= Scalar particle-antiparticle tree scattering
{title2=$\mathcal M=e^2[(u-t)/(s+i0)-(s-u)/(t+i0)]$}
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 is
$$
\mathcal M=e^2\left[\frac{(p_1-p_2)\cdot(p_3-p_4)}{s+i0}-\frac{(p_1+p_3)\cdot(p_2+p_4)}{t+i0}\right].
$$
Equal 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.
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