- #1

spaghetti3451

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## Homework Statement

Consider the Euclidean classical action ##S_{cl}[\phi] = \int d^{4}x (\frac{1}{2}(\partial_{\mu} \phi)^{2} + U(\phi))##, with ##U(\phi) = \frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)##.

(a) Show that, in four-dimensional space-time, the mass dimensions of the couplings are ##[\lambda]=0##, ##[a]=1##, and ##[\epsilon]=4##.

(b) Assume that ##a>0##, ##\lambda>0##, and ##\epsilon>0##. Hence, show that the minima of the potential are at ##\phi_{\pm} = \pm a (1 \pm \frac{\epsilon}{2 \lambda a^4}+ ...)##.

(c) Hence, show that ##U(\phi_{-})-U(\phi_{+})=\epsilon [1+ O(\frac{\epsilon}{\lambda a^4})]##. For ##\epsilon << \lambda a^4##, what is the physical interpretation of ##\epsilon##?

(d) Expanding the field ##\phi## about ##\phi_{-}## ##(\phi=\phi_{-}+\varphi)##, and keeping terms up to dimension four, show that the potential ##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}|\phi_{-}|##.

## Homework Equations

## The Attempt at a Solution

(a) The dimension of mass in the kinetic energy term is 0. Therefore, the dimension of mass in the potential energy term must also be 0. Therefore, it must be the case that ##[\lambda]=0##, ##[a]=0##, and ##[\epsilon]=0##.

Can someone provide a hint?