$[SU(2)]^3$ Dark Matter

An extra $SU(2)_D$ gauge factor is added to the well-known left-right extension of the standard model (SM) of quarks and leptons. Under $SU(2)_L \times SU(2)_R \times SU(2)_D$, two fermion bidoublets $(2,1,2)$ and $(1,2,2)$ are assumed. The resulting model has an automatic dark $U(1)$ symmetry, in the same way that the SM has automatic baryon and lepton $U(1)$ symmetries. Phenomenological implications are discussed, as well as the possible origin of this proposal.

Introduction : In the standard model (SM) of quarks and leptons, the choice of the gauge symmetry, i.e. SU (3) C × SU (2) L × U (1) Y , and the particle content, i.e. quarks and leptons: (u, d) L ∼ (3, 2, 1/6), u R ∼ (3, 1, 2/3), d R ∼ (3, 1, −1/3), (ν, l) L ∼ (1, 2, −1/2), l R ∼ (1, 1, −1), together with the one Higgs scalar doublet a discrete Z 2 parity, i.e. R = (−1) 3B+L+2j , may be used to distinguish some new particles from those of the SM automatically. The importance of this observation is that this parity is not imposed, as is necessary in the minimal supersymmetric standard model, or in models of dark matter [1] assuming only the SM gauge symmetry. Whereas this idea of an automatic R parity has been implemented in some recent studies [2,3,4,5,6], I look instead in this paper for a dark U (1) symmetry (and not just a dark parity) which is also unrelated to B or L, but on the same footing, i.e. its emergence as the result of gauge symmetry and particle content. In the following I show how it may be achieved by inserting an extra SU (2) D gauge factor to the well-known SU is a possible SU (6) generalization of the Pati-Salam SU (4) symmetry [7].
The charged gauge bosons W ± D have mass g 2 D v 2 D and does not mix with W ± L,R , the 2 × 2 mass-squared matrix of which is given by .
Since W + D takes ψ 1,3 to ψ 2,4 , it has charge +2 under U (1) D to conform with Eq. (10) and φ ++ D has charge +4. This shows that U (1) D is not broken by φ D . Note that the mass degeneracy is broken by a small finite radiative correction [8] through the exchange of neutral gauge bosons. Hence ψ − 3R decays to the invisible ψ 0 3R and a virtual W − R which may convert toūd. Its lifetime is presumably quite long and the outgoing lepton has rather low momentum because of the kinematics. This kind of signature may be searched for at the Large Hadron Collider (LHC) as already pointed out [8].
There are four neutral gauge bosons, i.e.
Let them be rotated to the following four orthonormal states: where 1 The mass terms are given by It is easily shown that the photon A is massless and decouples from Z, Z R , Z D as it should.
The 3 × 3 mass-squared matrix spanning the latter is given by To ensure that SU (2) L is broken at a scale significantly lower than that of SU (2) R or SU (2) D , it is assumed that Hence Z decouples effectively from Z R and Z D , with negligible mixing to the latter. In terms are neglected, then the 2 × 2 mass-squared matrix is of the form where There are two interesting limits.
• (1) B, C << A, then A and C are eigenvalues with Z R and Z D as eigenstates.
• (2) A << B, C, then B + C and AC/(B + C) are eigenvalues with Gauge Interactions : The neutral-current gauge interactions are given by In particular Z 2 couples to If v 2 D << v 2 R , then Z D is the much lighter mass eigenstate with mass given by Eq. (22).
It couples to quarks and leptons according to Eq. (29) with For the dark Dirac fermion ψ 3 /ψ 4 , At the LHC, Z D may be observed through its production by u and d quarks, with its subsequent decay to lepton pairs. The c u,d coefficients [9,10] used in the data analysis are where B is the Z D branching fraction to e − e + and µ − µ + . To estimate c u,d , let g D = g R = g L , Assuming that Z D decays to 3 copies of the dark fermions of Eq. (6)  Based on the 13 TeV LHC data from ATLAS [11], this translates to a bound of about 3.5 TeV on the Z D mass.
If v 2 R << v 2 D , then Z 2 is the much lighter mass eigenstate with mass given by and the neutral W 3D and the physical neutral scalar h D are trivial, which allow them to mix with the other neutral gauge bosons and scalar bosons. The dark Dirac fermion ψ is assumed to be dominantly composed of ψ 3R and ψ 4R . To be specific, the outgoing ψ 4R may be redefined as an incoming ψ 3L , in which case the Dirac fermion ψ has a vector coupling to The elastic scattering of ψ off nuclei in underground direct-search experiments is possible through Z D or Z 2 . The spin-independent cross section σ 0 is enhanced by coherence and depends only on their vector couplings to the u and d quarks. For Z D which couples to For Z 2 which couples to 0.547j 3R − 0.233(j 3D + j B ), The cross section σ 0 is then given by where µ is the reduced mass of the effective interaction and equal to the nucleon mass for large m ψ . In the case of Z D as the mediator, In the case of Z 2 as the mediator, Assuming m ψ = 150 GeV for example, σ 0 is bounded by the latest experimental result [12] to be below 2 × 10 −46 cm 2 . Using Z = 54 and A = 131 for xenon, this translates to M Z D > 7.8 TeV and M Z 2 > 9.0 TeV, which are stronger than the LHC bounds discussed earlier.
then it is easily shown from Eq. (29) that it couples to This means that ψ V = 0 and there would be no interaction through Z 3 with nuclei and no bound on the mass of Z 3 from direct-search experiments. In other words, if the lightest new neutral gauge boson has a dominant Z 3 component, its bound may be lowered to a value comparable to that from the LHC.
Consider now the relic abundance of ψ. Its annihilation cross section through any new neutral gauge boson is much below 1 pb for a gauge-boson mass greater than 3.5 TeV. Hence a different process is required. Consider then the Yukawa sector. Note first that there is no scalar singlet, so if the dark fermion ψ is composed of only ψ 0 3R and ψ 0 4R with the invariant mass term ψ 0 3R ψ 0 4R , it has noψψ coupling to any scalar. However, as pointed out already, there are also the allowedψ 0 3R (φ 0 Hence ψ annihilation to scalars is possible and it may remain in thermal equilibrium in the early Universe until the temperature drops below m ψ .
There are several diagrams for ψ annihilation to scalars. As an estimate, consider Fig. 1 which depicts the process ψψ → φ + φ − through ψ − exchange. The cross section × relative velocity is given by where f is theψ 0 ψ − φ + coupling and M is the mass of the exchanged ψ This points to the possible unity of matter with dark matter, as discussed previously [5,13,14].
The only other possible (and very intriguing) SU (6) assignment is where x i and y i have charges 1/2 and −1/2 respectively, and SU (2) D is unbroken. This is a realization of an idea proposed many years ago [15], where color SU (3) q for quarks is matched with a parallel color SU (3) l for leptons. Whereas SU (3) q is unbroken, SU (3) l is broken to SU (2) l , thereby confining only two components of the fundamental fermion triplet, leaving the third component free as the observed lepton. This notion of leptonic color may be unified [16] under [SU (3)] 4 , with interesting predictions [17] for a future e − e + collider.
It must of course be at least rank 7.
Concluding Remarks : The notion is put forward that dark matter is intimately related to matter and the global U (1) symmetry which allows it to be stable is an automatic consequence of gauge symmetry and particle content in the same way that baryon and lepton numbers are so in the standard model. A specific proposal is the addition of an SU (2)