The diamond rule for multi-loop Feynman diagrams

TL;DR

The paper addresses efficient reduction of dimensionally regularized Feynman integrals via IBP, by extending the triangle rule to a multi-loop diamond rule. It derives an explicit summation formula that avoids spurious poles and applies to massless propagator-type diagrams up to five loops. The key contribution is the (L+S)-loop diamond topology IBP identity and its pole-free recursion, enabling faster and more stable reductions. The work discusses implications for parametric reduction programs like Mincer and compares with Laporta-based methods, outlining practical impact and remaining questions.

Abstract

An important aspect of improving perturbative predictions in high energy physics is efficiently reducing dimensionally regularised Feynman integrals through integration by parts (IBP) relations. The well-known triangle rule has been used to achieve simple reduction schemes. In this work we introduce an extensible, multi-loop version of the triangle rule, which we refer to as the diamond rule. Such a structure appears frequently in higher-loop calculations. We derive an explicit solution for the recursion, which prevents spurious poles in intermediate steps of the computations. Applications for massless propagator type diagrams at three, four, and five loops are discussed.

Figures (3)

• Figure 1: A triangle subtopology where the loop momentum $k$ is assigned to the central line. $p_1$ and $p_2$ are external momenta. $a_1$, $a_2$, $b$, $c_1$, and $c_2$ represent the powers of their associated propagators.
• Figure 2: $(L+S)$-loop diamond-shaped diagram. $(L+1)$-lines have external connections and $S$-lines do not. Red with dashed lines, green with double lines, and blue with thick lines represent upper, lower, and external lines of the diamond, respectively. Label $T$ represents the top vertex, and $B$ the bottom vertex. $k_i$, $p_i$, and $l_i$ are momenta, and $a_i$, $b_i$, and $s_i$ are the powers of their associated propagators.
• Figure 3: Two topologies with highlighted diamond structures. Red with dashed lines, green with double lines, and blue with thick lines represent upper, lower, and external lines of the diamond, respectively. Label $T$ represents the top vertex, and $B$ the bottom vertex. (a) shows a four-loop topology which can be completely reduced. (b) shows the three-loop NO master topology, for which a modified form of the diamond rule can be applied to lower the power of line $p_1$ to 1. (c) shows five-loop topologies, which the diamond rule can be applied to.