Quartified Leptonic Color, Bound States, and Future Electron-Positron Collider

The $[SU(3)]^4$ quartification model of Babu, Ma, and Willenbrock (BMW), proposed in 2003, predicts a confining leptonic color $SU(2)$ gauge symmetry, which becomes strong at the keV scale. It also predicts the existence of three families of half-charged leptons (hemions) below the TeV scale. These hemions are confined to form bound states which are not so easy to discover at the Large Hadron Collider (LHC). However, just as $J/\psi$ and $\Upsilon$ appeared as sharp resonances in $e^-e^+$ colliders of the 20th century, the corresponding 'hemionium' states are expected at a future $e^-e^+$ collider of the 21st century.

Introduction : Fundamental matter consists of quarks and leptons, but why are they so different? Both interact through the SU (2) L × U (1) Y electroweak gauge bosons W ± , Z 0 and the photon A, but only quarks interact through the strong force as mediated by the gluons of the unbroken (and confining) color SU (3) gauge symmetry, called quantum chromodynamics (QCD). Suppose this is only true of the effective low-energy theory. At high energy, there may in fact be three 'colors' of leptons transforming as a triplet under a leptonic color SU (3) gauge symmetry. Unlike QCD, only its SU (2) l subgroup remains exact, thus confining only two of the three 'colored' leptons, called 'hemions' in Ref. [1] because they have ±1/2 electric charges, leaving the third ones free as the known leptons.
The notion of leptonic color was already discussed many years ago [2,3], and its incorporation into [SU (3)] 4 appeared in Ref. [4], but without full unification. Its relevance today is  2) The quartification scale determines the common gauge coupling for the SU (2) l symmetry. Its extrapolation to low energy predicts that it becomes strong at the keV scale, in analogy to that of QCD becoming strong at somewhat below the GeV scale. This may alter the thermal history of the Universe and allows the formation of gauge-boson bound states, the lightest of which is a potential warm dark-matter candidate [5]. (3) The hemions (called 'liptons' previously [3]) have ±1/2 electric charges and are confined to form bound states by the SU (2) l 'stickons' in analogy to quarks forming hadrons through the SU (3) C gluons. They have been considered previously [6] as technifermions responsible for electroweak symmetry breaking. Their electroweak production at the LHC is possible [7] but the background is large. However, in a future e − e + collider (ILC, CEPC, FCC-ee), neutral vector resonances of their bound states (hemionia) would easily appear, in analogy to the observations of quarkonia (J/ψ, Υ) at past e − e + colliders.
following, the mass terms from electroweak symmetry breaking, i.e.x L x Rφ 0 andȳ L y R φ 0 , will be assumed negligible.
Gauge coupling unification and the leptonic color confinement scale : The renormalizationgroup evolution of the gauge couplings is dictated at leading order by where b i are the one-loop beta-function coefficients, The number of families N F is set to three, and the number of Higgs doublets N Φ is set to two, as in the original BMW model. Here we make a small adjustment by separating the three hemion families into two light ones at the electroweak scale M Z and one at a somewhat higher scale M X . We then input the values [8] α C (M Z ) = 0.1185, where α Y has been normalized by a factor of 2 (and b Y by a factor of 1/2) to conform to We then use b l to extrapolate back to M Z and obtain α l (M Z ) = 0.0469. Below the electroweak scale, the evolution of α l comes only from the stickons and it becomes strong at about 1 keV. Hence 'stickballs' are expected at this confinement mass scale. Unlike QCD where glueballs are heavier than the π mesons so that they decay quickly, the stickballs are so light that they could decay only to lighter stickballs or to photon pairs through their interactions with hemions.
Thermal history of stickons : At temperatures above the electroweak symmetry scale, the hemions are active and the stickons (ζ) are in thermal equilibrium with the standard-model particles. Below the hemion mass scale, the stickon interacts with photons through ζζ → γγ scattering with a cross section The decoupling temperature of ζ is then obtained by matching the Hubble expansion rate to [6ζ(3)/π 2 ]T 3 σv . Hence where For M ef f = 110 GeV and g * = 92.25 which includes all particles with masses up to a few GeV, T ∼ 6.66 GeV. Hence the contribution of stickons to the effective number of neutrinos at the time of big bang nucleosynthesis (BBN) is given by [9] ∆N ν = 8 7 (3) 10.75 92.25 compared to the value 0.50 ± 0.23 from a recent analysis [10]. The most recent PLANCK measurement [11] coming from the cosmic microwave background (CMB) is However, at the time of photon decoupling, the stickons have disappeared, hence N ef f = 3.046 as in the SM. This is discussed in more detail below.
Formation and decay of stickballs : As the Universe further cools below a few keV, leptonic color goes through a phase transition and stickballs are formed. If the lightest stickball ω is stable, it may be a candidate for warm dark matter. It has strong self-interactions and the 3 → 2 process determines its relic abundance. Following Ref. [12] and using Ref. [5], we estimate that it is overproduced by a factor of about 3. However, ω is not absolutely stable.
It is allowed to mix with a scalar bound state of two hemions which would decay to two photons. We assume this mixing to be f ω m ω /M xy , so that its decay rate is given by where Setting m ω = 5 keV to be above the astrophysical bound of 4 keV from Lyman α forest observations [13] and M ef f = 150 GeV, its lifetime is estimated to be 4.4 × 10 17 s for f ω = 1. This is exactly the age of the Universe, and it appears that ω may be a candidate for dark matter after all. However, CMB measurements constrain [14] a would-be dark-matter lifetime to be greater than about 10 25 s, and x-ray line measurements in this mass range constrain [15] it to be greater than 10 27 s, so this scenario is ruled out. On the other hand, if m ω = 10 keV, then the ω lifetime is 1.4 × 10 16 s, which translates to a fraction of 2 × 10 −14 of the initial abundance of ω to remain at the present Universe. Compared to the upper bound of 10 −10 for a lifetime of 10 16 s given in Ref. [14], this is easily satisfied, even though ω is overproduced at the leptonic color phase transition by a factor of 3.
At the time of photon decoupling, 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. In this scenario, ω is not dark matter. However, there are many neutral scalars and fermions in the BMW model which are not being considered here. They are naturally very heavy, but some may be light enough and stable, and be suitable as dark matter.
Revelation of 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.
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) 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 assumed to be composed of the singlet hemions x R and y R with invariant mass term x 1R y 2R − x 2R y 1R (case A). Hence Γ ee = 43 eV. If Ω comes instead from x L and y L with invariant mass term x 1L y 2L − x 2L y 1L (case B), then the factor (− sin 2 θ W /4) in C V and C A is replaced with (cos 2 θ W /4) and Γ ee = 69 eV. Similar expressions hold for the other fermions of the Standard Model (SM).
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 in case A. Because of the accidental cancellation of the two terms in the above, C W turns out to be very small. Hence Γ W W = 3.2 eV. In addition to the s−channel decay of Ω to W − W + through γ and Z, there is also a t−channel electroweak contribution in case B because x L and y L form an electroweak doublet. Replacing (− sin 2 θ W /4) with (cos 2 θ W /4) in C W , and adding this contribution, we obtain where Thus a much larger Γ W W = 190 eV is obtained. 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 ) in case A, with sin 4 θ W replaced by cos 4 θ W in case B. Hence Γ ZZ is negligible in case A and only 2.5 eV in case B.
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 given by should be feasible to make this observation. Table 2 summarizes all the partial decay widths. 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. In case A, the cross section is given by Using m = 100 GeV and s = (250 GeV) 2 as an example, we find these cross sections to be 0.79 and 0.44 pb respectively. In