# Feynman rules for vector bosons

I'm learning QFT by reading Lewis Ryder's book, so my question in short is: how he arrives at Eq 7.57 and Eq 7.58? If you don't have the book, the question is: why are there Minkowski metric terms in the Feynman rules for a gauge field coupling to itself? If the answer is complicated, simply tell me where I can find it please, I have Peskin's and Weinberg's book, too.

For me it seems impossible to proceed after Chapter 7 with this book, it doesn't have a systematic way of introducing Feynman diagrams, and the Feynman rules for vector bosons seem to be pulled out of a hat, is this a suitable book for a man learning QFT for the first time on his own?

Thank you.

## Answers and Replies

tiny-tim
Science Advisor
Homework Helper
in Weinberg?

Hi wangjiaji! I don't have Ryder's book, but I do have Weinberg's Quantum Theory of Fields Volume I …

which page number, and which equations, is it in Weinberg? To derive the precise form of the Feynman rule, it would be best for you to work it out starting from the LSZ reduction formula for the scattering processes $gg\rightarrow g$ and $gg\rightarrow gg$. Essentially, the Minkowski metric arises from needing to expand the 4-vector dot products between the vector fields in the interacting Lagrangian before proceeding to contract fields:
$$A_\mu A^\mu=g_{\mu\nu}A^\mu A^\nu$$​

Physically speaking, the role of the Minkowski metric in the Feynman rules for the 3- and 4- vector boson vertices is to select, depending on the polarization of the incoming and outgoing vector bosons, the appropriate Clebsch-Gordon coefficient arising from the coupling of two spin-1 systems.

Avodyne
Science Advisor
Try Srednicki's book. A draft copy is available free at his web page.