For incoming four-momenta, the momentum-space Feynman rules in the chosen Minkowski metric convention are:
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 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 , and . These are the , and scattering channels; each bubble diagram has symmetry factor . External-branch insertions are excluded by the one-particle irreducible requirement.
Figure 1.
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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