Cross-Order Integral Relations from Maximal Cuts

TL;DR

This work analyzes the ABDK/BDS relation through maximal generalized unitarity for two-loop amplitudes in N=4 SYM, using multivariate residues to localize contributions on global poles. By systematically studying shared global poles and the existence of nonhomologous integration contours, the authors show how to reconstruct coefficients of the squared one-loop box and pentabox contributions, and how to separate overlapping cut surfaces to derive the four- and five-point ABDK relations. The key insight is that maximal and near-maximal cuts, together with contour choices, reveal integral identities beyond conventional dual conformal arguments, suggesting a general method to infer identities among loop integrals from their leading singularities. The results provide a concrete framework for extending unitarity-based approaches to multi-loop, multi-leg amplitudes and for exploring new integral relations in planar MSYM.

Abstract

We study the ABDK relation using maximal cuts of one- and two-loop integrals with up to five external legs. We show how to find a special combination of integrals that allows the relation to exist, and how to reconstruct the terms with one-loop integrals squared. The reconstruction relies on the observation that integrals across different loop orders can have support on the same generalized unitarity cuts and can share global poles. We discuss the appearance of nonhomologous integration contours in multivariate residues. Their origin can be understood in simple terms, and their existence enables us to distinguish contributions from different integrals. Our analysis suggests that maximal and near-maximal cuts can be used to infer the existence of integral identities more generally.

Figures (19)

• Figure 1: The three basic types of two-loop planar integrals, labeled by the number of legs attached to each internal line of the vacuum diagram: (a) $P_{n_1,n_2}$, (b) $P^*_{n_1,n_2}$, (c) $P^{**}_{n_1,n_2}$.
• Figure 2: An example of a global pole shared between the horizontal and vertical double-box integrals. The loop momentum labeling is chosen cunningly. At the global pole, the nonvanishing heptacut propagator momentum in each double box corresponds to that of a quadruply cut one-loop box. The white and black blobs indicate chiral (MHV) and antichiral ($\overline{\text{MHV}}$) vertices, respectively, as in the conventions (for example) of ref. ExternalMasses.
• Figure 3: The horizontal and vertical double boxes. All internal lines are on-shell.
• Figure 4: A second example of how a global pole could be shared between the horizontal and vertical double-box integrals.
• Figure 5: A third example of how a global pole could be shared between the horizontal and vertical double-box integrals.