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.

For two real scalar fields with the displayed field interaction term, Fourier factors $e^{-ip\cdot x}$ make each derivative contribute $-i{p}_{\mu }$. The two identical differentiated field slots cancel $1/2$, giving the Feynman vertex $-igp\cdot q$ when $p,q$ are the incoming four-momenta of the two $\varphi$ legs. With incoming $\Phi$ four-momentum $k=-p-q$, this is also $-ig(k^2-p^2-q^2)/2$. This identity is valid off shell; substituting squared masses for internal squared four-momenta is generally invalid.

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 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.

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

Differentiate $1/(AB)={\int }_0^1[xA+(1-x)B{]}^{-2}dx$ with respect to $B$ to obtain the displayed identity. The squared denominator supplies the factor $2(1-x)$. Changing the parameter to $1-x$ also changes which denominator carries each weight; its use must be consistent with the shifted mass parameter.

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.