Baikov representations, intersection theory, and canonical Feynman integrals

TL;DR

This paper introduces a generalized Baikov representation and intersection theory approach to construct canonical Feynman integrals, improving the calculation of complex scattering amplitudes in quantum field theory.

Contribution

It presents a new constructive method using generalized Baikov representation and intersection theory to find canonical bases of Feynman integrals, applicable to multi-scale scattering amplitudes.

Findings

Constructed $d$-dimensional $d ext{log}$-form integrands from the generalized Baikov representation.

Clarified the relation between Baikov representation and Feynman integrals using intersection theory.

Demonstrated the method's applicability to complex multi-scale scattering processes.

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 introduce the concept of 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.