Why Do Single Source Diagrams Matter in Vacuum Expectation Value Calculations?

[Higgsy]

Srednicki page 65 it says "Let us compute the vacuum expectation value of the field $$\phi(x)$$ which is given by $$\langle 0| \phi (x)|0 \rangle = \frac{\delta}{\delta J(x)} Z_{1}(J) |_{J=0}$$ This expression is then the sum of all diagrams that have a single source, with the source removed."

Why only the single source diagrams? Is it because it's for a single field?

 

[eljose79]

The reason why only the single source diagrams are considered in this expression is because it is specifically calculating the vacuum expectation value of the field $$\phi(x)$$. This means that we are only interested in the contribution of the field itself, not any interactions with other fields. By considering only single source diagrams, we are isolating the effect of the field $$\phi(x)$$ and not including any additional interactions. Additionally, this expression is for a single field, so it makes sense to only consider diagrams with a single source. Including diagrams with multiple sources would result in the calculation of a different quantity.