Complex scalar quartic contact vertex (${\mathcal{L}}_I=-\lambda ({\varphi }^{†}\varphi {)}^2/4,V=-i\lambda$)

For the connected tree scattering amplitude of one particle and one antiparticle of a complex scalar field, the Dyson series inserts $-i\lambda {\varphi }^{†2}{\varphi }^2/4$. There are $2!$ assignments of the particle and antiparticle external attachments to each pair of identical field factors, giving $2!2!=4$ and hence the displayed Feynman vertex. With relativistic normalization of a one-particle state, integrating the vertex position gives $i\mathcal{T}(2\pi {)}^4{\delta }^4(p_1+q_1-p_2-q_2)$ with $\mathcal{T}=-\lambda$. Vacuum diagrams and one-particle self-energy insertions are separate disconnected contributions; normal ordering eliminates the single-vertex vacuum and tadpole contractions.

For $\mathcal{L}=|\partial \varphi {|}^2-m^2|\varphi {|}^2-{\mu }^{ϵ}g|\varphi {|}^4/4$ in $d=4-ϵ$, the directed tadpole diagram has symmetry factor one. Its divergent Feynman contribution to $i{\Gamma }^{(2)}$ is $ig{m}^2/(8{\pi }^2ϵ)$. A mass insertion $-i{\delta }_{m^2}$ therefore gives the displayed minimal subtraction scheme counterterm. Differentiating at fixed bare mass and coupling gives the displayed ${\gamma }_m=\frac{1}{2}\mu d\log m^2/d\mu$. The equivalent two-real-field formulation has vertex coefficients $3g/2$ for a pure species and $g/2$ for two of each, giving total tadpole weight $(3g/2+g/2)/2=g$.