Introduction

In this chapter we study the mesonic spectrum of various QCD2 theories. The main idea is to use the current algebra of the underlying ungauged theories. In addition we combine the bosonization techniques developed in Chapter 6 with that of a large N expansion of Chapter 7 and a light-front quantization as in Chapter 10. We will focus our attention on the massive mesonic spectrum of conformal field theories coupled to non-abelian gauge fields. In particular massless multi-flavor fundamental quarks and adjoint quarks that will be shown to correspond to the particular case of Nf = Nc.

First a universality theorem, that states that the massive mesonic spectrum does not depend on the representation of the matter field but rather only on its ALA level, will be derived, following Kutasov and Schwimmer.

We then present a detailed determination of the massive mesonic spectrum using a 't Hooft-like equation for the wave functions of “currentballs” states. We will discuss in particular the special cases of Nf = 1, Nf = Nc and Nf » Nc. The last section is devoted to the spectrum of states built by the action of a single current creation operator on the adjoint vacuum. In both cases it will be shown that the bosonization approach leads to the introduction of current quanta as the basic degrees of freedom.

Conformal symmetry was shown to be extremely powerful in two-dimensional field theories and obviously also in string theories. This is due to the fact that the conformal algebra is infinite dimensional and hence supplies a set of infinitely many conserved charges. In four dimensions the conformal algebra is finite and therefore less powerful. The purpose of this chapter is to explore the use in 4d conformal field theories of notions and tools that we encountered in two-dimensional CFT, such as primary fields, conformal operator expansion, conformal anomalies and Ward identities.

Free massless theories are obviously scale and conformal invariant. However, field theories that describe the elementary particles of nature and their interactions, are interacting field theories. The question is thus, to what extent can one apply the techniques of CFT to those theories and in particular QCD in four dimensions? QCD with massless quarks is a prototype model of theories which are classically conformal invariant. In fact even for theories with masses and other dimension-full parameters, in certain cases these can be neglected in the high energy and high momentum transfer regime of the theory. However, even in the massless case and with only dimensionless couplings, it is easy to realize that the quantum picture lacks conformal invariance. This follows from the fact that one has to introduce dimension-full parameters such as UV cutoff, which turns into a scale where the coupling is defined, after renormalization.

In Chapter 5 we saw that solutions of the classical equations of motion, which are characterized by a topological number, play an important role in two-dimensional QFT. Derick's theorem (5.36) forbids scalar field soliton solutions in higher than two-dimensional space-time. However, for gauge fields one can bypass the theorem, and indeed, as we have seen in Chapter 21, there are solitons in the form of magnetic monopoles in four-dimensional gauge theories. The topic of this chapter will be solutions of the Yang–Mills theory defined on a Euclidean space-time which have finite action and are topological in their nature, the instantons. We will start with a description of the basic properties of one instanton solution including the topological charge that characterizes it. We then describe the construction of multi-instanton solutions and the moduli space of instantons including its dimension, complex nature, singularities and symmetries. When Wick rotated to Minkowski space-time the instanton describes a tunneling process between different vacua. We will elaborate on this phenomenon in the context of the four-dimensional YM theory. Various properties of QCD and hadron physics were thought to be related to instantons. In certain cases like confinement, the relation to instantons is still a mystery. One case where the role of instantons is clear is the U(1A) problem. This will be described in the last section of this chapter.

Integrability was discussed in Chapter 5 in the context of two-dimensional models. In particular solutions of spin chain models based on the Bethe ansatz approach were described in some detail. Integrable models are characterized by having the same number of conserved charges as the number of physical degrees of freedom. Furthermore, the scattering processes of those models always involve conservation of the number of particles.

A natural question at this point is whether integrability is a property of only two-dimensional models or whether one can also identify systems in four dimensions that admit integrability. Four-dimensional gauge theories have generically infinite numbers of degrees of freedom and their interactions do not conserve the number of particles. Thus four-dimensional gauge theories like the YM theory are not integrable theories. However, it turns out, as will be shown in this chapter, that various sectors of certain four-dimensional gauge theories, which are derived upon imposing certain limits, do admit integrability.

The two-dimensional integrable models discussed in Chapter 5 were nonconformal ones and were characterized by a scale and hence also with particles and an S-matrix. On the other hand the integrable sectors of four-dimensional gauge theories that we are about to describe are conformal invariant. The main idea is that these special conformal invariant sectors can be mapped into two-dimensional spin chains that were described in Section 5.14.

Introduction

Solitons play an important role in non-perturbative two-dimensional fields as we have seen in the first part of this book. They are intimately related to non-trivial topology, they are an essential ingredient in integrable models, and they enable the phenomenon of fermion boson duality-bosonization. When passing to four-dimensional field theories the topology may be even richer and thus we would anticipate having topological solitons as static solutions also in four-dimensional space-time. As we have seen in Section 5.3, Derrik's theorem does not permit the existence of solitons of scalar field theory in space dimensions higher than one, however, they are not prohibited in theories that include higher spin fields, in particular in theories of scalar fields coupled to non-abelian gauge fields. Indeed as we will see in this section certain theories of this type that admit spontaneous symmetry breaking, admit soliton solutions. These configurations carry a conserved topological charge which guarantees their stability against decay to the vacuum. As it will turn out this charge is in fact a magnetic charge and hence these solitons are magnetic monopoles, or in the more general case dyons with both magnetic and electric charge. The construction of dyons from static solutions will be the analog process of building up breathers from two-dimensional solitons.

In the first part of the book we have developed several non-perturbative tools for analyzing two-dimensional field theories. These include methods associated with conformal invariance, with affine Lie algebras and in particular the WZW model, techniques of integrable massive theories including solitons, S-matrix and the Yang–Baxter equation, sets of infinitely many conserved charges, the thermal Bethe ansatz, methods of bosonization and the large N limit approximation.

In this second part of the book the main idea is to implement those methods in the context of two-dimensional gauge dynamics aiming at extraction of the mesonic and baryonic spectra of two-dimensional QCD, decoding the confining behavior, versus a screening one, and analyzing models with other quark representations and models of generalized QCD.

The most important tool that will be used for this purpose will be bosonization. It will enable us to easily solve the Schwinger model, derive the baryonic spectrum of QCD2 using a strong coupling limit, determine the screening nature of massless QCD models and compute the string tension for the massive ones. Using the large N approximation we will extract the mesonic spectrum of QCD.

In two respects this part is an intermediate stage on the way to four-dimensional gauge dynamics. Firstly, as was just mentioned, we will gain experience from applying non-perturbative methods on the simpler two-dimensional models, which will serve us when using them in the context of “real” physical systems.