Dark SU(2) Antecedents of the U(1) Higgs Model

The original spontaneously broken U(1) gauge model with one complex Higgs scalar field has been known in recent years as a possible prototype dark-matter model. Its antecedents in the context of SU(2) are discussed. Three specific examples are described, with one dubbed"quantum scotodynamics".

Introduction : Consider the addition of the U(1) D Higgs model [1] to the standard SU(3) C × SU(2) L × U(1) Y gauge model (SM) of quarks and leptons. The former may be used for dark matter [2,3,4,5,6,7] because it has the built-in Z 2 symmetry where the massive gauge boson Z D after spontaneous symmetry breaking is odd and the one physical real scalar boson h D is even. However, U(1) D may mix kinetically [8] with U(1) Y , in which case the above Z 2 symmetry would be violated. To avoid this problem, it is suggested here that U(1) D be replaced with an SU(2) antecedent, with an enriched dark-matter sector. Three explicit examples will be discussed. Note that this version of dark SU(2) requires that it be broken to U(1), in contrast to the case where a local or global SU(2) dark symmetry remains [9]. SU(2) D with Scalar Doublet and Triplet : To break SU(2) D to U(1) D , the simplest choice is a real scalar triplet with χ 3 = v 3 . In that case, the vector gauge bosons acquire mass given by m 2 W D = 2g 2 D v 2 3 . Note that the superscript ± refers to dark charge, the details of which will be discussed later.
To break U(1) D in the context of SU(2) D so that D 3 = Z D acquires mass, a complex scalar doublet which prevents the coupling of χ to the triplet φ i ǫ ij σ jk φ k . The scalar potential consisting of χ and Φ is then given by Note that the triplet combination of two identical real scalar triplets is zero. The minimum of V admits a solution where v 2 is assumed real without any loss of generality, and provided that As a result and the 2 × 2 mass-squared matrix spanning A global residual symmetry remains, under which This comes from I 3D + S Φ , where S Φ = 1/2 for Φ and zero for all other fields. It is possible because of the imposed global U(1) Φ symmetry. Whereas φ 2 = v 2 / √ 2 breaks both I 3D and S Φ , the linear combination I 3D + S Φ is zero for φ 2 , so it remains as a residual dark symmetry.
An important consequence of this structure is the emergence of a dark charge conjugation symmetry as in the original Higgs model [1], i.e.
This comes from the gauge-invariant terms It means that Z D is stable if its mass is less than twice that of φ 1 , in complete analogy to the U(1) D model of Ref. [7]. This makes it possible in principle to implement the inception of self-interacting dark matter, i.e. φ 1 or W D of order 100 GeV with Z D as the light stable mediator of order 10 to 100 MeV, to explain [10] the observed core-cusp anomaly in dwarf galaxies [11]. If Z D is unstable and decays to SM particles, as is the case for the light mediator proposed in most models, then very strong constraints exist [12] from the cosmic microwave background (CMB) which basically rule out [13] this scenario. On the other hand, h D must also be light and decay quickly through its mixing with the SM Higgs boson h before big bang nucleosynthesis (BBN). In that case, the elastic scattering of W D or φ 1 off nuclei through h D exchange is much too large to be acceptable with present data. In Ref. [7], this is not a problem because the dark matter is a Dirac fermion which couples to As it is, this specific SU(2) D antecedent of the U(1) Higgs model may still be a model of dark matter without addressing the core-cusp anomaly in dwarf galaxies. Assuming that W D is heavy enough to decay into φ 1 h D and Z D heavy enough to decay into φ 1 φ * 1 , then the complex scalar φ 1 may be considered dark matter. Assuming that h D is lighter than φ 1 , the annihilation cross section of φ 1 φ * 1 at rest × relative velocity is given by where Assuming as an example m φ 1 = 150 GeV and m h D = 100 GeV, the above may be set equal to 4.4 × 10 −26 cm 3 /s for λ 2 = 0.126.
There is always the allowed quartic λ 2h coupling between the SU(2) D Higgs doublet and the SU(2) L × U(1) Y Higgs doublet of the SM, so that φ 1 interacts with quarks through the SM Higgs boson h in direct-search experiments. Using present data [14], it has been shown [15] that λ 2h < 4.4 × 10 −4 . This is also the mixing bewteen h D and h. Even with this limit on λ 2h , it can still be large enough so that h D decays promptly to bb in the early Universe. This interaction [7] also keeps h D in thermal equilibrium with the particles of the SM.

SU(2) D with Fermion Doublet
: Consider the addition of a fermion doublet to the SU(2) D model discussed in the previous section. It has the allowed interactions For the benchmark value of σ/m φ 1 ∼ 1 cm 2 /g for self-interacting matter, this is satisfied for example with This low-energy effective theory consisting of ψ, Z D and h D may be dubbed quantum scotodynamics, from the Greek 'scotos' meaning darkness.
Consider now the annihilation of ψψ → Z D Z D . Since Z D is much lighter than ψ, this cross section × relative velocity is given by For m ψ = 100 GeV, and setting σv rel = 4.4 × 10 −26 cm 3 /s, is obtained, which implies from Eq. (11) that As shown in Ref. [7], the light mediator Z D is stable but annihilates quickly to h D which decays. The cross section × relative velocity is given by Assuming m h D = 21 MeV as an example so that r = 0.25, the above is equal to 2 × 10 −18 cm 3 /s, which is orders of magnitude greater than what is required for Z D to be a significant component of dark matter. It may re-emerge at late times by φ 1 φ * 1 annihilation through Sommerfeld enhancement, but its fraction as dark matter remains negligible. Since Z D is stable, it would also not disturb [12,13] the cosmic microwave background (CMB).
As for h D , it is allowed to mix with the SM Higgs boson h in the 2 × 2 mass-squared matrix where v h = 246 GeV and m h = 125 GeV. For then θ 2h = 3.8 × 10 −5 and the h D lifetime for e − e + decay is given by which is short enough not to affect big bang nucleosynthesis (BBN). The decay of the SM Higgs boson to h D h D is given by which is less than 25% of the SM width of 4.12 MeV and allowed by present data. Note that  Eq. (4), is necessary for obtaining a dark symmetry. Hence the latter is not predestined [16], i.e. not the automatic consequence of gauge symmetry and particle content. To have a predestined dark Z 2 symmetry, the simpler scalar triplet is now replaced with a scalar quintet. This is analogous to having a fermion quintet [17] in the SM for minimal dark matter.
Consider thereby the real scalar quintet with ζ 0 = v 5 , then W ± D obtains a mass given by m 2 W D = 6g 2 D v 2 5 from absorbing ζ ± . This leaves ζ ±± as physical scalar bosons with two units of dark charge, interacting with Z D . The scalar potential consisting of ζ and Φ is then given by where V 3 contains the one cubic invariant formed out of 3 scalar quintets and V 4 contains two quartic invariants. Note that the triplet combination of two identical real scalar quintets is zero. As a result, this scalar potential automatically has an extra U(1) Φ symmetry, so that I 3D + S Φ remains unbroken as φ 2 acquires a vacuum expectation value v 2 / √ 2 as explained previously.
Assuming that then W + D decays to φ 1 h D , Z D decays to φ 1 φ * 1 , but both φ 1 and ζ are stable. Hence this is an explicit example of two-component dark matter under one dark U(1) symmetry. Let then using Eq. (16) for σ 1 (φ 1 φ * 1 → h D h D )v rel and the analogous where r 2 = m 2 h D /m 2 ζ and r 3 = m 2 φ 1 /m 2 ζ , the condition for the correct relic abundance is roughly given by It has for example the reasonable solution λ 5 = λ 2 = 0.173, in which case φ 1 is 53% and ζ 47% of dark matter. Again the mixing of ζ with the SM Higgs boson h must be small as it is for φ 1 to satisfy direct-search limits as discussed previously.
In this scenario, the addition of the fermion doublet of Eq. (17) could also provide a lowenergy effective theory of quantum scotodynamics with light Z D and h D . In that case, ζ ±± would decay into W ± D W ± D , φ 1 would decay into W + D h D , and W ± D would decay into ψ 1 ψ 1 /ψ 2 ψ 2 .
Concluding Remarks : Exploring the possible SU(2) antecedents of the famous U(1) Higgs model for a nontrivial application to dark matter, three interesting examples have been identified and discussed. The minimal version with one real scalar triplet χ and one complex scalar doublet Φ admits φ 1 as dark matter, but a global U(1) symmetry has to be imposed.
With the addition of a fermion doublet ψ, the inception of self-interacting dark matter may be implemented successfully, avoiding all potential astrophysical and laboratory constraints. A third example replaces χ with the real scalar quintet ζ, in which case the dark U(1) symmetry becomes predestined, i.e. automatic from the gauge symmetry and particle content.