Dark Revelations of the $[SU(3)]^3$ and $[SU(3)]^4$ Gauge Extensions of the Standard Model

Two theoretically well-motivated gauge extensions of the standard model are $SU(3)_C \times SU(3)_L \times SU(3)_R$ and $SU(3)_q \times SU(3)_L \times SU(3)_l \times SU(3)_R$, where $SU(3)_q$ is the same as $SU(3)_C$ and $SU(3)_l$ is its color leptonic counterpart. Each as three variations, according to how $SU(3)_R$ is broken. It is shown here for the first time that a built-in dark $U(1)_D$ gauge symmetry exists in all six versions, and may be broken to discrete $Z_2$ dark parity. The available dark matter candidates in each case include fermions, scalars, as well as {\it vector gauge bosons}. This work points to the unity of matter with dark matter, the origin of which is not {\it ad hoc}.

Introduction : To extend the SU(3) C × SU(2) L × U(1) Y gauge symmetry of the standard model (SM) of quarks and leptons, there are many possibilities. We focus in this paper on two such theoretically well-motivated ideas. The first [1,2] is SU(3) C × SU(3) L × SU(3) R , and the second [3,4,5] is SU(3) q × SU(3) L × SU(3) l × SU(3) R , where SU(3) q is the same as SU(3) C and SU(3) l is its color leptonic counterpart. It has been known for a long time that [SU(3)] 3 has three distinct variations, according to how SU(3) R is broken to SU(2) R .
• (A) (u, d) R is a doublet, which corresponds to the conventional left-right model. [6,7,8,9,10,11], where h is an exotic quark with the same charge as d, which corresponds to the alternative left-right model. [12,13,14,15], which implies that the vector gauge bosons of this SU(2) R are all neutral.
Note that in the early days of flavor SU(3) for the u, d, s quarks, these SU(2) subgroups are called T, V, U spins. The same three versions are obviously also possible for [SU (3] where the I 3L values from left to right are (−1/2, 1/2, 0) and the Y L values from left to right where the I 3R values from top to bottom are (1/2, −1/2, 0) and the Y R values from top to bottom are (1/3, 1/3, −2/3). The electric charge operator is given by Since (d c , u c ) and (e c , ν c ) are SU(2) R doublets, this reduces to the conventional left-right model. Consider now The [Q, D A ] assignments of q, λ, and q c are then given by This shows that u, u c , d, d c , ν, ν c , e, e c have D A = −1 (odd), whereas h, h c , N, N c , E, E c , S have even D A charges, i.e. 2 and −4. Let us define a parity [16] using the particle's spin j: Since j = 1/2, R A is even for u, u c , d, d c , ν, ν c , e, e c and odd for h, h c , N, N c , E, E c , S, thereby allowing the latter to be considered as belonging to the dark sector, as long as U (1) The electric charge is given as before by Eq. (4), but the dark charge is now Hence D q remains the same as in Eq. (6), but D λ and D q c are now given by Again using R B = (−1) D B +2j , we find it to be even for u, u c , d, d c , ν, ν c , e, e c and odd for i.e.
The electric charge and dark charge are now given by Hence Again using R C = (−1) D C +2j , we find it to be even for u, u c , d, d We assume that they survive to just above the electroweak scale with equal couplings (g) for SU(2) L and SU(2) R and a different where and the massless photon given by This implies If g ′ = g (which is valid at the unification scale), then sin 2 θ W = 3/8 as expected. Now v 31 breaks SU(2) R without breaking SU(2) L , so its value may be greater than the elctroweak scale. Its associated gauge boson Z ′ is given by Hence the SM Z boson is now The (Z, Z ′ ) mass-squared matrix is given by To avoid Z − Z ′ mixing so as not to upset precision electroweak measurements, M 2 ZZ ′ may be chosen to be negligible in the above.
The electric charge and dark charge in (A) are given by Hence They have thus the same would-be [Q, D] assignments. They are not responsible for fermion masses, but are required to break leptonic color SU(3) l to SU(2) l . Now φ L 33 has D A = 2 which may be used to break SU(3) l × SU(2) L to SU(2) l × SU(2) L × U(1) Y l +Y L . To break SU(2) R as well without breaking R A , we use the same trick as before by assigning φ R an odd parity under Z 2 as in [SU(3)] 3 for η. To preserve the R A parity for the gauge bosons, we may again define φ R i1 , φ R i2 to be even, and φ R i3 to be odd. Now φ R 31 breaks SU(3) l to SU(2) l , but it also breaks SU(2) R without breaking SU(2) L . It allows thus the separation of the SU(2) R scale without breaking the dark parity R A .
In the second variation (B), the electric charge is again the same as in (A) and the dark charge is the same as in (B) of [SU(3)] 3 , i.e. Eq. (11). Using the same changes in the pattern of symmetry breaking as discussed before, a model with dark Z 2 symmetry is again achieved. Here φ R 33 breaks SU(3) l × SU(3) R to SU(2) l × SU(2) R × U(1) Y l +Y R and separates the SU(2) l scale from the breaking of SU(2) R by φ 31 . This is the analog of the alternative left-right model in the [SU(3)] 3 case. Applying φ L 33 as well, the residual U(1) symmetry is now Y L + Y R + Y l , exactly as needed for the electric charge of Eq. (28). In the third variation (C), the electric charge is and the dark charge is the same as D C of Eq. (14). It also results in a model with dark to the unity of matter with dark matter, the origin of which is not ad hoc. Other possible candidates are SU(6) [19,20] and SU(7) [20]. Future more detailed explorations are called for.