Question

How to relate the number of bosons (fermions) with the number of vertices and external (internal) bosons (fermion) lines?

Answer 1

I am following the QCD book: "QCD: Renormalisation for the Practitioner" by P. Pascual, R. Tarrach. In chapter 3, page 64, equation III.13 they relate the following quantities:

Let us consider a diagram with nini vertices of the type associated with LiLi, BE(BI)BE(BI) being the number of external (internal) Boson lines, FE(FI)FE(FI) the number of external (internal) fermion lines and V=∑iniV=∑ini the total number of vertices. The quantities can be related by

2BI+BE=∑inibi,2FI+FE=>∑inifi2BI+BE=∑inibi,2FI+FE=>∑inifi

I would like to proof these two relations. I tried to look look at the Feynman diagrams I can produce from the QCD Lagrangian, but have not figured out on how to exactly derive these the above relations.

Could anybody give me an idea or a reference on how to proceed

Answer 2

Let us consider a diagram with vertices of the type associated with ,
being the number of external (internal) Boson lines, the number of external (internal) fermion lines and the total number of vertices. The quantities can be related by
I would like to proof these two relations. I tried to look look at the Feynman diagrams I can produce from the QCD Lagrangian, but have not figured out on how to exactly derive these the above relations.