Alternative $[SU(3)]^4$ Model of Leptonic Color and Dark Matter

The alternative $[SU(3)]^4$ model of leptonic color and dark matter is discussed. It unifies at $M_U \sim 10^{14}$ GeV and has the low-energy subgroup $SU(3)_q \times SU(2)_l \times SU(2)_L \times SU(2)_R \times U(1)_X$ with $(u,h)_R$ instead of $(u,d)_R$ as doublets under $SU(2)_R$. It has the built-in global $U(1)$ dark symmetry which is generalized $B-L$. In analogy to $SU(3)_q$ quark triplets, it has $SU(2)_l$ hemion doublets which have half-integral charges and are confined by $SU(2)_l$ gauge bosons (stickons). In analogy to quarkonia, their vector bound states (hemionia) are uniquely suited for exploration at a future $e^-e^+$ collider.


Introduction
To venture beyond the Standard Model (SM) of quarks and leptons, there have been many trailblazing ideas. One is the notion of grand unification, i.e. the embedding of the SM gauge symmetry SU (3) C × SU (2) L × U (1) Y in a single larger symmetry such as SU (5) ∼ E 4 , SO(10) ∼ E 5 , or E 6 . There are indeed very many papers devoted to this topic. Less visited are the symmetries [SU (3)] N , where N = 3, 4, 6 have been considered [1,2,3,4,5,6,7,8].
Another idea is that the SU (2) R quark doublet may not be (u, d) R but rather (u, h) R where h is an exotic quark of charge −1/3. This was originally motivated by superstring-inspired E 6 models [9,10] and later generalized to nonsupersymmetric models [11,12,13,14], but is easily implemented in [SU (3)] N models. A third idea is quark-lepton interchange symmetry [15,16] which assumes SU (3) l for leptons in parallel to SU (3) q for quarks, but with SU (3) l broken to SU (2) l × U (1) Y l . This is naturally embedded in [SU (3)] 4 [4] and implies that only one component of the color lepton triplet is free, i.e. the observed lepton, whereas the other two color components (with half-integral charges) are confined in analogy to the three color components of a quark triplet. Finally a fourth idea has been put forward recently [6,17] that a dark symmetry may exist within [SU (3)] N itself or perhaps [SU (3)] N × U (1). This new insight points to the possible intrinsic unity of matter with dark matter [18,19,20].
In this paper, all four of the above ideas are incorporated into a single consistent framework based on the symmetry SU At M R of order a TeV, SU (2) R × U (1) X is broken to U (1) Y of the SM, with particle content of the SM plus possible light particles transforming under the leptonic color SU (2) l symmetry. We will discuss their impact on cosmology as well as their possible revelation at a future e − e + collider, following closely our previous work [5] on the subject. We will also consider the phenomenology associated with the SU (2) R gauge symmetry and the possible dark-matter candidates of this model.

Fermion Content and Dark Symmetry
All fermions belong to bitriplet representations (3, 3 * The dark symmetry we will consider is the fermion content of our model is then given where u has charge 2/3, d, h have charge −1/3, x, z have charge 1/2, y has charge −1/2, ν, n have charge 0, and e has charge −1. Using we see that u, u c , d, d c , ν, ν c , e, e c , z, z c are even, and h, h c , x, x c , y, y c , n, n c are odd. Further, the gauge bosons which take h to u, d in SU (3) L and h c to u c , d c in SU (3) R are odd, as well as the corresponding ones in SU (3) l , and the others even, including all those of the SM.
Hence R D would remain a good symmetry for dark matter provided that the scalar sector responsible for the symmetry breaking obeys it as well.
The scalar bitriplets responsible for the masses of the fermions in Eqs.

Symmetry Breaking Pattern
We consider the breaking of [SU (3)] 4 at M U by two scalar bitriplets, one transforming as φ L+ ∼ (1, 3, 3 * , 1) ∼ l, belonging to a chain in parallel to the fermions, the other transforming as φ R− ∼ (1, 1, 3, 3 * ) ∼ l c , belonging to a chain with an additional overall imposed assignment of odd R D , i.e. an additional Z 2 factor [6]. This preserves the relative R D among its components, but prevents it from coupling to the fermions. Using φ L+ which also has even Assuming also that all the particles of the chain associated with φ R− are superheavy, the low-energy theory with the residual gauge symmetry SU Since there are three fermion chains, and five scalar chains, the b coefficients for the renormalization-group running of each SU (3) gauge coupling are all given by This shows that we have a possible finite field theory [3] above M U .
At M R , the SU (2) R × U (1) X gauge symmetry is broken to U (1) Y of the SM, where Y = I 3R − X, by an SU (2) R doublet whose neutral component is a linear combination of χ 0 from φ (1) , the conjugate ofχ 0 fromφ (2) , and φ R+ 31 from the (1, 1, 3, 3 * ) component of the chain containing φ L+ discussed previously. From the allowed antisymmetric trilinear term l c l c φ R+ , the mass term x c 1 y c 2 − x c 2 y c 1 is then obtained. Note that the correponding mass term x 1 y 2 − x 2 y 1 is superheavy because it comes from φ L+ 33 . Note also that the corresponding term l c l c φ R− is forbidden because of the overall assignment of odd R D for φ R− . Finally the

Renormalization-Group Running of Gauge Couplings
The renormalization-group evolution of the gauge couplings is dictated at leading order by where b i are the one-loop beta-function coefficients. From M U to M R , we assume that all fermions are light except the three families of (x, y) hemions. As for the scalars, we assume that only the following multiplets are light under SU (2) L × SU (2) R × U (1) X : 1 copy of (1, 2, −1/2), 6 copies of (2, 2, 0), 3 copies of (2, 1, −1/2), and 4 copies of (2, 1, 1/2). This choice requires fine tuning in the scalar sector as in other models of grand unification such as SU (5) and SO (10). As a result, the five b coefficients are given by From M R to M Z , we assume the SM quark and lepton content together with 1 copy of (x c , y c ) hemions and two SU (2) L Higgs scalar doublets. The massless SU (2) l stickons are of course included but they affect only α l . The four b coefficients are then where a factor of 1/2 has been inserted to normalize b Y . The boundary condition at M R for We then obtain Using the experimental inputs where a factor of 2 has been used to normalize α Y , we find This implies M R 600 GeV and M U 10 14 GeV, as shown in Fig we obtain α R (M R ) = 0.0290. Using it is a few keV [4,5].

Low-Energy Particle Content
The particles of this model at or below a few TeV are listed in Table 1 The SU (2) L × SU (2) R scalar bidoublet contains the SU (2) L doublets η = (η 0 , η − ) and before SU (2) R breaking, even though the corresponding gauge symmetry has been broken.
Whereas both S and I 3R are broken by χ 0 , the combination is unbroken. Although this idea was used previously [11,12], the important observation . This is expected because W + R takes h R to u R and l R to n R . Consider next the Yukawa terms allowed by the gauge symmetry and S, i.e.
and the scalar trilinear terms It is easily confirmed from the above that I 3R + S is not broken by φ 0 1,2 and χ 0 . Note that in the familar case of SU (5) grand unification, neither B nor L is part of SU (5)
Consider now the masses of the gauge bosons. The charged ones, W ± L and W ± R , do not mix because the latter have dark charge ±1. Their masses are given by Since Q = I 3L + I 3R − X, the photon is given by where where g −2 Y = g −2 R + g −2 X , then the 2 × 2 mass-squared matrix spanning (Z, Z ) is given by .
Their neutral-current interactions are given by where g 2 Z = g 2 L + g 2 Y and sin 2 θ W = g 2 Y /g 2 Z . Since Z − Z mixing is constrained by experiment to be less than 10 −4 or so, we assume (g 2 The new gauge boson Z may be produced at the Large Hadron Collider (LHC) through their couplings to u and d quarks, and decay to charged leptons (e − e + and µ − µ + ). Hence current search limits for a Z boson are applicable. Using α R (M R ) = 0.0290 and α X (M R ) = 0.0163, the c u,d coefficients [22,23] used in the data analysis for our model are where B is the branching fraction of Z to e − e + and µ − µ + . Assuming that Z decays to all the particles listed in Table 1, except for the scalars which become the longitudinal components of the various gauge bosons, we find B = 0.044. Based on the 2016 LHC 13 TeV data set [24], this translates to a bound of about 3 to 4 TeV on the Z mass.

Scalar Sector
Consider the most general scalar potential consisting of Φ L = (φ 0 Note that The minimum of V satisfies the conditions The 3 × 3 mass-squared matrix spanning √ 2Im(φ 0 1 , φ 0 2 , χ 0 ) is then given by and that spanning √ Hence there are two zero eigenvalues in M 2 The other two scalar bosons are much heavier, with suppressed mixing to H, which may all be assumed to be small enough to avoid the constraints from dark-matter direct-search experiments.
The dark scalars are λ 0 , χ ± , and (η 0 , η − ). Whereas χ ± become the longitudinal components of W ± R , the other scalars have the interaction The 2 × 2 mass-squared matrix linking (λ,η) to (λ, η) is given by We assume µ 2 to be very small so that there is negligible mixing, with λ 0 as the lighter particle which is our dark-matter candidate. Note of course that η 0 is not a suitable candidate because it has Z 0 interactions.

Dark Matter Interactions
Consider the scalar singlet λ 0 as our dark-matter candidate. Let its coupling with the SM Higgs boson be f λH √ 2v H , then it has been shown [14] that for m λ = 150 GeV, f λH < 4.4 × 10 −4 from the most recent direct-search result [25]. With such a small coupling, the λ 0 annihilation cross section in the early Universe through the SM Higgs boson is much too small for λ 0 to have the correct observed relic abundance. Hence a different process is required.
Consider then the Yukawa sector. As noted in Eq. (36), the interactions f x λ 0z L x R and scalar. The cross section × relative velocity is then given by (59) As an example, let m λ = 150 GeV, m x = 100 GeV, and m z = 600 GeV, then σv rel = 1 pb is obtained for f x = 0.385. The xx final states remain in thermal equilibrium through the photon, with their confined bound states (which are bosons with even R D ) decaying to SM particles as described in a following section.

Leptonic Color in the Early Universe
As discussed in our earlier paper [5], the SU ( where m is the mass of the one light x R y R hemion of this model. The decoupling temperature of ζ is then obtained by matching the Hubble expansion rate to [6ζ(3)/π 2 ]T 3 σv . Hence For m = 100 GeV and g * = 92.25 which includes all particles with masses up to a few GeV, T ∼ 9 GeV. Hence the contribution of stickons to the effective number of neutrinos at the time of big bang nucleosynthesis (BBN) is given by [26] ∆N ν = 8 7 (3) 10.75 92.25 compared to the value 0.50 ± 0.23 from a recent analysis [27].
As the Universe further cools below a few MeV, leptonic color goes through a phase transition and stickballs are formed. However, they are not stable because they are allowed to mix with a scalar bound state of two hemions which would decay to two photons. For a stickball ω of mass m ω , we assume this mixing to be f ω m ω /m, so that its decay rate is given by Using m ω = 1 MeV as an example with m = 100 GeV as before, its lifetime is estimated to be 1.0 × 10 7 s for f ω = 1. This means that it disappears long before the time of photon decoupling, so the SU (2) l sector contributes no additional relativistic degrees of freedom.
Hence N ef f remains the same as in the SM, i.e. 3.046, coming only from neutrinos. This agrees with the PLANCK measurement [28] of the cosmic microwave background (CMB), i.e.
10 Leptonic Color at Future e − e + Colliders Unlike quarks, all hemions are heavy. Hence the lightest bound state is likely to be at least 200 GeV. Its cross section through electroweak production at the LHC is probably too small for it to be discovered. On the other hand, in analogy to the observations of J/ψ and Υ at e − e + colliders of the last century, the resonance production of the corresponding neutral vector bound states (hemionia) of these hemions is expected at a future e − e + collider (ILC, CEPC, FCC-ee) with sufficient reach in total center-of-mass energy. Their decays will be distinguishable from heavy quarkonia (such as toponia) experimentally.
As discussed in Ref. [5], the formation of hemion bound states is analogous to that of QCD. Instead of one-gluon exchange, the Coulomb potential binding a hemion-antihemion pair comes from one-stickon exchange. The difference is just the change in an SU (3)  GeV. In analogy to the hydrogen atom, its binding energy is given by and its wavefunction at the origin is Since Ω will appear as a narrow resonance at a future e − e + collider, its observation depends on the integrated cross section over the energy range √ s around m Ω : where Γ tot is the total decay width of Ω, and Γ ee , Γ X are the respective partial widths.
Since Ω is a vector meson, it couples to both the photon and Z boson through its constituent hemions. Hence it will decay to W − W + , qq, l − l + , and νν. Using the Ω → e − e + decay rate is given by where In the above, Ω is composed of the singlet hemions x R and y R with invariant mass term For Ω → W − W + , the triple γW − W + and ZW − W + vertices have the same structure.
The decay rate is calculated to be where r = 4M 2 W /m 2 Ω and Because of the accidental cancellation of the two terms in the above, C W turns out to be very small. Hence Γ W W = 10 eV. For Ω → ZZ, there is only the t−channel contribution, i.e.
where r Z = 4M 2 Z /m 2 Ω and D Z = g 2 Z sin 4 θ W /4(m 2 Ω − 2m 2 Z ). Hence Γ ZZ is negligible. The Ω decay to two stickons is forbidden by charge conjugation. Its decay to three stickons is analogous to that of quarkonium to three gluons. Whereas the latter forms a singlet which is symmetric in SU (3) C , the former forms a singlet which is antisymmetric in SU (2) l .
However, the two amplitudes are identical because the latter is symmetrized with respect to the exchange of the three gluons and the former is antisymmetrized with respect to the exchange of the three stickons. Taking into account the different color factors of SU (2) l versus SU (3) C , the decay rate of Ω to three stickons and to two stickons plus a photon are Hence Γ ζζζ = 39 eV and Γ γζζ = 7 eV. The integrated cross section for X = µ − µ + is then 1.2 × 10 −32 cm 2 -keV. For comparison, this number is 7.9 × 10 −30 cm 2 -keV for the Υ(1S).
At a high-luminosity e − e + collider, it should be feasible to make this observation. Table 2 summarizes all the partial decay widths. There are important differences between QCD and QHD (quantum hemiodynamics). In the former, because of the existence of light u and d quarks, it is easy to pop up uū and dd pairs from the QCD vacuum. Hence the production of open charm in an e − e + collider is described well by the fundamental process e − e + → cc. In the latter, there are no light hemions. Instead it is easy to pop up the light stickballs from the QHD vacuum. As a result, just above the threshold of making the Ω resonance, the many-body production of Ω + stickballs becomes possible. This cross section is presumably also well described by the fundamental process e − e + → xx, i.e.
where x W = sin 2 θ W and s = 4E 2 is the square of the center-of-mass energy. Using m = 100 GeV and s = (250 GeV) 2 as an example, we find this cross section to be 0.79 pb.
In QCD, there are qq bound states which are bosons, and qqq bound states which are fermions. In QHD, there are only bound-state bosons, because the confining symmetry is SU (2) l . Also, unlike baryon (or quark) number in QCD, there is no such thing as hemion number in QHD, because y is effectivelyx. This explains why there are no stable analog fermion in QHD such as the proton in QCD.

Concluding Remarks
Candidates for dark matter are often introduced in an ad hoc manner, because it is so easy to do. There are thus numerous claimants to the title. Is there a guiding principle? One such is supersymmetry, where the superpartners of the SM particles naturally belong to a dark sector. Another possible guiding principle proposed recently is to look for a dark symmetry are accessible in a future e − e + collider, as described in this paper.