= Solution
Set $C_{\mu\nu}=A_\mu B_\nu-A_\nu B_\mu$, so $\mathcal G=G+ieC$. Expanding the minimally coupled <Lagrangian density> gives
$$
\mathcal L_{\rm int}=\frac{ie}{2}(C_{\mu\nu}^*G^{\mu\nu}-G_{\mu\nu}^*C^{\mu\nu})-\frac{e^2}{2}C_{\mu\nu}^*C^{\mu\nu}.
$$
Thus there is a derivative cubic $ABB^*$ <interaction vertex> of order $e$, and a quartic $AABB^*$ <interaction vertex> of order $e^2$. There is no pure $B^4$ contact term in this <minimal coupling> theory, nor an Abelian three-photon <interaction vertex>.
\Image[/past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2022-charged-vector-interactions.png]
{title=Cubic and quartic charged-vector vertices, and the two photon-exchange channels for identical charged vector bosons}
The straight arrowed lines in the original schematic denote charged massive <vector bosons>; the wavy lines denote <photons>. For two identical positively charged <vector bosons>, the leading scattering amplitude has the order-$e^2$ $t$-channel photon-exchange <Feynman diagram>, together with its $u$-channel interchange required by identical <bosons>. A quartic $AABB^*$ <interaction vertex> cannot directly attach four external massive <vector bosons>. For a particle-antiparticle incoming pair, the leading channels instead are $t$-channel exchange and $s$-channel photon annihilation. The shown arrows track charge flow; an outgoing charged particle corresponds to the conjugate field at a vertex.
Back to article page
