Parameter Integration of Bubble Integral

In summary, the conversation discusses the use of Mathematica's Integrate command to solve a loop integral, which does not give the expected result. The integrand is known to diverge for certain values of the Feynman parameter λ, so the method used involves finding the roots of the expression in the denominator. It is suggested that a deformation of a complex contour may be used to integrate around the branch cut of the logarithm. The solution eventually involves factoring the quadratic argument of the logarithm and integrating by parts.
  • #1
Elmo
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TL;DR Summary
Cant figure out how this PV bubble integral has been solved.
Referring to this link : https://qcdloop.fnal.gov/bubg.pdf
Using Mathematica Integrate command to solve it does not give the result stated here but I am unclear as to how they got to the result in the 4th line.
It is clear that the integrand (1st line) can diverge for certain values of the Feynman parameter λ and this is presumably why they find the roots of the expression in the denominator. I just dont know what they did to solve this loop integral and express the result in terms of the roots.
 
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  • #2
Maybe they used some deformation of a complex contour for the ##\lambda## integral integrating somehow around the branch cut of the ln?
 
  • #3
Just factor the quadratic argument of the logarithm as ##(\lambda-\lambda_1)(\lambda - \lambda_2)##, expand the logarithm into ##\log(\lambda - \lambda_1)+\log(\lambda - \lambda_2)## (modulo constant prefactors) and integrate by parts.
 

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