Dimensional Regularization of Feynman Integrals

[nigelscott]

I am looking at Appendix A Equation 52 (Loop Integrals and Dimensional Regularization) in Peskin and Schroeder's book.

∫d^dk/(2π)^d1/(k² - Δ)² = Γ(2-d/2)/(4π)²(1/Δ)^2-d/2 = (1/4π)²(2/ε - logΔ - γ + log4π)

Can somebody explain how this equation is derived? I would also like to know what the equivalent expression is for integrals of the type 1/(k² - m²) and k²/(k² - m²)². Thanks.

 

[vanhees71]

Have a look at my QFT manuscript:

http://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

 

[nigelscott]

Thanks. There's a lot of heavy math in there but I think I got the main idea. It seems that the following 3 expansions are critical to the derivations. To first order, omitting factors of π etc and putting Δ = m²:

a. Γ(ε) = 1/ε - γ

b. Γ(ε - 1) = -1/ε + γ - 1 (from WolframAlpha)

c. (1/Δ)^2-d/2 = 1 - (1 - d/2)logΔ.

Therefore, for (c) we get (1/Δ)^ε = 1 - εlogΔ and for 1 - d/2 we get (1/Δ)^ε-1 = (1/Δ)^ε(1/Δ)^-1 = (1 - εlogΔ)Δ

When I multiply these terms together in the right combinations I seem to get the expressions that I was looking for. Is my reasoning correct? Thanks again.

 

[vanhees71]

It's a bit lenghty to post the full calculation here in the forums, but you find a (so I hope) pretty self-contained derivation of the formulae in Appendix C of my manuscript in Section 5.3 (including a review about the $\Gamma$ and $\mathrm{B}$ functions needed to do the integrals. Together with Feynman parametrization that's all you need to evaluate simple Feynman diagrams as also demonstrated in the manuscript.

 

[nigelscott]

Thanks again. I think C.14 and C.15 in the Appendix are basically what I was looking for. C.15 would seem to correspond to Peskin's A.51. My intention here was to justify the form of each equation rather that replicate the exact proofs. I think the expansion of Γ(∈) and Γ(∈ - 1) were the missing pieces.