In this chapter, we provide a detailed demonstration of how the application of the PT algorithm at the level of the conventional Schwinger–Dyson series leads to a new, modified Schwinger–Dyson equation (SDE) for the gluon propagator endowed with very special truncation properties. In particular, because of the QED-like Ward identities from the fully dressed Green's functions entering into the gluon SDE, the transversality of the gluon self-energy is guaranteed at each level of the dressedloop expansion. This result constitutes one of the main objectives of the PT program, namely, the device of a gauge-invariant truncation scheme for the equations governing the nonperturbative dynamics of non-Abelian Green's functions.

Of course, like any propagator SDE, this equation depends on the full three- and four-point gluon vertices, and these in turn depend on infinitely many other Green's functions. We have suggested, in the last chapter, how to truncate the SDE for the PT propagator by using the gauge technique to approximate the three- and four-point PT Green's functions as functionals only of the PT proper self-energy, while maintaining the exact Ward identities demanded by the PT. This approximation can only be useful in the infrared; the gauge technique PT Green's functions clearly fail to be exact at large momenta (although this failure is quantitative, not qualitative, so it should not change the fundamental findings from PT SDEs, except in the numerics).

For the purposes of this book, it would be too much to study thoroughly all the ramifications of combining the gauge technique, which, in its most general form, is quite complicated, with the pinch technique in the all-order SDEs.

Introduction to nexuses and junctions

So far, it may appear that center vortices are embedded Abelian objects. But center vortices can be extended to non-Abelian objects in several ways. We describe two: the first we call junctions, representing the merging and branching of vortex lines (or sheets, in d = 4) without the necessity of monopole-like objects called nexuses; the second are nexuses themselves, which are modifications of 't Hooft–Polyakov monopoles but with their magnetic flux bundled into tubes that are parts of center vortices. The most interesting property of nexuses is that, along with center vortices, they admit the formation of quantum lumps of nonintegral topological charge. Nexuses do not change the picture of confinement given in this book in any material way, although this is not completely obvious. But they enter in a crucial way into a reinterpretation of Polyakov's discussion of confinement in the d = 3 Georgi–Glashow model, as we indicate at the end of this section.

Note that we continue to use the notation of Chapter 7. All sections are in Euclidean space, except for Section 8.3.2, which is in Minkowski space.

Junctions

Junctions are thick points (d = 3) or lines (d = 4) where a vortex can branch into other vortices. It is easy to draw them in d = 3, where they look like vacuum Feynman graphs. Figure 8.1 shows a simple example with four junctions in SU(3), where three lines meet at each junction (up to N lines can meet at an SU(N) junction, each associated with a distinct flux matrix Qj).

Non-Abelian gauge theories (NAGTs) have dominated the world of experimentally accessible particle physics for more than three decades in the form of the standard model with its SU(2) × U(1) (electroweak theory) and SU(3) (quantum chromodynamics: QCD) components. NAGTs are also the ingredients of grand unified theories and technicolor theories and play critical roles in supersymmetry and string theory. It is no wonder that thousands of papers have been written on them. But many of these papers violate the principle of gauge invariance, resulting in calculations of propagators, vertices, and other off-shell form factors that are valid only in the particular gauge chosen. Until these are combined into a gauge-invariant expression, they have very limited, if any, physical meaning. The reason for such violation of gauge invariance is that standard and widely used Feynman graph techniques generate gauge-dependent Green's functions (proper self-energies, three-point vertices, etc.) for the gauge bosons.

Of course, there is one combination of off-shell Green's functions – the on-shell S-matrix – that is gauge invariant no matter which gauge is used for the propagators and vertices that go into it. Thus, authors who (correctly) insist on calculating only gauge-invariant quantities often restrict themselves to dealing only with the S-matrix. This is fine as long as the question at hand can be answered with perturbation theory, but to calculate gauge invariantly a nonperturbative feature using only S-matrix elements is not easy.

In this chapter, we study several more advanced aspects of the pinch technique (PT), still sticking to one-loop processes, mostly in perturbation theory but with some discussion of one-dressed-loop effects related to gluon mass generation. One of these applications of the pinch technique also has nonperturbative consequences, coming from the invocation of a gauge-field condensate; it allows us to conclude, as we show in this chapter, that the dynamical gauge-boson mass in QCD vanishes like q−2, modulo logarithms, at large momentum. Finally, we introduce one of the main themes of the rest of the book: the pinch technique is realized to all orders by calculating conventional Feynman graphs in the background-field Feynman gauge. The subjects covered include the following:

1. The pinch technique and the operator product expansion (OPE) at one loop, where we see how only gauge-invariant condensates such as 〈Tr GµνGµν〉 arise in PT Green's functions and how this condensate governs the vanishing at large momentum of dynamically generated gauge-boson mass in QCD.

2. Uses of the pinch technique in studying gauge-boson mass generation, both dynamic in QCD (no symmetry breaking, whether by Higgs–Kibble fields or other mechanisms) and with spontaneous symmetry breaking.

3. The background field method and the effective action.

4. The one-loop equivalence between the pinch technique and the background field method in the Feynman gauge.

The pinch technique and the operator product expansion: Running mass and condensates

As mentioned more than once, the pinch technique is essential to unveiling the nonperturbative effects that are vital in understanding confinement in QCD.

Introduction

NAGTs in three dimensions have valuable applications in their own right because they are the high-temperature limit of d = 4 NAGTs with infrared slavery (see Chapter 11 for more details). They also lead to important insights into d = 4 NAGTs at zero T, and in many ways, d = 3 QCD is more interesting to study to gain this insight than the far more often-invoked two-dimensional theories. It is not a free-field theory (as is a d = 2 pure-gauge NAGT), and it has many features strongly analogous to those of d = 4 NAGTs that are best understood by applying the pinch technique. In particular, although a d = 3 NAGT cannot be asymptotically free (because it is superrenormalizable, not possessing the usual renormalization group), it is still very much infrared unstable, with even worse singularities than those in d = 4. Although this d = 3 infrared slavery had been strongly suspected before the pinch technique on the basis of conventional Feynman graph calculations, it took the pinch techniqe to settle the issue and demonstrate the existence of infrared slavery in d = 3 NAGTs.

Because a d = 3 NAGT is the critical nonperturbative part of the high-temperature behavior of its d = 4 counterpart, infrared slavery prevents the use of perturbation theory (beyond) in understanding all the phenomena of high temperature, including generation of a so-called magnetic mass, which vanishes identically to all orders of perturbation theory.

In this chapter, we give a general overview of how the pinch technique (PT) is modified in the case of a theory with spontaneous (tree level) symmetry breaking (Higgs mechanism), using the electroweak sector of the standard model as the reference theory.

The application of the pinch technique in the electroweak sector brings about significant conceptual and practical advantages. First, from the purely theoretical point of view, it is important to know that the PT construction is sufficiently general to encompass theories other than massless Yang–Mills. Though the required technical manipulations in the electroweak sector turn out to be fairly cumbersome, the basic underlying principles are practically the same; what increases the complexity is not the principle itself but rather the proliferation of fields and vertices involved. In addition, the PT algorithm exposes systematically the vast number of cancellations that take place when the (nonrenormalizable) Green's functions of the unitary gauge are put together to form physical amplitudes. In fact, one may start directly from the unitary gauge and derive the same PT Green's function constructed in the context of the (renormalizable) Rξ gauges. The application of the pinch technique provides a deeper understanding of the connection between the unitary gauge and the optical theorem and analyticity. Moreover, in the context of a theory with symmetry breaking, one gains new, important insights on the connection between the pinch technique and the background field method (BFM).

In this chapter, we present in detail the pinch technique (PT) construction at one loop for a QCD-like theory, where there is no tree-level symmetry breaking (no Higgs mechanism). The analysis applies to any gauge group (SU(N), exceptional groups, etc.); however, for concreteness, we will adopt the QCD terminology of quarks and gluons.

This introductory chapter and Chapter 2 go into both conventional technology and the pinch technique only at the one-loop level. Here, the reader will find an almost self-contained guide to the one-loop pinch technique with many calculational details plus some hints at the nonperturbative ideas used in later chapters (where nonperturbative effects will be studied by dressing the loops, i.e., using a skeleton expansion).

A brief history

Non-Abelian gauge theories (NAGTs) had been around for a long time when the pinch technique came into play. Their first use was in defining the oneloop PT gauge-boson propagator as a construct taken from some gauge-invariant object by combining parts of conventional Feynman graphs while preserving gauge invariance and other physical properties. The term pinch technique was introduced later, in a paper that extended the one-loop pinch technique to the three-gluon vertex. The name comes from a characteristic feature of the pinch technique, in which the needed parts of some Feynman graphs look as though a particular propagator line had been pinched out of existence. In all these early papers, only one-loop phenomena were studied, including a one-dressed-loop Schwinger–Dyson equation for the PT propagator.

In spite of the simple appearance of its Lagrangian, QCD is an extremely complicated and rich theory. As a theory of strong interaction, it is subject to various constraints posed by observations. Most important among them are scaling and confinement, which appear to have conflicting implications for the nature and characteristics of QCD. It also has many observational implications, such as the logarithmic violations of scaling, the spectrum of charmonium, and the two-jet and three-jet structure in the electron–positron annihilations. The successes in satisfying these constraints and in explaining and predicting observational events in the early to mid 1970s immediately after the proposal of QCD had provided QCD with much needed experimental justifications and consolidated its status as an acceptable theory in the particle physics community.

As a non-abelian gauge theory, QCD is also subject to some conceptual constraints posed by its symmetry structure, such as the U(1) anomaly (see Sections 8.2 and 8.5 below). The solution of the U(1) anomaly relied on the further explorations of the theoretical structure of QCD, which revealed its richness and great potential for theorizing the complexity of the sub-hadronic world, such as the existence of instantons and the related theta vacuum state. The fruitful explorations in this direction had shown that QCD, in Lakatos's terminology, was a progressive research program, which keeps opening new territory for explorations, and thus was conceived by the majority of the community as worth pursuing.