IBP Relations in Feynman integrals

[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!

 

[AndyW]

As a scientist familiar with using IBP identities in the evaluation of Feynman diagrams, I may be able to offer some insight into the equation (4.8) on page 8 of the paper you mentioned.

Firstly, it is important to note that IBP identities are a powerful tool in simplifying complicated integrals involving Feynman diagrams. They allow us to express a given integral in terms of simpler integrals, which can then be evaluated more easily.

In the case of the equation (4.8) in the paper, the authors are using the IBP identity $$\int p_2^{\mu} \cdot \frac{\partial}{\partial k^{\mu}} \frac{1}{k^2(k-p_2)^2 (k-q)^2} = 0$$ to simplify the integral on the left-hand side, which is denoted as $G(1,1,0,2)$. This integral corresponds to a triangle diagram, which can be related to two bubble diagrams using the IBP identity.

The equation (4.8) expresses $G(1,1,0,2)$ in terms of simpler integrals, namely $G(0,2,0,1)$, $G(1,0,0,2)$, $G(2,0,0,1)$, and $G(0,1,0,2)$. These integrals correspond to different bubble diagrams, which can be evaluated more easily.

To further simplify the terms on the right-hand side, the authors may have used other IBP identities or other techniques. Without further information, it is difficult to say exactly how they simplified the terms.

I hope this helps to clarify the equation (4.8) and the use of IBP identities in evaluating Feynman diagrams. If you have any further questions, please do not hesitate to ask.