Higgs Potential Algebra
One convention for Higgs potential
\[ V(\phi) = - \mu^2\phi^2+ \lambda \phi^4 \]
Find the minimum by computing
\[ \frac{\partial V}{\partial \phi} = - 2 \mu^2\phi+ 4\lambda \phi^3 = \phi ( 4\lambda \phi^2 - 2 \mu^2) \]
Minimum when \(\frac{\partial V}{\partial \phi} = 0\) => \(\phi = 0\) or
\[ \phi^2 = \frac{\mu^2}{2\lambda} \] or
\[ \phi = \pm \sqrt{\frac{\mu^2}{2\lambda}} \]
Alternative convention with factors of 2
\[ V(\phi) = - \frac{\mu^2}{2}\phi^2+ \frac{\lambda}{4} \phi^4 \] then
\[ \frac{\partial V}{\partial \phi} = - \mu^2\phi+ \lambda \phi^3 = \phi (\lambda \phi^2 - \mu^2) \] and min at \(\phi = 0\) or
\[ \phi = \pm \sqrt{\frac{\mu^2}{\lambda}} = \pm \frac{\mu}{\sqrt{\lambda}} \]
Follow-ups
Links:
Physics [BROKEN LINK: 6J7K8L9M-0N1O-2P3Q-4R5S-6T7U8V9W0X1Y]
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