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A Dimensional Regularization of Feynman Integrals

  1. Apr 28, 2017 #1
    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.
     
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  3. Apr 29, 2017 #2

    vanhees71

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  4. Apr 30, 2017 #3
    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.
     
  5. Apr 30, 2017 #4

    vanhees71

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    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.
     
  6. May 1, 2017 #5
    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.
     
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