= Invariant potential of one SU2 scalar doublet
{title2=$V(\phi)=F(\phi^\dagger\phi)$}
The <special unitary group> $SU(2)$ acts transitively on every sphere in its defining $\mathbb C^2$: for $|u|^2+|v|^2=1$, the <matrix> with columns $(u,v)^T$ and $(-\bar v,\bar u)^T$ is in $SU(2)$. Thus any invariant real <polynomial> in a single commuting doublet and its <complex conjugate> depends only on $\phi^\dagger\phi$. Through total degree four it is $V_0+m^2\phi^\dagger\phi+\lambda(\phi^\dagger\phi)^2$. A holomorphic <polynomial invariant> of positive degree does not exist for one doublet: $\operatorname{Sym}^d\mathbf2$ is a nontrivial <irreducible representation> for every $d>0$.
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