How Does Pauli-Villars Regularization Simplify Quantum Field Theory Integrals?

[Diracobama2181]

TL;DR Summary

I am currently trying to regularize integrals by using the Pauli-Villars scheme, and wanted to check if my setup is correct

For the below integral in Euclidean space,

$$\int d^4k_E k_E^2 \frac{1}{(k_E^2+M(\Delta))^2}= 2\pi^2\int dk_E k_E^5 \frac{1}{(k_E^2+M(\Delta))^2}$$
we find
$$2\pi^2\int_0^{\infty} dk_E k_E^5 \frac{1}{(k_E^2+M(\Delta))^2}\xrightarrow{}2\pi^2\int_0^{\infty} dk_E k_E^5(\frac{1}{k_E^2+M}-\frac{1}{k_E^2+\Lambda^2})(\frac{1}{k_E^2+M}-\frac{1}{k_E^2+\Lambda^2})\\
=2\pi^2(\Lambda^2-M)^2\int_0^{\infty} dk_E k_E^5\frac{1}{(k_E^2+M)^2(k_E^2+\Lambda)^2}$$
Is this setup correct? Thank you

 

[eljose79]

Yes, this setup is correct. By using the partial fraction decomposition, we can express the initial integral in terms of simpler integrals that can be easily evaluated. The result you have obtained is also correct, as it accounts for the two poles of the integrand at $k_E^2 = -M$ and $k_E^2 = -\Lambda^2$. The final integral can then be evaluated using standard techniques, such as substitution or integration by parts. Overall, your approach is a valid and efficient way to solve this integral.