Operator approach to analytical evaluation of Feynman diagrams

Abstract

The operator approach to analytical evaluation of multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of massless Feynman integrals, such as the integration by parts method and the method of "uniqueness" (which is based on the star-triangle relation), can be drastically simplified by using this operator approach. To demonstrate the advantages of the operator method of analytical evaluation of multi-loop Feynman diagrams, we calculate ladder diagrams for the massless ${\varphi }^3$ theory (analytical results for these diagrams are expressed in terms of multiple polylogarithms). It is shown how operator formalism can be applied to calculation of certain massive Feynman diagrams and investigation of Lipatov integrable chain model.

16 pages. To appear in "Physics of Atomic Nuclei" (Proceedings of SYMPHYS-XII, Yerevan, Armenia, July 03-08, 2006)

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• Three-Body Effective Potential in General Relativity at Second Post-Minkowskian Order and Resulting Post-Newtonian Contributions
• Yangian Bootstrap for Massive Feynman Integrals
• Mirror channel eigenvectors of the $d$-dimensional fishnets
• Ladder and zig-zag Feynman diagrams, operator formalism and conformal triangles
• Conformal Triangles and Zig-Zag Diagrams
• Mellin$-$Barnes Transformation for Two-Loop Master-Diagram