- #1

- 63

- 0

Hi!

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

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

Last edited by a moderator: