= Complex scalar quartic contact vertex
{title2=$\mathcal L_I=-\lambda(\phi^\dagger\phi)^2/4,\quad V=-i\lambda$}
For the connected tree <scattering amplitude> of one particle and one <antiparticle> of a <complex scalar field>, the <Dyson series> inserts $-i\lambda\phi^{\dagger 2}\phi^2/4$. 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.
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