# Feynman rule for 4-point fermion vertex

Suppose we have a Lagrangian with a fermion of mass $$m$$ and a 4-point interaction of the form

$$\mathcal L=\bar{\psi}(i\partial\!\!\!/-m)\psi-g\bar{\psi}\partial\!\!\!/\psi\bar{\psi}\partial\!\!\!/\psi$$

where $$g$$ has mass dimension $$-4$$. I'd like to determine the Feynman rule for this vertex, and the usual procedure is differentiating the action

$$i\frac{\delta^4 S}{\delta\psi\delta\bar\psi\delta\psi\delta\bar\psi}$$

and then transforming into momentum space, which gives for a vertex like

a rule of

$$ig(p\!\!\!/_{1\,ac}p\!\!\!/_{2\,bd}+p\!\!\!/_{1\,bd}p\!\!\!/_{2\,ac}-p\!\!\!/_{1\,ad}p\!\!\!/_{2\,bc}-p\!\!\!/_{1\,bc}p\!\!\!/_{2\,ad}).$$

Great, this follows the Pauli principle and all that jazz. But the problem is, if I had differentiated the action in some other order, maybe

$$i\frac{\delta^4 S}{\delta\psi\delta\bar\psi\delta\bar\psi\delta\psi},\,\mathrm{or}\,i\frac{\delta^4 S}{\delta\bar\psi\delta\psi\delta\bar\psi\delta\psi},$$

the Feynman rule would have a different sign.

If I had different interactions in the Lagrangian (maybe I added that interaction to QED), which order of differentiation should I use to make things consistent?