Constructing d-log integrands and computing master integrals for three-loop four-particle scattering

TL;DR

This work develops and publicizes a refined algorithm to construct dlog integrands with purely logarithmic singularities, enabling a canonical differential equations treatment of multi‑loop Feynman integrals. It applies the method to massless three‑loop four‑point scattering, organizing the problem into nine integral families, and computes all master integrals via ε‑expanded differential equations with harmonic polylogarithms. The approach integrates cut/leading singularity techniques with unitarity concepts, classifies integrals by infrared behavior, and provides a comprehensive dlog basis with a publicly available implementation (DlogBasis). The resulting master integrals complete the virtual three‑loop corrections program for di‑jet and di‑photon production at the LHC, marking a significant advance in high‑order perturbative QCD calculations.

Abstract

We compute all master integrals for massless three-loop four-particle scattering amplitudes required for processes like di-jet or di-photon production at the LHC. We present our result in terms of a Laurent expansion of the integrals in the dimensional regulator up to 8$^{\text{th}}$ power, with coefficients expressed in terms of harmonic polylogarithms. As a basis of master integrals we choose integrals with integrands that only have logarithmic poles - called $d$log forms. This choice greatly facilitates the subsequent computation via the method of differential equations. We detail how this basis is obtained via an improved algorithm originally developed by one of the authors. We provide a public implementation of this algorithm. We explain how the algorithm is naturally applied in the context of unitarity. In addition, we classify our $d$log forms according to their soft and collinear properties.

Figures (5)

• Figure 1: The nine integral families needed to describe all master integrals for three-loop massless four-particle scattering. The external legs are associated with the momenta $p_1$, $p_3$, $p_4$ and $p_2$ in clockwise order starting with the top left corner.
• Figure 2: The integrand of the triangle shown in (a) is an example of a dlog form. The integrand of the Feynman integrals shown in (b) and (c) has a (hidden) double pole in four dimensions.
• Figure 3: Workflow of the dlog algorithm
• Figure 4: Sectors corresponding to the cuts used in the dlog analysis of the planar double box family. Sectors corresponding to the three cuts in equation \ref{['eq:cuts']} are $c_7, c_{13}, c_{17}$. Labels with an asterisk represent sectors that can be obtained by flip symmetries and are not explicitly shown.
• Figure 5: To reveal the $\epsilon^{-6}$ pole we take the following consecutive limits: 1) $\,k_1$ collinear to $p_1$, 2) $k_2$ collinear to $p_1$, 3) $k_2$ collinear to $p_3$, 4) $k_1$ collinear to $p_3$, 5) $k_3$ soft, 6) $k_3$ collinear to $p_2$. The soft limit in $k_3$ contributes a pole only if applied after the series of four collinear limits in $k_1$ and $k_2$.