Seagull vertex

The scalar-QED term $e^2{A}_{\mu }{A}^{\mu }{\varphi }^{\ast }\varphi$ produces a local two-photon two-scalar vertex, conventionally called the seagull vertex.

The two identical photon fields at a scalar electrodynamics seagull give a vertex multiplicity two. Joining them into one closed photon line gives Feynman-diagram symmetry factor $1/2$. Directly expanding $-e^2{A}^2{\varphi }^{†}\varphi$ gives only one photon Wick contraction, independently checking the compensation. In a transverse gauge the remaining tensor trace is $d-1$; a massless photon tadpole is a scaleless integral in dimensional regularization.

With zero quartic coupling, a proper one-loop vertex with two scalar and two labelled photon legs has two boxes, five triangles and two bubbles. A directed scalar chain connects the charged external ends; the internal photon joins its endpoints. Cubic vertices have one photon slot, seagulls two. All nine graphs have Feynman-diagram symmetry factor one with external labels fixed. At zero external momentum a transverse internal photon kills any graph with a cubic endpoint, leaving only the two seagull bubbles.