Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Dimensional Regularization of an Integral

  1. Jul 14, 2008 #1

    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) 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
  2. jcsd
  3. Jul 14, 2008 #2
    I'm a bit confused. whats 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.
  4. Jul 16, 2008 #3

    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 [tex] \int \frac{d^3 \bf{k} }{\left( 2 \pi \right)^3 } \frac{T}{\epsilon^2_z} [/tex]

    I introduced a integration over [tex] \kappa [/tex] to receive an integral over 4 real-valued momenta.

    [tex] \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} [/tex]

    with [tex] \sqrt{Z^2 \bf{k}^2 +M^2 } \equiv \epsilon_z [/tex]

    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...
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Dimensional Regularization of an Integral