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. 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. There are many papers in which various scalar and fermionic DM models have been adequately studied. 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. Otherwise, renormalizable interactions should be considered. If extra dimensions exist, the first KaluzaKlein mode of the U(1)Y gauge boson could be a well motivated vector DM candidate. 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 or the Brout-Englert-Higgs mechanism. 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.

Focusing on fundamental DM particles, a simple vector DM model can be constructed by introducing a dark U(1) gauge group with a dark Higgs field providing the gauge boson mass and a portal to the SM sectors. A Z2 symmetry is required to guarantee the stability of the vector DM candidate. More complicated extensions involve non-abelian gauge groups, such as SU(2), SU(2) ⊗ U(1), SU(2) ⊗ SU(2), SU(3), and general SU(N). Typically, some discrete or global continuous symmetries remain after gauge symmetry breaking, stabilizing the vector DM particle.

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. 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. 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. On the other hand, two dark Higgs triplets would be able to generate masses for all the eight SU(3) gauge bosons. 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.

Now, Zhang and colleagues in their recent paper studied a vector DM model with a dark SU(2)D gauge symmetry 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 (A¹, A² and A³) can obtain different masses. An accidental Z’2 symmetry remains after the spontaneous symmetry breaking. Under this Z’2 symmetry, one heavier guage boson, “A¹” is even, while the two lighter gauge bosons (A² and A³) are odd under a remaining Z2 symmetry, and the lightest one (A²) is stable, playing the role of the DM candidate. No mass degeneracy appeared 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, h0, h1, h2, and h3, where h0 acts as the 125 GeV SM-like Higgs boson. These Higgs bosons provide a portal to the SM particles for the dark gauge bosons or you can say 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.

They have randomly scanned the 14-dimensional parameter space to investigate phenomenological constraints from collider measurements of the 125 GeV Higgs boson, the DM relic density, and the direct/indirect DM detection experiments. They 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.

In the parameter points, they have found significant effects due to the h0, h1, and h2 resonances for m_A² ∼ m_hi /2, as well as a remarkable h1h1 threshold effect for m_A² ∼ m_h1 . Because of these effects, σ (Stat. SI, Sys. N) and {σ_ann v}0 ( ‘_’ means base) for some parameter points could be sufficiently small, evading the current direct and indirect detection constraints.

According to authors, 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 h0 generally deviate from the SM couplings. Future Higgs precision measurements at e^+e^− colliders, such as CECP, FCC-ee, and ILC, will provide further tests on this model.

They 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.

**Reference***: Zexi Hu, Chengfeng Cai, Yi-Lei Tang, Zhao-Huan Yu, Hong-Hao Zhang, “Vector dark matter from split SU(2) gauge bosons”, ArXiv, pp. 1-23, 2021. https://arxiv.org/abs/2103.00220*

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