A brief summary of d = 3 NAGTs

Abstract

A brief summary of d = 3 NAGTs IntroductionNAGTs 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 O(g 4 3 )) 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. Just as we have already seen at zero temperature, the magnetic mass, found from the PT Schwinger-Dyson equations, cures the otherwise intractable infrared singularities of high-temperature d = 4 gauge theories. We study here only the d = 3 NAGT part of finite-temperature d = 4 NAGTs, In this chapter, we continue to use the notation introduced in Chapter 7. Also in the present chapter, g 3 is the d = 3 NAGT coupling, and g continues to be the d = 4 coupling.