Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals
R. N. Lee
TL;DR
This work reframes integration-by-parts (IBP) identities as a closed Lie algebra acting on loop-momentum structures, enabling systematic reduction of the otherwise enormous IBP set used in multiloop integral calculations. By proving that Lorentz-invariance (LI) identities are not independent of IBP and introducing a Weyl-algebra operator representation, the authors derive practical criteria to discard redundant identities and identify zero sectors. They propose a reduced operator basis and a sequence-based reduction strategy that blends division-with-remainder techniques with Laporta-like approaches on lower-dimensional sectors, thereby lowering computational complexity. The results suggest a path toward proving finiteness of master integrals in general settings and provide a coherent framework for more efficient, algebraically grounded reduction of multiloop integrals.
Abstract
The excessiveness of integration-by-part (IBP) identities is discussed. The Lie-algebraic structure of the IBP identities is used to reduce the number of the IBP equations to be considered. It is shown that Lorentz-invariance (LI) identities do not bring any information additional to that contained in the IBP identities, and therefore, can be discarded.