The Feyn-Structure of Yangian Symmetry

TL;DR

This paper demonstrates that Yangian symmetry constrains tree-level position-space Feynman diagrams and extends to multi-loop integrals, revealing a fundamental algebraic structure underlying Feynman integrals beyond integrability.

Contribution

It proves Yangian level-one momentum symmetry for all tree-level position-space Feynman diagrams with distinct external points and extends this to certain multi-loop integrals, generalizing previous symmetry results.

Findings

Yangian symmetry applies to all tree-level position-space Feynman diagrams.

The symmetry extends to multi-loop integrals without loops of loops.

Results suggest new differential equations for bootstrapping complex Feynman integrals.

Abstract

Yangian-type differential operators are shown to constrain Feynman integrals beyond the restriction to integrable graphs. In particular, we prove that all position-space Feynman diagrams at tree level feature a Yangian level-one momentum symmetry as long as their external coordinates are distinct. This symmetry is traced back to a set of more elementary bilocal operators, which annihilate the integrals. In dual momentum space, the considered Feynman graphs represent multi-loop integrals without `loops of loops', generalizing for instance the family of so-called train track or train track network diagrams. The extension of these results to integrals with massive propagators on the boundary of the Feynman graph is established. When specializing to the dual conformal case, where propagator powers sum up to the spacetime dimension at each position-space vertex, the symmetry extends to the full dual conformal Yangian. Hence, our findings represent a generalization of the statements on the Yangian symmetry of Feynman integrals beyond integrability and reveal its origin lying in a set of more elementary bilocal annihilators. Previous applications of the Yangian suggest to employ the resulting differential equations for bootstrapping multi-loop integrals beyond the dual conformal case. The considered bilocal constraints on Feynman integrals resemble the definition of conformal partial waves via Casimir operators, but are based on a different algebraic structure.