 Higgs field potential

For ${\mu }_h^2>0$ and $\lambda >0$, the minimum has ${\varphi }^{†}\varphi =v^2/2$ with $v^2={\mu }_h^2/\lambda$. In unitary gauge, $\varphi =(0,v+H{)}^T/\sqrt{2}$ gives $V=\text{constant}+\lambda v^2{H}^2+\lambda v{H}^3+\lambda H^4/4$. 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 $r=\sqrt{2{\varphi }^{†}\varphi }$. The same Higgs potential becomesThe 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.

For a chosen model, not fitted physical constants, take $v=240\mathrm{GeV}$ and $\lambda =1/8$ in units $\hslash =c=1$. 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 $\kappa =\lambda$.