The number of master integrals is finite
A. V. Smirnov, A. V. Petukhov
TL;DR
The paper proves that for a fixed Feynman graph, the set of master integrals obtained via IBP relations is finite, formalizing this through a reformulation in terms of Lie algebra actions and holonomic D-modules. It shows that the relevant quotient space is finite-dimensional under a finite-orbit condition, thereby establishing finiteness for standard Feynman integrals. The result provides a foundational, non-constructive justification for the empirical observation of finiteness, while noting limitations for exotic integrals and highlighting potential for future work to derive practical reduction bounds. Overall, the work connects Feynman integral reduction to deep algebraic-geometry techniques, offering a rigorous basis for the master integral paradigm.
Abstract
For a fixed Feynman graph one can consider Feynman integrals with all possible powers of propagators and try to reduce them, by linear relations, to a finite subset of integrals, the so-called master integrals. Up to now, there are numerous examples of reduction procedures resulting in a finite number of master integrals for various families of Feynman integrals. However, up to now it was just an empirical fact that the reduction procedure results in a finite number of irreducible integrals. It this paper we prove that the number of master integrals is always finite.