Baikov representations, intersection theory, and canonical Feynman integrals

Abstract

The method of canonical differential equations is an important tool in the calculation of Feynman integrals in quantum field theories. It has been realized that the canonical bases are closely related to d-dimensional d log-form integrands. In this work, we explore the generalized loop-by-loop Baikov representation, and clarify its relation and difference with Feynman integrals using the language of intersection theory. We then utilize the generalized Baikov representation to construct d-dimensional d log-form integrands, and discuss how to convert them to Feynman integrals. We describe the technical details of our method, in particular how to deal with the difficulties encountered in the construction procedure. Our method provides a constructive approach to the problem of finding canonical bases of Feynman integrals, and we demonstrate its applicability to complicated scattering amplitudes involving multiple physical scales.

Module Intersection and Uniform Formula for Iterative Reduction of One-loop Integrals

“…To determine the number of master integrals, we can consider the maximum cut of this sector in Baikov representation, i.e., setting z i = 0 in G. If G| z=0 is a nonzero constant, the number of master integrals is just one by counting critical points [33,58,59]. But if G| z=0 = 0, there is no master integral in this sector.…”

Intersection Numbers in Quantum Mechanics and Field Theory

“…In this work, we elaborate on the role of the intersection numbers of twisted cocycles, which is an operation based on the interplay of Stokes' and Residues' theorems, originally introduced in a purely mathematical context, for the study of hypergeometric integrals [6], and whose importance in physics has been made manifest more recently in the context of scattering amplitudes and Feynman integrals evaluation [24-29, 32, 33], recently discussed also in [34][35][36][37][38][39][40] (see also [41]). These applications triggered the development of refined methods for the evaluation of intersection numbers [26,28,30,31,[42][43][44], and other interesting physics applications [45][46][47][48][49][50][51].…”

IBP reduction via Gröbner bases in a rational double-shift algebra

“…More recently, new ideas towards a more direct reduction procedure have been developed. They are mostly based on syzygy equations [18][19][20][21], algebraic geometry [22][23][24], and intersection numbers [25][26][27][28][29][30][31][32]. In these proceedings we report on work in progress [33], where we choose an approach to IBP reduction that is based on Gröbner bases and hence leaves the propagator powers parametric.…”