Scalar potential ($V(\varphi )$)

A scalar potential is the derivative-free energy density $V(\varphi )$ of a scalar field. Its derivative supplies the force term in the field equation, and its local minima describe candidate vacuum field values.

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 $d>0$.

In vacuum analysis, an exact flat direction of a scalar potential is a nonconstant continuous family of degenerate stationary vacua. The scalar potential stays constant along this family and its gradient vanishes there. An ordinary level set does not suffice: $V(x,y)=x^2+y^2$ is constant on every circle, but its vacuum at the origin has no flat direction. A vanishing eigenvalue of the quadratic Hessian matrix also does not suffice: $V(x)=x^4$ has zero quadratic mass at zero but no family of nearby vacua. Exact flat directions can be lifted by corrections absent from the classical scalar potential.