For massless <phi-fourth theory> in four dimensions, the momentum-space rules are
$$
\text{propagator: }\frac{i}{p^2+i\epsilon},
\qquad
\text{quartic vertex: }-i\lambda.
$$
Writing
$$
\mathcal L_{\mathrm{ct}}
=\frac12\delta_Z(\partial\phi)^2
-\frac12\delta m^2\phi^2
-\frac{\delta\lambda}{4!}\phi^4,
$$
the two-point counterterm insertion is $i(\delta_Zp^2-\delta m^2)$ and the four-point counterterm is $-i\delta\lambda$.

The renormalized <one-particle-irreducible correlation function> $\Gamma_4$ through order $\lambda^2$ contains the tree quartic vertex, the quartic counterterm, and three one-loop bubble diagrams. The bubbles are the $s$, $t$, and $u$ channels and each has symmetry factor $1/2$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 2 / a / Solution

{title2=$\Gamma^{(n)}$}

An $n$-point one-particle-irreducible correlation function is obtained by differentiating the <quantum effective action> $n$ times. Its diagrammatic expansion contains the amputated one-particle-irreducible diagrams with $n$ external legs.

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