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?
