The Feyn-Structure of Yangian Symmetry

TL;DR

This work demonstrates that position-space Feynman diagrams at tree level are constrained by a Yangian level-one momentum symmetry, with bilocal annihilators serving as the building blocks for the full symmetry. By expressing the level-one generator \\widehat{P}^\\mu in terms of bilocal densities and employing a recursive scheme of evaluation parameters, the authors show that all such graphs — including stars, tracks, and track networks — are annihilated by \\widehat{P}^\\mu without imposing dual conformal constraints on propagator powers. They also extend these results to massive boundary propagators and provide a dual momentum-space interpretation, linking the x-space Yangian to a momentum-space conformal structure. The findings generalize Yangian-invariance beyond integrable cases and suggest Yangian-based bootstrap approaches for multi-loop integrals, with potential connections to conformal partial waves and GKZ geometry. Overall, the paper uncovers a hierarchical, bilocal origin of Yangian symmetry in Feynman integrals and lays groundwork for systematic bootstrap and geometric explorations.

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.

Figures (5)

• Figure 1: Generic position ($x$-)space tree diagram (black) and its dual momentum ($p$-)space graph (green) related via the duality transformation $p_j=x_j-x_{j+1}$. The findings of the present paper require that internal propagators are massless (dashed lines), while external propagators can be either massive or massless (solid lines).
• Figure 2: Left hand side: Train track diagram in $x$-space (black) and $p$-space (green). Right hand side: Train track network diagram.
• Figure 3: Example of a two-point symmetry annihilating a two-loop graph. All two-point level-one momentum densities $\mathrm{\widehat{P}}_{jk}^\mu$ with legs $j$ and $k$ attached to the same integration vertex provide annihilators of the integral.
• Figure 4: Part of the boundary of a position space Feynman graph. The evaluation parameters $s_j$ entering into the Yangian level-one generator are defined recursively. In particular, $s_{j+1}$ is obtained from $s_j$ by adding the terms on top of the arrow connecting the two subsequent legs. Here, we distinguish external propagators (solid lines) and internal propagators (dashed lines), which connect the two neighboring external points $j$ and $j+1$.
• Figure 5: The symmetries discussed in this paper can be considered in different spaces. Here, the duality transformation on the left edge may turn local symmetries into non-local symmetries or vice versa (cf. Section \ref{['sec:MomSpace ']}), while the Fourier transform on the right edge may turn first order differential operators into second order differential operators, e.g. the special conformal generator.