Simplified differential equations approach for Master Integrals

TL;DR

This paper presents a simplified differential-equations approach to Master Integrals (MI) that yields analytic expressions in Goncharov polylogarithms. By parameterizing kinematics with a variable x and solving DE using integration-by-parts and an integrating factor, the authors obtain GP representations for MI at one and two loops. They demonstrate the method on cases including the massless one-loop pentagon and various two-loop topologies, with numerical cross-checks validating the results. A key finding is that, in many instances, the DE itself fixes boundary contributions, suggesting a path toward automated NNLO MI computations in GP form, while highlighting areas for future work such as masses and physical-region analyses.

Abstract

A simplified differential equations approach for Master Integrals is presented. It allows to express them, straightforwardly, in terms of Goncharov Polylogarithms. As a proof-of-concept of the proposed method, results at one and two loops are presented, including the massless one-loop pentagon with up to one off-shell leg at order epsilon.

Figures (6)

• Figure 1: The one-loop graph with three off-shell legs.The label $1,2,3$ refer to the denominators $-(k_1)^2,-(k_1+x p_1)^2,-(k_1+p_1+p_2)^2$, Eq. \ref{['1l3m']}.
• Figure 2: The one-loop pentagon graph with one off-shell leg, $G_{11111}$. The labels refer to the denominators in Eq. \ref{['1l5p']}.
• Figure 3: Two-loop MI with three off-shell legs, $G_{0101011}$: labels are used to identify the propagators, for instance label $2$ refer to $-(k_1+x p_1)^2$, Eq. \ref{['3mtop']} ($i$ for $a_i=1$).
• Figure 4: Two-loop box MI with two off-shell legs: the easy one, $G_{010010111}$. See in the text for more details. Labels as in Fig. \ref{['fig:2l3m']} with respect to Eq. \ref{['box2pie']}.
• Figure 5: Two-loop box MI with two off-shell legs: the hard one, $G_{001011011}$. See in the text for more details. Labels as in Fig. \ref{['fig:2l3m']} with respect to Eq. \ref{['box2pih']}.