Perturbative interactions in scalar electrodynamics ($D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$)

Expanding the covariant kinetic term of scalar electrodynamics gives a one-photon two-scalar vertex and a two-photon two-scalar seagull vertex. Their tree amplitudes and Ward identities depend on a consistent choice of charge-flow and gauge covariant derivative convention.

At order $e^2$, charged-scalar particle-antiparticle scattering has $s$ annihilation and $t$ photon exchange. For the convention $D=\partial +ieA$ and incoming momenta $p_1,p_2$, outgoing $p_3,p_4$, its amplitude isEqual external masses turn the numerators into $u-t$ and $s-u$. The scalar electrodynamics three-point vertex gives conserved currents, allowing the Coulomb-gauge propagator between conserved currents identity to replace each propagator by the covariant one. The seagull vertex has only two scalar legs and does not give a four-scalar tree at this order.

Expanding the covariant scalar kinetic term gives $ie{A}_{\mu }(\varphi {\partial }^{\mu }{\varphi }^{\ast }-{\varphi }^{\ast }{\partial }^{\mu }\varphi )$. With $D=\partial +ieA$, the vertex for an incoming charged particle of momentum $p$ and an outgoing one of momentum $p^{\prime }$ is $-ie(p+p^{\prime }{)}^{\mu }$. The antiparticle has the opposite charge-flow sign. The quadratic term $e^2{A}^2{\varphi }^{\ast }\varphi$ gives the separate seagull vertex $2i{e}^2{\eta }^{\mu \nu }$. On equal-mass external shells, contracting the three-point momentum with the photon momentum gives zero, the tree scalar quantum electrodynamics Ward identity.

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.