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Homework Help: Evalutation of loop-integrals with momentum cutoff

  1. Nov 3, 2011 #1
    1. The problem statement, all variables and given/known data

    I want to calculate the integral:

    [itex]\int dk^4 \frac{1}{k^2-m^2}[/itex]

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

    [itex]-i2\pi^2\int^\Lambda_0 d\bar k \frac{\bar k^3}{\bar k^2+m^2}[/itex]

    Now, the integral is trivial to calculate,

    [itex]\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))[/itex]

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

    The result should look like
    [itex]\sim\Lambda^2+m\log\left(\frac{\Lambda^2+m^2}{m^2}\right)[/itex]
     
  2. jcsd
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