Speaker
Description
In recent years it has become clear that the critical points of the "twist" play a crucial role in the study of relations between dimensionally regulated Feynman integrals. There is a great deal of understanding of these points both in the Lee-Pomeransky counting of master integrals and in the application of intersection theory to Feynman integrals. However, their role in the structure of integration by parts relations remains less well understood.
In this talk, building on insights from intersection theory, we work in the syzygy formalism for relations between Feynman integrals and analyze its large-$\epsilon$ limit. Remarkably, we find that the critical locus of the twist naturally emerges in this setting. Moreover, the large-$\epsilon$ limit singles out a subset of syzygies, which we call "critical syzygies". We use techniques from commutative algebra to show that, when the critical locus is isolated and a multiplicity criterion is satisfied, critical syzygies generate all relations among Feynman integrals. This analysis provides a new perspective on the Lee-Pomeransky integral counting.
Finally, we discuss concrete applications at both one and two loops to demonstrate the relevance of critical syzygies to Feynman integral reduction in cutting-edge LHC scattering processes.
