Vector dark matter from a gauged $SU(2)$ symmetry

We propose a scenario of dark sector described by a hidden $SU(2)$ gauge symmetry which is broken by a vacuum expectation value(VEV) of a scalar multiplet. We discuss a general mass relation among $SU(2)$ gauge bosons after spontaneous symmetry breaking which is determined by weight of gauge group representation associated with a scalar multiplet developing VEV. Then a model with quintet and triplet scalar fields is discussed in which hidden gauge boson can be dark matter(DM) stabilized by remnant discrete $Z_2$ symmetry and resonant dark matter annihilation is realized by mass relation between DM and mediator. We estimate relic density and spin independent DM-nucleon scattering cross section searching for allowed parameter region.


I. INTRODUCTION
Explanation of dark matter(DM) nature is one of the outstanding issues which requires physics beyond the standard model(SM). One promising candidate of DM is a stable particle weakly interacting with the SM particles where its relic density in current Universe can be explained by thermal production. Interactions among DM and the SM particles have been tested by direct detection, indirect detection and collider experiments. However we have no clear signal of DM in such experiments indicating that interaction among DM and the SM particle would be sufficiently weak to escape these searches.
In describing DM physics, one attractive scenario is introduction of dark sector that is described by hidden gauge symmetry as the SM sector is also described by gauge symmetry; here we indicate hidden gauge symmetry under which the SM fields are neutral. In such a scenario we can explain stability of DM by a remnant of hidden gauge symmetry after spontaneous symmetry breaking [1,2]. In particular a case of non-Abelian group is interesting since an unbroken discrete symmetry can be naturally preserved; in Abelian case we might need to tune charge assignment since its choice is more arbitrary compared to non-Abelian case. Moreover hidden gauge boson(s) can be DM candidate depending on a structure of remnant symmetry. Here we focus on the simplest non-Abelian gauge group SU (2) as hidden gauge symmetry. In fact, various approaches applying a hidden SU (2) gauge symmetry have been discussed in literatures, for examples, a remaining Z 2,3,4 symmetry with a quadruplet(quintet) in ref. [3][4][5][6][7], Z 2 × Z 2 symmetry [8], a custodial symmetry in refs. [9,10], an unbroken U (1) from SU (2) in refs. [11][12][13], a model adding hidden U (1) h [14], a model with classical scale invariance [15], Baryogengesis [16] and electroweak phase transition [17].
In this paper, we consider a scenario of dark sector described by hidden SU (2) gauge symmetry which is broken by a vacuum expectation value(VEV) of scalar multiplet(s). Then we show general mass relation among SU (2) gauge bosons after spontaneous symmetry breaking which is determined by weight of gauge group representation associated with a scalar multiplet developing VEV. It is found that hidden gauge bosons can be DM and mediator Z with mass ratio of 1 : 2 realizing resonant DM annihilation when SU (2) is broken by scalar quintet VEV. In our scenario Z from hidden SU (2) can interact with SM particles via kinetic mixing coming from higher dimensional operator with triplet VEV. We then discuss relic density and DM-nucleon scattering cross section in such a scenario. This paper is organized as follows. In Sec. II, we show general relation among SU (2) gauge bosons after spontaneous symmetry braking. In Sec. III we consider a model realizing resonant annihilation of DM and estimate relic density of DM and spin independent nucleon-DM scattering cross section. We then conclude and discuss in Sec. IV.

II. GENERAL ASPECT
In this section we discuss SU (2) gauge symmetry which is spontaneously broken by a VEV of a scalar field, and show relation among massive gauge boson masses after symmetry breaking in general way.
Let us start from the definition of a scalar field Φ belonging to SU (2) multiplet whose components are described by , m, where numbers and m respectively correspond to the highest weight and weight (eigenvalue of diagonal generator T 3 ). Then, we symbolize it as which is often used to explain the active Majorana neutrino mass matrix via type-II seesaw mechanism. In this case, we have Φ = φ The upper indices coincide with electric charge again due to Y = and then each of the charges is found to be 2, 1 and 0 that are consistent with the component of Higgs triplet. Notice here that we consider SU (2) is a hidden symmetry, thus nonzero VEV can be found in the components with nonzero charge.
Next, let us consider the kinetic term of Φ ,m , which is written by 2 and the following relations are found . Then, we find the following mass terms for vector bosons when only φ Taking the mass ratio between X ± µ and X 3 µ , one finds m 2 .
Here we consider only one scalar field develops a VEV but it is straitforward to extend more general cases in which multiple scalar fields and/or more than one components in a multiplet develop non-zero VEVs.
In case of = |m| = 2, especially, one finds the mass ratio of gauge bosons as follows: Here = 2 case corresponds to a quintet scalar field Φ[φ 2,1 , φ 2,2 has non-vanishing VEV 1 . Remarkably Eq. (II.9) indicates that X ± µ annihilates into pairs of SM particles around the resonant point when X ± µ is DM and X 3 µ is mediator decaying into SM fields. Note also that a component with even and odd m have respectively Z 2 parity even and odd where this remnant Z 2 can stabilize X ± since it has m = ±1. We discuss a model with a quintet scalar field realizing resonant DM annihilation scenario in next section.

III. A MODEL WITH RESONANT DARK MATTER ANNIHILATION
In this section we consider a model with hidden SU (2) X gauge symmetry introducing real quintet and triplet scalar fields denoted by Φ and Φ = φ , respectively. In this setup resonant DM annihilation is realized by mass relation between DM and mediator gauge boson, and interaction between dark sector and the SM sector is obtained via kinetic mixing between SU (2) X and U (1) Y gauge fields. In our model scalar potential is written by where H is the SM Higgs, V (H) is scalar potential including only H, each term is contracted to be SU (2) X singlet implicitly, subscript i(a) distinguishes different contraction of SU (2) X representation indices. In our analysis we do not discuss details of scalar potential and assume our VEV alignments are obtained by turning parameters in potential.

A. Mass spectrum and relevant interactions
Firstly, we discuss our gauge symmetry breaking pattern when φ 1,−1 ) = (−, +, −) under U T 3 transformation. It suggests that we have a remnant Z 2 symmetry that plays a role in stabilizing the DM candidate with odd parity as discussed below. Note also that if we introduce a field having half integer (without developing a VEV), a remnant symmetry becomes Z 4 ; for example a ( = 1/2) doublet field Next step is to find how X 3 µ interacts with the SM fields. In order to realize the interaction, we introduce an SU (2) X triplet scalar field ϕ which is given by where the elements in the most right side correspond to our standard form 2 . Then we can write the following term: The relevant term for these kinetic mixing is dim-5 operator [7,18]: where Λ indicates the cut off scale and X a µν = ∂ µ X a ν − ∂ ν X a µ + abc g X X b µ X c µ andB µν are the gauge field strength for SU (2) X and U (1) Y , respectively (g X and abc are SU (2) X gauge coupling and anti-symmetric tensor). After ϕ developing VEV, we obtain the following kinetic mixing term: where we define sin χ ≡ √ 2C ϕ v ϕ /Λ as a new kinetic mixing parameter.
The kinetic terms for X 3 µ andB µ can be diagonalized by the following transformations (we neglect refractive phenomena contributed by a forwardly scattering off the thermal bath, i.e. plasma effect [19][20][21][22][23][24]):B µ = B µ − tan χX 3 µ , (III.6) (III.7) In our analysis, we take a limit of χ 1 and gauge fields are approximately written bỹ Here we denote dark gauge bosons associated with X 1,2,3 µ field as X 1,2,3 henceforth. After quintet and triplet scalar fields develop nonzero VEVs, we obtain mass terms for SU (2) X gauge fields and SM Z boson field including kinetic mixing effect such that The relations between ϕ a and components in our standard form; φ ( +m) 1,m , are given by φ 1,0 = ϕ 3 . whereZ is Z boson field in the SM, g and g B are gauge couplings of SU (2) and U (1) B , and v is the SM Higgs VEV. DiagonalizingZ and X 3 mass terms, mass eigenstate and mixing angles are given by where we approximate m Z m Z SM and m Z m X 3 for tiny χ. The mass eigenstates are In Fig. 1, we show sin θ ZZ as a function of m X ± for χ = 10 −2 and 10 −3 where we fix 2v 2 ϕ /v 2 Φ = 0.01. We find that sin θ ZZ 10 −3 for χ ≤ 10 −2 and m X ± ≥ 100 GeV, and thus it is safe from current constraints [25].
Finally interactions among dark gauge fields are also given by where abc is the structure constants of SU (2) X and a = 1, 2, 3. Since m Z m X 3 m X ± the four point gauge interaction in Eq. (III.14) does not provide dominant contribution to DM annihilation process. We thus focus on the three point interaction which is given by X ± and Z such that where C ZZ ≡ cos θ ZZ . Then Z interacts with the SM particles via Z-Z mixing. The Z interaction with SM fermions f is given by where T 3 is diagonal generator of SU (2) L , Q is the electric charge, S ZZ ≡ sin θ ZZ and C ZZ ≡ cos θ ZZ .

B. Dark matter physics
Here we discuss dark matter physics such as relic density and direct detection by carrying out numerical analysis. In our analysis we write Z mass by where we ignored tiny kinetic mixing effect and R M ≡ v 2 ϕ /v 2 Φ representing deviation from the relation m Z = 2m X ± . We expect R M << 1 so that we can explain the sizable relic density by resonant enhancement. Note that if the hidden gauge coupling g X is very large around order 1, the resonant point would not be needed to explain the correct relic density. Performing numerical analysis, we search for parameter region satisfying observed relic density of Ωh 2 ∼ 0.12 [26].
We calculate relic density of DM with micrOMEGAs 5 [27] implementing relevant interactions. In our analysis, we randomly scan parameters {m X ± , g X , R M } within the following where we consider two values of kinetic mixing parameter as χ = 10 −2 and 10 −3 . Each point in Fig. 2 shows parameter sets which realize relic density of DM in approximated observed region of 0.11 ≤ Ωh 2 ≤ 0.13; here we fix χ = 10 −3 but the behavior will be the same if we Next we discuss direct detection of DM calculating nucleon-DM scattering cross section for allowed parameter sets which explain observed relic density. In our case nucleon and DM interact via Z and Z exchanging process due to Z-Z mixing effect. Spin independent nucleon-DM scattering cross section is obtained by where m N is nucleon mass (N = p, n). The nucleon-Z(Z ) vector coupling C V N N Z(Z ) is given by (III.20) In Fig. 3, we showσ N X ≡ (σ pX + σ nX )/2 as a function of DM mass for allowed parameter respectively. We also compare our result with experimental constraints by XENON1T [28,29] and neutrino scattering limit [30]. For χ = 10 −3 , the scattering cross sections are much less than current bound. They are even smaller than neutrino scattering limit except for light DM region m X ± 250 GeV. For χ = 10 −2 , the scattering cross sections are above current bound in m X ± 150 GeV region, between neutrino scattering limit and current bound in 150 GeV m X ± 1200 GeV, and below neutrino scattering limit in m X ± 1200 GeV.
Thus DM signal could be detected for χ = 10 −2 and m X ± 1200 GeV in future experiments while it is difficult to test DM signal by direct detection for small kinetic mixing parameter χ.
Before closing this section we comment on DM production at collider experiments. Our DM candidate X ± can be produced at collider via Z and/or scalar portal when we have Z-Z mixing and/or mixing between new scalar from Φ ( ) and SM Higgs. From Z-Z mixing, we would have qq → Z → X + X − process where we need extra jet/photon to detect at collider experiments. From scalar mixing, we would have gg → φ → X + X − process where φ is a scalar boson which is mixture of SM Higgs and a new scalar boson. Detailed study of collider signal is beyond the scope of this paper and it will be given in future works. IV.

CONCLUSIONS AND DISCUSSIONS
We have proposed a scenario of dark sector described by hidden SU (2) gauge symmetry which is broken by a VEV of scalar multiplet. Firstly we have discussed general mass relation among SU (2) gauge bosons after spontaneous symmetry breaking which is determined by weight of gauge group representation associated with a scalar multiplet developing VEV.
Interestingly, when SU (2) is broken by scalar quintet VEV, hidden gauge bosons can be DM and mediator with mass ratio of 1 : 2 which realizes resonant DM annihilation. In addition Z 2 symmetry stabilizing DM is naturally obtained in this case.
Then we have discussed a model introducing hidden SU (2) X quintet and triplet scalar fields in which resonant DM annihilation is realized. In this model dark sector and the SM sector interact through kinetic mixing between SU (2) X and U (1) Y gauge fields obtained from higher dimensional operator through the VEV of SU (2) X triplet scalar. We have shown the kinetic mixing term and formulated Z-Z mixing to give interaction between DM and the SM particles. After formulating interactions, we estimated DM relic density and DM-nucleon scattering cross section. We have found that DM relic density can be explained for SU (2) X gauge coupling in the range of ∼ [0.01, 1]. In addition, most of parameter sets giving observed relic density is safe from direct detection constraints and some region can be tested in future experiment.