Derivative Terms of Effective Action

[TriTertButoxy]

I know that the effective action can be written as a double expansion in derivatives and loop (h-bar). For example, take the effective action for a real scalar field:

\Gamma[\phi]=\int d^4x\left[V_\text{eff}(\phi)+\frac{1}{2}Z(\phi)\partial_\mu\phi\partial^\mu\phi+\mathcal{O}(\partial^4)+\ldots\right]
=\int d^4x\left[V^{(0)}(\phi)+\hbar V^{(1)}(\phi)+\ldots+\frac{1}{2}(1+\hbar Z^{(1)}(\phi)+\ldots)\partial_\mu\phi\partial^\mu\phi+\mathcal{O}(\partial^4)+\ldots\right]
​

I am familiar with the techniques of getting the terms with zero derivatives (effective potential) to the desired order in the loop expansion (path integral method, tadpole method, etc...) -- this is fairly standard.

But, are there techniques of reliably getting closed-form expressions for the Z-functions in front of derivative terms? (preferably using path integrals)

 

[RedX]

By effective action do you mean the quantum action, i.e., an action whose tree diagrams give the loop diagrams of the real action? Or is this the Wilson effective action?

Generically would you would take your theory, calculate the propagator and n-point vertices to some order using the Feynman rules, plug it into this action:

\Gamma(\phi)= \frac{1}{2} \int \frac{d^4k}{(2\pi)^4}\phi(-k)(k^2+m^2-\Pi(k^2))\phi(k)+\Sigma_n \frac{1}{n!}\int \frac{d^4k_1}{(2\pi)^4}...\frac{d^4k_n}{(2\pi)^4} (2\pi)^4 \delta(k_1+...+k_n)V_n(k_1,...,k_n)\phi(k_1)...\phi(k_n)}

and then inverse Fourier-transform everything, and collect the terms into the form of the effective action you have above?