From Quarks to Pions

Chiral Symmetry and Confinement

Michael Creutz

• Publisher: WSPC
• Year: 2018
• ISBN:
  9789813229259
• Topic: Physical Sciences
• Subtopic: Nuclear Physics

At a fundamental level, the interaction of quarks with gluon fields lies at the heart of our understanding of the strong nuclear force. Experimentally, however, we only observe physical hadrons such as protons and pions. This book explores the fascinating physics involved in the path between these contrasting pictures of the world. Along the way, the book discusses symmetries, which play a crucial role in understanding the parameters of the theory, and details of the spectrum of physical particles.

This would be the first book to elaborate on the detailed connections between confinement and chiral symmetry, with an emphasis on a unified treatment of the non-perturbative nature of these phenomena. As such, it should be a valuable title on any particle theorist's bookshelf, containing extensive pedagogical material for scientists at the graduate level and above.

Contents:

• QCD
• Perturbation Theory is Not Enough
• Lessons From Two Dimensions
• The Two-Dimensional Ising Model
• Path Integrals and Statistical Mechanics
• Quark Fields and Grassmann Integration
• Lattice Gauge Theory
• Monte Carlo Simulation
• Renormalization and the Continuum Limit
• Classical Gauge Fields and Topology
• Zero Modes and the Dirac Operator
• Chiral Symmetry
• The Chiral Anomaly
• Massive Quarks and the Theta Parameter
• Quark Masses in Two-Flavor QCD
• Lattice Fermions
• Actions Preserving Chiral Symmetry
• Quantum Fluctuations and Topology
• The Standard Model
• Final Remarks

Readership: Advanced students and researchers in high-energy physics.
Quark Confinement;Chiral Symmetry;QCD;Quantum Field Theory Key Features:

• This would be the first book to elaborate the detailed connections between confinement and chiral symmetry

Chapter 1

QCD

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.¹ 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.

This is in striking contrast with the electro-weak sector of the Standard Model. There, the basic electric charge is small and perturbation theory works to incredible accuracy. Nevertheless, the fundamental definition of these interactions remains fuzzy because we know the perturbative series must ultimately diverge.

The treatment of ultraviolet divergences is central to the perturbative approach, requiring a renormalization scheme to obtain physical observables. This is handled rather differently in a non-perturbative scheme such as the lattice. There, the theory is defined as a limiting procedure as the lattice spacing is taken to zero. For QCD, the phenomenon of asymptotic freedom tells us precisely how to take this limit, which occurs as the coupling approaches zero.

Infrared divergences are another annoyance of a perturbative treatment. But QCD, at least with massive quarks, is expected to have a mass gap. Thus, infrared issues should not be relevant. The non-perturbative generation of a mass gap introduces a new scale into the theory. This interplays with the perturbative mass parameters in a non-trivial way, intimately related to issues of gauge field topology.

Perturbation theory often hides crucial qualitative features. Two properties of QCD, confinement and chiral symmetry breaking, stand out as being particularly intractable. Non-perturbative phenomena enter the theory in a fundamental way at both the classical and quantum levels. Over the years, a coherent qualitative picture of the interplay between chiral symmetry, quantum mechanical anomalies, and the lattice has emerged, which forms the theme of this book.

1.1. Why quarks

Although an isolated quark has not been seen, we have many reasons to believe in the reality of quarks as the basis for this next layer of matter. First, quarks provide a rather elegant explanation of certain regularities in low energy hadronic spectroscopy. It was the successes of the Eight-Fold way [2, 3] which originally motivated the quark model. Two “flavors” of low mass quarks lie at the heart of isospin symmetry in nuclear physics. Adding the somewhat heavier “strange” quark gives the celebrated multiplet structure described by representations of the group SU(3).

Second, the large cross sections observed in deeply inelastic leptonhadron scattering suggest non-trivial structure within the proton at distance scales of less than 10⁻¹⁶ centimeters, whereas the overall proton electromagnetic radius is on the order of 10⁻¹³ centimeters. Furthermore, the angular dependencies observed in these experiments indicate that any underlying charged constituent carries a half-integer spin [4].

Yet, a further piece of evidence for compositeness lies in the excitations of the low-lying hadrons. Particles differing in angular momentum fall neatly into place along the famous “Regge trajectories” [5]. Families of states group together as orbital excitations of an underlying extended system. The sustained rising of these trajectories with increasing angular momentum points toward strong long-range forces between the constituents.

Finally, the idea of quarks became incontrovertible with the discovery of heavier quark species beyond the first three. The intricate spectroscopy of the charmonium and upsilon families is admirably explained via potential models for non-relativistic bound states. These systems represent what are sometimes thought of as the “hydrogen atoms” of elementary particle physics [6]. The fine details of their structure have since provided a major testing ground for quantitative predictions from lattice techniques.

1.2. Gluons and confinement

Despite its successes, the quark picture raises a variety of puzzles. For the model to work so well, the constituents must not interact so strongly that they lose their identity. The question arises as to whether it is possible to have objects display point-like behavior in a strongly interacting theory. The phenomenon of asymptotic freedom, discussed in more detail later, turns out to be crucial to realizing this picture.

Perhaps the most peculiar aspect of the theory relates to confinement. These basic constituents of matter do not copiously appear as free particles emerging from high energy collisions. This is in marked contrast to the empirical observation in hadronic physics that anything which can be created will be. Only processes prevented by symmetries do not occur. The difficulty in producing quarks has led to the concept of exact confinement. It may be simpler to have a constituent which can never be produced than an approximate imprisonment relying on an unnaturally small suppression factor. This is particularly true in a theory like the strong interaction, which is devoid of any large dimensionless parameters.

But how can one ascribe any reality to an object which cannot be produced? Is this just some sort of mathematical trick? Remarkably, gauge theories potentially possess a simple physical mechanism for giving constituents infinite energy when in isolation. In this picture a quark-antiquark pair experiences an attractive force which remains non-vanishing even for large separations. This linearly-rising long distance potential energy is central to essentially all models of confinement.

For a qualitative description of the mechanism, consider coupling the quarks to a conserved “gluo-electric” flux. In usual electromagnetism the electric field lines spread and give rise to the inverse square law Coulombic field. If one can somehow eliminate massless fields, then a Coulombic spreading will no longer be a solution to the field equations. A Gauss’ law constraint states that quarks are the sources of electric fields. If, in removing the massless fields we do not destroy this constraint, the electric lines would be unable to diverge and must form into tubes of conserved flux, schematically illustrated in Fig. 1.1. These tubes begin and end on the quarks and their antiparticles. The flux tube is meant to be a real physical object carrying a finite energy per unit length. This is the storage medium for the linearly-rising inter-quark potential. In some sense, the reason we cannot have an isolated quark is the same as the reason that we cannot have a piece of string with only one end. In this picture, a baryon would require a string with three ends. It is the group theory of non-Abelian gauge fields that allows this peculiar state of affairs.

Figure 1.1: A tube of gluonic flux connects quarks and anti-quarks. The strength of this string is 14 tons.

Of course a piece of real string can break into two, but then each piece itself has two ends. In the QCD case, a similar phenomenon occurs when there is sufficient energy in the flux tube to create a quark-antiquark pair from the vacuum. This is qualitatively what happens when a rho meson decays into two pions.

One model for this phenomenon is a type II superconductor containing magnetic monopole impurities. Because of the Meissner effect [7], a super-conductor does not admit magnetic fields. However, if we force a hypothetical magnetic monopole into the system, its lines of magnetic flux must go somewhere. Here the role of the “gluo-electric” flux is played by the magnetic field, which will bore a tube of normal material through the superconductor until it either ends on an anti-monopole or it leaves the boundary of the system [8]. Such flux tubes have been experimentally observed in real superconductors [9].

Another example of this mechanism occurs in the bag model [10]. Here the gluonic fields are unrestricted in the bag-like interior of a hadron, but are forbidden by ad hoc boundary conditions from extending outside. In attempting to extract a single quark from a proton, one would draw out a long skinny bag carrying the gluo-electric flux of the quark back to the remaining constituents.

The above models may be interesting phenomenologically, but they are too arbitrary to be considered as the basis for a fundamental theory. In their search for a more elegant approach, theorists have been drawn to non-Abelian gauge fields [11]. This dynamical system of coupled gluons begins in analogy with electrodynamics, with a set of massless gauge fields interacting with the quarks. Using the freedom of an internal symmetry, the action also includes self-couplings of the gluons. The bare massless fields are all charged with respect to each other. The confinement conjecture is that this input theory of massless charged particles is unstable to a condensation of the vacuum into a state in which only massive excitations can propagate. In such a medium, the gluonic flux around the quarks should form into the flux tubes needed for linear confinement. While this has never been proven analytically, strong evidence from lattice gauge calculations indicates that this is indeed a property of the theory.

The confinement phenomenon makes the theory of the strong interactions qualitatively rather distinct from the theories of the electromagnetic and weak forces. The fundamental fields of the Lagrangian do not manifest themselves in the free particle spectrum. Physical particles are all gauge singlet bound states of the underlying constituents. In particular, an expansion about the free field limit is inherently crippled at the outset. This is perhaps the prime motivation for the lattice approach.

In the quark picture, baryons are bound states of three quarks. Thus, the gauge group should permit singlets to be formed from three objects in the fundamental representation. This motivates the use of SU(3) as the underlying group of the strong interactions. This internal symmetry must not be confused with the broken SU(3) represented in the multiplets of the Eight-Fold way. Ironically, one of the original motivations for quarks has now become an accidental symmetry, arising only because three of the quarks are fairly light. The gauge symmetry of importance to us now is hidden behind the confinement mechanism, which only permits observation of singlet states.

The presentation here assumes, perhaps too naively, that the nuclear interactions can be considered in isolation from the much weaker effects of electromagnetism, weak interactions, and gravitation. This does not preclude the possible application of the techniques to the other interactions. Indeed, unification may be crucial for a consistent theory of the world. At normal laboratory energies, however, it is only for the strong interactions that we are forced to go beyond well-established perturbative methods. Hence, we frame the discussion around quarks and gluons.

¹If you prefer not to confuse this with the 4000 Angstroms typical of color, you could regard this as an acronym for Quark Confining Dynamics.

Chapter 2

Perturbation theory is not enough

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?

The lattice is a mathematical trick. It provides a cutoff removing the ultraviolet infinities so rampant in quantum field theory. On a lattice, it makes no sense to consider momenta with wavelengths shorter than the lattice spacing. As with any regulator, it must be removed via a renormalization procedure. Physics can only be extracted in the continuum limit, where the lattice spacing is taken to zero. As this limit is taken, the various bare parameters of the theory are adjusted while keeping a few physical quantities fixed at their continuum values.

But infinities and the resulting need for renormalization have been with us since the beginnings of relativistic quantum mechanics. The program for electrodynamics has had immense success without recourse to discrete space. Why reject the time-honored perturbative renormalization procedures in favor of a new cutoff scheme?

Perturbation theory has long been known to have shortcomings in quantum field theory. In a classic paper, Dyson [14] showed that electrodynamics could not be analytic in the coupling around vanishing electric charge. If it...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Preface
• Contents
• 1 QCD
• 2 Perturbation theory is not enough
• 3 Lessons from two dimensions
• 4 The two-dimensional Ising model
• 5 Path integrals and statistical mechanics
• 6 Quark fields and Grassmann integration
• 7 Lattice gauge theory
• 8 Monte Carlo simulation
• 9 Renormalization and the continuum limit
• 10 Classical gauge fields and topology
• 11 Zero modes and the Dirac operator
• 12 Chiral symmetry
• 13 The chiral anomaly
• 14 Massive quarks and the Theta parameter
• 15 Quark masses in two-flavor QCD
• 16 Lattice fermions
• 17 Actions preserving chiral symmetry
• 18 Quantum fluctuations and topology
• 19 The Standard Model
• 20 Final remarks
• Bibliography
• Index