Analytic representation of all planar two-loop five-point Master Integrals with one off-shell leg

TL;DR

This work delivers fully analytic expressions for all planar two-loop five-point Master Integrals with one off-shell leg, using the Simplified Differential Equations framework to obtain canonical differential equations and express results in Goncharov polylogarithms. Boundary conditions are determined via an efficient x→0 analysis combined with IBP and expansion by regions, enabling ε-expansion up to ε^4. The results cover three planar families P1, P2, and P3 and are accompanied by extensive ancillary data, with thorough numerical validation against existing literature and FIESTA. The study advances NNLO calculations for 2→3 processes at the LHC and sets the stage for completing the full two-loop five-point integral basis, including non-planar topologies and massless limits.

Abstract

We present analytic expressions in terms of polylogarithmic functions for all three families of planar two-loop five-point Master Integrals with one off-shell leg. The calculation is based on the Simplified Differential Equations approach. The results are relevant to the study of many $2\to 3$ scattering processes of interest at the LHC, especially for the leading-color $W+2$ jets production.

Figures (3)

• Figure 1: Diagrammatic representation of the planar and non-planar families with one external massive leg (double line). In the first row, $P_1$ (left), $P_2$ (middle) and $P_3$ (right) planar families are shown. In the second and third row, $N_1$ (top left), $N_2$ (top middle), $N_3$ (top right), $N_4$ (bottom left), $N_5$ (bottom right) non-planar families are shown. All internal particles are massless.
• Figure 2: The two-loop diagrams representing the top-sector of the planar pentabox family $P_1$, $P_2$ and $P_3$. All external momenta are incoming.
• Figure 3: The two-loop diagram representing the decoupling basis element.