Multi-loop Feynman integrals and conformal quantum mechanics

TL;DR

This paper introduces an algebraic method to simplify the analytical calculation of multi-loop Feynman integrals, connecting them to Green functions in conformal quantum mechanics, demonstrated through ladder diagrams in massless $\,\phi^3$ theory.

Contribution

The paper presents a novel algebraic approach that simplifies multi-loop Feynman integral evaluations by linking them to quantum mechanical Green functions, especially in conformal cases.

Findings

Algebraic method simplifies multi-loop Feynman integral calculations.

Evaluation of ladder diagrams reduces to conformal quantum mechanics Green functions.

Demonstrated advantages over traditional integration by parts and star-triangle methods.

Abstract

New algebraic approach to analytical calculations of D-dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts and star-triangle relation methods, can be drastically simplified by using this algebraic approach. To demonstrate the advantages of the algebraic method of analytical evaluation of multi-loop Feynman diagrams, we calculate ladder diagrams for the massless $\phi^3$ theory. Using our algebraic approach we show that the problem of evaluation of special classes of Feynman diagrams reduces to the calculation of the Green functions for specific quantum mechanical problems. In particular, the integrals for ladder massless diagrams in the $\phi^3$ scalar field theory are given by the Green function for the conformal quantum mechanics.