Alphabet of one-loop Feynman integrals

TL;DR

This paper uncovers the universal structure of the alphabet of one-loop Feynman integrals using the Baikov representation, facilitating the construction of differential equations and the bootstrap of solutions.

Contribution

It introduces a universal structure for the alphabet of one-loop Feynman integrals, expressed via Gram determinants, and links convergent and divergent cases through limits.

Findings

Letters expressed as Gram determinants

Divergent letters obtained from convergent cases

Facilitates construction of differential equations

Abstract

In this paper, we present the universal structure of the alphabet of one-loop Feynman integrals. The letters in the alphabet are calculated using the Baikov representation with cuts. We consider both convergent and divergent cut integrals and observe that letters in the divergent cases can be easily obtained from convergent cases by applying certain limits. The letters are written as simple expressions in terms of various Gram determinants. The knowledge of the alphabet enables us to easily construct the canonical differential equations of the $ d\log $ form and aids in bootstrapping the symbols of the solutions.