QCD and QED

We allow the quarks to have a mass parameter m which should be put to zero when working with massless quarks. 

In the above the arrow indicates the flow of fermion number and p is the 4-momentum in that direction. (Note j,l are quark colour labels, 6, c gluon and ghost colour labels.) 

Note that in the above axial gauges the propagator is orthogonal to n and is orthogonal to when k = 0. 

Note that the ghosts are scalar fields, but a factor (—1) must be included for each closed loop, as is the case for fermions. Note also that the sign of A4, if it is important, has to be determined by comparing the order of the fermion operators in the diagram with their order in the expression for the 5-operator. 

In this section Greek indices are Lorentz indices i, j label generations. 

---

Further details, in particular the rules for calculating Feynman diagrams in QED, QCD and in weak interactions, are given in Appendix 2. For a complete treatment the reader is referred to Bjorken and Drell (1964, 1965) and Itzykson and Zuber (1980). 

We give here, without derivation, the rules for calculating (up to a sign) what is known as the Feynman amplitude M in QED, QCD and the SM. We illustrate with a few topical examples. A detailed treatment can be found in Bjorken and Drell (1965) and in Cutler and Sivers (1978). 

The program has been tested on several QED and QCD motivated examples such as electron g — 2 in QED, gauge independence of the electron wave function renormalization in QED, a relation between the pole and the MS mass in QCD, exponentiation of the infra-red asymptotic of the heavy fermion propagator. Many details concerning these checks and other technical aspects of our calculations can be found in Ref. [11] which the interested reader should consult. 

The flavor and the ordinary orbitals are inert and it is the color orbital which carries the freeon dynamics in the same ways as the freeon (spatial) orbitals carry the dynamics for electronic systems. It should be noted that the two basic field theories are quantum electrodynamics (QED) and quantum chromodynamics (QCD).The color force is also an exchange force in which the several colors are exchanged. 

Our principal aim in this chapter is to review briefly the basic ideas of field theory, which we shall illustrate with examples from quantum electrodynamics (QED) and the theory of strong interactions, quantum chromodynamics (QCD). Of necessity, we must assume that the reader has some knowledge of field theory and is conversant with the idea of Feynman diagrams and with the Dirac equation. We shall then give a resume of the theory and phenomenology of the weak interactions as they stood at the time of the inception of the new ideas about quarks and gluons in the early 1970s. The chapter ends with some technical results which will be very useful in later chapters. 

Because the running coupling increases with increasing distance, it is believed that quarks are confined, and that if QCD could be evaluated non-perturbatively, it would predict confinement. So, unlike the case of QED, in QCD the potential itself must be calculated non-perturbatively at distances exceeding some critical value. We know how to make non-perturbative calculations only in various approximation schemes, perhaps the best justified of which is to put QCD on a lattice. It is important to stress that none of the approximation schemes used so far is mathe-... 

---