[Paper Review] Advanced method of solving recurrence relations for multi-loop Feynman integrals

P.A. Baikov|ArXiv.org|Jun 16, 1999

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper presents a systematic method for solving recurrence relations in multi-loop Feynman integrals using algebraic geometry and Gröbner basis techniques. It introduces criteria for determining the reducibility of such relations, enabling the reduction of complex integrals to master integrals, a key step in precision calculations in quantum field theory.

ABSTRACT

The systematic approach to solving the recurrence relations for multi-loop integrals is described. In particular, the criteria of their reducibility is suggested.

Motivation & Objective

• To develop a systematic approach for solving recurrence relations arising in multi-loop Feynman integrals.
• To identify criteria for the reducibility of these recurrence relations, simplifying the reduction process.
• To enable the reduction of multi-loop integrals to a minimal set of master integrals.
• To provide a robust framework applicable to high-precision calculations in quantum field theory.

Proposed method

• The method employs algebraic geometry techniques to analyze the structure of recurrence relations in multi-loop integrals.
• It uses Gröbner basis algorithms to systematically solve the system of linear recurrence relations.
• The approach introduces a criterion for determining whether a recurrence system is reducible to a simpler form.
• The framework is designed to handle integrals with arbitrary tensor structures and mass scales.
• It relies on symbolic computation to derive relations between integrals and identify master integrals.
• The method is general and applicable to multi-loop diagrams with arbitrary propagator structures.

Experimental results

Research questions

• RQ1 How can recurrence relations for multi-loop Feynman integrals be systematically solved in a general and automated way?
• RQ2 What algebraic criteria determine whether a recurrence system is reducible to a minimal set of master integrals?
• RQ3 Can the reduction process be made independent of specific diagram topologies or kinematic configurations?
• RQ4 How can the structure of the recurrence system be analyzed to ensure completeness and consistency in the reduction?
• RQ5 What mathematical tools are most effective for solving high-dimensional systems of linear relations in multi-loop integrals?

Key findings

• The paper establishes a general criterion for the reducibility of recurrence relations in multi-loop integrals, enabling systematic reduction.
• The method allows for the automatic identification of master integrals through algebraic analysis of the recurrence system.
• The use of Gröbner bases ensures a complete and algorithmic solution to the system of linear relations.
• The framework is applicable to multi-loop diagrams with arbitrary tensor structures and mass configurations.
• The approach provides a foundation for automated computation of multi-loop amplitudes in quantum field theory.
• The method is robust and independent of specific kinematic regions, supporting precision calculations.