Considering the scalar field, we have the effective action $$\tag 1 \Gamma[\phi_{cl}]=\int d^4x\mathcal{L}_1[\phi_{cl}]+\frac{i}{2}\log\det\left[-\frac{\delta^2\mathcal{L}_1}{\delta\phi\delta\phi}\right]+\int d^4x\delta\mathcal{L}[\phi_{cl}]+...,$$ where the determinant plays an important role. In Peskin & Schroeder's book, the authors perform an explicit calculation when $$-\frac{\delta^2\mathcal{L}_1}{\delta\phi\delta\phi}=\partial^2+m^2$$ which is a case where the operator is independent of $\phi_{cl}$. But in general (such as cases of interacting theories), the operator will depend on $\phi_{cl}$ and the calculation becomes complicated. Do anyone know of some references to which I can refer for such calculations?
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You can have a look at Jean Zinn-Justin's book "Quantum Field Theory and Critical Phenomena", which is far from being fun to read, but has all the technical details you may want.
