# Homework Help: Evalutation of loop-integrals with momentum cutoff

1. Nov 3, 2011

### salparadise

1. The problem statement, all variables and given/known data

I want to calculate the integral:

$\int dk^4 \frac{1}{k^2-m^2}$

with a momentum cutoff. I know what the result should look like, but I'm not sure how to get it.

3. The attempt at a solution

I understand one needs to perform a Wick rotation, for evaluating the integral in spherical coordinates.

So: $k^0\rightarrow ik_E^0$, $k^2=-k_E^2$ and $\bar k=\sqrt{(k_E^0)^2+|\vec k|^2}$. And we obtain for the integral

$-i2\pi^2\int^\Lambda_0 d\bar k \frac{\bar k^3}{\bar k^2+m^2}$

Now, the integral is trivial to calculate,

$\int d\bar k \frac{\bar k^3}{\bar k^2+m^2}=\frac{1}{2}(\bar k^2-m^2\log(\bar k^2+m^2))$

but my problem is that I don't know how to apply the boundary values, and "revert" the Wick rotation. I assume that $\Lambda^2=|\vec k|^2$...

The result should look like
$\sim\Lambda^2+m\log\left(\frac{\Lambda^2+m^2}{m^2}\right)$

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