A Dimensional Regularization of Feynman Integrals

1. Apr 28, 2017

nigelscott

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

∫ddk/(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.

2. Apr 29, 2017

vanhees71

3. Apr 30, 2017

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.

4. Apr 30, 2017

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.

5. May 1, 2017

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.