Chiral symmetry,the 1/N expansion and the SU(N) thirring model

In the two-dimensional SU(N) Thirring model, the 1/N expansion seems to predict spontaneous breaking of the continuous chiral symmetry. This is impossible in two-dimensions. Reasoning along the lines of Berezinski, Kosterlitz and Thouless for the two-dimensional XY model, we argue that, in fact, rather than showing long-range order, $\text{〈}\text{ψ}\text{ψ(x)}\text{ψ}\text{ψ(0)〉}$ vanishes in this model as |x|^?1/N at large |x|. The 1/N expansion is, in fact, a rather good guide to the properties of this model.     

Non-perturbative effects in the O(N) gross-neveu model

The construction of an ordered lagrangian for the O(N) Gross-Neveu model is presented. The procedure introduces a mass gap into the theory, prior to the development of perturbation theory. The results of the standard large N analysis of the theory, including mass generation, can then succesfully be produced. The renormalization of the ordered lagrangian at next to leading order is discussed in detail, and the large N corrections to the β-function are established.     

Magnetic susceptibility of the O(N) non-linear sigma model

We investigate a non-perturbative method suited to the study of asymptotically free theories. We apply it to the two-dimensional O(N) non-linear sigma model to extract the value of the magnetic susceptibility. This scheme is shown to be exact when N goes to infinity. We compare our results with Monte Carlo data.     

Perturbation around the migdal transformation for the non-linear O(n) sigma model in two dimensions

The perturbation around the Migdal transformation proposed by Martinelli and Parisi, has been computed to first order in the shifting parameter for the O(n) sigma model in two dimensions, in the weak coupling limit. The correction has the correct sign, but to first order heavily overshoots the exact result of the momentum-space perturbation theory.     

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.     

The Martinelli-Parisi expansion for the non-linear sigma model

The Martinelli-Parisi expansion around the Migdal-Kadanoff transformation is computed to second order in the shift parameter for the non-linear sigma model on a triangular lattice. The computation is performed in a completely analytical way in the region of weak coupling which is the most crucial region for checking the goodness of the expansion. The value of the beta function improves definitely with respect to the zero- and first-order results, reducing to 13% the error with regard to the exact value of continuum perturbation theory.     

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.     

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.     

Monte Carlo simulations for the two-dimensional O(3) non-linear sigma model

We report on the results of Monte Carlo simulations for the two-dimensional O(3) non-linear sigma model. The estimates based on the combined use of the renormalization group and of the high temperature expansion are found to be in agreement with our “data”. We present good experimental evidence for the absence of any phase transition, as expected on theoretical grounds.     

Modification of the topological expansion of the O(3) sigma model

We show that the one-instanton sector moduli-space divergence of the O(3) sigma model leads to an unacceptable dependence of Green functions on the arbitrary way that the field is split into a quantum fluctuation about a classical background. In particular the rotational symmetry of the model is spoilt. Since the divergence is associated with the degeneration of field configurations to those of the zero-instanton sector this arbitrariness may be cancelled by modifying the zero-instanton functional integral. We explicitly construct this modification.     

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.     

Migdal approximation and 1/N expansion in QCD

We show that the Migdal approximation for the renormalization group in gauge theories in exact in the leading order in $\text{1}\text{N}$, above a critical value of the coupling constant. A systematic expansion in $\text{1}\text{N}$ is proposed. The string tension is calculated in the leading order in $\text{1}\text{N}$ in a theory without fermions.     

Non-perturbative effects in two-dimensional lattice O(N) models

Non-abelian analogues of Kosterlitz-Thouless vortices may have important effects in two-dimensional lattice spin systems with O(N) symmetries. Renormalization group equations which include these effects are developed in two ways. The first set of equations extends the renormalization group equations of Kosterlitz to O(N) spin systems, in a form suggested by Cardy and Hamber. The second is derived from a Villain-type O(N) model using Migdal's recursion relations. Using these equations, the part played by topological excitations in the crossover from weak to strong coupling behavior is studied. Another effect which influences crossover behavior is also discussed: irrelevant operators which occur naturally in lattice theories can make important contributions to the renormalization group flow in the crossover region. When combined with conventional perturbative results, these two effects may explain the observed crossover behavior of these models.     

On-shell O(N) supergravity in superspace

We show that general considerations of the properties of free on-shell O(N) superfield strengths lead directly to linearized on-shell O(N) supergravity. We consequently obtain a formulation of the fully non-linear on-shell theories in which all fields are contained in the covariant derivatives.     

Coupling constant renormalization due to instantons in the O(3) non-linear sigma model

The renormalized lattice coupling constant for the O(3) non-linear sigma model is calculated, including instanton effects, and the correlation length estimated. The results are in good agreement with the Monte Carlo simulation of Shenker and Tobochnik. The topological charge density is also discussed in the light of recent Monte Carlo simulations.     

The Gross-Neveu model at finite temperature at next-to-leading order in the 1/N expansion

We present new results on the Gross-Neveu model at finite temperature and at next-to-leading order in the 1/N expansion. In particular, a new expression is obtained for the effective potential which is explicitly invariant under renormalization group transformations. The model is used as a playground to investigate various features of field theory at finite temperature. For example we verify that, as expected from general arguments, the cancellation of ultraviolet divergences takes place at finite temperature without the need for introducing counterterms beyond those of zero temperature. As well known, the discrete chiral symmetry of the (1+1)-dimensional model is spontaneously broken at zero temperature and restored, in leading order, at some temperature T_c; we find that the 1/N approximation breaks down for temperatures below T_c: as the temperature increases, the fluctuations become eventually too large to be treated as corrections, and a Landau pole invalidates the calculation of the effective potential in the vicinity of its minimum. Beyond T_c, the 1/N expansion becomes again regular: it predicts that in leading order the system behaves as a free gas of massless fermions and that, at the next-to-leading order, it remains weakly interacting. In the limit of large temperature, the pressure coincides with that given by perturbation theory with a coupling constant defined at a scale of the order of the temperature, as expected from asymptotic freedom.     

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.     

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.     

Hopf term,fractional spin and soliton operators in the O(3) non-linear sigma model

We re-examine three issues, the Hopf term, fractional spin and the soliton operators, in the 2+1 dimensional O(3) non-linear sigma model based on the adjoint orbit parametrization (AOP) introduced earlier. It is shown that the Hopf term is well defined for configurations of any soliton charge Q if we adopt a time-independent boundary condition at spatial infinity. We then develop the Hamiltonian formulation of the model in the AOP and thereby argue that the well-known Q² formula for fractional spin holds only for a restricted class of configurations. Operators that create states of given classical configurations of any soliton number in the (physical) Hilbert space are constructed. Our results clarify some of the points that are crucial for the above three topological issues and yet have remained obscure in the literature.