Feynman rules

We gave in Chapter 6 the Lagrange function for the fermions and the gauge bosons. In the previous chapter we defined the physical bosons with definite masses. It is now a straightforward exercise to rewrite the Lagrange density in terms of the physical bosons and read off the Feynman rules. For the rules, it is necessary to introduce quantized fields in order to keep track of the combinatorics and other factors, especially for diagrams with closed loops. The canonical quantization method in terms of Wick's theorem does not work for non-Abelian gauge theories because there are ambiguities that arise from gauge transformations. The appropriate discussion at this point is the quantization in the path-integral formalism. This will be a long digression and will delay us from arriving at physical results. We adopt a compromise. We consider the fermionic part of the Lagrange function in terms of the physical fields and read off the relevant vertices. The interested reader can compare this method with the procedure used in textbooks of quantum electrodynamics. In this way we obtain an extensive set of Feynman rules for vertices and propagators, in terms of which we discuss many physical processes.

Later on, we repeat this procedure for other parts of the Lagrangian, which include Higgses and gauge bosons. The rules that we obtain suffice when we calculate tree diagrams to any order. Difficulties occur when loop diagrams are computed, beginning with one-loop diagrams. The difficulties are solved by introducing additional diagrams with scalar particles: the Faddeev–Popov ghosts.

We saw in the previous chapter that the neutral gauge fields mix among themselves.