Resummation methods for Master Integrals

TL;DR

This paper introduces two resummation techniques based on the Simplified Differential Equations approach for master integrals, enabling easier boundary condition determination and basis reduction for multi-loop Feynman integrals.

Contribution

It presents novel resummation methods that leverage residue matrices and boundary relations to simplify solving canonical bases of master integrals.

Findings

Successfully applied methods to three-loop ladder-box integrals

Derived a canonical basis for massless three-loop ladder-box

Achieved solutions for complex multi-loop integrals

Abstract

We present in detail two resummation methods emerging from the application of the Simplified Differential Equations approach to a canonical basis of master integrals. The first one is a method which allows for an easy determination of the boundary conditions, since it finds relations between the boundaries of the basis elements and the second one indicates how using the $x \rightarrow 1$ limit to the solutions of a canonical basis, one can obtain the solutions to a canonical basis for the same problem with one mass less. Both methods utilise the residue matrices for the letters $\{0,1\}$ of the canonical differential equation. As proof of concept, we apply these methods to a canonical basis for the three-loop ladder-box with one external mass off-shell, obtaining subsequently a canonical basis for the massless three-loop ladder-box as well as its solution.