What do we take the functional determinant of in computing the effective action in the Background field method?

Question

I have some schematic notes on computing the effective action and I would like someone to help me fill the gaps.

We start with
\begin{equation*}
\int{}\mathcal{D}\phi\,e^{-iS[\phi]}
\end{equation*}
employing the background field method we write
\begin{equation*}
\phi=\phi_0+\Delta\phi
\end{equation*}
so we have
\begin{equation*}
\int{}\mathcal{D}(\Delta\phi)\,e^{-iS[\phi_0+\Delta\phi]}
\end{equation*}
Taylor expanding around $\phi_0$ 

$$S[\phi_0+\Delta\phi]=S[\phi_0]+\int{}d^4x_1\,\frac{\delta{}S}{\delta\phi(x_1)}\Delta\phi(x_1)$$
$$+\frac{1}{2}\int{}d^4x_1d^4x_2\frac{\delta^2S}{\delta\phi(x_1)\delta\phi(x_2)}\Delta\phi(x_1)\Delta\phi(x_2)+$$
$$\frac{1}{3!}\int{}d^4x_1d^4x_2d^4x_3\frac{\delta^3S}{\delta\phi(x_1)\delta\phi(x_2)\delta\phi(x_3)}\Delta\phi(x_1)\Delta\phi(x_2)\Delta\phi(x_3)+\ldots$$
since $\phi_0$ satisfies the equations of motion the linear term in $\Delta\phi$ vanishes. Then we have

$$e^{-iS[\phi_0]}\int{}\mathcal{D}(\Delta\phi)e^{-i\frac{1}{2}\int{}d^4x_1d^4x_2\frac{\delta^2S}{\delta\phi(x_1)\delta\phi(x_2)}\Delta\phi(x_1)\Delta\phi(x_2)+\ldots}$$

from here on my notes neglect terms cubic,quartic... in $\Delta\phi$. Can anybody tell me why?.

Also, after this it is written
$$e^{-iS[\phi_0]}det(\ldots)$$
where the dots represent (I think) a functional determinant of something. Can anybody tell me what goes inside the determinant, and where this comes from?

Answer 1

Once you throw out terms higher than quadratic (this is just your approximating $\Delta \phi$ to be small), you get an integral of the sort

$\int d^nx \exp(-\frac{1}{2} x^T A x)$.

This is a Gaussian integral equal to $(2\pi)^{n/2} / \sqrt{det A}$.

Comments

I would thank you a lot if you could please be more explicit on what goes exactly inside the determinant using the notation I have used.

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@silvrfuck: To tell ''what goes exactly inside the determinant'' you should either say what is in your notes in place of the ..., or if the ... are in the notes themselves, what the author concludes from it.