= Solution
The <One-particle-irreducible Feynman diagram>[one-particle-irreducible] <Scalar field>[scalar] <two-point function> has three <Loop diagram>[one-loop] topologies: a scalar-photon exchange loop built from two <Interaction term>[cubic vertices]; a <Tadpole diagram>[photon tadpole] at one <seagull vertex>; and a <Tadpole diagram>[scalar tadpole] at one <Quartic interaction>[quartic] <vertex>. The last is absent when $\lambda=0$. Reducible chains of <self-energy> insertions belong to the <Dyson series>[Dyson expansion] and are not additional <Loop diagram>[one-loop] <self-energy> graphs. The <scalar-QED photon-tadpole symmetry factor> is $1/2$ because its two identical <photon> legs are joined; the oriented complex-scalar <Tadpole diagram>[tadpole] has no such identical-real-line factor.
\Image[/past-exam-of-the-advanced-quantum-field-theory-for-particle-physics-course-of-the-university-of-oxford/2017-scalar-self-energy-and-quark-annihilation.png]
{title=<Loop diagram>[One-loop] <scalar self-energy> topologies and <different-flavour quark annihilation into a massive pair>[tree-level quark-antiquark annihilation]}
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