New $[SU(3)]^4$ Realization of Lepton/Dark Symmetry

Extending the well-known $SU(3)_C \times SU(3)_L \times SU(3)_R$ model of quarks and leptons to include a fourth $SU(3)_N$ gauge factor, a new realization is obtained, different from leptonic color, which contains a lepton/dark symmetry with the help of an input $Z_4$ symmetry. It is seen to encompass a previous extension of the standard model to $SU(2)_N$ lepton symmetry.

Introduction : It is well-known that the standard model (SM) of quarks and leptons may be embedded in SU (5) with the fermions as 5 * and 10 representations per family. Adding the right-handed neutrino, they form a single 16 representation of SO (10). It is also well-known that this 16 may be embedded in the 27 of E 6 . This last set of fermions has an interesting realization, using the maximal SU(3) C × SU(3) L × SU(3) R subgroup of E 6 , i.e. where The columns in the 3 × 3 fermion matrices denote 3 representations of SU (3) transforms as f , with allowed Yukawa couplings To generalize the above, [SU(N)] k in a moose chain, i.e. fermions of the form (N, N * , 1...1), (1, N, N * , ...1), ... to k copies, may be considered. In general, supersymmetric [SU(N)] k has the intriguing property that it is a finite field theory [1] with three families, for any N or k.
In this paper, a new choice of the fourth SU (3) is studied for an [SU(3)] 4 model. It will be shown that it contains a previously proposed [13,14] SU(2) N lepton symmetry. Furthermore, residual conserved global baryon number B and lepton number L may be defined, and dark symmetry is derivable [15,16] from lepton symmetry, with vector gauge bosons [17,18,19] in the dark sector [20].
Model : The gauge symmetry is with the electric charge given by of Refs. [13,14] is clearly embedded in l, l c .
The scalars transform as The allowed Yukawa terms are Two other scalars are added: Hence Eq. (11) remains valid and λ ′ L,R do not couple to l, l c . Allowed trilinear scalar couplings The absence of these terms will lead to a residual lepton/dark symmetry as discussed in the next section.  Table 1. From the allowed Yukawa couplings it is seen that the u quarks get masses from v 2 , and the d, h quarks get diagonal masses from v 1 and u 0 , with mixing terms from v L,R . The l and l c fermions have the allowed Yukawa It is seen that the charged leptons get masses from v 2 , whereas S 1,2 and E 0 get masses from The λ 0 scalars all have L = 0.
It is also clear that the complex vector gauge bosons in SU (3) Of the nine VEVs, four (v 1,2,3,L ) contribute to the mass of W 3L . They must be small compared to the other five VEVs, from which four of the five vector fields (W 3N , W 3R , W 8L , W 8N , W 8R ) obtain mass. If v 6 is missing, only three would do so. As it is, one linear combination is the analog of the U(1) Y gauge boson of the SM and would get a mass from v 1,2,3,L . It mixes with W 3L to form the photon and the SM Z boson in the usual way. In the limit g L = g R = g N , this state is given by ( whereas Dark Sector : With the conservation of lepton number L as defined in the previous sections, a dark parity may be derived [15], i.e. π D = (−1) L+2j . This means that ν, e, S 1 are even, but (N, E) 1,2 , S 2 , E 0 are odd. The scalars in λ 0 are even, together with Φ 1 , χ 1 , Φ 5 , χ 5 , in restricted to the simplified SU(2) N sector, see Ref. [14].
Since the SU(3) N gauge bosons couple only to l, l c fermions and λ L,R , λ ′ L,R scalars, they are not easily produced. The highest energy of the e + e − LEP II collider was 209 GeV. Hence the W 3N and W 8N bosons should be heavier than this value.
There are three SU(2) L scalar doublets Φ 1,2,3 in λ L and three SU(2) R scalar doublets Φ 4,5,6 in λ R . They have different L values as shown in Eq. (17), and are connected by A similar pattern exists also for λ ′ L,R . Hence this model predicts many more scalars beyond the lone Higgs boson of the SM.
Concluding Remarks : A new realization of [SU(3)] 4 gauge symmetry is proposed, embedding the SM quarks and leptons as shown in Eqs. (6) and (7). The new SU(3) N symmetry has a neutral SU(2) N subgroup which identifies with the non-Abelian lepton symmetry pro-posed before [13,14]. It is shown how all fermions in q, q c , l, l c may acquire mass with the breaking of [SU(3)] 4 to the SM gauge symmetry, then to SU(3) C × U(1) Q . With the help of a Z 4 symmetry which applies to q, q c , l, l c fermions and the λ 0,L,R , λ ′ L,R scalars, it is shown that two conserved residual symmetries remain. One is the usual baryon number B; the other is generalized lepton number L, as shown in Eqs. (16), (17), (19), and (20). Hence two complex vector gauge bosons (one neutral and one charged) have L = 0. The former may be dark matter, as discussed in Ref. [14], with dark parity π D = (−1) L+2j . As most presumed candidates of dark matter are either scalar or fermion, this possibility should not be overlooked.