# Finding symmetry factors

Suppose we have a two-loop graph that looks like two balloons next to each other or stacked on top of each other. What are the symmetry factors of these graphs?

Note that I'm trying to compute a two-point function, so I don't need to consider the symmetry of the external lines. My intuition tells me that these both should have symmetry factor $$4$$. For the first graph we have a factor $$2$$ for both loops. But why can't we swap the loops (i.e. switching the 'balloons' but not the 'strings')? For the second graph we have factor $$2$$ from the upper loop and a factor $$2$$ from the lower loop (switching these internal lines). However, why do we also not consider the permutation of the middle two vertices?

• The symmetry factor is 4 for both graphs. I counts automorphisms of the graph and the latter must keep the external legs individually fixed. That's why we can't swap the loops in the first graph, nor switch the middle vertices in the second graph. Commented Mar 26, 2023 at 17:13
• Hi Geigercounter. Welcome to Phys.SE. For your information, it is considered rude to delete a question after it is answered. Commented Mar 26, 2023 at 18:29
• @Qmechanic I wanted to edit the post but clicked delete instead. Couldn't figure out how to get it back. Thanks for un-deleting it! Commented Mar 26, 2023 at 18:37
• Same issue here and here? Commented Mar 26, 2023 at 18:40
• @Qmechanic no, those questions lacked some information in my opinion. They also were very close to each other in subject, so I opted to delete them. I didn't have time to really write everything out that I had found Commented Mar 26, 2023 at 18:43