Fermion Doubling Research Articles

Theoretical and phenomenological constraints on preon models and roles of supergroups

Models of massless composite quarks and leptons are proposed on the basis of the following requirements: 1. (1) Confining hypercolor gauge forces which are asymptotically free with a scale Λ H in the TeV range. 2. (2) Anomaly free gauge sector. 3. (3) Anomaly matching between preons and composite fermions for consistency of an unbroken chiral symmetry. 4. (4) Decoupling of heavy preons via a parity doubling of the composite fermions that contain them. 5. (5) Massless composite fermions contain the minimal number of valence preons that make a hypercolor singlet. Exotics are excluded or assumed to be massive. 6. (6) Indices of composites do not exceed 1. 7. (7) Pauli principle is satisfied with a completely symmetric spatial wave function. 8. (8) The preonic chiral symmetries contain SU(3) × SU(2) × U(1). When the symmetry is broken down to this group, the only remaining massless fermions fall just into the patterns of observed families. 9. (9) Unobserved processes such as proton decay, e +e −→ μ ±e ∓, eN→ μN, υ→e γ, K L→ μ ±e −+, K +→ π + ±μ ∓, Δm(K L−K S), and anomalous magnetic moments of e and μ are suppressed or do not occur. 10. (10) A mechanism for SU(2) w breaking that generates masses for quarks and leptons exists. We find that requirement (4) always imposes the structure of representations of supergroups SU( N/ M). Unlike supersymmetric models, the grading here is between left-handed and right-handed fermions rather than between bosons and fermions. Only the even subgroup of the supergroup is a symmetry, but preons and composite fermions must be classified in irreducible representations of SU( N/ M) as if the full supergroup were a symmetry. Using supertableaux techniques, we find and classify all models containing 3 preons in a composite fermion. We study models in which the preons fall into one of these structures with respect to hypercolor: (i) A single irreducible representation R of any hypercolor, group (including direct products) with R × R × R ∼ 1, (ii) Two irreducible representations R 1, R 2 of hypercolor with R 1R 2 ∗R 2 ∗ ∼ 1, and (iii) Three irreducible representations satisfying R 1R 2R 3 ∼ 1. We treat the hypercolor group G H and the representations R i as unknowns and follow a strategy for finding constraints on G H and R i which lead to all possible models consistent with conditions (1)–(5), then (6)–(7) and finally (8)–(10). Explicitly we construct many models that satisfy conditions (1)–(6). One model of type (i) satisfies requirement (7) but no model of type (i) satisfies the additional requirements (8)–(10). Only two classes of models of type (ii) and one class of type (iii) are found to pass tests (1)–(7). Among then we find an SU(3) × SU(2) × U(1) embedding which satisfies the remaining physical requirements (8)–(10). Δm(K L − K S) provides the most severe bounds on the hypercolor scale Λ H.

Eliminating lattice fermion doubling

The problem of fermion doubling on a lattice is analyzed. Two new solutions are found in one space dimension. The first fails in the usual manner to generalize to three dimensions. The second solution generalizes to eliminate doubling in any number of dimensions.

Homology theory of lattice fermion doubling

I give a general discussion of the phenomenon of spectrum degeneracy in transcribing continuum field equations to the lattice, using concepts of homology theory. This leads to a topological understanding of the problems in transcribing the Dirac equation and a unified treatment of the many lattice fermion schemes in the literature. The connection between spectrum degeneracy and chiral symmetry is explained geometrically without appealing to quantum effects such as anomalies.

A remnant of chiral symmetry on the lattice

A new criterion for chiral symmetry in lattice theories of fermions is derived within a block-spin formalism. This "remnant" symmetry criterion properly incorporates the Adler Bell-Jackiw anomaly and avoids the fermion-doubling problem of other lattice fermion methods. Some obstacles to implementing this approach in the presence of fully dynamical gauge fields are discussed.

Doubling of Fermions?

Although the baryons possess strong interactions, in addition to the electromagnetic and weak ones, it is conceivable that whatever the cause of the doubling of electrons into light and heavy ones might also be responsible for a doubling of baryons. Considerations are given to the interactions which could possibly give rise to the doublets and to the characteristics which may help distinguish the unknown particle from the known. (T.B.A.)