Yang-Mills theory coupled to an arbitrary scalar representation ($\mathcal{L}=-B(\mathcal{F},\mathcal{F})/(4g^2)+h(D\Phi ,D\Phi )-V$)

For a compact simple gauge group, use $B=-\kappa$ on its algebra and an averaged positive invariant Hermitian form $h$ on any supplied finite-dimensional complex representation. With anti-Hermitian connection and signature $(+---)$, $\mathcal{L}=-B({\mathcal{F}}_{\mu \nu },{\mathcal{F}}^{\mu \nu })/(4g^2)+h(D_{\mu }\Phi ,D^{\mu }\Phi )-V(\Phi )$ is gauge-invariant whenever $V$ is invariant. A real scalar representation instead has a symmetric form and kinetic prefactor $1/2$.

A real potential $V$ for a scalar field is invariant when $V(R(U)\Phi )=V(\Phi )$ for every gauge transformation. In a unitary representation, $m^{2}h(\Phi ,\Phi )+\lambda h(\Phi ,\Phi {)}^2$ is always an example. Particular representations may admit additional invariant interactions.