Maximal Cuts in Arbitrary Dimension

TL;DR

The paper develops a systematic procedure to compute maximal unitarity cuts of multiloop Feynman integrals in arbitrary dimensions using the Baikov representation. It shows that maximal cuts inherit IBP and dimension-shift identities from the uncut integrals and that the cut data across real integration regions form a Wronskian matrix for the differential equations on the maximal cut; explicit examples across planar and nonplanar topologies (including sunset and double boxes with varying masses) yield closed-form cut functions expressed through Gamma and hypergeometric functions. The key contributions are (i) a complete framework for region-wise maximal cuts in $D$ dimensions, (ii) demonstration that the maximal-cut functions span the master-cut space and satisfy the same linear relations as the full integrals, and (iii) a pathway to epsilon-form differential equations via the Wronskian basis, enabling streamlined canonicalization and potential extensions to elliptic sectors. This work provides a principled link between generalized unitarity at maximal cuts and the differential-equation approach, with practical implications for efficient multiloop computations and analytic understanding of cut structures.

Abstract

We develop a systematic procedure for computing maximal unitarity cuts of multiloop Feynman integrals in arbitrary dimension. Our approach is based on the Baikov representation in which the structure of the cuts is particularly simple. We examine several planar and nonplanar integral topologies and demonstrate that the maximal cut inherits IBPs and dimension shift identities satisfied by the uncut integral. Furthermore, for the examples we calculated, we find that the maximal cut functions from different allowed regions, form the Wronskian matrix of the differential equations on the maximal cut.

Figures (13)

• Figure 1: Two-loop box triangle diagrams
• Figure 2: The massless sunset diagram.All momenta flow to the right.
• Figure 3: Subregions for the integration of the sunset diagram on the maximal cut.
• Figure 4: Massless double box diagram. This graph is produced by the package AzuriteGeorgoudis:2016wff.
• Figure 5: Subregions for the integration of the massless double box diagram on the maximal cut. This plot is for the kinematic configuration, $s=1$ and $\chi=1/3$. The two axes are the ISPs, $z_8$ and $z_9$. These subregions are symmetric under the flip of two axes.