Dimensional Regularization of an Integral

[Sunset]

Hi!

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

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

$$a>0$$

I can only find literature which deals with integrands $$f \left(p\right)$$, i.e. the components of $$p=(p_0,p_x,p_y,p_z)$$ 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" ) a general prescription how to do DimReg, but I guess I cannot apply it in my case.

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

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?

Martin

 

[per.sundqvist]

I'm a bit confused. what's the difference between M and p0? I guess its some four-dim relativistic stuff you are doing?

you could at least integrate d^3p=4*pi*p^2*dp and integrate out that analytic first, but then you get p0 left, which is hard to integrate out.

 

[Sunset]

Hi!

sorry, I wasn't able to read & write until today, due to the server problems.

The integral is a divergent part of a thermal integral (finite temperature field theory). I carried out the Matsubara Sum and I received the contribution $$\int \frac{d^3 \bf{k} }{\left( 2 \pi \right)^3 } \frac{T}{\epsilon^2_z}$$

I introduced a integration over $$\kappa$$ to receive an integral over 4 real-valued momenta.

$$\int \frac{d^3 \bf{k} }{\left( 2 \pi \right)^3 } \frac{T}{\epsilon^2_z} = \int \frac{d^3 \bf{k} }{\left( 2 \pi \right)^3 } \frac{T Z}{\pi} \int\limits_{-\infty}^{\infty} d \kappa \ \frac{1}{Z^2 \kappa^2 + \epsilon^4_z}$$

with $$\sqrt{Z^2 \bf{k}^2 +M^2 } \equiv \epsilon_z$$

Leibbrandts prescription is also for such integrals but he assumes that the integral is made of propagators. Although my integrand looks similar to a propagator, it differs from it because of the +signs instead of the -signs from the euclidean metric (I guess that caused your confusion)

My question is mainly: why does the prescription of Leibbrandt assume that there are always propagators in the integrals you want to dimReg-ularise...? There are definitely others! Where can such a prescription be found? Or how can Leibbrandts be applied to others?

I managed to renormalize the above integral with a CT-scheme, but I would like to know how it is with DimReg...