Hi!(adsbygoogle = window.adsbygoogle || []).push({});

I want to renormalize the following UV-divergent integral using Dimensional Regularization:

[tex] \int_{- \infty}^{\infty} \frac{d^4 p}{\left(2 \pi \right)^4} \frac{1}{a p_0^2 +\left(a p_x^2+a p_y^2+a p_z^2 +M^2\right)^2} [/tex]

[tex] a>0 [/tex]

I can only find literature which deals with integrands [tex]f \left(p\right) [/tex], i.e. the components of [tex] p=(p_0,p_x,p_y,p_z) [/tex] do not appear isollated as in my integral above. I found for example on p.854 (M>0) resp. p.862 (M=0) of "Leibbrandt: Introduction into the technique of Dimensional regularization" (see http://prola.aps.org/abstract/RMP/v47/i4/p849_1" [Broken]) a general prescription how to do DimReg, but I guess I cannot apply it in my case.

Maybe step (ii) would be [tex]\frac{1}{a p_0^2 +\left(a p_x^2+a p_y^2+a p_z^2 +M^2\right)^2} = \int_{0}^{\infty} e^{ - \alpha \left(a p_0^2 + \left[a p_x^2+a p_y^2 + a p_z^2 + M^2\right]^2 \right) } [/tex]

But step (iii) I cannot perfom because I don't have a generalised gaussian integral. Unfortunately I haven't found a corresponding formulae for my case...

Maybe someone of you has an idea how the prescription in the URL can be generalised, or where to look for a hint how to cope with the integral?

Best regards, Martin

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Dimensional Regularization of an Integral

**Physics Forums | Science Articles, Homework Help, Discussion**