So I was learning about the Faddeev-Popov method and my instructor briefly told to use it on the following action:
$S = \int \frac12 \partial_{\mu}\phi_1 \partial^{\mu}\phi_1+ \frac12 \partial_{\mu}\phi_2 \partial^{\mu}\phi_2 + m \chi (\phi_1^2+\phi_2^2)$$$S = \int d^dx\left( \frac12 \partial_{\mu}\phi_1 \partial^{\mu}\phi_1+ \frac12 \partial_{\mu}\phi_2 \partial^{\mu}\phi_2 + m \chi (\phi_1^2+\phi_2^2)\right),$$
where $\phi_1,\phi_2,\chi$ are scalar fields. (Note: No kinetic term for $\chi$ field). Now I can see this is like a linear sigma model so there is a rotation symmetry $\phi_i \rightarrow R_{ij} \phi_j$. But I am not able to see how to use the method. All examples I have seen are for gauge fields. There is one for a 2D Gaussian integral, but except for that I haven't seen it being applied anywhere else. Any hint will be useful!
