 Solution

Set $C_{\mu \nu }=A_{\mu }{B}_{\nu }-A_{\nu }{B}_{\mu }$, so $\mathcal{G}=G+ieC$. Expanding the minimally coupled Lagrangian density givesThus there is a derivative cubic $AB{B}^{\ast }$ interaction vertex of order $e$, and a quartic $AAB{B}^{\ast }$ 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.

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 $AAB{B}^{\ast }$ 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.