Dark matter physics in dark SU(2) gauge symmetry with non-Abelian kinetic mixing

We investigate a model of dark sector based on non-Abelian SU(2)D gauge symmetry. This dark gauge symmetry is broken into discrete Z2 via vacuum expectation values of two real triplet scalars, and an SU(2)D doublet Dirac fermion becomes Z2−odd particles whose lighter component makes stable dark matter candidate. The standard model and dark sector can be connected via the scalar mixing and the gauge kinetic mixing generated by higher dimensional operators. We then discuss relic density of dark matter and implications to collider physics in the model. The most unique signatures of this model at the LHC would be the dark scalar (Φ (′) 1 ) productions where it subsequently decays into : (1) a fermionic dark matter (χl) and a heavy dark fermion (χh) pair, Φ (′) 1 → χ̄lχh(χ̄hχl), followed by χh decays into χl and non-Abelian dark gauge bosons (Xi’s) which decays into SM fermion pair f̄SMfSM resulting in the reaction pp → Φ ′) 1 → χ̄hχl(χ̄lχh) → fSM f̄SMχlχ̄l, (2) a pair of Xi’s followed by Xi decays into a DM pair or the SM fermions resulting in the reaction, pp→ Φ ′) 1 → XiXi → χ̄lχlfSM f̄SM or even number of fSM f̄SM pairs.


I. INTRODUCTION
The standard model (SM) of particle physics has been very successful in describing phenomenology observed in various experiments. However the existence of dark matter (DM) cannot be explained in the SM framework, and it would be described as a new particle associated with physics beyond the SM. The nature of DM is an open question and there are many experimental searches for interactions among DM and the SM particles such as in direct detection, indirect detection and collider experiments. No clear evidence of DM would indicate a dark sector which is hidden from current observations. As the SM is described by local gauge symmetries, it is plausible that the dark sector is also ruled by a hidden/dark gauge symmetry. Moreover stability of DM indicates necessity of a symmetry to protect it from decay, and it can be a remnant of dark gauge symmetry (see Ref. [1] for a review along this line). Thus, it is an attractive scenario that dark gauge symmetry is spontaneously broken to a symmetry stabilizing DM candidate. To realize this concept, we are especially interested in the extension of the SM introducing a new SU (2) D gauge symmetry where all the SM fields are singlet under it. The interesting properties of a model with local SU (2) D group is that an unbroken discrete symmetry can be naturally preserved after the spontaneous breaking of the SU (2) D gauge symmetry comparing with a hidden local U (1) case in which the U (1) charge has to satisfy some artificial tuning [2].
In this paper, we discuss a simple scenario in which SU (2) D gauge symmetry is broken into discrete Z 2 symmetry by vacuum expectation values (VEVs) of two SU (2) D triplet real scalar fields ( φ and φ ), and an SU (2) D doublet Dirac fermion χ is Z 2 -odd DM candidate. We also introduce higher dimensional operators that induce gauge kinetic mixing terms between SU (2) D and U (1) Y gauge fields after SU (2) D symmetry breaking by nonzero VEV's of φ and φ as mediators between the dark gauge sector and the SM sector. After fixing our model, we formulate particle mass spectra and their interactions in the dark sector and the portals to the SM sector. Then relic density of our DM candidate is estimated taking into account constraint from direct detection of DM. Furthermore we discuss implications to  collider physics considering the scalar portal and the kinetic mixing as connections between dark sector and the SM. This paper is organized as follows. In Sec. II, we show our model of SU (2) D dark sector formulating mass spectra and interactions. In Sec. III, we analyze DM relic density and discuss the allowed parameter region. In Sec. IV, we discuss implications to collider physics.
Finally We conclude and discuss in Sec. V.

II. MODEL AND FORMULAS
In this section we summarize the setup for our model. We introduce a dark sector which is controlled by a non-Abelian SU (2) D dark gauge symmetry, with two real scalar fields φ and φ , and one Dirac fermion χ as summarized in Table I. In components, φ( φ ) and χ are written as where the indices for triplet scalars correspond to three SU (2) D generators.
The SU (2) D dark gauge symmetry is spontaneously broken by nonzero VEV's of two real scalar triplets φ and φ . In our scenario, we assume VEV alignments of two scalar triplets When SU (2) D is broken by the VEVs of the triplets, the vacuum is invariant under the trans- where X a µν (a = 1, 2, 3) is the field strength of SU (2) D gauge field, and H is the SM Higgs doublet field written as Here v H is the VEV of the SM Higgs doublet, H, and G ± and G Z are Nambu-Goldstone(NG) bosons absorbed by W ± and Z bosons.

A. Scalar sector
Here we consider the scalar sector of the model. Firstly we consider conditions to get VEV alignment in Eq. (II.2). From the stationary conditions ∂V /∂φ i = 0 and ∂V /∂φ i = 0, we obtain following non-trivial conditions (or vanishing tadpole conditions): The mass terms of scalar fields are given by the quadratic terms in the scalar fields in the Lagrangian: where we used the last equation of Eq. (II.6) to substitute λ 5 . Notice that mass terms associated with φ 2 and φ 2 are absent and they are identified as Nambu-Goldstone(NG) bosons which are absorbed by the two massive gauge bosons in the dark sector.
From now on, we shall assume λ 4 1 to simplify the scalar mass terms. Then Eq. (II.7) becomes ignoring terms with the λ 4 coupling. The terms for φ 1 and φ 3 can be organized as We then find that the mass eigenstate (cos αφ 1 + sin αφ 3 ) has the mass eigenvalue λ 3 (v 2 φ + v 2 φ )/2, whereas its orthogonal state (− sin αφ 1 + cos αφ 3 ) corresponds to the NG boson absorbed by SU (2) D dark gauge boson.
Finally, the mass matrix for (h, φ 3 , φ 1 ) is given by (II.10) Thus φ 3 and φ 1 can mix with the SM Higgs field and interact with SM particle via mixing effects. In our phenomenological analysis, we discuss the following two simplified cases.
Scenario (1): λ Hφ , λ 6 → 0 In this case,h and φ 3 mix while φ 1 is almost the mass eigenstate without mixing. Then squared mass terms for {h, φ 3 } are given by This squared mass matrix can be diagonalized by an orthogonal matrix, and the resulting mass eigenvalues are given by The relevant mass eigenstates h and Φ 1 are also given by where α is the mixing angle, and h is identified as the SM-like Higgs boson. Also we rewrite φ 1 as an approximate mass eigenstate such that The scalar mixing is constrained by Higgs precision measurements as sin α 0.3 when the SM Higgs does not decay into particles in the dark sector [12,13] We investigate the bound for sin α when the SM Higgs decays into dark gauge bosons below.

B. Gauge sector
The dark and the SM sectors can interact through terms in the potential associated with the SM Higgs in Eq. (II.4) that is called the Higgs portal. In addition the dark gauge sector and the SM gauge sector can be concocted via kinetic mixings between SU (2) D and U (1) Y after SU (2) D gauge symmetry breaking by nonzero VEVs of φ and φ [14]. The relevant terms for these kinetic mixings are two dim-5 operators: where Λ indicate the cut off scale and B µν is the gauge field strength for U (1) Y . After φ and φ developing VEVs, we obtain the following kinetic mixing terms: The kinetic terms for X 1,3 µ and B µ can be diagonalized by the following transformations: where δ is defined as sin δ ≡ − tan δ 1 tan δ 3 . In our analysis, we take a limit of δ 1 1 and δ 3 1 and gauge fields are approximately written by Then we denote dark gauge bosons associated with X 1,2,3 µ field as X 1,2,3 henceforth. Note that mixing with Z boson is suppressed unless dark gauge boson and the SM Z boson masses are not close enough. In our analysis, we assume a dark gauge boson mainly mixes with photon field.
After two triplet scalar fields develop nonzero VEVs, SU (2) D gauge bosons obtain masses from kinetic term such that Here we have ignored kinetic mixing effects since it is negligibly small in our scenario. We thus find the masses of dark gauge bosons be Note that the X 2 is always the heaviest one. In addition, three-point interactions among scalar and gauge bosons are given by where φ 3 and φ 1 can be written as mass eigenstates using Eqs. (II.13) and (II.14) for the case (1) and using Eqs (II. 16) and (II.17) for the case (2) described in previous subsection.
Finally interactions among dark gauge fields are also written → given by where abc is the structure constants of SU (2) D and a = 1, 2, 3. The heaviest gauge boson X 2 would decay into X 1 X 3 through the three point gauge interaction, where the X 1 and/or X 3 transition will be off-shell due to the mass relation among dark gauge bosons and both of them will eventually decay into the SM particles through kinetic mixings, Eq. (II.19).

C. Fermions in the dark sector
The mass terms of SU (2) D doublet fermion are given by where we assumed all coefficients are real. The mass splitting and the mass mixings between χ 1 and χ 2 are induced by the y χφ and y χφ respectively, in the last line of Eq. (II.4). The mass eigenvalues and eigenstates are obtained in a straight forward manner as where m χ l < m χ h by definition. The mixing angle θ χ is given by Furthermore we obtain interactions among scalar fields and mass eigenstates of dark fermions such that where φ 1,3 and φ 1,3 are substituted to mass eigenstates as discussed in previous subsection.  [15]. Therefore in the particle spectra of this model, there will be Z 2 string which is a topological object. One Z 2 vortex is topologically nontrivial, but two of them can be deformed smoothly into the vacuum, thereby being topologically trivial. These Z 2 string can contribute to the dark matter of the current Universe to some extent, but detailed study of this issue is beyond the scope of this paper. In the following, we shall simply ignore topological Z 2 strings assuming their contribution to the Universe is negligible.

III. DARK MATTER
In this section, we discuss DM phenomenology in our model including DM relic density. In our scenario, DM is the lightest lightest dark fermion χ l which is stabilized by the remnant  (2), we prefer small mixing, |θ χ | << 1, since DM couples with Higgs via φ 1 .

Then relic density of DM is determined by gauge interactions in dark sector in our scenarios
where we assume dark scalars are heavier than DM.
Then, the relevant interaction terms are where J µ EM is electromagnetic current and Q f SM is the electric charge of the SM fermions f SM . Then we implement these interactions in micrOMEGAs 4.3.5 [16] to estimate relic density.
In our analysis we take the dark fermion mixing angle for each scenario as and m X 2 = m 2 X 1 + m 2 X 3 . Also we require interaction between DM and the lightest dark gauge boson not to be suppressed by the dark fermion mixing effect. In the following, we shall focus on the scenario (1) since we just obtain similar results by replacing the role of X 1 and X 3 for the scenario (2).
In addition, we take into account DM -nucleon scattering via Z boson exchanging process.
The cross section for this process is calculated in non-relativistic limit as We then assume δ 1 10 −5 to avoid direct detection constraints such as XENON1T [18] and PandaX-II [17] which provide upper limit of ∼ 10 −46 cm 2 for DM mass of ∼ 100 GeV.
In Fig. 1, we show thermal relic density of DM, adopting dark gauge boson masses {m X 1 , m X 3 } as {200, 500} GeV and {10, 30} GeV as reference values, m χ h = 1.5m χ l , δ 1,3 = 10 −5 , and some relevant values of gauge coupling g D . We find that relic density is decreased when χ lχl → X 1 X 1 and χ lχl → X 2 X 2 processes are kinematically allowed. Then larger gauge coupling is required for larger m X 1 mass to accommodate with observed thermal relic density of DM. We can also explain relic density around resonance 2m Xχ l ∼ m X 1 when X 1 mass is relatively light while the relic density tends to be larger than observed one for heavier dark gauge boson case due to small kinetic mixing parameter.
Next we scan free parameters fixing δ 1,3 = 10 −5 to avoid direct detection constraint. The where masses of χ h and X 3 are respectively determined by those of χ l and X 1 for simplicity.
We then search for the parameter region which provide observed DM thermal relic density approximately in the range of 0.11 < Ωh 2 < 0.13. In the left and right panels of Fig. 2, we show allowed parameter region on {m DM , m X 1 } plane for the region I and II where color gradient indicates values of g D . We find large allowed region when χ lχl → X 1 X 1 (X 2 X 2 ) processes are kinematically allowed. On the other hand, for m X 1 > m χ l , we need some fine tuning around m X 1 2m χ l to obtain resonant enhancement of annihilation cross section.

A. Constraint from the SM Higgs boson decay
Firstly we discuss constraints from the SM Higgs decay process, h → X 1,2,3 X 1,2,3 → + − + − where denotes electron or muon. This multi-lepton decay channel is strongly constrained by the search for Higgs boson decaying into extra gauge boson which can decay into charged leptons [19] because of little background. The decay h → X 1,2,3 X 1,2,3 is induced via scalar mixing between the dark sector and the SM Higgs sector.
For the scenario (1), we obtain the decay widths as For the scenario (2), we also obtain the decay widths as TeV.
In Fig. 3, we show branching ratio (BR) for the process h → X 1 X 1 → + − + − in the scenario (1) where we consider 2m X 1 < m h and 2m X 2,3 < m h for simplicity; for the scenario (2) we obtain the same result replacing the role of X 1 and X 3 . We also show the upper limit on the BR as a dashed horizontal line. It is then found that the scalar mixing angle and/or the gauge coupling g D should be suppressed when the SM Higgs can decay into dark gauge boson decaying into charged leptons.

B. Scalar boson production
Here we discuss Φ 1 (Φ 1 ) production processes at the LHC. The scalar boson can be produced by gluon fusion process gg → Φ 1 (Φ 1 ) through the mixing with the SM Higgs boson parametrized by mixing angle α(α ). The relevant effective interaction for the gluon fusion is written as [20] L φgg = α s 16π where G a µν is the field strength for gluon and We obtain this effective interaction fromttΦ 1 (Φ 1 ) coupling via the mixing effect where we take into account only top Yukawa coupling since the other contributions are subdominant. In Fig. 4, we show the production cross section for scalar boson as a function of its mass with √ s = 14 TeV adopting several values of sin α(α ). We find that a sizable scalar mixing is required to obtain observable cross section. Thus we consider parameter region of 2m X 1 > m h in our discussion of collider physics since the scalar mixing is constrained for 2m X 1 < m h as shown in previous subsection.

C. Branching ratio of extra particles
Here we estimate BRs of particles in dark sector. The decay widths for the Φ 1 [Φ 1 ] → χ a χ b (a(b) = l, h) processes are given by The dark scalar bosons also decay into dark gauge bosons. For the scenario (1), we obtain For the scenario (2), we also obtain In Fig. 5, we show BR for Φ 1 → χ l χ h as functions of m χ h and m X 1 in the scenario (1) where we have scanned coupling as y χφ (g D ) ∈ [0.5, 2.5]([1.5, 2.5]) and fixed some parameters sin α = 0.1, θ χ = π/4, m χ l = 200 GeV, m X 2 m X 3 = 500 GeV and m Φ 1 = 600 GeV.
We find the BR for Φ 1 → χ l χ h is maximally 1.2 × 10 −2 and dominant decay mode is the Φ 1 → X 1 X 1 mode where the other modes are suppressed. For the scenario (2), we obtain similar result by replacing X 1 and X 3 and the corresponding plot is omitted here.

D. Signal at the LHC
Here we discuss signature of our model at the LHC based on decay modes of extra scalar bosons which are produced through gluon fusion process via scalar mixing. As we discuss in Sec. IV A, scalar mixing cannot be sizable when the SM Higgs decays into dark gauge bosons. Thus dark gauge boson masses are assumed to be heavier than half of Higgs mass to realize observable signals from extra scalar production. We summarize possible signature of the model in the following.
decay mode: For m X 1 [3] < 2m χ l , X 1 [3] dominantly decays into SM fermions induced by kinetic mixing. The BR of this decay chain of Φ 1 (Φ 1 ) is dominant when it is kinematically allowed, m Φ 1 [Φ 1 ] > 2m X 1 [3] , and provide sizable cross section. The most clear signal is four charged lepton final states which can be well tested at the LHC.
For m X 1 [3] > 2m χ l , X 1 [3] dominantly decay into DM since SM fermion mode is suppressed by small kinetic mixing. In this case, the final state becomes transverse missing energy and we need additional jet/photon for tagging.
For m X 2 < m χ l + m χ h , our signal is eight SM fermions coming from decay chain of X 2 → X 1 X 3 (X 1,3 →f SM f SM ). The BR of this decay mode of Φ 1 (Φ 1 ) can be sizable when it is kinematically allowed and masses among dark gauge bosons are not hierarchical. For m X 2 > m χ l + m χ h , X 2 dominantly decays intoχ h χ l (χ l χ h ).
Then χ h decays as χ h → X ( * ) 1 [3] χ l where dark gauge boson is off-shell or on-shell depending on mass hierarchy. In this case, we obtain signal of four SM fermions with missing transverse momentum from the decay chain.
We indicate dominant decay mode of Φ 1 for some mass relations in scenario (1)

V. SUMMARY AND DISCUSSIONS
We have discussed a model of dark sector described by SU (2) D gauge symmetry in which two triplet real scalar fields and one doublet Dirac fermion are introduced. In our scenario, SU (2) D symmetry is broken to discrete Z 2 symmetry by VEVs of two triplet scalar fields.
Then remaining Z 2 symmetry guarantees stability of DM candidate which is the lighter component from doublet fermion χ l . In the gauge sector, we consider kinetic mixing term between SU (2) D and U (1) Y which is assumed to be generated via 5-dimensional operators.
Then we have investigated dark gauge sector which provides three massive dark gauge bosons X 1,2,3 , two of which can mix with SM gauge boson via the kinetic mixings.
We have estimated relic density of our DM candidate where the observed value is explained via gauge interactions in dark sector with kinetic mixing effect as a portal to the SM sector. Then we have explored parameter region satisfying observed relic density. We have found that the relic density is explained by the process, χ lχl → X 1,2,3 X 1,2,3 , in large parameter region while we need fine tuning to obtain resonant enhancement for the process, χ lχl → f SMfSM , via dark gauge boson exchange with kinetic mixing.
Implications to collider physics have been discussed such as h → Z Z decay, and extra scalar production and its possible signals at the LHC. We have found that the constraint from h → Z Z branching ratio restricts scalar mixing with the SM Higgs and SU (2) D gauge coupling severely when the mode is kinematically allowed. Extra scalar boson can be produced by gluon fusion process through scalar mixing associated with the SM Higgs. For extra scalar production, we obtain some specific signatures depending on mass relation of dark sector particles.