IBP Relations in Feynman integrals

1. Nov 1, 2015

CAF123

I am wondering if anyone has experience in using IBP( Integration by parts) identities in the evaluation of Feynman diagrams via differential equations?

My question is that I can't seem to understand where equation (4.8) on P.8 of this paper: http://arxiv.org/pdf/hep-ph/9912329.pdf comes from. They are relating a triangle diagram to two bubble diagrams. I can write the l.h.s of 4.8 as being of the form $$\frac{1}{k^2 (k-p_2)^2(k-q)^4}$$ which in the family of integrals associated to the box diagram is denoted as $G(1,1,0,2)$ where one of the propagators is squared and the other contracted. Then find an IBP using relation $$\int p_2^{\mu} \cdot \frac{\partial}{\partial k^{\mu}} \frac{1}{k^2(k-p_2)^2 (k-q)^2} = 0$$ which I think gives rise to the equation $$G(1,1,0,2) = \frac{1}{s+t}\left(G(0,2,0,1) - G(1,0,0,2) - G(2,0,0,1) + G(0,1,0,2\right))$$ But the problem is I now need to reduce the terms on the r.h.s down to simpler diagrams and all attempts so far have gone in the direction of introducing higher exponents.

I don't have high hopes of getting a reply but I just thought I'd ask anyway to see if anyone is familiar with these identities and could help.

Many thanks!

2. Nov 6, 2015