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.