# In the semiclassical approximation, should I expand the generating functional around saddles of the sourced or the unsourced action?

Consider a Euclidean path integral say in a real scalar field theory. $$\int d[\phi]\exp(-I[\phi])$$ In the semiclassical approximation, we consider stationary points of the action and expand around them. Now, consider I want to make a semiclassical expansion of the generating functional $$Z[J]=\int d[\phi]\exp\bigg(-I[\phi]-\int d^4x\,J\phi\bigg)$$ I have a doubt, should I consider saddles of $$I$$ or those of all the sourced action? $$I_J[\phi]\equiv I[\phi]+\int d^4x\,J\phi$$ Naively I would guess that I gotta take the saddles of the whole exponent, but my biggest concern then is that if I take saddles of the sourced action, the stationary field configurations will, in general, have $$J$$ dependence and thus after expanding the action around these stationary points $$\phi_s$$, taking functional derivatives of $$Z$$ with respect to $$J$$ will be very dirty since I will have $$J$$ dependence in every place I have a $$\phi_s$$.

So, saddles of the sourced or of the unsourced action?

As you noticed this will be obtained for configurations depending on the source in a messy way. This dependence however is precisely what you are interested in: it will bring to the properties of the generating functional (if you derive it with respect with J you get expectation values of powers of $$\phi$$), even if in general this is way easier to see without making the actual computation of the partition function.