrium value. We derive the thermally averaged effective annihilation cross section times velocity σ ann v FO at T = T f for the parameter points, as shown in Fig. 1. All the points in the figure predict the relic density Ω DM h 2 within the 3σ range of the Planck measured value 0.1200 ± 0.0012 [61].
Besides the most prominent points with the basically stable σ ann v so that σ ann v FO is close to the standard annihilation cross section σ ann v sd 2 × 10 −26 cm 3 s −1 , there are enormous points with significantly temperature-dependent σ ann v , so their σ ann v FO deviate from the standard value. Part of the deviations is due to the Breit-Wigner resonance effects [60].
In Fig. 1, there is a significant resonant structure lying around m A 2 ∼ m h 0 /2 62.5 GeV due to the A 2 A 2 annihilation mediated by the s-channel SM-like Higgs boson h 0 . For the points with σ ann v FO < σ ann v sd , the invariant mass m inv of the A 2 A 2 pairs is typically higher than the resonance center m h 0 at the T f , and then decreases to approach the m h 0 as the temperature drops, lifting the σ ann v to consume more DM particles in the later period. Therefore, a smaller σ ann v FO is needed for the observed relic density. On the other hand, some other points with σ ann v FO > σ ann v sd probably have the m inv equal to or slightly lower than the m h 0 at the T f , Lowering the temperature will therefore increase the distance between m h 0 and the invariant mass of the annihilating A 2 A 2 pairs, suppressing the annihilation at lower temperatures. Hence, the observed relic density requires a larger σ ann v FO .
Besides the SM-like Higgs boson, the exotic Higgs bosons h 1 , h 2 , and h 3 can also give rise to resonance effects. To see this clearly, we change the horizontal axis to m A 2 /m h 1,2,3 and plot the points in Fig. 2. Remarkable structures due to resonance effects appear at m A 2 /m h 1 ∼ 1/2 in Fig. 2(a) and at m A 2 /m h 2 ∼ 1/2 in Fig. 2(b). Nevertheless, Fig. 2(c) does not show an obvious resonance structure, because h 3 is the heaviest exotic Higgs boson, and m A 2 is probably too small to approach m h 3 /2 in our scan.
In Fig. 2(a), there is a "peak" structure near m A 2 /m h 1 ∼ 1. This originates from the h 1 h 1 threshold effect [60]. If m A 2 is slightly smaller than m h 1 , the A 2 A 2 → h 1 h 1 annihilation channel would open at sufficiently high temperatures, and then would be kinematically prohibited as the temperature drops, leading to a rapid shrink of σ ann v to cease further annihilation, requiring a larger σ ann v FO to clear out the redundant dark matter in advance.
To see both the resonance and the threshold effects clearly, we select four benchmark points (BMPs) and plot their σ ann v as functions of x ≡ m A 2 /T in Fig. 3. The properties of the BMPs are listed in Table I. BMP1 is a normal benchmark point without these effects, leading to roughly constant σ ann v from x = 10 to x = 10 4 . For BMP2 with m A 2 ∼ m h 1 /2, the h 1 resonance effect is important. As a result, σ ann v is larger than σ ann v sd at the freeze-out epoch, but suddenly decline since then. BMP3 with m A 2 ∼ m h 0 /2 is affected by the h 0 resonance, leading to σ ann v FO < σ ann v sd . After x f = m A 2 /T f , its σ ann v is elevated by the more effective resonance, and then peaks at x ∼ 200, and finally decreases because the invariant masses of the A 2 pairs drop to pass through the m h 0 value to reduce the resonant effect again. The h 1 h 1 threshold effect manifests for BMP4 with m A 2 ∼ m h 1 ,   and hence σ ann v decreases as the temperature decreases until x ∼ 400.

B. Direct and indirect detection
In this subsection, we calculate the predictions for direct and indirect detection and compare them with the experimental data.
In the model, the DM particle A 2 interacts with quarks (q = d, u, s, c, b, t) via interchanging all the Higgs bosons. The relevant Higgs-portal interaction terms can be expressed as where The A 2 -quark interactions induce A 2 -nucleon interactions. Thus, the spin-independent (SI) cross section for A 2 scattering off a nucleon N = p, n is given by [5] with the A 2 -N reduced mass defined as µ A 2 ,N = m A 2 m N /(m A 2 + m N ) and the effective coupling f N q are the nucleon form factors for quarks [4]. In this model, σ A 2 n is slightly different from σ A 2 p . In order to take this difference into account, we should calculate the normalized-to-nucleon SI cross section [62] for A 2 scattering off nuclei A I with Z protons and A I − Z neutrons. The index I denotes isotopes, and η I is the fractional number abundance of the isotope A I in nature.
We can compare σ SI N predicted by the model with the 90% C.L. upper limits on σ SI N from the direct detection experiment XENON1T [63]. The detection material in XENON1T is  [62]. We compute σ SI N for the parameter points and plot the result in Fig. 4(a). The dot-dashed line indicates the XENON1T bound, which excludes a fraction of the parameter points. This model can also be tested in future direct detection experiments. For an example, the proposed LZ detector is designed to reach a sensitivity of ∼ 3 × 10 −48 cm 2 for σ SI N [64], which is expected to be relevant to some parameter regions of the model. We also plot the LZ sensitivity as a dotted line in Fig. 4(a). Now we discuss indirect detection. A 2 A 2 annihilation can produce high energy γ rays, which might be received by the Fermi Large Area Telescope (Fermi-LAT). It has long been known that the dwarf spheroidal galaxies surrounding the Milky Way Galaxy are dominated by dark matter, as their mass-to-light ratios are of a hundred or greater [66]. A global  In the left panel, the dot-dashed line denotes the 90% C.L. upper limit on σ SI N from XENON1T [63], while the dotted line demonstrates the 90% C.L. projected sensitivity of LZ [64]. In the right panel, the dot-dashed line indicates the 95% C.L. upper limit from the Fermi-LAT γ-ray observations of dwarf galaxies [65]. The blue points are excluded by XENON1T, the red points excluded by Fermi-LAT, the purple points are excluded by both, and the green points survive from the two bounds.
analysis from γ-ray observations of 27 dwarf galaxies using 11 years of the Fermi-LAT data [65] offers a stringent bound on DM annihilation.
In Fig. 1, we project the parameter points in the m A 2 -σ ann v 0 plane, where σ ann v 0 is the A 2 A 2 annihilation cross section in the low velocity limit. The dot-dashed line denotes the 95% C.L. upper limit on the DM annihilation cross section in the bb channel derived by the analysis of Fermi-LAT data. Since the annihilation channels into W + W − , ZZ, tt, and a pair of Higgs bosons induce γ-ray spectra similar to that from the bb channel [67], we can approximately compare σ ann v 0 with the Fermi-LAT bb constraint.
In Figs. 1, 3, and 4, we have denoted the parameter points with colors. The blue and red points are excluded by XENON1T and Fermi-LAT, respectively, while the purple points are excluded by both. The green points survive from the two constraints. Comparing Fig. 4(b) with Fig. 1, one may notice that σ ann v 0 σ ann v FO holds for many points. As mentioned above, when the temperature drops, the A 2 A 2 invariant mass decreases and the distance to a resonance center changes. Consequently, for the parameter points with m A 2 ∼ m h 0 /2, which are related to the h 0 resonance, σ ann v at low temperatures significantly decreases for m A 2 < m h 0 /2, but could remarkably increase for m A 2 > m h 0 /2. This is clearly shown in Fig. 4(b), where the parameter points in the latter case lead to too large σ ann v 0 , usually excluded by the Fermi-LAT. For the former case, both σ SI N and σ ann v 0 could be small enough to evade the XENON1T and Fermi-LAT constraints, as demonstrated in Figs. 4(a) and 4(b).
The resonances of the exotic Higgs bosons have similar effects. To illustrate it, we compare for R ∈ O(N ) D , so such a term is P D -odd. Therefore, only the global SO(N ) D symmetry is respected for the N Higgs multiplets. However, we will prove by contradiction that no P D -odd potential term arises for the N − 1 Higgs multiplets. This can be done by verifying that O(N ) D vectors V 1 , V 2 , · · · , V n with n < N cannot form any P D -odd scalar. Assume that V 1 , V 2 , · · · , V n (n < N ) can form a P D -odd scalar. Then, similarly as the Levi-Civita symbol, there must exist a "constant O(N ) D tensor" E a 1 a 2 ···an leading to a P Dodd scalar E a 1 a 2 ···an V a 1 1 V a 2 2 · · · V an n , where a i (i = 1, 2, · · · , n) are O(N ) D indices. The tensor components E a 1 a 2 ···an are constants because they are the combinations of the corresponding Clebsch-Gordan coefficients which stay constantly as the coordinates rotate. That is to say, the tensor E a 1 a 2 ···an is invariant under any SO(N ) D transformation, so we have for any R ∈ SO(N ) D with det(R) = 1. If a number i appears in the indices {a 1 , a 2 , · · · , a n } for odd times, then E a 1 a 2 ···an = 0. This can be understood as follows. Since n < N , we can always find an integer number j (1 ≤ j ≤ N ) that doses not appear in {a 1 , a 2 , · · · , a n } due to the pigeonhole principle. Construct a diagonal matrix R ∈ SO(N ) D with the elements R ii and R jj being −1 but the rest diagonal elements being 1. Applying Eq. (62), the R ii = −1 appears for odd times in the right hand side and collectively contributes a −1 factor, leading to E a 1 a 2 ···an = −E a 1 a 2 ···an , and hence E a 1 a 2 ···an must vanish. Therefore, any integer number i (1 ≤ i ≤ N ) can only appear for even times in the indices of a nonzero E a 1 a 2 ···an . This implies that a nonzero E a 1 a 2 ···an tensor must have an even rank, i.e., an even n.
If N is odd, one can adopt P D = diag(−1, −1, · · · , −1). Applying this on E a 1 a 2 ···an V a 1 1 V a 2 2 · · · V an n introduces a (−1) n = 1 factor because n is even. Thus, E a 1 a 2 ···an V a 1 1 V a 2 2 · · · V an n remains unchanged and is a P D -even scalar, contradicting our assumption.
If N is even, usually P D = diag(−1, 1, 1, · · · , 1) is adopted without loss of generality. Under the P D transformation, V 1 i → −V 1 i and V j i → V j i (j = 1). Since 1 must appear for even times in the indices of a nonzero E a 1 a 2 ···an , we can still prove that E a 1 a 2 ···an V a 1 1 V a 2 2 · · · V an n is P D -even. This again contradicts with the assumption.
In summary, there is no way to construct a P D -odd scalar by contracting n vectors for n < N , so the generic scalar potential constructed from the N − 1 Higgs multiplets Φ i has an accidental global O(N ) D symmetry.
If all Φ i develop nonzero VEVs Φ i , which are linearly independent, then the global O(N ) D symmetry is spontaneously broken completely. We are now going to seek for the remaining Z 2 symmetry. Similarly with what we have done in subsection II B, we can always rotate the axes such that all the first VEV components Φ 1 i = 0. In fact, this VEV configuration is preserved under the P D = diag(−1, 1, 1, · · · , 1) transformation, which is actually the Z 2 symmetry we want. Expanding the Higgs fields around the VEVs, Φ i = Φ i +φ i , we find that all φ 1 i (i = 1, 2, · · · , N −1) are P D -odd, while the rest φ a i (a = 2, · · · , N ) are P D -even.
There are N (N − 1)/2 gauge bosons in such a SO(N ) D gauge model. We denote the gauge fields as A ab µ (a, b = 1, 2, · · · , N ), where A ab µ = −A ba µ . The trilinear gauge interaction terms are proportional to A ab µ Φ a i ∂ µ Φ b i . Taking the VEVs, we have A ab µ Φ a i ∂ µ Φ b i , which only vanishes for a = 1. Therefore, the N − 1 gauge fields A a1 µ (a > 1) are P D -odd, while all the other A ab µ (a = b) are P D -even. The number of the P D -odd gauge bosons is equal to the number of the P D -odd scalar bosons φ 1 i , implying that all these scalar bosons become the Goldstone bosons eaten by the P D -odd gauge bosons. The lightest mass eigenstate of the P D -odd gauge bosons serves as a DM candidate. In this way, our model setup has been generalized to the SO(N ) D cases.

V. CONCLUSIONS AND FUTURE PROSPECT
In this paper, we have discussed a vector DM model based on a dark SU(2) D gauge theory. Two real SU(2) D dark Higgs triplets are introduced to completely break the SU(2) D gauge symmetry, leading to three massive dark gauge bosons A 1 , A 2 , and A 3 with a split spectrum. An accidental Z 2 symmetry remains after the spontaneous symmetry breaking. Under this Z 2 symmetry, A 1 is even, while both A 2 and A 3 are odd. Since A 2 is generally lighter than A 3 , it becomes a stable vector DM candidate. In the unitary gauge, the physical degrees of freedom in the dark Higgs triplets and the SU(2) L Higgs doublet form four Higgs bosons h 0 , h 1 , h 2 , and h 3 , where h 0 acts as the 125 GeV SM-like Higgs boson. These Higgs bosons provide a portal to the SM particles for the dark gauge bosons.
We have randomly scanned the 14-dimensional parameter space, taking into account the constraints from current 125 GeV Higgs measurements, direct searches for exotic Higgs bosons, and electroweak precision measurements. The DM relic density has been calculated, including the A 2 A 3 coannihilation effect. We have found the parameter points predicting the observed relic density. These parameter points have been further tested by the bounds from the XENON1T direct detection experiment and the Fermi-LAT indirect detection experiment.
Resonance and threshold effects in DM annihilation could lead to important differences between σ ann v FO and σ ann v 0 . In the parameter points, we have found significant effects due to the h 0 , h 1 , and h 2 resonances for m A 2 ∼ m h i /2, as well as a remarkable h 1 h 1 threshold effect for m A 2 ∼ m h 1 . Because of these effects, σ SI N and σ ann v 0 for some parameter points could be sufficiently small, evading the current direct and indirect detection constraints.
There are numerous parameter points remaining after all the above constraints are considered. The future LZ direct detection experiment is expected to test some of them. Moreover, the interactions between the dark Higgs triplets and the SU(2) L Higgs doublet induce mixings among the Higgs bosons. As a result, the couplings of the SM-like Higgs boson h 0 generally deviate from the SM couplings. Future Higgs precision measurements at e + e − colliders, such as CECP [68], FCC-ee [69], and ILC [70], will provide further tests on this model.
We have proved that the way to construct this model can be generalized to the general SO(N ) D cases. For a SO(N ) D gauge model, N − 1 real Higgs multiplets in the SO(N ) D fundamental representation can be introduced to break the gauge symmetry, with a remaining Z 2 symmetry ensuring the stability of a dark gauge boson. Thus, more vector DM models can be similarly constructed.