= Solution
For incoming <four-momenta>, the momentum-space <Feynman rules> in the chosen <Minkowski metric> convention are:
* Each internal line contributes the <scalar propagator> $i/(p^2-m^2+i0)$.
* Each quartic <interaction vertex> contributes $-i\lambda_0$ and $(2\pi)^D\delta^D(\sum p)$ for <momentum conservation>.
* Each independent <loop momentum> is integrated with $d^Dk/(2\pi)^D$; after eliminating redundant <Dirac delta functions>, one overall conservation factor remains.
* Each graph is divided by its <symmetry factor> $S$, with external labels held fixed.
There is one connected one-loop <one-particle irreducible> two-point graph: a single quartic <vertex> with two external lines and one <tadpole diagram>. Its <symmetry factor> is $S=2$ from interchange of the two loop half-edges.
There are three connected one-loop <one-particle irreducible> four-point graphs: two quartic <vertices> joined by two parallel internal lines, with external partitions $(12|34)$, $(13|24)$ and $(14|23)$. These are the $s$, $t$ and $u$ <scattering channels>; each <bubble diagram> has <symmetry factor> $S=2$. External-branch insertions are excluded by the <one-particle irreducible> requirement.
\Image[/past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2016-quartic-one-loop-diagrams.png]
{title=One-loop one-particle-irreducible quartic scalar graphs: the two-point tadpole and the three labelled four-point bubble channels, each with symmetry factor S=2}
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