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.