Towards a Basis for Planar Two-Loop Integrals

TL;DR

The paper constructs explicit finite bases for planar two-loop integrals by separating an all-orders (D-dimensional) basis from a regulated four-dimensional (O(ε^0)) basis and by developing integration-by-parts (IBP) strategies that avoid doubled propagators. It introduces Gram-determinant identities and Gröbner-basis techniques to generate irreducible, ε-controlled relations, enabling constructive reductions of high-multiplicity tensor integrals to a minimal set of master integrals. The authors illustrate the approach with detailed analyses of the massless double box, various massive double boxes, the pentabox, and a six-point double pentagon, demonstrating how to compute IBP-generating vectors and perform reductions, including ε-dependent refinements. They further relate their framework to generalized unitarity, offering a conceptual bridge between algebraic reductions and on-shell methods, and discuss extensions to non-planar and higher-loop cases. The work provides a practical, scalable pathway toward automation of two-loop planar amplitude computations and lays groundwork for broader applicability to massive propagators and higher-order loops.

Abstract

The existence of a finite basis of algebraically independent one-loop integrals has underpinned important developments in the computation of one-loop amplitudes in field theories and gauge theories in particular. We give an explicit construction reducing integrals to a finite basis for planar integrals at two loops, both to all orders in the dimensional regulator e, and also when all integrals are truncated to O(e). We show how to reorganize integration-by-parts equations to obtain elements of the first basis efficiently, and how to use Gram determinants to obtain additional linear relations reducing this all-orders basis to the second one. The techniques we present should apply to non-planar integrals, to integrals with massive propagators, and beyond two loops as well.

Figures (8)

• Figure 1: The basis of scalar integrals: (a) box, (b) triangle, (c) bubble, and (d) tadpole. Each corner can have one or more external momenta emerging from it. The tadpole integral (d) vanishes when all internal propagators are massless.
• Figure 2: The three basic types of two-loop planar integrals, labeled by the number of legs attached to each leg of the vacuum diagram: (a) $P_{n_1,n_2}$, (b) $P^*_{n_1,n_2}$, (c) $P^{**}_{n_1,n_2}$.
• Figure 3: The non-trivial two-loop vacuum diagram.
• Figure 4: Two-loop integrals which are products of one-loop integrals, labeled by the number of legs attached to each leg of the vacuum diagram: (a) $I_{n_1,n_2}$, (b) $I^*_{n_1,n_2}$.
• Figure 5: The double box $P^{**}_{2,2}$.