For the $\lambda \phi^4$ theory, I checked the connected Feynman diagrams arising in correlation functions for a number of cases; and the following summarises my observation. Let us denote by $E, V$ and $L$ the number of sources (external lines), vertices and loops respectively in a Feynman diagram.
\begin{array}{|c|ccc|ccc|ccc|ccc|} \hline E & & 0 & & & 2 & & & 4 & & & 6 & \\ \hline V & 0 & 1 & \cdots & 1 & 2 & \cdots & 2 & 3 & \cdots & 3 & 4 & \cdots\\ \hline L & 1 & 2 & \cdots & 1 & 2 & \cdots & 1 & 2 & \cdots & 1 & 2 & \cdots\\ \hline \end{array}
This clearly shows that the formula to obtain the number of loops in a connected Feynman diagram with a given number of vertices and external lines is the following.
$$ L = 1 + V - E/2 $$
So, here are my questions.
- How do I prove this formula?
- How do I generalise to any arbitrary theory? For example, what is the corresponding formula for $$ \mathcal L = \frac1{2}(\partial \phi)^2 - \frac1{2}\mu_0\phi^2 -\frac{\lambda}{n!} \phi^n \quad ?$$
MOTIVATION: I am looking into this because if the formula stands true, then there is a simple relation between the number of external lines and the superficial degree of divergence $D$ of a Feynman diagram, namely:
$$ D = 4 - E \,.$$
This is so because we know, by definition, that $D = 4L - 2I$ where we denote the number of internal lines by $I,$ which is now given by: $I=L+V-1.$ Combining all the information together, we get the above result.