= Higgs field potential
{c}
{title2=$V(\phi)=-\mu_h^2\phi^\dagger\phi+\lambda(\phi^\dagger\phi)^2$}
= Higgs potential
{c}
{synonym}
For $\mu_h^2>0$ and $\lambda>0$, the minimum has $\phi^\dagger\phi=v^2/2$ with $v^2=\mu_h^2/\lambda$. In <unitary gauge>, $\phi=(0,v+H)^T/\sqrt2$ gives $V=\text{constant}+\lambda v^2H^2+\lambda vH^3+\lambda H^4/4$. The canonically normalized <Higgs boson> therefore has $m_H^2=2\lambda v^2$. The angular modes produce the <Goldstone bosons> absorbed in the <Higgs mechanism>.
Set $r=\sqrt{2\phi^\dagger\phi}$. The same <Higgs potential> becomes
$$
V(r)=-\frac{\mu_h^2r^2}{2}+\frac{\lambda r^4}{4},\qquad
V'(r)=r(-\mu_h^2+\lambda r^2),\qquad
V''(v)=2\lambda v^2.
$$
The origin has negative radial curvature, while $r=v$ minimizes the <scalar potential>. Changing the field's angular coordinates at fixed $r$ costs no potential <energy>; changing its radial magnitude does. This geometric distinction explains why the surviving physical <scalar field> has a mass even though the angular directions are flat in the <scalar potential>. Local angular variations still enter the <gauge-covariant kinetic term>.
\Image[/higgs-radial-mode-and-gauge-mass-count.png]
{title=Radial curvature, angular gauge directions and conserved degrees of freedom}
{description=Original two-coordinate slice through the four real <Higgs doublet> components. The full vacuum orbit has three angular directions. Its points represent gauge-related descriptions; the radial <Higgs boson> changes the field magnitude. The plotted <Higgs potential> uses the normalization of this section.}
For a chosen model, not fitted physical constants, take $v=240\,\mathrm{GeV}$ and $\lambda=1/8$ in units $\hbar=c=1$. Then $\mu_h^2=7200\,\mathrm{GeV}^2$ and $m_H=120\,\mathrm{GeV}$. Direct expansion gives a quadratic coefficient $7200\,\mathrm{GeV}^2$, cubic coefficient $30\,\mathrm{GeV}$ and quartic coefficient $1/32$. The quadratic coefficient is $m_H^2/2$, not $m_H^2$. Keeping the same named $\lambda$ in the <Higgs radial mass for a half-normalized quartic potential> would instead give $m_H^2=7200\,\mathrm{GeV}^2$; these are different potentials. The <shifted quartic Higgs potential normalization> agrees with this section when $\kappa=\lambda$.
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