Dark Revelations of the [SU(3) ] 3 and [SU(3) ] 4 Gauge Extensions of the Standard Model
Corey Kownacki, Ernest Ma, Nicholas Pollard, Oleg Popov, Mohammadreza Zakeri
aa r X i v : . [ h e p - ph ] O c t UCRHEP-T585Oct 2017
Dark Revelations of the [ S U (3)] and [ S U (3)] Gauge Extensions of the Standard Model
Corey Kownacki, Ernest Ma, Nicholas Pollard, Oleg Popov, andMohammadreza Zakeri
Physics and Astronomy Department,University of California, Riverside, California 92521, USA
Abstract
Two theoretically well-motivated gauge extensions of the standard model are SU (3) C × SU (3) L × SU (3) R and 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. Each has threevariations, according to how SU (3) R is broken. It is shown here for the first time thata built-in dark U (1) D gauge symmetry exists in all six versions, and may be brokento discrete Z dark parity. The available dark matter candidates in each case includefermions, scalars, as well as vector gauge bosons . This work points to the unity ofmatter with dark matter, the origin of which is not ad hoc . ntroduction : To extend the SU (3) C × SU (2) L × U (1) Y gauge symmetry of the standardmodel (SM) of quarks and leptons, there are many possibilities. We focus in this paper ontwo 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 sameas SU (3) C and SU (3) l is its color leptonic counterpart. It has been known for a long timethat [ SU (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. • (B) ( u, h ) R is a doublet [6, 7, 8, 9, 10, 11], where h is an exotic quark with the samecharge as d , which corresponds to the alternative left-right model. • (C) ( h, d ) R is a doublet [12, 13, 14, 15], which implies that the vector gauge bosons ofthis 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 arecalled T, V, U spins. The same three versions are obviously also possible for [ SU (3] .Whereas these structures have been known for a long time, an important property of thesemodels has been overlooked, i.e. the existence of a built-in dark U (1) D gauge symmetryalready present in [ SU (3)] and [ SU (3)] under which the SM particles are distinguishedfrom those of the dark sector. We will identify this symmetry in all six cases and discusshow it may fit into a viable extension of the SM. Dark Symmetries in [ SU (3)] : The fermion assignments under SU (3) C × SU (3) L × SU (3) R are q ∼ (3 , ∗ , ∼ d u hd u hd u h , (1)2here the I L values from left to right are ( − / , / ,
0) and the Y L values from left to rightare ( − / , − / , / λ ∼ (1 , , ∗ ) ∼ N E c νE N c eν c e c S , (2)where the I L values from top to bottom are now (1 / , − / ,
0) and the Y L values from topto bottom are (1 / , / , − / I R values from left to right are ( − / , / ,
0) and the Y R values from left to right are ( − / , − / , / q c ∼ (3 ∗ , , ∼ d c d c d c u c u c u c h c h c h c , (3)where the I R values from top to bottom are (1 / , − / ,
0) and the Y R values from top tobottom are (1 / , / , − / Q = I L − Y L I R − Y R . (4)Since ( d c , u c ) and ( e c , ν c ) are SU (2) R doublets, this reduces to the conventional left-rightmodel. Consider now D A = 3( Y L − Y R ) . (5)The [ Q, D A ] assignments of q , λ , and q c are then given by Q q = − / / − / − / / − / − / / − / , D q = − − − − − − , (6) Q λ = − −
10 1 0 , D λ = −
12 2 − − − − , (7) Q q c = / / / − / − / − / / / / , D q c = − − − − − −
12 2 2 . (8)3his shows that u, u c , d, d c , ν, ν c , e, e c have D A = − 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 : R A = ( − D A +2 j . (9)Since j = 1 / 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 , therebyallowing the latter to be considered as belonging to the dark sector, as long as U (1) D isbroken only by two units, in analogy to the breaking of B − L in models of neutrino mass,where lepton parity ( − L remains conserved.To break [ SU (3)] , a scalar bitriplet φ ∼ (1 , , ∗ ) is used. It transforms exactly as λ and has the same [ Q, D ] assignments. Now h φ i breaks SU (3) L × SU (3) R to SU (2) L × SU (2) R × U (1) Y L + Y R . The U (1) D symmetry is broken by 4 units at the same time. Thisgives masses to the exotic fermions h, N, E . Two other neutral scalars φ , φ have D A = 2.Their vacuum expectation values would break SU (2) L × SU (2) R to U (1) I L + I R , and U (1) D by 2 units, allowing mass terms for uu c , dd c , ee c , νν c , N S , and N c S . At this point, it lookslike a dark residual Z symmetry is still possible. However this is not a viable scenario,because the SU (2) L and SU (2) R breaking are now at the same scale, contrary to what isobserved. Furthermore, both I L + I R and Y L + Y R are still unbroken. Whereas Q is alinear combination of the two, there remains another unbroken U (1) gauge symmetry. Tosolve these problems, the usual procedure is to allow φ and φ to acquire nonzero vacuumexpectation values as well, thus breaking SU (2) R and SU (2) L separately. However, sincethey have D A = − R A ), the dark symmetry is lost.To save the dark symmetry, we insert another bitriplet η ∼ (1 , , ∗ ) with an extra Z symmetry under which it is odd and all other fields are even. This extra symmetry prevents η from coupling to the quarks and leptons, so that the absolute R A values of the η componentsare not fixed by them as in φ . However their relative R A values are still fixed by the gaugebosons. Using Eqs. (5) and (9), we see that of the eight SU (3) L and eight SU (3) R gauge4osons, the four gauge bosons which take u and d to h , and the corresponding ones which take u c and d c to h c are odd under R A , and the others are even. We can now choose h η i 6 = 0 and h η i 6 = 0 to break SU (3) L × SU (3) R to just U (1) Q and preserve R A , because η , η , η , η may be defined to be even and the other components odd without breaking R A .Of the 27 fermion fields for each family, 16 are in the visible sector ( R A even), i.e. u, u c , d, d c , ν, ν c , e, e c , and 11 are in the dark sector ( R A odd), i.e. h, h c , N, N c , E, E c , S . Ofthe 24 gauge bosons, 16 are visible, i.e. the 8 gluons, W ± L , W ± R , the photon, Z , and two otherheavier neutral ones, a linear combination of which couples to the dark charge D A , and 8 aredark, i.e. those with odd R A . The scalars are also divided into sectors with even and odd R A .This is thus a model with possible fermion, scalar, and vector dark-matter candidates. Theirexistence is not an ad hoc invention, but a possible outcome of the postulated theoreticalframework beyond the standard model.Consider next the alternative left-right model, i.e. variation (B), where d c is switchedwith h c and ( ν, e, S ) are switiched with ( N, E, ν c ), i.e. q c ∼ h c h c h c u c u c u c d c d c d c , λ ∼ ν E c Ne N c ES e c ν c . (10)The electric charge is given as before by Eq. (4), but the dark charge is now D B = 3( Y L + I R + Y R . (11)Hence D q remains the same as in Eq. (6), but D λ and D q c are now given by D λ = − − − − − , D q c = − − − − − − . (12)Again using R B = ( − D B +2 j , we find it to be 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 . Choosing φ , φ , φ to have nonzero vacuum expectation values, thesymmetry breaking pattern is as in (A), only that the SU (2) subgroup of SU (3) R is now5ifferent. It suffers from the same problems as in (A), which may be solved again by adding η , with h η i 6 = 0 and h η i 6 = 0.In the third variation (C), u c is switched with h c , and ( ν, e, S ) are switched with ( E c , N c , e c ),i.e. q c ∼ d c d c d c h c h c h c u c u c u c , λ ∼ N ν E c E e N c ν c S e c . (13)The electric charge and dark charge are now given by Q = I L − Y L Y R , D C = 3( Y L − I R + Y R . (14)Hence Q λ = − − , D λ = − − − − − , (15) Q q c = / / / / / / − / − / − / D q c = − − −
12 2 2 − − − . (16)Again using R C = ( − D C +2 j , we find it to be 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 . Choosing φ , φ , φ to have nonzero vacuum expectation values, thepattern of symmetry breaking is the same as in (A) and (B), but the SU (2) R subgroup isdifferent from either. It suffers from the same problems as the two previous cases, and theyare again solved by adding η , with h η i 6 = 0 and h η i 6 = 0. However, in contrast to thevariations (A) and (B), the φ and η entries are not neutral, so it is not possible to preserve SU (2) L × SU (2) R as a low-energy subgroup. Gauge Boson Masses in (B) : Consider the breaking of SU (3) L × SU (3) R by a very large h η i = v . Of the 8 vector gauge bosons W Li of SU (3) L and the 8 vector gauge bosons W Ri of SU (3) R , 9 become very heavy. The remaining 7 are the 3 of SU (2) L , the 3 of SU (2) R ,and the one linear combination W V = ( W L + W R ) / √
2. We assume that they survive to just6bove the electroweak scale with equal couplings ( g ) for SU (2) L and SU (2) R and a differentone ( g ′ ) for Y L + Y R . Let h η i = v , h φ i = v , h φ i = v , h φ i = v , then M ( W R , ) = g v + v + v ] , (17)where ( W R ∓ iW R ) / √ W ± R are the charged SU (2) R gauge bosons with odd R B . Theother gauge bosons have even R B with M ( W L , ) = g v + v + v ] , (18)and the massless photon given by A = eg ( W L + W R ) − eg ′ s W V . (19)This implies e g ′ = 32 (1 − θ W ) . (20)If g ′ = g (which is valid at the unification scale), then sin θ W = 3 / v breaks SU (2) R without breaking SU (2) L , so its value may be greater than the elctroweakscale. Its associated gauge boson Z ′ is given by Z ′ = √ gW R + √ g ′ W V q g + 3 g ′ = 1cos θ W [ q − θ W W R + sin θ W W V ] . (21)Hence the SM Z boson is now Z = cos θ W W L − tan θ W [sin θ W W R − q − θ W W V ] . (22)The ( Z, Z ′ ) mass-squared matrix is given by M ZZ = g θ W [ v + v + v ] , (23) M Z ′ Z ′ = g " cos θ W − θ W v + 1 − θ W cos θ W ( v + v ) + 2 tan θ W v , (24) M ZZ ′ = g tan θ W q − θ W [sin θ W v − (1 − θ W )( v + v )] . (25)7o avoid Z − Z ′ mixing so as not to upset precision electroweak measurements, M ZZ ′ maybe chosen to be negligible in the above.In this alternative left-right model, ( u, h ) R and ( S, e ) R are SU (2) R doublets with h and S odd under R B . The mass terms for u and ν come from v , those for d and e from v ,those for h , E from v , and the 3 × N, N c , S ) from all three. As such, itcontains the necessary ingredients for a consistent model of built-in dark matter. In variation(C), it has already been noted that SU (2) L × SU (2) R cannot be maintained as a low-energysubgroup. Hence the associated dark sector must be very heavy and does not lead to arealistic model. In variation (A), whereas SU (2) L × SU (2) R may emerge as a low-energysubgroup, the dark sector consists of singlets under this symmetry and must also be veryheavy. Dark Symmetries in [ SU (3)] : The notion of leptonic color [17, 18] is based on quark-lepton interchange symmetry. Postulating SU (3) l to go with SU (3) q , leptons have threecolor components to begin with, but SU (3) l is broken to SU (2) l which remains exact, sothat two of these leptonic color fields are confined in analogy to the three color quarksbeing confined. The third unconfined component is the observed lepton of the SM. The newparticles of this model are not easily produced and observed at the Large Hadron Collider,but will have unique signatures in a future lepton collider, as recently discussed [5]. Under SU (3) q × SU (3) L × SU (3) l × SU (3) R , q ∼ (3 , ∗ , ,
1) as in Eq. (1) and q c ∼ (3 ∗ , , , SU (3] . As for the leptonic sector, l ∼ (1 , , ∗ , ∼ x x νy y ez z n (26)is the same in all three variations, in analogy to q , whereas l c has three variations to match8 c , i.e. l c ∼ (1 , , , ∗ ) ∼ x c y c z c x c y c z c ν c e c n c , z c y c x c z c y c x c n c e c ν c , x c z c y c x c z c y c ν c n c e c . (27)The electric charge and dark charge in (A) are given by Q = I L − Y L I R − Y R − Y l , D A = 3( Y L − Y R ) . (28)Hence Q l = / / − / − / − / / , Q l c = − / / − / − / / − /
20 1 0 , (29)and D l = − D q c of Eq. (8), D l c = − D q of Eq. (6), i.e. u, u c , d, d c , ν, ν c , e, e c , x, x c , y, y c have D A = 1 (odd), whereas h, h c , n, n c , z, z c have D A = − R A = ( − D A +2 j ,then the former group of fermions is even and the latter odd, i.e. belonging to the darksector if U (1) D is broken only by two units.The breaking of SU (3) L × SU (3) R by a scalar bitriplet φ ∼ (1 , , , ∗ ), which couples alsoto the fermions, proceeds as before. It has the same problems as discussed in the [ SU (3)] case. However, there are now two additional scalar bitriplets [4] in [ SU (3)] with nonzerovacuum expectation values, i.e. φ L ∼ (1 , , ∗ , ∼ l, φ R ∼ (1 , , , ∗ ) ∼ l c . (30)They have thus the same would-be [ Q, D ] assignments. They are not responsible for fermionmasses, but are required to break leptonic color SU (3) l to SU (2) l . Now φ L has D A = 2which 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 anodd parity under Z as in [ SU (3)] for η . To preserve the R A parity for the gauge bosons, wemay again define φ Ri , φ Ri to be even, and φ Ri to be odd. Now h φ R i breaks SU (3) l to SU (2) l ,9ut 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 darkcharge is the same as in (B) of [ SU (3)] , i.e. Eq. (11). Using the same changes in thepattern of symmetry breaking as discussed before, a model with dark Z symmetry is againachieved. Here h φ R i breaks SU (3) l × SU (3) R to SU (2) l × SU (2) R × U (1) Y l + Y R and separatesthe SU (2) l scale from the breaking of SU (2) R by h φ i . This is the analog of the alternativeleft-right model in the [ SU (3)] case. Applying h φ L i as well, the residual U (1) symmetry isnow 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 Q = I L − Y L Y R − Y l , (31)and the dark charge is the same as D C of Eq. (14). It also results in a model with dark Z symmetry. However, as with its [ SU (3)] analog, it is not possible to preserve SU (2) L × SU (2) R as a low-energy subgroup. Note that sin θ W = 1 / SU (3)] which is of order 10 GeV for a nonsupersymmetric model [4, 5].
Concluding Remarks : The existence of a dark sector is easily implemented by adding anew symmetry and new particles to the standard model. There are indeed numerous suchproposals. As a guiding principle, supersymmetry is a well-known and perhaps the onlyexample, where superpartners of all SM particles belong to the dark sector. In this paper,we suggest another, i.e. that such a dark symmetry may have a gauge origin buried inside acomplete extended theoretical framework for the understanding of quarks and leptons. Theinevitable consequence of this hypothesis is to divide all fermions, scalars, as well as vectorgauge bosons into two categories. One includes all known particles of the SM and some newones; the other is the dark sector. They are however intrinsically linked to each other asessential components of the unifying framework.10e consider as first examples [ SU (3)] and [ SU (3)] , and identified the exact nature ofthis dark symmetry in three variations of the above two unified symmetries. We have shownhow this dark gauge symmetry is broken to the discrete Z dark parity which stabilizes darkmatter. Whereas all these models contain dark matter, only variation (B) in either [ SU (3)] or [ SU (3)] allows it to be such that it exists at or near the electroweak scale. They mayserve as the prototypes for a deeper understanding of the origin of dark matter as a built-insymmetry of a theoretically motivated extension of the Standard Model. Our study pointsto the unity of matter with dark matter, the origin of which is not ad hoc . Other possiblecandidates are SU (6) [19, 20] and SU (7) [20]. Future more detailed explorations are calledfor. Acknowledgement : This work was supported in part by the U. S. Department of EnergyGrant No. de-sc0008541.
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