Syzygies for integration-by-parts reductions from Laplace expansion

Authors: Janko Böhm, Alessandro Georgoudis, Kasper J. Larsen, Mathias Schulze, Yang Zhang

Preprint number: UUITP-44/17

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a constraint on the total derivatives which takes the form of a specific type of polynomial equation, known in algebraic geometry as syzygy equations. We present an explicit generating set of solutions to the encountered syzygy equation, valid for any number of loops and external momenta. The generating set of solutions is obtained from the Laplace expansion of the Gram determinant formed of the loop momenta and independent external momenta. In particular, no S-polynomial computations are required in order to obtain the syzygies. We moreover show how to obtain the syzygies needed for integration-by-parts identities evaluated on a generalized unitarity cut.