Critical points and number of master integrals

By Roman N. Lee, A. Pomeransky. 2013

188 citations recorded by the source corpus.

A bstractWe consider the question about the number of master integrals for a multiloop Feynman diagram. We show that, for a given set of denominators, this number is totally determined by the critical points of the polynomials entering either of the two representations: the parametric representation and the Baikov representation. In particular, for the parametric representation the corresponding polynomial is just the sum of Symanzik polynomials. The relevant topological invariant is the sum of the Milnor numbers of the proper critical points. We present a Mathematica package Mint to automatize the counting of the master integrals for the typical case, when all critical points are isolated.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5c409969aacf55f4c8732215af977b2608791821

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Citations in the corpus

A quasi-finite basis for multi-loop Feynman integrals (2014, background): “It is well-known that, for a given topology and its subtopologies, there is a finite number of linearly independent integrals [37, 38], which are usually referred to as master integrals.”
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph (2017, unknown): “05 46”
Decomposition of Feynman integrals on the maximal cut by intersection numbers (2019, methodology): “[59], it was proposed that the number of master integrals in a given sector equals the number of critical points of the Baikov polynomial on the cut corresponding to this sector: we refer to it as the Lee–Pomeransky (LP) criterion.”
Cuts of Feynman Integrals in Baikov representation (2017, methodology): “In this contribution we present results [65] on FI based on Baikov representation [66, 67, 68, 69, 70, 71].”
Feynman integrals and intersection theory (2018, methodology): “Early signs of this fact were found by Lee and Pomeransky, who applied Morse theory to relate the number of MIs to counting of the critical points of B D−D∗ 2 [33].”
Collider physics at the precision frontier (2020, background): “From a more formal side, a very promising alternative to IBP reduction has been suggested based on the intersection theory of differential forms [518, 519, 500, 520–523], for which the Baikov representation [524, 525] is particularly suited.”
Mathematical aspects of scattering amplitudes (2014, background): “While it can be proven that the number of master integrals is always finite [48], and their number can be predicted from the topology [49], there are of course different ways of choosing the set of master integrals.”