Non-Abelian Gauge Lepton Symmetry as the Gateway to Dark Matter

Following a previous proposal, lepton number is considered as the result of a spontaneously broken non-Abelian gauge $SU(2)_N$ symmetry. New fermions are added to support this new symmetry, the spontaneous breaking of which allows these new fermions to be part of the dark sector, together with the vector gauge boson which communicates between them and the usual leptons. A byproduct is a potential significant contribution to the muon anomalous magnetic moment.

Introduction : Lepton number is an automatic global U(1) symmetry in the minimal standard model (SM) of quarks and leptons. If right-handed neutrino singlets are added, B − L (baryon number minus lepton number) may then become a gauge U(1) symmetry. Many other possible variants have been studied [1,2,3,4,5,6,7,8,9,10], but all involve U (1) symmetries. The first hint that leptons may be extended to a non-Abelian SU (2) N symmetry was based [11] on E 6 , where only the left-handed lepton doublet is involved. It was then realized [12,13,14] that the vector gauge boson in SU (2) N and the new fermions which it links with the SM leptons could be dark matter. To go one step further, the lepton singlet e R was also proposed [15] to be part of a doublet under SU (2) N , together with an added ν R .
Thus SU (2) N acts as the gateway to a dark sector, reaffirming the notion that leptons are the key [7,16] to understanding dark matter.
In this framework [15], the SM leptons transform under SU (2) N together with their partners (N, E), N , E , which will be shown to belong to the dark sector, after spontaneous breaking of SU (2) N . The residual conserved symmetry is generalized global lepton number, under which with the two sectors connected through the SU (2) N gauge analog (call it X) of the W boson of the SM. This means that X also has lepton number and belongs to the dark sector.
The scalar sector is very minimal, consisting only of the SM doublet, and a corresponding SU (2) N doublet. Just as the former results in heavy W ± , Z gauge bosons, the latter yields heavy X 1,2,3 guage bosons. The residual physical scalars are then just the SM Higgs boson h and the corresponding H of SU (2) N .
In the following, the consequences of this new extension of the SM will be discussed, regarding to dark-matter phenomenology [17], as well as its contributions to the muon anomalous magnetic moment [18,19,20] as an example.
Model : Under SU (2) L × U (1) Y × SU (2) N , the SM leptons ν, e and their SU (2) N partners N, E and N , E transform as It is easy to see that this gauge extension is free of anomalies because each new left-handed fermion is balanced by an appropriate right-handed counterpart. The scalar sector is minimally simple. It has just the usual SM doublet Φ plus an SU (2) N doublet χ: The allowed Yukawa couplings are Let φ 0 = v and χ 1 = u, then whereas the 2 × 2 mass matrices for (E, E ) and (N, N ) are and W 3 mixing with the U (1) Y gauge boson to form the neutral Z and photon.
Gauge and Scalar Sector : The SU (2) L × U (1) Y gauge sector is as in the SM. The SU (2) N gauge sector is separate but is analogous to just SU (2) L alone. The X 1,2,3 gauge bosons obtain masses from χ 1 = u, resulting in The simple Higgs potential is In terms of the physical Higgs bosons h = Assuming that λ 3 is small, then h and H are almost mass eigenstates with m 2 h = 2λ 1 v 2 and m 2 H = 2λ 2 u 2 .
The fermion interactions with X and Z are easily read off from Eqs. (2)-(4), i.e. [The N fermion may also be considered [15], but its mixing in Eq. (8) to N must be suppressed so that it has negligible coupling to the SM Z boson to avoid direct-search constraints.] Since the coupling of XX to Z vanishes for X,X at rest, whereas the XXH coupling does not, its relic abundance comes from XX → HH annihilation [21], assuming H to be lighter than X, as shown in Fig. 1 in the last one) for XX annihilation at rest is where 1,2 are the polarization vectors of X,X and k is the momentum of H.

The annihilation cross section times relative velocity is
where r = m 2 H /m 2 X and Once produced, H decays through its very small mixing with h to SM particles. For m H /m X = 0.8, the typical value of 3 × 10 −26 cm 3 /s for the correct relic abundance is obtained for m X /g 2 N = 352.5 GeV. Assuming m X = 210 GeV to be above the highest energy of the LEP II e + e − collider, g N = 0.77 is obtained.
The interactions of X with the SM leptons are shown in Eq. (12), and through the Z gauge boson. In underground direct-search experiments using nuclear recoil, only the Higgs exchange is applicable, which occurs through h − H mixing. The spin-independent cross section for elastic scattering off a xenon nucleus is where [22] f p m p = 0.075 For m X = 210 GeV, and m Xe = 122.3 GeV, the upper limit on σ 0 is [23] 2 × 10 −46 cm 2 .
Using m h = 125 GeV and m H = 168 GeV, the h − H mixing parameter λ 3 is then less than 1.27 × 10 −4 .
Muon Anomalous Magnetic Moment : Since Z and X couple to leptons as shown in Eq. (12), there are contributions [20] to the muon anomalous magnetic moment. The Z contribution is If E and E do not mix, the X contribution is where m E = m E = m X has been assumed for convenience. Using g N = 0.77 and m X = m Z = 210 GeV which imply u = 385.7 GeV, their sum is 7.9×10 −10 . This is of the right sign, but not large enough to account for the observed discrepancy [18,19] of 25.1 ± 5.9 × 10 −10 .
Consider now E − E mixing for the muon. Neglecting m µ in theĒ L E R term, the 2 × 2 mass matrix shown in Eq.
This is diagonalized by two unitary matrices, one on the left and one on the right. For illustration and simplicity, let m 2 0 = m 2 E + m 2 , then the mass-squared eigenvalues are corresponding to the eigenstates E 1,2 where with r = m /m 0 . Whereas the X couplings to E(E ) are purely left(right)-handed, the fact that they are no longer mass eigenstates results in an additional contribution to ∆a µ which is enhanced [20,24] by m E 1,2 /m µ . However, the effect must be proportional to r because m = 0 is the limit of no mixing.
The key factor is where x is an integration variable in the formula for ∆a µ , and r << 1 is assumed with m X m 0 . The contribution from E − E mixing is then fully the muon g − 2 discrepancy.
Concluding Remarks : A simple SU (2) N gauge symmetry is studied, under which leptons transform. Together with new fermions (N, E), N , E and an SU (2) N Higgs doublet which renders the X 1,2,3 gauge bosons of SU (2) N heavy, this model results in a dark sector, so that X = (X 1 − iX 2 )/ √ 2 becomes vector dark matter. A numerical example is given with m X = 210 GeV, for which the various contributions to the muon anomalous magnetic moment may add up to the observed discrepancy.