The canonical differential equations of the one-loop-like integrals

TL;DR

This paper systematically studies one-loop-like integrals by establishing IBP relations, identifying canonical master integrals, and deriving their differential equations, advancing the understanding of high loop integral computations.

Contribution

It introduces a systematic method for analyzing one-loop-like integrals, including IBP relations, reduction techniques, and canonical differential equations, enhancing computational approaches.

Findings

Established IBP relations for the integral family

Identified canonical master integrals

Derived canonical differential equations

Abstract

Recently, a new approach for high loop integrals has been proposed in \cite{Huang:2024nij}, where the whole parameter integration has been divided into two parts: a one-loop-like integration and the remaining parameter integration. In this paper, we systematically study the one-loop-like integrals. We establish the IBP relations for the integral family and show how to complete the reduction. We find the canonical master integrals and write down the corresponding canonical differential equations.