Cubic and quartic Higgs couplings to electroweak gauge bosons ($HWW$, $HHWW$, $HZZ$, $HHZZ$)

The gauge-covariant kinetic term of the Higgs doublet gives $M_W^2(1+H/v{)}^2{W}_{\mu }^{+}{W}^{-\mu }+M_Z^2(1+H/v{)}^2{Z}_{\mu }{Z}^{\mu }/2$. Differentiation with respect to the external fields yields Feynman rules $2i{M}_V^2{g}_{\mu \nu }/v$ for one Higgs boson and two matching massive electroweak gauge bosons, and $2i{M}_V^2{g}_{\mu \nu }/v^2$ for two Higgs bosons and the same pair. Identical neutral fields supply combinatorial factors. The neutral radial Higgs boson has no corresponding elementary coupling to photons at tree level.