Left-handed Fermions Research Articles

TWO-DIMENSIONAL CHIRAL GAUGE THEORIES IN THE SCHRÖDINGER REPRESENTATION

We present a quantization, in the Schrödinger representation, of a two-dimensional theory consisting of an Abelian gauge field coupled to an arbitrary number of right- and left-handed fermions. A consistent and unitary quantum theory is obtained. The gauge symmetry is broken but Lorentz invariance survives quantization. The theory contains a series of chiral massless particles and a massive mode, which can be interpreted as collective excitations of the fields that we have in the initial Lagrangian. Our formalism is quite transparent and allows a clear understanding of the consistency of the model.

Consistent quantization and symmetry structure of a non-Abelian chiral gauge theory.

The SU(N) chiral Schwinger model with a Wess-Zumino term is studied by use of non-Abelian bosonization, the Becchi-Rouet-Stora formalism, and a dual transformation, and it is confirmed that this model is a sensible quantum theory in a certain range of the anomaly parameter a. The SU(N) gauge symmetry restored by the inclusion of the Wess-Zumino term gets spontaneously broken and the gauge field becomes massive. Left-handed fermions are found to be confined while right-handed fermions remain free and massless. For the specific value a=2, the symmetry of the model enlarges [to a U(N)\ifmmode\times\else\texttimes\fi{}U(N) Kac-Moody symmetry]. It is shown by fermionization of the Wess-Zumino field that for a=2 this model is equivalent to massless two-dimensional QCD (${\mathrm{QCD}}_{2}$) in the sense that they share the same gauge field and the same left-handed fermions. A dual transformation is used to cast the model into an equivalent nonlinear system of scalar fields only, which reveals the particle spectrum of the model.

Charged Massless Left-Handed Fermions

A model Hamiltonian is proposed to describe the interacting massless left-handed fermions and U(l) gauge field. It contains a deep interaction which is assumed to bring about an effective momentum cutoff without affecting the low energy properties. Investigation of the eigenstates of the Hamiltonian shows that there appear massive leftand right-handed fermions which satisfy the free Dirac equation.

The general massive Schwinger model

The massless Schwinger model, with left-handed and right-handed fermions coupled differently to an abelian gauge field, is consistent if a parameter exceeds a critical number. With the fermions given a gauge-invariant mass, the critical number is twice as large. The low-energy theory is of the sine-Gordon form in the strong-coupling limit.

Fermions in the chiral Schwinger model.

We derive an operator solution for the fermion in the chiral Schwinger model with a Wess-Zumino term and study the quantum structure of the model in a manifestly covariant operator formalism. The U(1${)}_{\mathrm{L}}$ gauge symmetry restored by the inclusion of the Wess-Zumino term gets spontaneously broken and the gauge field becomes massive. The left-handed fermion is found to be confined. The right-handed fermion, on the other hand, remains a massless free field in spite of the fact that the left- and right-handed sectors of the model are coupled through the anomaly. This massless fermion is interpreted as the Nambu-Goldstone mode associated with the spontaneous breakdown of the global U(1${)}_{\mathrm{R}}$ symmetry.

Why do the knownleft-handed fermions have weak isospin [formula omitted]? A possible lattice gauge theory explanation

We suggest a possible explanation, based on properties of the (zero-temperature) chiral transition in lattice gauge-Higgs theories, for why the known left-handed fermions have weak isospin 1 2 .

Chiral fermions on two-dimensional fractal structures

Fermion propagation is studied on two-dimensional fractal structures. It is shown that these structures select fermions with a particular chirality: only (say) left-handed Weyl fermions survive. More, due to the self-similarity of the space, there is no chiral symmetry restoration in the continuum limit. Hence, it is not only possible to get continuum chiral fermions in our model, but what is more, it is impossible to keep chiral symmetry unbroken.

Topological terms induced by finite temperature and density fluctuations.

In (3 + 1)-dimensional finite-temperature and -density SU(2) gauge theories with left-handed fermions, the three-dimensional Chern-Simons term (topological mass) can be induced by radiative corrections. This result is derived by use of a family's index theorem which also implies that in many other quantum field theories various additional lower-dimensional topological terms can be induced. In the high-temperature limit these terms dominate the partition function, which suggests applications to early-Universe cosmology.

Excited fermions and low-energy parity-violating phenomena

A calculation is made of the parity-violating contributions of hypothetical excited quarks and leptons to two examples of low-energy phenomena. In particular, the contributions to polarized-electron--deuteron scattering and the parity-violating nuclear potential are studied. The parity-violating contributions are induced by the magnetic coupling of photons or gluons with virtual excited fermions and left-handed ordinary fermions. It is concluded that the mass of an excited quark, if it exists, must be larger than approx.3 TeV. The restriction on the mass of an excited lepton is less stringent.

Precocious unification in simple grand unified theories.

We investigate how simple grand unified theories with a precocious-unification mass can be constructed. The maximal local symmetry of 2mn left-handed fermions and their left-handed antiparticles is shown to be a viable candidate, where m and n are the number of families and colors, respectively. The predictions of such a model are studied.

Path-integral measure and anomalies for left-handed fermions

A path-integral measure for left-handed fermions is constructed for non-Abelian gauge theories, and its properties under unitary transformations are discussed.

Flavor unification in SU(8)

Flavor unification is proposed in the SU(8) gauge group. This model imbeds the Georgi-Glashow S${\mathrm{U}}_{\mathrm{gg}}$(5) as a subgroup. It is necessary to modify Georgi's second postulate for grand unification so that the left-handed fermion representations are complex with respect to $\mathrm{S}{\mathrm{U}}_{c}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{S}{\mathrm{U}}_{W}(2)\ifmmode\times\else\texttimes\fi{}{\mathrm{U}}_{Y}(1)\ifmmode\times\else\texttimes\fi{}{\mathrm{U}}_{J}(1)$. There are three families of light quarks, one right-handed isospin-singlet quark with charge $\ensuremath{-}\frac{1}{3}$ three ordinary lepton families, and one lepton doublet without a right-handed partner.

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.

An SU(2) anomaly

A new restriction of fermion quantum numbers in gauge theories is derived. For instance, it is shown that an SU(2) gauge theory with an odd number of left-handed fermion doublets (and no other representations) is mathematically inconsistent.

Anomalies and the lattice Schwinger model: Paradigm not paradox

This paper shows that, contrary to statements extant in the literature, it is possible to introduce fermions into a lattice gauge theory in such a way as to preserve the continuous chiral symmetries of the massless theory and the physics of the axial anomaly. The particular model discussed is the lattice Schwinger model and the methods used are based upon the nonperturbative gauge-invariant variational techniques introduced by Horn and Weinstein. It is demonstrated that the physics of the anomaly, and its relation to the angles appearing in the exact solution to the continuum model, appears in a simple and elegant way. The generalization of the model to several sets of independent fermions is discussed at the end of the paper. Some brief remarks are made about what happens if one attempts to gauge an anomalous current. These results are of interest, since the Drell-Weinstein-Yankielowicz prescription is the only known way of writing down purely lattice gauge theories with only left-handed fermions.

Fermionic composite models from complementarity

Composite models for (in principle massless) quarks and leptons without fundamental scalars are constructed with the aim of providing for fermionic realizations of models which include elementary bosons (by Abbott and Farhi, Casalbuoni and Gatto and Barbieri, Mohapatra and Masiero). The models use one confining unitary (subcolor) group (with left-handed fermions in the fundamental, in its conjugate, and either in the adjoint, or in the symmetric, or in the antisymmetric representation of subcolor) or two confining groups. Families may arise from discrete symmetries.

Adler-Bell-Jackiw anomaly and fermion-number breaking in the presence of a magnetic monopole

In (V - A) theories, fermion number is broken in the presence of the 't Hooft-Polyakov magnetic monopole through the Adler-Bell-Jackiw anomaly. An exactly solvable zeroth-order approximation for evaluating Green functions of zero-angular-momentum fermions in the presence of a monopole is developed in the case of an SU(2) model with massless left-handed fermions. Within this approximation the density of the fermion-number breaking condensate is calculated. This density is found to be O(1), i.e. to be independent of the coupling constant and of the vacuum expectation value of the Higgs field. The corrections to the approximation are estimated. It is argued that the above effect can give rise to the strong baryon-number breaking in monopole-fermion interactions in SU(5) grand unified theory.

Superweak CP violation and right-handed horizontal interactions

A horizontal extension of the Weinberg-Salam electroweak theory by a right-handed O(3) RH gauge symmetry for the three-fermion families is studied. By an appropriate choice of the Higgs scalar fields the CP symmetry of the lagrangian is spontaneously broken, but the mixing of the left-handed fermion states, and hence the Kobayashi-Maskawa mixing matrix, remains real. The CP violation is manifested in the superweak horizontal gauge interactions, which are suppressed by the large mass of the corresponding gauge bosons. It is, however, possible that the horizontal boson acting on the second and third families can be considerably lighter than the two,s implying an interesting phenomenology of the related CP violation effects and flavour-changing neutral currents.

Flavour and superunification

We consider the possibility of a non-trivial embedding of ( 10 + 5 ) SU(5) families of spin if 1 2 left-handed fermions in a combination of irreducible massless supermultiplets of N extended supersymmetry. We demand the whole spectrum of spin 1 2 states to be anomaly free with respect to SU( N). This turns out to be a necessary condition for the absence of anomalies at the SU(5) level. We find two classes of models, with spin 1 2 fermions in SU( N) representations associated to one- and two-column Young tableaux, respectively, in which each irreducible massless multiplet occurs at most once. These two classes of models lead to a nontrivial family generation due to supersymmetry. For N = 8 extended supersymmetry, they give at most three and five families, respectively. The first class of models is more natural in the way it excludes SU(5) exotics. The same analysis is extended to the massless multiplets that can be obtained from bilinear composite fields of the (preonic) elementary fields of N extended supergravity. We prove that the generation of families requires the repetition of massless multiplets and that ( 10 + 5 ) SU(5) families can only be generated in pairs. General properties of multilinear composite operators of the preonic fields are given and the rôle of massive representations to classify towers of operators with definite spin is pointed out.

Unifying theSU 5 grand unification theory

One left-handed fermion field χ in the fundamental decuplet representation ofSU5 is proposed as the common building block of the phenomenological fields of theSU5 grand unification theory. In a first step, bilinears of the χ field can form the gauge and the Higgs bosons. In a second step, the quarks and leptons can arise as bound states of the composite Higgs field and the basic χ field. TheSU5 symmetry will be broken spontaneously by a condensate of the Higgs and of the gauge bosons. If the dimension 1/2 is assigned to the basic field χ, the correct canonical dimensions for the phenomenological fields will arise.