Vector SIMP dark matter with approximate custodial symmetry

We consider a novel scenario for Vector Strongly Interacting Massive Particle (VSIMP) dark matter with local $SU(2)_X\times U(1)_{Z'}$ symmetry in the dark sector. Similarly to the Standard Model, after the dark symmetry is broken spontaneously by the VEVs of dark Higgs fields, the approximate custodial symmetry determines comparable but split masses for $SU(2)_X$ gauge bosons. In this model, we show that $U(1)_{Z'}$-charged gauge bosons of $SU(2)_X$ ($X_\pm$) becomes a natural candidate for SIMP dark matter, annihilating through $3\rightarrow 2$ or forbidden $2\rightarrow 2$ annihilations due to gauge self-interactions. On the other hand, the $U(1)_{Z'}$-neutral gauge boson of $SU(2)_X$ achieves the kinetic equilibrium of dark matter through a gauge kinetic mixing between $U(1)_{Z'}$ and Standard Model. We present the parameter space for the correct relic density in our model and discuss in detail the current constraints and projections from colliders and direct detection experiments.


Introduction
Despite indirect evidences of the presence of dark matter in our Universe [1], its nature is still elusive. The absence of signal in direct detection experiments like LUX [2], PANDAX [3] or more recently XENON1T [4] questions the WIMP (Weakly Interacting Massive Particle) paradigm. The simplest extensions, involving minimal ingredients as Higgs-portal [5,6], Z-portal [7] or even Z -portal [8], etc, are already excluded for Beyond the Standard Model (BSM) scale below ∼ 3 TeV, once we combine cosmological, accelerators and direct detection constraints (see [9] for a recent review on the subject). In this context, one needs to develop extensions of the Standard Model beyond the WIMP paradigm.
One possibility for going beyond the WIMP paradigm is to modify drastically the thermal history of the dark matter, allowing for extremely feeble couplings between the visible and dark sectors. The correct relic abundance is then ensured through the freezein mechanism [10][11][12]. Another possibility proposed recently is to allow for large selfinteracting couplings generating 3 → 2 or 2 → 2 forbidden channels [13][14][15][16][17][18], which can then provide solutions to some of small-scale problems at galaxy scales [19,20]. These Strongly Interacting Massive Particle (SIMP) scenarios, naturally present in the case of non-abelian dark matter, open up a complete new range of parameter space that still needs to be explored.
We propose in this work to analyse in detail a simple SU (2) X × U (1) Z extension of the Standard Model (SM), where the charged non-abelian vector boson X ± is the dark matter candidate, while the Z µ boson plays the role of the portal between the visible and hidden sectors through its kinetic mixing with the SM hypercharge gauge boson B µ . Our construction provides a novel and efficient mechanism for maintaining VSIMP dark matter in kinetic equilibrium during freeze-out, being consistent with the observed relic density. We will discuss the interplay between the correct relic density and the kinetic equilibrium in constraining the parameter space, which can be tested at current and future experiments.
Our paper is organized as follows. We describe the model in the first section and compute its spectrum in Section 3. Section 4 is devoted to the computation of the dark matter abundance through the 3 → 2 and 2 → 2 forbidden channels while a combined analysis including constraints from direct detection and accelerators searches is presented in Section 5. We then conclude. There are two appendices dealing with general masses for gauge and Higgs bosons in the dark sector.

Model
We consider models with local SU (2) X × U (1) Z symmetry in the dark sector, "dark" in the sense that the Standard Model particles are not charged under these transformations. The dark Higgs sector is composed of a singlet scalar S and a nontrivial representation scalar H X under SU (2) X , with U (1) Z charges given by q S and q H X , respectively. Without loss of generality, we choose q H X = I 3 where I 3 is the dark isospin of H X .
The Lagrangian for the dark sector in our model is then given by where the field strength tensors are X µν = ∂ µ X ν − ∂ ν X µ + g X ( X µ × X ν ) and Z µν = ∂ µ Z ν − ∂ ν Z µ . The scalar part of the Lagrangian containing the SM Higgs doublet H is given by where the covariant derivatives for S and H X are with t i (i = 1, 2, 3) being SU (2) X generators satisfying [t i , t j ] = i ijk t k , V mix the scalar potential including mixing quartic couplings between dark Higgs fields 1 , and q Z = q S or q H X . We also introduce a gauge kinetic mixing between Z and hypercharge gauge bosons, This mixing will play the role of the portal between the dark sector and the Standard Model for the thermalization process. For completeness, we allow for Higgs portal couplings between the dark scalars and the SM Higgs, Moreover, developing eq. (1), we extract self-interacting interactions of dark gauge bosons [18], with 1 Notice that we can setλ H X = 0 when H X is a doublet.
where X µ ≡ (X 1,µ + iX 2,µ )/ √ 2 and its complex conjugate, After SU (2) X × U (1) Z is broken by the VEVs of dark scalars, the dark gauge bosons have nonzero masses, and X 1,2,µ gauge bosons are combined to be a complex gauge boson X µ with nonzero charge under U (1) Z . Thus, X 1,2,µ gauge bosons, if the lightest particle with U (1) Z charge in the dark sector, can be a dark matter candidate. Indeed, as Standard Model is neutral under U (1) Z , no decay modes of X µ into the visible sector are allowed due to the DM Z 2 symmetry after the spontaneous breaking of U (1) Z . On the other hand, as it happens in the Standard Model with W ± and Z 0 , X 3µ has a different mass from X 1,2µ due to the VEV of H X charged under the U (1) Z , and it can couple to the SM particles through the dark Weinberg angle in combination with the gauge kinetic mixing, as it will be discussed later. Our situation is a mirror case of the Standard Model, except for the presence of an extra-singlet S avoiding the presence of a massless gauge boson.

Dark gauge boson masses
In this section, we discuss the approximate custodial symmetry in the SU (2) X gauge sector with general dark Higgs representations and show the effect of the Z gauge boson on the mass splitting between dark-neutral and charged gauge bosons.

Dark custodial symmetry and its breaking
We considered the expansions of dark Higgs fields about nonzero VEVs by a singlet S = 1 √ 2 (v S + s), and a Higgs field H X in several representation of SU (2) X : a doublet Φ = (0, 1 We present general masses for dark gauge bosons in appendix A, while the expression for the dark Higgs masses and the mixing between the dark Higgs and the SM Higgs are given in appendix B. First, due to the VEV of the dark Higgs H X in a nontrivial representation of SU (2) X , such as Φ, T , Q 4 or Q 5 , masses of dark-charged gauge bosons, X µ , X † µ , are given by with I = 1 2 , 1, 3 2 , 2 respectively. For a vanishing Z charge of the dark Higgs H X or in the limit of a vanishing g Z , the dark-neutral gauge boson has mass leading to the mass relations due to dark custodial symmetry, Notice that we recover the M Z = M W relation of the Standard Model with a Higgs doublet (I = 1 2 ) in the case of a null Weinberg angle. As a result, if I > 1 2 , the dark-charged gauge boson X can be the lightest gauge boson in the dark sector, becoming a candidate for non-abelian dark matter. On the other hand, once taking into consideration the VEV of S, Z gauge boson can decouple from the X spectrum due to the contribution of v S to its mass. To maintain the thermal equilibrium of X µ with the Standard Model bath, one needs to consider first its coupling to Z .
Indeed, in order to communicate between non-abelian dark matter and SM by the renormalizable couplings, it is sufficient to consider the dark Higgs H X with nonzero Z charge and take a sizable g Z . Then, the mass matrix for neutral gauge bosons in the basis (Z µ , X 3µ ), receives a correction term violating the dark custodial symmetry, as follows, where m 2 X 3 ≡ (g 2 X + g 2 Z )I 2 v 2 I = m 2 X 3 ,0 + g 2 Z I 2 v 2 I , c X ≡ cos θ X and s X ≡ sin θ X , with sin θ X = g Z / g 2 X + g 2 Z , and Unlike in the SM, there exists a singlet scalar S contributing to to the Z mass. The most general dark gauge boson masses with VEVs of Higgs fields in arbitrary representations are given in the appendix A.
In the absence of the gauge kinetic mixing, the mass matrix (11) can be diagonalized explicitly by introducing a dark Weinberg angle as in the SM. Performing a rotation of dark gauge fields to mass eigenstates 2 ,Z µ ,X 3µ , as we obtain the mass eigenvalues for dark gauge bosons, Therefore, from the results in eqs. (8)- (16), we can keep the hierarchy of masses, as far as In this case, the dark charged gauge boson is still the lightest gauge boson in the dark sector, so that the 2 → 2 annihilation of X µ , X † µ is forbidden while 3 → 2 processes with gauge self-interactions become dominant for determining the relic density of X µ , X † µ .
We note that if m X > mX 3 , the 2 → 2 annihilation of X µ , X † µ into aX 3 pair is open, dominating the relic abundance calculation and leading to an interesting possibility for WIMP dark matter. However, in this work, we are interested in the production of light dark matter below sub-GeV scale, so henceforth we focus on the case with m X < mX 3 . Due to the mixing between Z µ and X 3µ , self-interactions for SU (2) X gauge bosons in eq. (6) become, in the basis of mass eigenstates, with Moreover, in the basis of mass eigenstates for dark gauge bosons, for instance, from eq. (B.8) in the case of the triplet dark Higgs, we can also obtain the interactions between dark Higgs and gauge bosons. Concerning the self-interacting processes for dark-charged gauge bosons, we can use the above interactions by ignoring the mixing between dark Higgs and SM Higgs bosons and the mixings between dark-neutral gauge bosons and SM neutral gauge bosons.

Split dark gauge bosons
In the limit of α 1, the Z boson decouples from X andX 3 . From eq. (14), we get tan θ X ≈ − 1 α tan θ X , leading to the approximate gauge boson masses, leading to Then, in order for the dark-charged gauge boson X to be a viable dark matter candidate, we require mX 3 > m X , which means I > 1 2 , that is, at least a triplet Higgs field with nonzero VEV must be introduced. Moreover, in order for SIMP processes XXX → XX 3 (which should be the dominant one in the relic abundance calculation) to be kinematically allowed, we also require mX 3 < 2m X , which means I ≤ 2. Thus, we need m X < mX 3 < 2m X in order to realize a viable SIMP dark matter in our model. From eq. (24), we get the general expression for the mass splitting parameter in the limit of large α: which will be relevant for forbidden channels in the later section. As a result, the mass relations between dark gauge bosons are dictated by dark custodial symmetry at the leading order, but up to small corrections due to the mixing with Z . For instance, we get ∆ ≈ √ 2− 1, √ 3 − 1, 1 for I = 1, 3 2 , 2, respectively. Then, for the triplet Higgs case with ∆ ≈ √ 2 − 1, we need to consider the 2 → 2 forbidden channels for dark matter annihilations. On the other hand, for quadruplet with I = 3 2 or quintuplet with I = 2, the mass splitting being too large, it is the XXX → XX 3 process which is dominant. For general dark Higgs VEVs, we can make ∆ to vary continuously from ∆ ≈ 0 to ∆ ≈ 2. We keep this fact in mind for scanning the parameter space for the correct relic density with general X 3 masses in the next section.

Degenerate dark gauge bosons
A quick look at eqs. (8) and (16) shows that there is a possibility of having degenerate mass terms in the dark sector, mX 3 ≈ m X provided that tan θ X ≈ − 1 2I (2I − 1) cot θ X which gives, once combined with eq. (14) : . Then, we obtain the abelian dark gauge boson mass as In this case, in the limit of tan θ X 1 or g Z g X for α ≈ 2I 2I−1 > 0, we can havẽ Z decoupled. This case is possible for a nonzero VEV of the Higgs field with I = 1, 3 2 , 2. Then, all the non-abelian gauge bosons of SU (2) can participate in the full 3 → 2 processes, without Boltzmann suppression. But, in this case, the mixing angle between dark neutral gauge bosons become suppressed due to | tan θ X | 1. Therefore, DM-SM elastic scattering through the kinetic mixing would be suppressed, so we need to rely on Higgs portal coupling for kinetic equilibrium.
If one considers multiple Higgs fields with different isospins, in particular, doublet Φ and triplet T , from eq. (A.11), we get the approximate mass difference in the limit of a largeZ mass : Then, we can tune dark Higgs VEVs, v Φ and v T such that m 2X 3 ≈ m 2 X . In this case, as discussed in the Appendix, the mixing angle between dark neutral gauge bosons becomes Therefore, even in this case, the mixing angle in the dark gauge sector is small so Higgs portal coupling would be more relevant for kinetic equilibrium as in the previous case.

Dark matter annihilations with self-interactions
We consider the Boltzmann equations for determining the dark matter relic density mainly by the DM self-interactions, in particular, by taking 3 → 2 channels and forbidden 2 → 2 channels to be dominant. We also present the DM self-scattering cross sections and impose current astrophysical limits on them together with the condition for a correct relic density.

Boltzmann equations
For CP conserving interactions of dark matter, we get n X + = n X − . Then, the total DM number density, n X = n X + + n X − , is governed by the following Boltzmann equations, where Here, in the last line on right in eq. (30), we have already used the detailed balance condition for forbidden channels to obtain the cross section for forbidden 2 → 2 annihilation, X + X − →X 3X3 , as follows, Here, we assumed thatX 3 keeps in thermal equilibrium with the SM during freeze-out, and the equilibrium abundances for X andX 3 are given by with x ≡ m X /T . We have also included the SM 2 → 2 annihilations, X + X − → ff , in the Boltzmann equations (30), as will be discussed in the next section.

3 → 2 annihilations
The effective 3 → 2 annihilation cross section in eq. (30) is given by the channel, X + X + X − → X +X3 , with the corresponding cross section given by  Figure 1: Parameter space for m X vs α X ≡ g 2 X /4π for the case with quadruplet dark Higgs. The correct relic density is satisfied in red region. Contours for DM self-scattering cross section with σ self /m X = 0.1, 1, 10 cm 2 /g are shown in dotted, dashed and dot-dashed lines, respectively. We also indicated the favored parameter space in green line to explain the off-set of one of the galaxies in Abell 3827 [21][22][23] and the region disfavored by Bullet cluster in gray [24]. where Then, assuming that dark matter annihilates mainly due to 3 → 2 channels, we can determine the relic density [15,16] as where α eff is defined from σv 2 rel 3→2 ≡ α 3 eff /m 5 X . In Fig. 1, we showed the parameter space for m X and α X ≡ g 2 X /(4π), satisfying the relic density condition in red region, dominantly due to SIMP channels. We took the VEV of quadruplet Higgs to get masses for non-abelian gauge bosons, for which mX 3 ≈ √ 3 m X . Forbidden channels become dominant only for small m X or α X . We also drew the contours with σ self /m X = 0.1, 1, 10 cm 2 /g in dotted, dashed and dot-dashed lines. We note that the observed off-set of one of the galaxies of Abell 3827 [21,22] indicates a large self-scattering cross section for dark matter, σ self /m X ∼ 0.68 cm 2 /g [23], so we showed the parameter space favored by Abell 3827 in green line. The gray region is disfavored by Bullet cluster bound [24], σ self /m X 1 cm 2 /g.
In Fig. 2, we also depicted the parameter space for m X and mX 3 , satisfying the correct relic density in red regions. We took different values for α X = 0.1, 0.5, 1, from left to right. We included both the SIMP and forbidden 2 → 2 channels for determining the relic density. Forbidden channels are important for mX 3 1.6m X , as will be discussed in the next subsection. In each plot of Fig. 2, the SIMP channels are kinematically closed in the upper purple region because mX 3 > 2m X , whereas the 2 → 2 annihilations such as X + X − →X 3X3 are open and dominant in the lower green region. For comparison, the mass relation, mX 3 ≈ √ 3m X , is satisfied along the blue dashed line. Below the blue dashed line, the forbidden channels tend to contribute to the DM annihilations dominantly. But, above the blue dashed line, the SIMP channels are dominant, and the relic density is saturated close to mX 3 ∼ 2m X , due to the t-channel diagrams. This behavior is regularized by a relatively large velocity of dark matter during freeze-out.

Forbidden 2 → 2 annihilations
For the inverse process of 2 → 2 annihilation, i.e.X 3X3 → X + X − , in the dark sector, the corresponding cross section is given by For dominance with forbidden channels in the Boltzmann equation (30), the solution for the DM abundance [16] becomes . Figure 3: Relic density as a function of ∆ = (mX 3 − m X )/m X . SIMP channels only are in red and forbidden channels only are in green, while both channels are included in black.
where λ ≡ (s(m X )) 2 /H(m X ) with s(m X ) and H(m X ) being entropy density and Hubble parameter, respectively, evaluated at T = m X , and and ∆ ≡ (mX 3 − m X )/m X . Then, forbidden channels can determine the relic abundance [16] as follows, where β eff is defined from f (∆, x f ) ≡ m 2 X /β 2 eff and we took x f 15 in the exponent. As a consequence, as far as ∆ 0.6, the forbidden channels can be efficient enough even for a small self-interaction to produce a correct relic density. This is the case with the triplet dark Higgs, for which ∆ ≈ √ 2 − 1. In cases with quadruplet and quintuplet dark Higgs fields, ∆ is larger so that SIMP channels are dominant in determining the relic density.
In Fig. 3, we showed the relic density as a function of the mass difference, ∆ = (mX 3 − m X )/m X in black dotted line. In each plot, SIMP channels only are assumed in red line and forbidden channels only are assumed in green line. We have included both SIMP and Figure 4: Relic density as a function of m X for the case with quadruplet dark Higgs. SIMP channels only are in red and forbidden channels only are in green, while both channels are included in black.
forbidden channels in black line. In all the cases for m X ∼ 100 − 500 MeV, SIMP channels become dominant for ∆ 0.6 or mX 3 1.6m X . Therefore, from eq. (25), ∆ √ 2I − 1, so we need that dark Higgs fields with I = 3 2 or 2 determine the SU (2) X gauge boson masses dominantly. Otherwise, the relic density is determined mainly by forbidden channels.
In Fig. 4, we drew similar plots as in Fig. 3, but now for the relic density as a function of m X for the fixed mass relation, mX 3 = √ 3 m X , for the nonzero VEV of quadruplet Higgs. In this case, as we increase α X from 0.01 to 0.1, 0.5 and 1, clockwise, showing the dominance of SIMP channels in determining the relic density for α X 0.1. For a correct relic density, we need a sizable α X as in the plots in the lower panel, in which case the SIMP channels are dominant.

DM self-scattering
The effective DM self-scattering cross section is given by with Then, the DM self-scattering with σ self /m X = 0.1 − 10 cm 2 /g can explain small-scale problems at galaxies such as core-cusp problem, too-big-to-fail problem, etc. On the other hand, we also note that the DM self-scattering cross section is constrained to be σ self /m X 1 cm 2 /g by either Bullet cluster [24] or halo shapes/ellipticity [25].

Z portal couplings for dark matter
Dark matter, if its abundance is determined mainly by SIMP processes, must be in kinetic equilibrium with another species in the dark or visible sectors for the structure formation.
In order not to introduce the temperature of dark matter as an independent parameter, we consider the case that dark matter is equilibriated by the elastic scattering between dark matter and light particles in the SM.
In this section, we discuss the conditions for kinetic equilibrium in the presence of a gauge kinetic mixing between Z and SM hypercharge gauge bosons, and consider various constraints on the model, coming from the consistency of SIMP scenarios to experimental bounds such as direct detection and collider searches.

General current interactions with Z portal
In the presence of a gauge kinetic mixing, we need to consider the full basis of neutral gauge bosons, including neutral dark gauge bosons and those in the SM, i.e. B µ and W 3µ . For a zero dark Weinberg angle, we can easily diagonalize the gauge kinetic terms and mass terms for neutral gauge bosons only by the 4 × 4 rotation matrix, O W , as in usual Z portal models. Even with a nonzero dark Weinberg angle, we can still diagonalize the mass matrix by approximate rotations with O W , followed by the 2 × 2 rotation matrix, O X , in the limit of a small gauge kinetic mixing. Then, the gauge bosons in the interaction basis, (B µ , W 3µ , Z µ , X 3µ ), can be written in terms of approximate mass eigenstates (Ã µ ,Z µ ,Z µ ,X 3µ ), as follows, Here, c ξ ≡ cos ξ, t ξ ≡ tan ξ, c W ≡ cos θ W , s W ≡ sin θ W , and ζ is the approximate mixing angle between Z and Z bosons, given [14] by In the limit of m 2 Z m 2 Z and |ξ| 1, we obtain ζ ≈ −s W ξ. Moreover, the approximate mass eigenvalues for Z-like and Z -like gauge bosons are given [14] by The current interactions in the interaction basis are given by where J µ X 3 and J µ Z are dark-neutral and Z currents, and J µ EM and J µ Z are electromagnetic and neutral currents. Then, using eq. (48), we can rewrite the above current interactions in the basis of mass eigenstates, at first order in ε ≡ c W t ξ ≈ c W ξ, as Therefore, we find that dark-neutral gauge bosons,Z andX 3 , couple to electromagnetic currents and dark-neutral currents as well as Z currents. SinceX 3 has mass comparable to dark matter mass by the approximate custodial symmetry in the dark sector, it can play an important role for the kinetic equilibrium of dark matter by elastic scattering.
As a result,Z can decay into either a lepton pair or a pair of dark matter particles. Then, the total decay width ofZ is given by On the other hand,X 3 has mass mX 3 < 2m X for SIMP processes to be kinematically open. So,X 3 decays dominantly into a pair of leptons with f being electron or muon, with the decay rate given by On left of Fig. 5, we depicted contours of the ratio of visible to invisible decay rates ofZ gauge boson in the parameter space for ε vs α X ≡ g 2 X /(4π). Here, we have fixed m X = 100 MeV, mZ = 1 GeV and sin(2θ X ) = 0.1. So, in most of the parameter space for α X 0.1 and ε 10 −2 , the visible decay ofZ is negligibly small. On right of Fig. 5, we also showed the ratio of decay rate to mass ofX 3 as a function of ε, for mX 3 = 500 MeV and sin(2θ X ) = 0.1. In this case, for m X < mX 3 < 2m X , mX 3 decays visibly into a lepton pair, so the decay rate is doubly suppressed by the dark Weinberg angle and the gauge kinetic parameter.

Kinetic equilibrium
We consider the elastic scattering between dark matter and electron through Z portal couplings to achieve a kinetic equilibrium for SIMP dark matter. In this case, the corresponding momentum relaxation rate during freeze-out is given by Them, for kinetic equilibrium during SIMP freeze-out, we need to require where T F m X /15 is the freeze-out temperature and H(T F ) is the Hubble parameter at freeze-out.
On the other hand, since dark-neutral gauge bosonX 3 has a similar mass as dark matter in our model, its abundance is not that suppressed around freeze-out temperature. Thus,X 3 plays an important role of keeping dark matter in thermal equilibrium through the SM by the decays into electron or muon pairs, the kinetic equilibrium for dark matter can be also achieved by the elastic scattering between dark matter andX 3 . Thus, in this case, we require n eq We find that kinetic equilibrium for dark matter can be easily achieved by the elastic scattering between dark matter and dark-neutral gauge bosonX 3 as far as the latter  Figure 6: Various constraints on ε vs m X for the case with quadruplet dark Higgs. The blue region and the region above the blue dashed line are excluded by invisible modes ofZ in BaBar [26] (observed) or Belle2 [27] (expected), respectively. The green region is excluded by visible modes ofX 3 in BaBar [28] (dilepton+monophoton: observed). Direct detection limit on DM-electron elastic scattering from XENON10 [29] (observed) and SENSEI-100 1yr [30] (expected) are shown in yellowish region and red dashed line, respectively, and contours with DM-electron elastic scattering cross section, σ e − DD = 10 −47 cm 2 , are also shown in orange dashed lines. Kinetic decoupling occurs in purple region and 2 → 2 visible annihilations would be dominant in red region. For comparison, the DM-electron scattering process becomes negligible for kinetic equilibrium in the region below the purple dashed line. The relic density is saturated along the dashed black vertical lines.  Figure 7: Similar constraints as in Fig. 6, but in the parameter space for ε vs mZ for the case with quadruplet dark Higgs. remains in kinetic equilibrium for a tiny gauge kinetic mixing ∼ 10 −6 in the parameter space of our interest.
In Fig. 6, we showed the parameter space for ε ≈ c W ξ vs m X with various constraints coming from the model consistency and experiments. We have assumed that the VEV of a quadruplet dark Higgs determines SU (2) X gauge boson masses and chosen different values for the DM self-coupling α X = 1, 0.5 in the top and bottom panels, respectively, varying mZ . We have similar plots for ε vs mZ in Fig. 7.
As for the theoretical constraints, dark matter becomes out of kinetic equilibrium already during freeze-out in purple region and 2 → 2 visible annihilations such as X + X − → X 3X3 would be dominant in red region. The relic density is saturated along the black dashed vertical line in Fig. 6, which is determined mostly by the SIMP processes. In Fig. 7, we have chosen the values of m X and α X such that the correct relic density is saturated.
We now turn to the experimental constraints on the model. AsZ decays mostly invisibly into a dark matter pair, the observed limit from BaBar [26] and the future projection from Belle2 [27] can rule out the parameter space in blue region and in the region above the blue dashed line. On the other hand,X 3 decays visibly into a lepton pair in the SM so there are constraints from the monphoton+dilepton searches in BaBar [28], but they are not as strong as the invisible searches forZ in BaBar. Direct detection from electron-recoil signals in XENON10 [29] has excluded the yellowish region, sometimes being comparable than or even stronger than the visible bound from BaBar. The future updated SENSEI-100 1yr [30] can reach the limit for lightZ mediator, even beyond the Belle2 projection. For comparison, we also showed the contours with DM-electron elastic scattering cross section, σ e − DD = 10 −47 cm 2 , in orange dashed lines.

SM 2 → 2 annihilations and direct detectiom
Light dark matter can also annihilate into a pair of leptons or mesons for m X 1 GeV. In particular, the annihilation cross section for X + X − → ff , with f being electron or muon, is before thermal average, The thermal average of the above annihilation cross section needs care near the resonances, except which, the DM 2 → 2 annihilations are velocity-suppressed so they are not constrained by indirect detection experiments [14].
TheX 3 andZ couple to both dark matter and electron so direct detection for light dark matter is relevant. For m e , m X , mZ , mX 3 p X m X v X , the DM-electron elastic scattering cross section is approximately given by Thus, there are two mediators contributing to the DM 2 → 2 annihilation and the DMelectron elastic scattering in our case, in particular, we can impose the direct detection constraints on the Z portal coupling with a single mediator by identifying 1/m 4 Z with (1/m 2X 3 − 1/m 2 Z ) 2 as we have discussed for Figs. 6 and 7.

Conclusions
We have proposed a new model for Vector SIMP dark matter in the context of dark SU (2) X × U (1) Z gauge theory with a similarity to the SM counterpart. The mass splitting between dark matter and neutral components of SU (2) X are predicted by an approximate custodial symmetry in the dark Higgs sector, playing a crucial role for the production mechanism of dark matter due to self-interactions. The kinetic equilibrium for VSIMP is maintained during the freeze-out thanks to a gauge kinetic mixing between U (1) Z and the SM, providing a testing ground for searches for light mediators of order GeV scale or below at current and future collider and direct detection experiments.
where m 2 X 3 ≡ (g 2 X +g 2 Z ) I I 2 v 2 I , c X ≡ cos θ X and s X ≡ sin θ X , with sin θ X = g Z / g 2 X + g 2 Z , and In the absence of the gauge kinetic mixing, the above mass matrix (A.3) can be diagonalized by introducing a dark Weinberg angle as in the SM. Performing a rotation of dark gauge fields to mass eigenstates,Z µ ,X 3µ , as we obtain the mass eigenvalues for dark gauge bosons, The results generalize our results in eqs. (8)- (16) with a singlet S and a triplet T in the main text.
For β 1, from eq. (A.6), we get tan θ X ≈ − 1 β tan θ X , leading to the approximate gauge boson masses, In this limit, the mass difference between the light gauge bosons is given by Therefore, for m 2X 3 > m 2 X , we need I > 1 2 , namely, at least a triplet dark Higgs with nonzero VEV.
For instance, ignoring the mass splitting due to the dark Weinberg mixing and keeping only one Higgs representation with I = 1 2 , 1, 3 2 , 2, we get m 2X 3 ≈ m 2 X , 2m 2 X , 3m 2 X , 4m 2 X . (A.12) Then, we get ∆ ≡ (mX 3 − m X )/m X for I = 1 2 , 1, 3 2 , 2, as follows, But, for general VEVs of all Higgs representations with 1 and the mixing angle is . (B.5) In the limit of a small λ T H , the mass eigenvalues of neutral scalars become Then, the vacuum stability bound, λ T >λ T , ensures the positive squared masses for neutral dark Higgs boson, h 1 ≈ h T , for a small mixing quartic coupling.
In order to ensure the consistency of the dark vacuum, we require m 2 h ++ > 0 or which naturally pushes the value of the quartic mixing to be λ T H 10 −5 for v T ∼ GeV, implying the mixing angle tan 2θ 10 −5 to be below the bound from the Higgs invisible decay.
Moreover, from the kinetic terms of the triplet complex field in eq. (2), we also derive the following interactions between dark Higgs and gauge bosons,

Quadruplet Higgs bosons:
For the quadruplet Higgs Q 4 = h (3) , h (2) , 0, 1 the masses of dark-charged and neutral Higgs bosons are As a result, there are similar consistent conditions on the mixing quartic couplings for m 2 h (2) > 0 and m 3 h (2) > 0, as in eq. (B.7). We can ignore the dark charged Higgs contributions in the later discussion, if they are heavy enough for λ Q 4 v 2 Q 4 ∼λ Q 4 v 2 Q 4 m 2 X . Moerover, a similar vacuum stability bound, λ Q 4 > 9 4λ Q 4 , ensures the positive squared masses for neutral dark Higgs boson h Q 4 , for a small mixing quartic coupling.

Quintuplet Higgs bosons:
For the quintuplet Higgs Q 5 = h (4) , h (3) , h (2) , 0, 1 the masses of dark-charged and neutral Higgs are As a result, there are similar consistent conditions on the mixing quartic couplings for m 2 h (a) > 0 with a = 1, 2, 3, as in eq. (B.7). Similarly, he dark charged Higgs contributions can be neglected in the later discussion, when λ Q 5 v 2 Q 5 ∼λ Q 5 v 2 Q 5 m 2 X . Moerover, a similar vacuum stability bound, λ Q 5 > 4λ Q 5 , ensures the positive squared masses for neutral dark Higgs boson h Q 5 , for a small mixing quartic coupling.