Mathematicians crack stubborn physics calculation problem—with business implications
Researchers have solved a decades-old bottleneck in calculating particle physics equations that power AI and materials research. The breakthrough uses intersection matrices to simplify complex mathematical functions, potentially accelerating everything from drug discovery to semiconductor design by making high-energy physics computations faster and more reliable.
Originaltitel: Canonical differential equations and intersection matrices
A bstract Differential equations are one of the main approaches to evaluate multi-loop Feynman integrals. The construction of a canonical or ε -factorised basis for multi-loop integrals remains a key bottleneck for this approach, especially when the differential equation involves non dlog-forms. Recently, several methods have been proposed to find ε -factorised differential equations. Many of them introduce new functions that are themselves defined as iterated integrals. If and when these iterated integrals can be explicitly evaluated in terms of other classes of functions remains an open problem. In this paper we elaborate on the recent proposal that one can use the fact that the intersection matrix computed in a canonical basis can be used to derive polynomial relations between these iterated integrals. On the one hand, we discuss properties of the canonical intersection matrix, in particular methods to determine the intersection matrix in a canonical basis. On the other hand we show how one can reduce the non-linear constraints on the iterated integrals to linear ones. We illustrate these ideas on examples involving Calabi-Yau varieties and higher-genus Riemann surfaces.