Non-Local Symmetries of Planar Feynman Integrals

TL;DR

The paper proves that planar scalar Feynman graphs are invariant under the Yangian level-one momentum symmetry $\hat{P}^\mu$ provided linear constraints on propagator powers hold, connecting momentum-space conformal invariance of planar mesh integrals to a non-local region-momentum symmetry. By planarizing simplex graphs and employing soft limits, it shows $\hat{P}^\mu I_n(x)=0$ for massless cases and extends to massive boundary propagators via a massive $\widehat{\mathbb{P}}^\mu$, unifying prior observations of Yangian invariance in fishnet and loom graphs. A precise relation between propagator powers and evaluation parameters is derived, including a recursive formula for $s_i$, and the results hold under broad classes of loop configurations. These findings reinforce the integrability of a wide class of scalar Feynman integrals and point toward possible super-symmetric extensions and deeper links to planar $\mathcal{N}=4$ SYM, with future questions about the origin of integrability in planarization vs non-planarity.

Abstract

We prove the invariance of scalar Feynman graphs of any planar topology under the Yangian level-one momentum symmetry given certain constraints on the propagator powers. The proof relies on relating this symmetry to a planarized version of the conformal simplices of Bzowski, McFadden and Skenderis. In particular, this proves a momentum-space analogue of the position-space conformal condition on propagator powers. When combined with the latter, the invariance under the level-one momentum implies full Yangian symmetry of the considered graphs. These include all scalar Feynman integrals for which a Yangian symmetry was previously demonstrated at the level of examples, e.g. the fishnet or loom graphs, as well as generalizations to graphs with massive propagators.

Figures (5)

• Figure 1: Planarizing a momentum-space Feynman graph.
• Figure 2: Left: A generic nine-point mesh integral in momentum space. Middle: Removing certain propagators from the mesh, one obtains a nine-point graph, which is dual to a position-space tree (blue). Right: A six-point integral dual to a position-space loop graph (blue), and obtained by removing certain propagators from the generic mesh and taking three external momenta soft; the latter three vertices are moved to the inside of the black graph after the soft limit.
• Figure 3: Proof of the massive level-one momentum symmetry in terms of momentum-space graphs. We start with a planar mesh with massive boundary, whose dual (tree) graph is invariant under $\mathbb{ \mathrm{\widehat{P}}}^\mu$ alias momentum-space $\mathbb{\bar{K}}^\mu$Loebbert:2024qbw. Next we use the induction of Bzowski:2020kfw, to extend the $\mathbb{\bar{K}}^\mu$ invariance to non-planar massive cycle mesh integrals. Finally, the latter are planarized via soft limits, which preserve the $\mathbb{ \mathrm{\widehat{P}}}^\mu$ symmetry and lead to (position-space loop) integrals with massive boundary.
• Figure 4: Left: Coincidence limit that can be removed for the $\mathrm{\widehat{P}}$ constraint $a'+b'+c'=D/2$ by virtue of the star-triangle relation. Right: Graph with a coincidence limit that cannot be removed using the star-triangle identity but features a $\mathrm{\widehat{P}}$ symmetry for $a'+b'+c'+d'=D$.
• Figure 5: Illustration on the prescription for reading off the evaluation parameters by going along the boundary of the graph in $x$-space (blue) or its dual (black).