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## Main Question or Discussion Point

Consider the potential ##U(\phi) = \frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)##, where ##\phi## is a scalar field and the mass dimensions of the couplings are: ##[\lambda]=0##, ##[a]=1##, and ##[\epsilon]=4##.

Expanding the field ##\phi## about the point ##\phi=\phi_{-}## (##\phi = \phi_{-}+ \varphi##) and keeping terms up to dimension four, we find

##U(\varphi)=\frac{m^{2}}{2}\varphi^{2}-\eta\varphi^{3}+\frac{\lambda}{8}\varphi^{4}##,

where ##m^{2}=\frac{\lambda}{2}(3\phi_{-}^{2}-a^{2})## and ##\eta = \frac{\lambda}{2}\lvert\phi_{-}\lvert##.

How do you derive this potential ##U(\varphi)## in explicit steps? Can you provide just the first two lines? I'll work out the rest for myself.

Expanding the field ##\phi## about the point ##\phi=\phi_{-}## (##\phi = \phi_{-}+ \varphi##) and keeping terms up to dimension four, we find

##U(\varphi)=\frac{m^{2}}{2}\varphi^{2}-\eta\varphi^{3}+\frac{\lambda}{8}\varphi^{4}##,

where ##m^{2}=\frac{\lambda}{2}(3\phi_{-}^{2}-a^{2})## and ##\eta = \frac{\lambda}{2}\lvert\phi_{-}\lvert##.

How do you derive this potential ##U(\varphi)## in explicit steps? Can you provide just the first two lines? I'll work out the rest for myself.