Large N saddle formulation of quadratic building block theories

I develop a large N saddle point formulation for the broad class of “theories of quadratic building blocks”. Such theories are those in which the sums over internal indices are contained in quadratic building blocks, e.g., φ² = Σ_{a = 1}^Nφ^aφ^a. The formulation applies as well to fermions, derivative coupling and non-polynomial interactions. In a related development, closed Schwinger-Dyson equations for Green functions of the building blocks are derived and solved for large N.     

Noncompact sigma-models: Large N expansion and thermodynamic limit

Noncompact SO(1,N)$\mathrm{SO}(1,N)$ sigma-models are studied in terms of their large N   expansion in a lattice formulation in dimensions d?2$d?2$. Explicit results for the spin and current two-point functions as well as for the Binder cumulant are presented to next to leading order on a finite lattice. The dynamically generated gap is negative and serves as a coupling-dependent infrared regulator which vanishes in the limit of infinite lattice size. The cancellation of infrared divergences in invariant correlation functions in this limit is nontrivial and is in d=2$d=2$ demonstrated by explicit computation for the above quantities. For the Binder cumulant the thermodynamic limit is finite and is given by 2/(N+1)$2/(N+1)$ in the order considered. Monte Carlo simulations suggest that the remainder is small or zero. The potential implications for “criticality” and “triviality” of the theories in the SO(1,N)$\mathrm{SO}(1,N)$ invariant sector are discussed.     

Duality for Z(N) gauge theories

The duality properties of simple Z(N) gauge theories are discussed. For N ? 4 we find self duality in four dimensions and we give the transition points. For N > 4 these systems are not self dual. Also, the order parameter is discussed. The general Z(N) gauge theory is found to be self dual for all N.     

Equivalent lagrangians in large N field theories

The auxiliary field method is generalized to any O (N)-invariant theory with non-polynomial interactions. In non-supersymmetric theories, two lagrangians with and without an auxiliary field are shown to be equivalent to the leading order of the 1/N expansion. In supersymmetric theories, these two lagrangians are equivalent to all orders of the 1/N expansion. The lagrangian with an auxiliary field is solvable in the 1/N expansion.     

Bound states in N = 8 supergravity and N = 4 supersymmetric Yang-Mills theories

We present a study of possible bound states in N = 8 supergravity. We find evidence for the existence of multiplets of two-body bound states and expect that many-body bound states may exist as well. Our study is based on a calculation of Regge trajectories in the two-body scattering amplitudes of the lagrangian field theory. We also study Regge trajectories in N = 4 Yang-Mills theory and find evidence for a possible spin zero, SU(4) and gauge singlet, massless bound state. If such a state actually exists and supersymmetry is not broken, it may be a member of a supersymmetric multiplet which includes the graviton.     

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.     

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 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.     

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.     

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.     

Dynamical supersymmetry breaking in two-dimensional N = 1 supergravity theories

We investigate a possible dynamical mechanism for spontaneous supersymmetrybreaking in N = 1 supergravity theories in 1 + 1 space-time dimensions. It will be shown that supersymmetry is never broken at the tree level, but it can be broken for a certain class of models by quantum effects due to trace anomalies of the energy-momentum tensor and the supercurrent.     

Two dimensional gases of weight charges and SU(N) fermino theories

We analyze two dimensional gases composed of particles interacting via a Coulomb or Yakawa potential through their “non-Abelian” charges. These charges are taken to be elementary weight or root vectors of SU(N). The grand partition function of these gases is shown to be equivalent to the generating functional of sine-Gordon models with weight vectors and hence to that of SU(N) fermion models. The fermion field creates or annihilates topological solitons which have elementary weight vectors as topological quantum numbers. Then, we discuss the confinement of fermions in the SU(N) Higgs models, where instantons (Z_N vortices) constitute a Yukawa gas of weight charges. We prove that fermions are confirmed by the effects of instantons in the SU(N) Higgs models in contrast with the Abelian Higgs model.     

Inequalities for order and disorder parameters in SU(N) gauge theories

We discuss inequalities of the Mack-Fröhlich type in SU(N) gauge theories and their possible implications for the various phases of those theories.     

The anomaly in the Slavnov identity for N = 1 supersymmetric Yang-Mills theories

We solve the consistency conditions of the BRS symmetry in a general N = 1 sypersymmetric Yang-Mills theory with semi-simple gauge group. As a result we find uniquely the supersymmetric extension of the chiral anomaly. Its coefficient is calculated in one loop and does not, in general, vanish. This corrects our earlier statement on the absence of this anomaly.     

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.     

Effective Field Theory for Fermi Systems in a Large N Expansion

A system of fermions with short-range interactions at finite density is studied using the framework of effective field theory. The effective action formalism for fermions with auxiliary fields leads to a loop expansion in which particle-hole bubbles are resummed to all orders. For spin-independent interactions, the loop expansion is equivalent to a systematic expansion in 1/N, where N is the spin-isospin degeneracy g. Numerical results at next-to-leading order are presented and the connection to the Bose limit of this system is elucidated.     

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.     

Kinematics of periodic structures in SU(N) gauge theories

We study periodicity by use of the holonomy group. For a general field configuration, which spans the whole SU(N) group, we show that periodicity of gauge-invariant quantities implies that either the flux is quantized or that there is no flux at all. Finally we show that in a special gauge, the quantized 't Hooft flux can be expressed as a line integral over one of the components of the vector potential. The other components then have “Higgs-like” zeros, with winding numbers related to the 't Hooft flux.     

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.