All planar three-loop Feynman integrals for the production of two vector bosons at hadron colliders

TL;DR

This work delivers the complete set of planar three‑loop four‑point Feynman integrals with two external massive legs, essential for the leading‑colour N3LO QCD corrections to diboson production at hadron colliders. The authors organize the integrals into nine families, construct pure master integral bases, and derive canonical differential equations that are solved numerically with generalised power series, aided by finite‑field IBP reductions. A key outcome is the identification of an enlarged alphabet of differential forms, including new square‑root structures, and robust numerical validation against AMFlow, providing ready‑to‑use results for future three‑loop amplitudes in diboson production. The ancillary files supply boundary conditions and scripts for evaluating the master integrals at generic physical points. Collectively, these results lay the groundwork for full N3LO predictions in the generalized leading colour approximation and advance our understanding of the analytic structure of planar three‑loop integrals with multiple massive external legs.

Abstract

We compute all the planar three-loop master integrals relevant for the leading colour N3LO QCD corrections to the production of two massive or off-shell vector bosons at hadron colliders. These integrals are organised into nine four-point integral families with massless internal propagators and two external massive legs. For each family, we construct a basis of pure master integrals and we reconstruct the corresponding canonical differential equations using finite field techniques. We evaluate the master integrals by solving the differential equations using generalised power series expansions.

Figures (5)

• Figure 1: The top-sectors of the two reducible integral families. Black lines represent massless particles, curled coloured lines correspond to the vector bosons. The particle with mass $m_4$ ($m_3$) is depicted in blue (red), respectively.
• Figure 2: The top-sectors of the three ladder-box integral families. The same conventions as in \ref{['fig:irreducibles']} for line styles and colours are used.
• Figure 3: The top-sectors of the four tennis-court integral families. The same colour and line conventions as in \ref{['fig:irreducibles']} are followed.
• Figure 4: The sector $\{0,1,0,0,0,0,1,1,1,0,0,1,1,1,0\}$ of superfamily $F_{123}$.
• Figure 5: The starting basis of integrals for the sector $\{0,1,0,0,0,0,1,1,1,0,0,1,1,1,0\}$ of superfamily $F_{123}$. Red dots indicate that the corresponding propagator is squared. The insertion of $D_i$ indicates the presence of an ISP.