Power of the Yang-Mills invariant ($\mathcal{L}=s^p,s=\kappa (F_{\mu \nu },F^{\mu \nu })$)

For a semisimple gauge algebra and positive integer $p$, this gauge-invariant density yields $D_{\mu }(s^{p-1}{F}^{\mu \nu })=0$. The curvature variation $\delta F_{\mu \nu }=D_{\mu }\delta A_{\nu }-D_{\nu }\delta A_{\mu }$ gives $\delta s=4\kappa (D_{\mu }\delta A_{\nu },F^{\mu \nu })$, and gauge-covariant integration by parts proves the equation. More generally a density $f(s)$ gives $D_{\mu }(f^{\prime }(s)F^{\mu \nu })=0$. The product form retains the equation at zeros of $s$; dividing by $s$ can discard solutions.