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

Question

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?

Answer 1

Since we want to deal with path integral by using the stationary phase approximation, you will need to take the stationary points of the sourced action: these regions contribute coherently to the path integral (cause the phase changes very little over a small region surrounding each stationary point) giving the main contribution to the value of the quantity you compute, like in a constructive interference.

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.