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:

[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]
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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:

[tex]\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)}[/tex]

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

 

[Entropia]

I can confirm that there are indeed techniques for obtaining closed-form expressions for the Z-functions in front of derivative terms in the effective action. These techniques involve using the path integral method, which is a powerful tool for calculating effective actions in quantum field theory.

One approach is called the heat kernel expansion, which involves expanding the path integral in terms of the heat kernel, which is a fundamental solution to the heat equation. This method can be used to obtain the Z-functions at any desired order in the loop expansion. Another approach is the background field method, which involves expanding the path integral around a classical background field and then calculating the effective action in terms of this background field. This method can also be used to obtain the Z-functions at any desired order.

It is important to note that obtaining closed-form expressions for the Z-functions can be a challenging and complex task, and it often requires advanced mathematical techniques. However, with the use of these methods, it is possible to calculate the Z-functions to the desired order and thereby obtain a more complete understanding of the effective action and its derivative terms.