Invariant potential of one SU2 scalar doublet ($V(\varphi )=F({\varphi }^{†}\varphi )$)

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 ${\varphi }^{†}\varphi$. Through total degree four it is $V_0+m^2{\varphi }^{†}\varphi +\lambda ({\varphi }^{†}\varphi {)}^2$. A holomorphic polynomial invariant of positive degree does not exist for one doublet: ${\mathrm{Sym}}^{d}2$ is a nontrivial irreducible representation for every $d>0$.