Working out some quantum field theory computations, I have to find out the value of the two-loop Feynman integral $$ I(p)=\int\frac{d^4p_1}{(2\pi)^4}\frac{d^4p_2}{(2\pi)^4}\frac{1}{(p_1^2+m_1^2)(p_2+m_2^2)[(p-p_1-p_2)^2+m_3^2]}. $$ This integral is rather common and so, its value at small $p$ should be already well-knwon. But I was not able to find out it in literature. I would also appreaciate to see all the procedure to get the right value with whatever regularization procedure one likes.

Thanks beforehand.

  • $\begingroup$ how do we understand this double integral ?? first integration over $ p1 $ keepen $ p2 $ constant and then integration over $p2$ as is made in calculus ? $\endgroup$ – Jose Javier Garcia Jul 26 '13 at 20:51

In the case of equal masses, there is an analytical solution (of this diagram known by the name "the two loop sunrise diagram" for the obvious reason) in terms of hypergeometric functions given by O.V. Tarasov (equation 4.32). There is also a numerical method given by: Pozzorini and Remiddi.

In the case of unequal masses Müller-Stach, Weinzierl and Zayadeh were able to obtain a second order differential equation for this graph.

  • $\begingroup$ Thanks a lot David. Müller-Stach et al. paper seems exactly what I am looking for. $\endgroup$ – Jon Aug 16 '12 at 16:39
  • $\begingroup$ wby couldn't we simply apply an iterated dimensional regularizaton ? first over variable $p_{1} $ and thenover the variable $ p_{2} $ like in integral calculus ? $\endgroup$ – Jose Javier Garcia Jun 15 '14 at 18:21

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