Gauge boson self-interaction

Gauge boson self-interactions arise from the quadratic-field term in a non-Abelian gauge field strength. Squaring that field strength in the Lagrangian yields cubic and quartic pure-gauge interaction vertices. Higher-point scattering can occur by connecting these vertices, even without a corresponding fundamental higher-valence vertex.

The non-Abelian electroweak gauge kinetic term contains a $W^{+}{W}^{-}{W}^3{W}^3$ interaction. Substituting $W^3=\sin {\theta }_{W}A+\cos {\theta }_{W}Z$ gives a mixed four-gauge-boson vertex with coupling magnitude $g^2\sin {\theta }_W\cos {\theta }_W=eg\cos {\theta }_W$. It can emit both a photon and a Z boson at one vertex. A one-loop flavour-changing quark graph with two quark-W vertices and this quartic vertex consequently has three vertices despite having four external particles.