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:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\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] [/tex]

[tex]=\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] [/tex]

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)

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# Derivative Terms of Effective Action

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