Graduate Pauli Villars for Quadratic Divergences

[Diracobama2181]

TL;DR

How would I do Pauli Villars Regularization for an integral of the form

$\frac{\int d^4k}{(2\pi)^4}\frac{k^2}{(k^2-m^2+i\epsilon)^2}$

My guess would be to do an integral of the form

$$\frac{\int d^4k}{(2\pi)^4}k^2(\frac{1}{(k^2-m^2+i\epsilon)}-\frac{1}{k^2-\Lambda_1^2+i\epsilon})(\frac{1}{(k^2-m^2+i\epsilon)}-\frac{1}{k^2-\Lambda_2^2+i\epsilon})$$

before Wick otating and integrating. Any help is appreciated. Thanks.

 

[vanhees71]

This looks promising. What's your specific question? I never liked Pauli-Villars regularization much, because it's pretty complicated compared to dimensional regularization ;-)).

 

[Diracobama2181]

Just wanted to check if I was on the right path. Thanks!