Multi-loop Feynman integrals and conformal quantum mechanics

TL;DR

Analytical evaluation of multi-loop Feynman integrals is challenging due to diagram proliferation in perturbative QFT. The authors develop an operator-based algebraic framework that recasts integration-by-parts and star-triangle relations as commutator identities and traces, transforming diagram calculations into Green-function problems. They explicitly treat $L$-loop ladder diagrams in massless $\phi^3$ theory, showing these integrals correspond to Green functions of $D$-dimensional conformal quantum mechanics and introducing a generating-function approach with symmetry-based reductions. The work bridges perturbative diagrammatics with conformal quantum-mechanical dynamics and hints at deeper connections to integrable systems, with potential extensions to massive propagators and applications to high-energy QCD-inspired models.

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 $φ^3$ 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 $φ^3$ scalar field theory are given by the Green function for the conformal quantum mechanics.