Integration-by-parts reductions from unitarity cuts and algebraic geometry

We show that the integration-by-parts reductions of various two-loop integral topologies can be efficiently obtained by applying unitarity cuts to a specific set of subgraphs and solving associated polynomial (syzygy) equations.

Figure 1

The massless double-box diagram, along with our labeling conventions for its internal lines, is shown on top. The lower part shows the subset of the master integrals in Eq. (21) with the property that their graphs cannot be obtained by adding internal lines to the graph of some other master integral. The corresponding cuts $\{2,5,7\}$, $\{1,4,7\}$, $\{2,4,6,7\}$ and $\{1,3,4,6\}$ are the cuts required for deriving complete IBP relations for the double-box diagram.