Symmetry conditions for coupled scalar field Lagrangian

[Dixanadu]

Homework Statement

Hey guys, so this is a follow up from my previous post.

So I have this Lagrangian for two coupled fields:

$\begin{split}<br /> \mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi_{1})(\partial^{\mu}\phi_{1})+\frac{1}{2}(\partial_{\mu}\phi_{2})(\partial^{\mu}\phi_{2})-\frac{m_{1}^{2}}{2}(\phi_{1}\phi_{1}) -\frac{m_{2}^{2}}{2}(\phi_{2}\phi_{2})-g(\phi_{1}\phi_{2})^{2}<br /> \end{split}$

I was given the transformation $\phi_{1}\rightarrow\phi_{1}+\epsilon\phi_{2}, \phi_{2}\rightarrow\phi_{2}-\epsilon\phi_{1}$, which (if I've done it right) yields the following:

$\begin{split}<br /> \mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi_{1})(\partial^{\mu}\phi_{1})+\frac{1}{2}(\partial_{\mu}\phi_{2})(\partial^{\mu}\phi_{2})-\frac{m_{1}^{2}}{2}(\phi_{1}^{2}+2\epsilon\phi_{1}\phi_{2}) \\<br /> -\frac{m_{2}^{2}}{2}(\phi_{2}^{2}-2\epsilon\phi_{1}\phi_{2})-g(\phi_{1}\phi_{2}-\epsilon\phi_{1}^{2}+\epsilon\phi_{2}^{2})^{2}<br /> \end{split}$Now I have to "Find the most general values for m_1, m_2 and g such that this transformation is a symmetry".

Homework Equations

Apparently there is no coordinate transformation, so I've been told that I need to impose the condition

$\mathcal{L}(\phi',\partial_{\mu}\phi')=\mathcal{L}(\phi,\partial_{\mu}\phi)$

The Attempt at a Solution

So err...am I just meant to equate both Lagrangians? If I do that I end up with the equations:
$\frac{m_{1}^{2}}{2}(\phi_{1}^{2}+2\epsilon\phi_{1}\phi_{2})=\frac{m_{1}^{2}}{2}(\phi_{1}\phi_{1})$
$\frac{m_{2}^{2}}{2}(\phi_{2}^{2}-2\epsilon\phi_{1}\phi_{2})=\frac{m_{2}^{2}}{2}(\phi_{2}\phi_{2})$
$g(\phi_{1}\phi_{2})^{2}=g(\phi_{1}\phi_{2}-\epsilon\phi_{1}^{2}+\epsilon\phi_{2}^{2})^{2}$

And I'm stuck lol...how do I use these to find the most general values for m_1, m_2 and g?

Thank guys

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?