From Quarks to Pions

The following sections are included:

• Preface
• Contents

The interaction of quarks with gluons is, in some ways, the best understood part of the Standard Model. This quantum field theory is known under the somewhat-whimsical name “Quantum Chromodynamics”, or QCD. As will be discussed in this book, we know precisely how to define QCD as the limit of a cutoff theory. On the other hand, the theory has no small parameter; so, perturbation theory, the standard technique of quantum field theory, applies at best only at the highest energies.

The best evidence for confinement in a non-Abelian gauge theory comes by way of Wilson’s [12, 13] formulation on a space-time lattice. This prescription seems a little peculiar at first because the vacuum is not a crystal. Indeed, experimentalists work daily with highly relativistic particles and see no deviations from the continuous symmetries of the Lorentz group. Why, then, have theorists spent so much time describing field theory on the scaffolding of a space-time lattice?

While not the real world, except to string theorists, field theories in one space dimension and one time direction provide a rich environment for studying field theories beyond perturbation. Many models are exactly solvable due to a fascinating equivalence between bosonic and fermionic theories. This “bosonization” process is possible because there is no spin in one space dimension. The difference between bosons and fermions lies in the transformation properties of the corresponding fields under boosts. With massless particles, chiral symmetry amounts to the fact that no boost can exceed the speed of light and thus a particle moving to, say, the “right”, will do so in all frames. In this chapter, we first review the surprising equivalence between a theory of a massless boson and a theory of a massless fermion. Then we consider adding various interactions to solve some models that appear non-trivial in one formulation but are free field theories in the other.

The two-dimensional Ising model [37] is remarkable in that despite having a non-trivial phase transition, the free energy can be calculated exactly. [22, 23]We discuss it here because it provides examples of a variety of tools, many of which are discussed elsewhere in this book. The solution begins with constructing the transfer matrix. This will also be discussed in Chapter 5, and provides the primary pathway between Hamiltonian and Euclidean formulations of quantum systems. The Ising model is converted into a fermion theory, with much paralleling the continuum discussion of Chapter 3. Finally, Fourier transform techniques allow us to diagonalize this operator, directly leading to an explicit expression for the free energy of the model.

Throughout much of the remainder of this book we will focus on the Euclidean path integral formulation of QCD. This approach to quantum mechanics reveals deep connections with classical statistical mechanics. Here we will explore this relationship for the simple case of a non-relativistic particle in a potential. Starting with a partition function representing a path integral on an imaginary time lattice, we will see how a transfer matrix formalism reduces the problem to the diagonalization of an operator in the usual quantum mechanical Hilbert space of square-integrable functions [39]. In the continuum limit of the time lattice, we obtain the canonical Hamiltonian. Except for our use of imaginary time, this treatment is identical to that in Feynman’s early work [40].

Of course, since we are dealing with a theory of quarks, we need additional fields to represent them. There are subtle complications in defining their action on a lattice; we will explore these in some detail in later chapters. Here we go into some of the more mathematical details of what Grassmann integrals mean.

Lattice gauge theory is currently the dominant path to understanding non-perturbative effects in QCD. As formulated by Wilson, the lattice cutoff is quite remarkable in preserving many of the basic ideas of a gauge theory. But just what is a gauge theory anyway? Indeed, there are many ways to think of what is meant by this concept.

As mentioned earlier, Monte Carlo methods have come to dominate work in lattice gauge theory. These are based on the idea that we need not integrate over all fields, since much information is available already in a few “typical configurations.” For bosonic fields these techniques work extremely well, while for fermions the methods remain rather tedious. Over the years, advances in computing power have brought many such calculations for QCD into the realm of possibility. Nevertheless, in situations where the path integral involves complex weightings, the algorithmic issues remain unsolved. In this section we review the basics of the method; this is not meant to be an extensive review, but only a brief introduction.

Asymptotic freedom is a signature feature of the theory of the strong interactions. Interactions between quarks decrease at very short distances. From one point of view this allows perturbative calculations in the high energy limit, and this has become an industry in itself. But the concept is also of extreme importance to lattice gauge theory. Indeed, asymptotic freedom tells us precisely how to take the continuum limit. This chapter reviews this crucial connection to the lattice and the renormalization group.

The renormalization group analysis demonstrates that non-perturbative effects are crucial to understanding the continuum limit of QCD on the lattice. However, the importance of going beyond the perturbation expansion was dramatically exposed from a completely different direction with the discovery of non-trivial classical solutions to the Yang-Mills theory. These are characterized by an explicit essential singularity at vanishing coupling. In this and the next chapter, we review these solutions and some rather interesting consequences for the Dirac operator.

A particularly important and intriguing aspect of these classical field configurations is that they support exact zero modes for the Dirac operator. We will shortly discuss the rigorous connection between the gauge field winding and the zero modes of the Dirac operator. Here, we verify this connection explicitly for the above solution. Thus, we look for a spinor field ψ(x) satisfying

Chirality refers to the parity symmetry under a mirror reflection. The name comes from Lord Kelvin [79] and is most frequently used in chemistry, where many molecules come in chiral pairs. These have the same chemical formula, but are transformed non-trivially into each other under parity. In particle physics, this symmetry is related to how massless particles are special under Lorentz transformations. If a particle moves at the speed of light, its spin along the direction of motion is frame invariant. Basically, one cannot overtake something moving at the speed of light.

The picture of pions as approximate Goldstone bosons is, of course, completely standard. It is also common lore that the anomaly prevents the η′ from being a Goldstone boson and leaves it with a mass on the order of Λ_QCD, even in the massless quark limit. The issue is that the effective potential V considered as a function of the fields in Eq. (12.23) must not be symmetric under an anomalous rotation between η′ and σ, that is,

In the next section, we discuss how this symmetry is lost, as well as its connection to the zero modes of the Dirac operator.

As discussed earlier and illustrated in Fig. 13.2, a quark mass term $m\overline{\psi }\psi \sim m\sigma$ is represented by a “tilting” of an effective potential. This selects one of the multiple minima in the σ, η′ plane as the true vacuum. When the quark masses are small in comparison to the basic scale of QCD, the other minima will persist as extrema, although some of them may become unstable under small fluctuations due to the flatness of flavor-non-singlet directions. Counting the minima sequentially with the true vacuum having n = 0, each is associated with excitations in the pseudo-Goldstone directions that have an effective mass of $m_{\pi }^2\sim m\cos \left(2\pi n/N_f\right)$. Note that when N_f exceeds four, there will be more than one meta-stable state. However, in the usual case of considering two or three quarks as light, only one minimum remains locally stable.

In the previous chapter we concentrated on degenerate quarks. In general, each species introduces another complex mass parameter. Using flavored chiral rotations, we can move the phases of the masses around arbitrarily, leaving only one overall phase, the Theta (Θ) parameter. Once the overall scale has been set, QCD depends on N_f + 1 parameters.

We now have a coherent picture of how the spectrum of pseudo-scalar mesons is connected with chiral symmetry in the continuum theory. The anomaly plays a crucial role in introducing the Θ parameter into the theory and contributing to the η′ mass. Throughout, we have assumed that we have a regulator to define the various composite fields, but have not been specific as to how that regulator is formulated. Early sections indicated the lattice should provide a natural route to a non-perturbative formulation, but we have postponed the details until some of the desired continuum features were elucidated.

The following sections are included:

• The Nielsen-Ninomiya theorem
• Minimal doubling
• Domain wall and overlap fermions
• Staggered fermions

We have seen how zero modes of the Dirac operator are closely tied to the chiral anomaly. And we have seen that for smooth classical fields, configurations that give zero modes for the continuum Dirac operator do indeed exist. However, when getting into more detail with defining a lattice Dirac operator, we found subtle issues about which operator to use. And way back in Chapter 5 we saw that typical fields in path integrals are non-differentiable. This leads to the question of uniqueness for the winding number of a given gauge configuration. Indeed, is something like the topological susceptibility of the vacuum a true physical observable?

So far, we have concentrated on the strong interactions. It is only for this sector of the Standard Model that perturbation theory fails so spectacularly. But the weak and electromagnetic interactions are crucial parts of the full Standard Model. For these interactions, it is also true that the perturbative expansion is not a convergent series. Because the underlying couplings are so small, this does not appear to be of any practical importance.

We have explored how non-perturbative physics is required in order to understand many features of QCD. This is particularly important for the phenomena of confinement and chiral symmetry breaking. Issues such as doubling and anomalies suggest at first that these two features do not mesh well. However, in this book we have argued that these actually combine into an elegant and coherent picture. Of particular interest is how chiral symmetry is broken in three rather different ways. The interplay of these mechanisms is the fascinating physics crucial to the final picture.

The following sections are included:

• Bibliography
• Index