Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation

TL;DR

The paper addresses the challenge of computing planar massless three-loop four-point Feynman integrals by recasting the differential equations for master integrals into a Knizhnik-Zamolodchikov form. By constructing a basis of pure functions with uniform transcendentality, the authors obtain a tractable epsilon-expanded solution expressed through harmonic polylogarithms, with boundary conditions fixed by physical constraints and a propagator integral. They explicitly formulate integral bases for two planar families (A and E), derive the corresponding one-variable DEs with three regular singularities, and provide weight-six results along with extensive checks and ancillary data. This work offers a practical route to complete analytic results for complex multi-loop planar integrals and sets the stage for extensions to non-planar cases and finite-epsilon analyses.

Abstract

We apply a recently suggested new strategy to solve differential equations for master integrals for families of Feynman integrals. After a set of master integrals has been found using the integration-by-parts method, the crucial point of this strategy is to introduce a new basis where all master integrals are pure functions of uniform transcendentality. In this paper, we apply this method to all planar three-loop four-point massless on-shell master integrals. We explicitly find such a basis, and show that the differential equations are of the Knizhnik-Zamolodchikov type. We explain how to solve the latter to all orders in the dimensional regularization parameter epsilon, including all boundary constants, in a purely algebraic way. The solution is expressed in terms of harmonic polylogarithms. We explicitly write out the Laurent expansion in epsilon for all master integrals up to weight six.

Figures (6)

• Figure 1: The triple box (A) and tennis court diagrams (E). Latin numbers refer to propagators associated to line parameters $a_{i}$, cf. eqs. (1.3) and (1.4). Lines associated to possible numerators are not shown in the figures.
• Figure 2: Integrating out propagator subintegrals related the basis choice at $(L+1)$ loops to the corresponding choice at $L$ loops, up to some trivial prefactors, and indices shifted by $\epsilon$.
• Figure 3: Master integrals for integral family A that have bubble subintegrals. Dots denote doubled propagators. An asterisk indicates that there are numerator factors not shown in the figure.
• Figure 4: Master integrals for integral family A without bubble subintegrals. Dots denote doubled propagators. An asterisk indicates that there are numerator factors not shown in the figure.
• Figure 5: Master integrals for integral family E that have bubble subintegrals. Dots denote doubled propagators. An asterisk indicates that there are numerator factors not shown in the figure.