Feynman Integral Reduction without Integration-By-Parts

TL;DR

The paper addresses the reduction of Feynman integrals without using integration-by-parts by exploiting equivalence of integration contours in Feynman parameterization. The authors extend the Cheng-Wu delta-function choice to a more general contour form and derive universal one-loop reduction formulas that apply to both irreducible and reducible sectors, including numerators and degenerate limits. They validate the approach with explicit one-loop examples and a two-loop sunrise case, highlighting the method's efficiency and its potential limitations at higher loops. The work opens avenues for deeper mathematical structures, such as intersection numbers of contours, to further elucidate reduction coefficients.

Abstract

We present an interesting study of Feynman integral reduction that does not employ integration-by-parts identities. Our approach proceeds by studying the equivalence relations of integral contours in the Feynman parameterization. We find that the integration contour can take a more general form than that given by the Cheng-Wu theorem. We apply this idea to one-loop integrals, and derive universal reduction formulas that can be used to efficiently reduce any one-loop integral. We expect that this approach can be useful in the reduction of multi-loop integrals as well.

Figures (5)

• Figure 1: The independence of the integral on the integration domain for the case of two variables. $S_{1}$ is the integration domain defined by $\mathcal{X}=1$ and $\mathcal{Y}=\alpha_{0}+\alpha_{1}$, while $S_{1}'$ is the integration domain defined by two functions $\mathcal{X}'(\alpha_0,\alpha_1)$ and $\mathcal{Y}'(\alpha_0,\alpha_1)$.
• Figure 2: The massless triangle with $p_1^2 = s$ and $p_2^2=p_3^2=0$.
• Figure 3: The bubble diagram with two masses.
• Figure 4: The pentagon diagram with three masses.
• Figure 5: The sunrise diagram with $p^2 = s$.