Scale invariance in the thirring model with SU(n) symmetry

It is shown that the condition for scale invariance of the Green's functions is satisfied for two sequences of the SU(n) coupling g_v. For all these solutions the c-number Schwinger term in the commutator of the SU(n) currents has a negative values different from the free field value.     

Nonconvergence of the 1/N expansion for SU(N) gauge fields on a lattice

We present specific examples that demonstrate the non-convergence of the 1/N expansion for the lattice theory of SU(N) gauge fields.     

Comparing O(N) and SU(N) × SU(N) spin systems in 1 + 1 dimensions to SU(N) gauge theories in 3 + 1 dimensions

We have computed the scale breaking Λ parameters of the euclidean and hamiltonian formulations of the lattice regulated O(N) and SU(N) × SU(N) spin systems in 1 + 1 dimensions in terms of the Λ_PV parameters of the Pauli-Villars regulated continuum models. Using lattice perturbation theory, the renormalized mass gap has been determined in terms of Λ_PV for each model. These results are compared to analogous calculations in SU(N) gauge theories.     

Explicit operator solution to thirring model with U(n) symmetry

Two explicit solutions to the Thirring model with U(n) symmetry are constructed following Klaiber's approach. Products of fields are obtained in terms of those for the free case and scale dimensions of composite operators are computed.     

Spinor-inverted solution to thirring model and its generalization to U(n) symmetry

Using the properties of massless free Fermi fields in (1-1) dimensions, it is shown that the spinor inverted form of Klaiber's operator solution to Thirring model is also a scale-invariant solution of the model. But unlike the former it admits a nonvanishing SU(n) current coupling in the generalization of the model to include U(n) symmetry. The value of this coupling constant is fixed and equals Dashen-Frishman number $\text{?4π}\text{(n + 1)}$. The general form of the 2m-point function is given and operates product expansions are exhibited in terms of composite local operators. Scale dimensions of all the bilinear and quadrilinear local operators with U(n) symmetry are computed and are found to depend on n. However, different parts of a composite local operator belonging to different irreducible U(n) representations have the same dimension.     

Coleman-Weinberg symmetry breaking in the chiral SU(n) × SU(n) linear σ model

The onset of symmetry breaking in the chiral SU(n) × SU(n) linear σ model is investigated. It is shown that the model possesses no stable fixed points in d = 4 ? ? dimensions and that the chiral symmetry of the theory can be broken via the CW mechanism when d = 4. The implications of these results are discussed.     

Non-perturbative effects and infrared renormalons within the 1/N expansion of the O(N) non-linear sigma model

We analyze the structure of the Borel transform of the two-dimensional O(N) non-linear σ model within its $\text{1}\text{N}$ expansion. We check the existence of IR singularities (IR renormalons) and the presence of non-perturbative terms which organize themselves in an operator expansion à la Shifman-Vainshtein-Zakharov. We prove that renormalons cancel between the different terms of the operator expansion, so that there is a well-defined resummation procedure of the perturbative series. We suggest that this mechanism provides a general solution of the IR renormalons problem for massless UV free field theories.     

Solution of the SU(N) massless thirring model

In the present work a method is proposed of constructing the eigenvectors and finding the quantum numbers of SU(N) symmetric massless theories with four-fermion interaction for an arbitrary value of N.     

Non-planar diagrams in the large N limit of U(N) and SU(N) lattice gauge theories

It is shown that the limit as N → ∞ with g²N fixed of the strong coupling expansion for the vacuum expectation values of a U(N) or SU(N) lattice gauge theory is not given by a sum of planar diagrams. This contradicts a result claimed by De Wit and 't Hooft.     

SU(N) invariant non-linear σ models

Two series of SU(N) invariant non-linear σ models are constructed using exceptional orbits of the adjoint representation of SU(N). Each of these models possesses infinitely many local as well as non-local conserved charges.     

The hidden SO(4) symmetry of general SU(2) thirring models

General four-fermion interactions in two dimensions with SU(2) invariance are shown to possess a hidden SO(4) symmetry. As a consequence physical states belong to irreducible representations of the two commuting O(3) subgroups and their interactions decouple accordingly. Two independnet stable trajectories of the renormalization group are shown to exist perturbatively and are consistently reproduced by abelian bosonization.     

SO(2, 1) algebra and the large N expansion in quantum mechanics

We discuss the large N expansion in quantum mechanics using an algebraic procedure based on a Holstein-Primakoff representation of the well-known SO(2, 1) algebra. Both spherically and axially symmetric potentials are studied. The method is explicitly illustrated for the family of potentials $\text{V =}\text{ω}{}_0{}^2\text{r}{}^2\text{2}\text{+ 2νr}{}^{\mathrm{2\nu }}$ as well as the hydrogen atom in a uniform magnetic field. In the latter case, the first non-trivial iteration of the present perturbative scheme yields accurate results for the energy levels, even for strong magnetic field intensities. Further generalizations and applications are outlined.     

More about the S-matrix of the chiral SU(N) Thirring model

We fix the bound state poles of the S-matrix of the chiral SU(N) Thirring model by general arguments. Avoiding an infrared problem by using a modified $\text{1}\text{N}$ expansion, the result is confirmed in leading order.     

The phase structure of SU(N)/Z(N) lattice gauge theories

The phase structure of pure SU(N)/Z(N) lattice gauge theories in four dimensions is discussed. The presence of ZN monopoles plausibly leads to a phase transition. A Monte Carlo simulation of SO(3) shows the presence of a very strong, may be first order, phase transition.     

Finiteness, 1/N expansion and supersymmetry breaking

The possibility of spontaneous supersymmetry breaking in the limit of large N, in models with an internal O(N) symmetry, is demonstrated by constructing an explicit example in two dimensions. The model is finite and this is shown to be important for the supersymmetry breaking. A general criterion for finiteness of scalar superfield theories in two dimensions is given. Finally, the generalization of our results to three dimensions, and their relevance to four-dimensional models, is discussed.     

Gauge hierarchies of SU(N) grand unification

The structure of spontaneous breaking of SU(N) gauge symmetry for grand unification is investigated. The results obtained are applied to the analysis of SU(8) symmetry for which possible ways of breaking and intermediate symmetries are considered. It is assumed that the SU(8) group unifies the subgroups of colour, standard electroweak and horizontal symmetries. We find conditions which it is necessary to impose on the vacuum expectation values of Higgs multiplets to provide an arbitrary breaking pattern of SU(N) symmetry and conserve any intermediate symmetry. If in the SU(8) models considered fermions and mirror fermions do not violate the (V-A) and (V+A) structure of weak interactions, then their masses should not be greater than ～10² GeV. It is also shown that the contributions of fermion and Higgs multiplets to the renormalization group equation for the coupling constant of any subgroup of SU(N) are identical. Renormalization group identities for the case of arbitrary SU(N) breaking are given where the contribution of Higgs multiplets have been taken into account (but they cancel each other). Using these identities one can calculate the mass values for the breaking of the intermediate symmetries in the SU(8) models, and also exclude part of the possible breaking patterns.     

Gaussian fluctuations to U(N) and SU(N) lattice gauge theories in the mean field approximation

The mean field can be considered as a classical solution of an appropriately reformulated version of lattice gauge theories. Axial gauge fixing renders it stable. The quadratic forms for the fluctuations in the gaussian approximation are analyzed. The gaussian correction to the mean field free energy is expressed for all U(N) and SU(N) in terms of structure functions that are explicitly calculated for U(N), SU(∞, and SU(∞) numerical calculations are performed for the phase transition point, its latent heat, and some correlation lengths that are characteristic for this kind of mean field approach.     

The effective potential,vacuum tunneling and the 1/N expansion

The problem of vacuum decay in quantum field theory, when the instability is the result of radiative corrections, is discussed. The large-N expansion of the O(N) quartic model in four (Euclideanized) dimensions is analysed in detail and it is shown that, although the effective potential has no lower bound, tunnelling solutions of the usual type (instantons) do not exist. This is shown to happen because that expansion of the action which begins with the effective potential is inappropriate for the kind of field configurations in question. The relevance of this result to the related problem in the Salam-Weinberg model is also discussed.     

All-loop group-theory constraints for color-ordered SU(N) gauge-theory amplitudes

We derive constraints on the color-ordered amplitudes of the L-loop four-point function in SU(N) gauge theories that arise solely from the structure of the gauge group. These constraints generalize well-known group theory relations, such as U(1) decoupling identities, to all loop orders.