I'm studying a very particular conformal field theory where unusual Feynman integrals appear when I'm trying to evaluate a two-loop correlator (in position space). These integrals are on the form $\int_0^1du\frac{\text{Log}(b^2 + (a^2 - b^2)u)}{\sqrt{b^2 + (a^2 - b^2)u}} \ , \ \int_0^1du\frac{\text{Log}[u(1 - u)]}{\sqrt{b^2 + (a^2 - b^2)u}}$ and $\int_\mathbb{R^3}\frac{dzdw}{(x - z)^2(y - w)^2(z - w)^{2n}} \ ,$ where I need to solve the last of these for three different cases: $n\in\{1,2,3\} \ .$ Any help on solving these integrals would be appreciated. Alternatively if you know a good reference where similar integrals are studied.

  1. I assume log(x) = ln(x). The first integral: enter image description here

  2. Unfortunately the second integral is divergent.

enter image description here

  • $\begingroup$ Thanks for your answer! I appreciate it. Do you know whether it's possible to regularize the second integral. If I'm able to isolate the divergence I can add a counterterm that removes it. $\endgroup$ – A.Dunder Apr 19 '18 at 10:33
  • $\begingroup$ @A.Dunder You're welcome. I edited my post above. $\endgroup$ – Székely Balázs Apr 19 '18 at 11:42
  • $\begingroup$ How did you find this? Did you use a program or did you do a couple of suitable variables changes? Edit: Nevermind I managed to do it with Mathematica. $\endgroup$ – A.Dunder Apr 19 '18 at 11:56
  • $\begingroup$ I did a few integration by parts. Please note: I don't use any program for calculation, it would kill all the fun, but feel free to check the results if you like. $\endgroup$ – Székely Balázs Apr 19 '18 at 12:07

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