Completing the experimental viability analysis of the Foot–Hernández extended-color two-Z model against precision electroweak data and modern collider searches

The Standard Model color group can be embedded in an enlarged color gauge factor $SU(5)_c$ whose low-scale breaking leaves ordinary QCD plus an unbroken confining $SU(2)'$, and produces, together with electroweak symmetry breaking, a second neutral gauge boson $Z'$, six fractionally charged gauge bosons $X^{\pm 1/6}$, and exotic electric-charge-$\pm 1/2$ fermions ('quirks') bound by the new $SU(2)'$ force. Glashow and Sarid (1990) confronted this Foot–Hernández 'two-Z' scenario with then-current data and found only the mild bound $M_{Z'}\gtrsim 100$ GeV, leaving six open issues: radiative corrections to the $Z$–$Z'$ mass eigenvalues, the exotic-fermion mass spectrum, $X^{\pm 1/6}$ collider phenomenology, nonperturbative $SU(2)'$ bound-state ('hueball') signals, flavor-changing neutral currents, and proposed $e^+e^-$ and hadron-collider tests. Later literature closed parts of this program — UA2 dijet data excluded $M_{Z'}\lesssim 280$ GeV, the recommended LEP/LEP2 precision tests were performed with null results, and quirkonium and kaon-mixing analyses restricted the exotic-fermion spectrum — but no dedicated analysis confronts the surviving parameter space with the complete modern data set. The open problem is to finish the original question: compute the model's radiative corrections and induced flavor-changing processes, recast LEP Z-pole and LEP2 fermion-pair measurements, Tevatron/LHC dijet-resonance and exotic-charged-particle searches onto this model's nonstandard (lepton-suppressed, dijet-dominated) signatures, and determine whether any region of the original scenario remains experimentally viable.

Background

The Standard Model (SM) of particle physics describes the strong interaction with the gauge group $SU(3)_c$ ('color'). A classic question of model building is whether the observed color group is instead the unbroken remnant of a larger gauge symmetry broken near the current collider frontier. Foot and Hernández (1990) proposed the minimal such 'extended color' scenario: the SM color group is embedded in $SU(5)_c\times SU(2)_L\times U(1)_{Y'}$, and a scalar field $\chi$ acquiring a vacuum expectation value breaks $SU(5)_c\to SU(3)_c\times SU(2)'$ at a scale of order a few hundred GeV to a TeV. Electroweak symmetry breaking then produces a second, heavier neutral gauge boson $Z'$ that mixes with the SM $Z$, six charged gauge bosons $X^{\pm 1/6}$ carrying both color and the new $SU(2)'$ charge, and exotic fermions of electric charge $\pm 1/2$ — in later literature called 'quirks' — that remain confined by the unbroken $SU(2)'$ interaction, whose glueball bound states (bound states of the gauge bosons of a confining interaction, analogous to the glueballs of the strong interaction) were dubbed 'hueballs'. The exotic-fermion masses arise from new, unknown Yukawa couplings to $\chi$ and are therefore free parameters; apart from them, the model's only new parameter is essentially the breaking scale, equivalently $M_{Z'}$.

This scenario evades the standard searches for extra neutral gauge bosons. The $Z'$ couples to quarks with strength $g_s/\sqrt{15}$, where $g_s$ is the coupling of quantum chromodynamics (QCD, the Standard Model strong interaction), and its coupling to charged leptons is so suppressed that the dimuon branching fraction is only of order $10^{-4}$. The cleanest and most widely used $Z'$ searches — resonances in dilepton final states at hadron colliders — therefore barely apply. The operative constraints instead come from dijet resonance searches, contact-interaction interpretations of inclusive jet and fermion-pair data (fits that reinterpret precision data as limits on heavy four-fermion interactions), precision electroweak observables, searches for exotic electrically charged particles, and flavor-changing neutral currents (FCNCs) — processes in which quarks change flavor without exchanging electric charge, induced in this model by box diagrams (a standard class of one-loop quantum corrections) with exotic fermions and $X^{\pm 1/6}$ bosons.

Glashow and Sarid (1990) performed the first systematic confrontation of this two-Z model with experiment and found only a mild bound, $M_{Z'}\gtrsim 100$ GeV (with a low-mass window around 36–48 GeV already removed by early $e^+e^-$ storage-ring data from the TRISTAN collider at KEK and the PETRA collider at DESY), because $Z$–$Z'$ mixing effects on leptonic observables are suppressed in this model. They identified six open issues: (1) the radiative corrections to the $Z$ and $Z'$ masses, which depend on the unknown exotic spectrum and were omitted from the mass-eigenvalue analysis; (2) the exotic-fermion masses, each exotic $SU(2)'$ doublet adding roughly 200 MeV to the $Z$ width and thus requiring exotic masses above $M_Z/2\approx 45$ GeV; (3) the collider phenomenology of the $X^{\pm 1/6}$ bosons; (4) quirkonium (quirk–antiquirk bound states, analogous to positronium) and hueball production, including anomalous electromagnetic energy deposition in exotic cascade decays; (5) the induced FCNC effects; and (6) the proposed experimental tests — the ratio $R$ of hadronic to muonic $e^+e^-$ cross-sections and forward–backward asymmetries $A_{FB}$ (the asymmetry between fermions emerging in the forward and backward hemispheres relative to the beam direction) for $M_{Z'}$ below about 200 GeV, and related hadron-collider measurements. A companion analysis of other two-Z models (Glashow and Sarid, Phys. Rev. Lett. 64, 725) predicted Z-pole modifications appearing as an excess in the effective number of light neutrino species.

Much of the recommended program was in fact carried out. The final measurements at LEP (CERN's Large Electron–Positron collider) and SLC (the Stanford Linear Collider) on the $Z$ resonance give $N_\nu=2.9840\pm 0.0082$ for the effective number of light neutrino species, excluding the companion two-Z class, and the combined LEP2 (the high-energy phase of LEP) fermion-pair cross-sections and asymmetries at center-of-mass energies of 130–209 GeV agree with the SM. Dijet data from UA2 (an experiment at CERN's proton–antiproton collider) require $M_{Z'}\gtrsim 280$ GeV, closing the 100–200 GeV window in which Glashow and Sarid had proposed to seek the $Z'$. Foot, Hernández and Rizzo (1990) computed the model's $Z'$ and $X^{\pm 1/6}$ hadron-collider production and decay phenomenology against the data then available, Hernández (1991) extended it to single-jet inclusive signatures, and Heyssler (1996) treated the generic leptophobic $Z'$; this collider-phenomenology part of the program therefore requires updating and recasting those computations onto the modern Tevatron and LHC data set rather than repeating them. Carlson, Hall, Sarid and Burton (1991) showed that cosmology and neutral-kaon mixing (via the FCNC box diagrams) forbid pushing all exotic charge-$1/2$ fermions above about 1 TeV, and that the kaon-mixing bound requires the exotic fermions to be roughly mass-degenerate (or their mixing matrix almost diagonal); they named these particles quirks and analyzed quirkonium and hueball signatures to the sensitivity of LEP2 and the Tevatron (Fermilab's proton–antiproton collider). The model then went dormant. Foot (2011) revived the extended-color idea, generalized to $SU(N_c)$, as an explanation of the Tevatron top-quark forward–backward asymmetry, found the required low breaking scale in tension with CDF (the Collider Detector at Fermilab, a Tevatron experiment) dijet constraints but not conclusively excluded, and explicitly deferred detailed LHC (CERN's Large Hadron Collider) predictions and the mapping of the allowed parameter space to future work. Modern offshoots reuse the extended-color construction in other sectors — the enlarged-color QCD axion (a hypothetical light pseudoscalar particle) of Gherghetta, Nagata and Shifman (2016), and the extended-color twin-Higgs models (mirror-sector constructions that keep the Higgs boson naturally light) of Batell, Cochran, Page and Verhaaren (2025) — without analyzing the two-Z phenomenology. What none of the works discussed above provides is the completion of the original question: a quantitative, model-specific determination of whether any region of the Foot–Hernández two-Z parameter space survives the complete modern data set, and how the surviving region could be tested.

Problem Statement

Determine the experimental viability of the Foot–Hernández extended-color two-Z model — the $SU(5)_c\times SU(2)_L\times U(1)_{Y'}$ gauge extension of the Standard Model whose low-scale breaking produces a $Z'$ boson, six $X^{\pm 1/6}$ charged gauge bosons, and exotic charge-$\pm 1/2$ fermions ('quirks') confined by an unbroken $SU(2)'$ — by completing the phenomenological program that Glashow and Sarid (1990) left open and confronting the model with the complete modern data set. Concretely: (a) compute the radiative corrections to the $Z$ and $Z'$ masses, widths, and mixing, including the loop contributions of the exotic fermions and scalars whose masses are set by unknown Yukawa couplings; (b) determine or bound the exotic-fermion spectrum consistently with the $Z$-width constraint (exotic masses above $M_Z/2$), the kaon-mixing (FCNC box) near-degeneracy requirement, and the roughly 1 TeV upper limit from kaon mixing and cosmology; (c) compute the production and decay phenomenology of the $X^{\pm 1/6}$ gauge bosons and the $Z'$ at hadron colliders, accounting for the suppressed couplings ($Z'$ quark coupling $g_s/\sqrt{15}$, dimuon branching fraction about $1.4\times 10^{-4}$) that make dijet signatures dominant; (d) analyze quirkonium formation and the $SU(2)'$ glueball ('hueball') emission — including the anomalous electromagnetic energy deposition in exotic cascade decays — to establish detectability; (e) evaluate the induced flavor-changing neutral currents beyond the kaon-mixing estimate; and (f) confront the resulting predictions with precision electroweak data (LEP Z-pole, LEP2 fermion-pair cross-sections and asymmetries at 130–209 GeV) and with Tevatron and LHC searches (dijet resonances, contact interactions, exotic charged particles), mapping the allowed parameter space in $M_{Z'}$ and the exotic-fermion masses — and in the $SU(N_c)$ generalization flagged by Foot (2011) — and establishing whether any region of the original scenario, beyond the windows already excluded by the earlier literature ($M_{Z'}\lesssim 100$ GeV from Glashow and Sarid, the 36–48 GeV window from TRISTAN/PETRA, and $M_{Z'}\lesssim 280$ GeV from UA2 dijet), remains experimentally viable, and how it could be tested. The overarching question is whether any region of the model's parameter space survives; items (a)–(f) are the components whose joint completion answers it, while the $SU(N_c)$ generalization flagged by Foot (2011) is an optional extension of the analysis.

References

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17. B. Batell, T. Cochran, L. Page and C. B. Verhaaren, 'Extended color twin Higgs', JHEP 11 (2025) 092. DOI: 10.1007/JHEP11(2025)092, arXiv:2508.10102, https://doi.org/10.1007/JHEP11(2025)092
18. T. G. Rizzo, 'Constraints on new gauge bosons from radiative corrections', preprint MAD-PH-626 (1990), unpublished; bibliographic record at https://inspirehep.net/literature/302857. This unrefereed preprint is cited for context only; it is not treated as resolving the radiative-correction analysis of item (a)