Wilczek and Zee's 'Families from Spinors' (Phys. Rev. D 25, 553 (1982)) proposed to explain why fermions come in repeated families by placing all quarks and leptons into a single spinor representation of a large orthogonal group such as SO(18), the largest asymptotically free choice when all fermions occupy one spinor. The price is an equal number of mirror ('conjugate') families carrying right-handed (V+A) weak currents, which must be removed from the low-energy spectrum. The paper's own analysis shows ordinary Higgs masses cannot do this - the mirror mass terms have no electroweak-singlet channel for the charged mirror fermions, so gapping the mirror families at a high scale breaks SU(2)xU(1) - and its proposed escape, confinement by an unbroken 'heavy color' subgroup of the horizontal SO(8) factor (the part of the group acting across families), fails in the fully gauged version because one-loop running forces a confinement scale near 10^10 GeV. The open question is whether such a picture admits a dynamical completion: a mechanism that selects the confining horizontal subgroup, and a nonperturbative demonstration that the mirror families decouple without breaking electroweak symmetry, together with consistency with established low-energy phenomenology.

Background

Particle physics observes that quarks and leptons come in three sequential copies ('families' or 'generations') with identical gauge quantum numbers; the Standard Model does not explain this repetition. Grand Unified Theories (GUTs) explain the quantum numbers of a single family by embedding it, plus a right-handed neutrino, into the 16-dimensional spinor representation of SO(10) - the 'spinor' being the special representation of orthogonal groups, of dimension 2n−12^{n-1} for SO(2n)SO(2n), built from the Clifford algebra rather than from tensors. 'Family unification' tries to derive the repetition itself: all fermions are placed in one irreducible representation RR of a larger group, such that when the group breaks to a low-energy group the single RR decomposes into several copies of the family representation. Simple unitary groups fail at this (unwanted extra representations, gauge anomalies, which are quantum inconsistencies of a gauge symmetry, or invariant bare masses that destroy chirality), but Wilczek and Zee (1982) showed that orthogonal-group spinors do it naturally: a chiral spinor - one of the two complex half-spinor representations, inequivalent to its conjugate, so that the fermions have a definite handedness - of SO(2n+2m)SO(2n+2m) decomposes under SO(2n)SO(2n) into 2m−12^{m-1} copies of each chiral spinor S+S_+ and S−S_- of SO(2n)SO(2n). Taking n=5n=5 (SO(10) families) and SO(18) - which their one-loop analysis identifies as the largest asymptotically free member of the SO(4N+2)SO(4N+2) series when all fermions sit in one spinor, 'asymptotic freedom' meaning that the gauge coupling decreases at high energy - the single 256-dimensional spinor contains, after the breaking SO(18)→SO(10)×SO(8)SO(18)\to SO(10)\times SO(8), the pieces (16+,8+)(16_+,8_+) and (16−,8−)(16_-,8_-): eight ordinary families with left-handed (V-A) weak currents and eight mirror families with right-handed (V+A) currents, which are not observed.

The mirror families are the central obstruction. They cannot simply be given Higgs masses: in the mirror mass term 16−×16−16_-\times 16_- of SO(10), no electroweak-singlet channel exists for the charged mirror fermions (only the neutral, Standard-Model-singlet components can pair), so any condensate (a vacuum expectation value formed by strong dynamics) that gaps the mirror families at a high scale necessarily breaks electroweak symmetry there, contrary to observation. The paper's proposal was 'heavy color' (HC): let a subgroup HC of the horizontal SO(8) (the factor acting across the family copies, as opposed to the SO(10) acting within each family) remain a strong, unbroken gauge interaction that confines (binds, like QCD) all components that are not HC singlets into heavy bound states, so that only HC-singlet families remain light. Different HC choices inside SO(8) - SU(4), Sp(4) (equivalently SO(5)), SU(2)×SU(2)SU(2)\times SU(2), or the diagonal SU(2) - leave respectively 2, 3, 4, or 5 light V-A families, so in the minimal logic the observed three families single out Sp(4). But nothing selects the choice, and the paper's own one-loop renormalization-group (RG) analysis of the fully gauged SO(18) chain places the HC confinement scale near 101010^{10} GeV, at which the confinement dynamics must break electroweak symmetry - so the straightforward implementation fails. The authors sketched escapes: gauge only SO(10) and HC separately, so that the confinement scale becomes a free parameter; or realize spinors as composite particles ('preon' models), for which they built an explicit SO(2N)SO(2N) spinor multiplet from fermion zero modes bound to a magnetic monopole (Jackiw-Rebbi modes in an SO(3) theory broken to U(1)) - but left open whether such composites can be fermionic, with realistic chiral spectra. Every one of these mechanisms was an input rather than derived dynamics, and what has remained unresolved ever since is a dynamical, nonperturbatively controlled completion of the picture.

Later work proposes concealment mechanisms rather than deriving them: orbifold compactifications (extra dimensions truncated by boundary conditions), Peccei-Quinn protection (a postulated global symmetry), and symmetric mass generation (giving fermions masses through gauge-invariant interactions rather than mass terms); the recent descendants of the program likewise assume confinement by a specific subgroup, Sp(4), as an input. The experimental questions raised by the source are settled in favor of the Standard Model - the b quark carries the left-handed charged current, and no additional or mirror families appear at accessible energies - which sharpens but does not close the theoretical core.

Problem Statement

Can spinor-based family unification be completed dynamically? Specifically, does there exist a realization - within the class of models in which all known fermions, and nothing else that remains light, are components of a single spinor representation of a large orthogonal group (or its Spin cover, the simply-connected double cover), possibly realized as composites - in which: (1) a dynamical mechanism, not a model-building postulate, determines the fate of the horizontal subgroup within the model's symmetry-breaking pattern down to the low-energy gauge group: which subgroup (if any) plays the confining heavy-color role, at what scale relative to the electroweak scale, and hence how many light families survive; (2) the conjugate mirror families (those with right-handed, V+A, weak currents) that group theory forces into the spinor are confined or otherwise decoupled by a nonperturbatively demonstrable mechanism, at a scale that leaves electroweak SU(2)xU(1) unbroken down to the weak scale; and (3) the completed model is consistent with established low-energy phenomenology: a consistent low-energy spectrum, agreement with the absence of additional or mirror families at accessible energies, and with the left-handed charged current of the b quark. These three requirements are facets of one objective - turning the group-theoretic explanation of family repetition into a defensible physical theory - and must be satisfied jointly by any candidate realization. A demonstrated impossibility for this model class (a no-go theorem) is the admissible negative resolution.

References

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