Feynman rule

Feynman rules translate each diagram into momentum-space factors for propagators, vertices, external states, momentum conservation and loop integration.

With interaction density $\mu {\varphi }^3/3!-\lambda {\varphi }^4/4!$, the Dyson series contains $i{\mathcal{L}}_{\mathrm{int}}$ and hence cubic vertex $i\mu$ and quartic vertex $-i\lambda$. Wick contractions into the labelled slots cancel each factorial. Position integration gives the four-momentum delta function, and equivalent unlabelled contractions leave the Feynman-diagram symmetry factor. The free line carries $i/(p^2-m^2+i0)$ in the mostly-minus convention.

An interaction ${\mathcal{L}}_{\mathrm{int}}=+\lambda {\varphi }^4/4!$ produces a four-leg momentum-space vertex $i\lambda$. The $4!$ assignments of external legs cancel the coefficient factorial. With a negative interaction sign the vertex is $-i\lambda$. Graph Feynman-diagram symmetry factors are a separate combinatorial issue.

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 Feynman parameter combines propagator denominators; for example,After a momentum shift, this often converts a loop integral into a rotationally symmetric one.

For $\mathrm{Re}A>0$ and $\mathrm{Re}v>0$, rescaling the defining Gamma function integral proves this representation, with compatible complex-power branches. It turns denominator powers into exponential factors, making Gaussian integrals possible. Combining two parameters by their sum and ratio gives the usual Feynman parameter identity. Outside its convergence domain a further continuation or pole prescription is needed.