Calculating connected loop-graphs of of two- and three-point functions

Question

I'm trying to find the connected one- and two-loop graphs of $\langle\phi(x_1)\phi(x_2)\rangle$ and $\langle\phi(x_1)\phi(x_2)\phi(x_3)\rangle$ for a specific interaction potential. I know how to look for these graphs, but how can I know that I have found them all? For example when looking at a potential $V(\phi) = \frac{g}{3!}\phi^3$ there are already a lot of connected graphs which have 2 loops. Is it just brute force writing everything out?

Answer 1

If you have to find all connected $L$-loop diagrams with $n$ external legs, short of working with the perturbative definition of the generating functional $$Z[J]~\sim~ \exp\left\{\frac{i}{\hbar} S_{\neq 2}\left[ \frac{\hbar}{i} \frac{\delta}{\delta J}\right] \right\} \exp\left\{- \frac{i}{2\hbar} J_k (S_2^{-1})^{k\ell} J_{\ell} \right\}$$ of all Feynman diagrams, here is 1 possible systematic strategy$^1$ that ensures that you don't miss a diagram:

• Firstly, find all connected trees with $n+2L$ external legs. (First list all connected trees with 3 external legs; Next find inductively for $3\leq k\leq n+2L$ all connected trees with $k$ external legs by adding a vertex to tree-diagrams with $<k$ external legs.)

• Secondly, connect 2 external legs in all possible ways on trees from above list to find all connected 1-loop diagrams with $n+2L-2$ external legs.

• thirdly connect 2 external legs in all possible ways to find all connected 2-loop diagrams with $n+2L-4$ external legs.

• and so forth.

In each step remember to weed out duplicates on your list. $\Box$

   Y  

$\uparrow$ Fig. 1 Example: All 3-valent connected trees with 3 external legs.

   H  

$\uparrow$ Fig. 2 Example: All 3-valent connected trees with 4 external legs.

   _|_|_|_ 

$\uparrow$ Fig. 3 Example: All 3-valent connected trees with 5 external legs.

   _|_|_|_|_          _|_Y_|_  

$\uparrow$ Fig. 4 Example: All 3-valent connected trees with 6 external legs.

   _|_|_|_|_|_         _|_Y_|_|_  

$\uparrow$ Fig. 5 Example: All 3-valent connected trees with 7 external legs.

   _|_|_|_|_|_|_         _|_Y_|_|_|_
||Y||           |_Y_Y|_

$\uparrow$ Fig. 6 Example: All 3-valent connected trees with 8 external legs.

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$^1$Another method uses the skeleton decomposition, cf. e.g this Phys.SE post.