What is the most general Feynman diagram?
Srednicki, in his Quantum Field Theory book, says:
The most general diagram consists of a product of several connected diagrams. Let $C_I$ stand for a particular connected diagram, including its symmetry factor. A general diagram $D$ can then be expressed as
$$ D = \frac{1}{S_D} \prod_I (C_I)^{n_I}$$
where $n_I$ is an integer that counts the number of $C_I$ ’s in $D$, and $S_D$ is the additional symmetry factor for $D$ (that is, the part of the symmetry factor
that is not already accounted for by the symmetry factors already included
in each of the connected diagrams). [see eqn (9.12)]
Can anyone please explain this for me? It would be very helpful if you exemplify this.
I have another question too. Can you give me a reference where phi-cubed theory is treated as I can use it as a reference while reading Srednicki.
Best Answer
To explain what Srednicki is doing:
$C_i$ labels the connected diagrams with symmetry factors associated with them (individual diagrams) included, $n_i$ represents the number of diagrams $C_i$ present in the disconnected diagram $D$ and $S_D$ is an extra symmetry factor for the entire disconnected diagram due to interchange of lines between different connected graphs. Note that because we have $n_i$ copies of $C_i$ in our disconnected diagram, not only can we interchange lines in the individual diagrams due to symmetries (which has already been taken into account), but we can also change lines between diagrams of the same type, this contributes to the overall symmetry factor of the diagram. As we have $n_i$ of each $C_i$ we find $S_D = \prod_i n_i!$, so each diagram $D$ can be written as: $$D = \prod_{i}\frac{1}{n_i!}{C_i}^{n_i}$$