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With non-interactive (free) Action of two scalar fields $\phi_1$ and $\phi_2$ I add an interaction term $\mathcal{L_{int}} = \mu^{2}\phi_1\phi_2 \ \ $ i.e.:

$$S = S^{free}[\phi_1, \phi_2] + S_{int} [\phi_1, \phi_2 ]= \int d^4 x ( \partial_{\mu}\phi_1 \partial^{\mu}\phi_1 +\partial_{\nu}\phi_2 \partial^{\nu}\phi_2 - m^{2}_1\phi^{2}_1 - m^{2}_2\phi^{2}_2 + \mu^2 \phi_1\phi_2)$$

It is clear to me how $\mu=0$ gives me Klein-Gordon Equations for both $\phi_1$ and $\phi_2$, but I am supposed to obtain the aforementioned Action as a non-interacting one, with the masses $m_1$ and $m_2$ only changed. What's the way here, or am I missing something?

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    $\begingroup$ Well try a field transformation that completes the square :) $\endgroup$ Jun 9 at 20:31

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It is obvious as AlmostClueless said that you can rewrite that lagrangian as a new one without interaction, with a simple change of variables in the fields to complete the square:

\begin{align} m_1^2 \phi_1^2 +m_2^2\phi_2^2 - \mu^2 \phi_1 \phi_2 =& \ m_1^2 \phi_1^2 - \mu^2 \phi_1 \phi_2 + a^2\phi_2^2 + (m_2^2-a^2) \phi_2^2 = \\ = & \ (m_1 \phi_1 - a \phi_2) ^2 + (m_2^2-a^2) \phi_2^2 \end{align}

where you chose $a$ such that it fullfills $-2m_1a=-\mu^2$.

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    $\begingroup$ What is the mass of the altered field configuration? I see $m^2_2 -a^2 = {m'}^2$ the mass term for $\phi_2$ but what is it for that superposition term of $m_1 \phi_2 - a\phi_1$? $\endgroup$ Jun 9 at 21:16
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    $\begingroup$ The new field term $m_1\phi_1 - a\phi_2$ will have a modulus $\sqrt{|m_1|^2+|a^2|} $, so the mass term will be that squared, which gives a final Lagrangian term: $(|m_1|^2+|a^2|) \phi_1'^2$ $\endgroup$
    – guiablo
    Jun 9 at 21:26
  • $\begingroup$ But will the kinetic energy term change accordingly? I don't think so, since mixed terms like $\partial^{\mu}{\phi_1}\partial_{\mu}{\phi_2}$ will come along. How do we deal with that? $\endgroup$ Jun 10 at 6:52
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    $\begingroup$ Oh yeah, sorry I might have missed some i (complex unit) somewhere, this normally gets solved with that, the resulting independent fields one being the complex of the other, I would love to help you more, but I have lot of work rn, I'm sure you will figure things out :). Check eq. 18 here: zaguan.unizar.es/record/97969/files/TAZ-TFG-2020-3231.pdf $\endgroup$
    – guiablo
    Jun 11 at 15:21

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