Feynman-diagram symmetry factor

The symmetry factor of a Feynman diagram divides out permutations of indistinguishable internal lines and vertices that leave the labeled diagram unchanged. It is the order of the corresponding automorphism group after external labels are fixed.

A closed Dirac loop gives a minus sign and a spinor trace. Replacing it by two distinguishable real scalar species removes both, with no extra interchange factor. One identical real scalar species instead has Feynman-diagram symmetry factor two. A cubic interaction $g\varphi {\chi }^2/2$ has vertex $ig$, whereas $g\varphi {\chi }^2$ has vertex $2ig$; those choices give different numerical coefficients even though their drawn loop topology is the same.

For interaction $-\lambda {\varphi }^2({\varphi }^{\ast }{)}^2/4$, the vertex factor is $-i\lambda$. A two-line bubble carrying both internal arrows in the same direction has symmetry factor two. Oppositely directed lines cannot be interchanged as an oriented graph, so the two mixed external channels have factor one. A directed tadpole inserted on a labelled external leg also has factor one. Thus the connected four-point function at second order includes three bubble attachments and four external-leg insertions; retaining only one-particle-irreducible bubbles would compute a different object. These factors do not depend on whether the mixed free kernel is $D_F$ or a consistently rescaled multiple of it.