Yangian Symmetry for Fishnet Feynman Graphs

TL;DR

This paper demonstrates that fishnet Feynman graphs possess a Yangian symmetry $Y(\mathfrak{so}(2,4))$ extending their conformal invariance, providing differential constraints for their all-loop integrals. The authors develop an RTT monodromy framework built from conformal Lax operators to prove Yangian invariance for scalar fishnets and show how to incorporate on-shell legs, fermions, and higher dimensions (3D and 6D) by adapting Lax representations and using a momentum-space formulation where needed. They derive explicit invariance conditions for the basic box integral $I_4$ and its $n$-point generalizations, and reveal a mechanism to deform propagators while preserving Yangian symmetry under a conformal constraint $\sum_k \alpha_k=4$. The results imply that integrability techniques, such as the Bethe Ansatz and QISM, may be applied to the otherwise intractable fishnet integrals, illuminate the role of double-scaled conformal field theories in the planar AdS/CFT correspondence, and motivate extending the framework to the full double-scaled model with richer field content.

Abstract

Various classes of fishnet Feynman graphs are shown to feature a Yangian symmetry over the conformal algebra. We explicitly discuss scalar graphs in three, four and six spacetime dimensions as well as the inclusion of fermions in four dimensions. The Yangian symmetry results in novel differential equations for these families of largely unsolved Feynman integrals. Notably, the considered fishnet graphs in three and four dimensions dominate the correlation functions and scattering amplitudes in specific double scaling limits of planar, gamma-twisted N=4 super Yang-Mills or ABJM theory. Consequently, the study of fishnet graphs allows us to get deep insights into the integrability of the planar AdS/CFT correspondence.

Figures (5)

• Figure 1: Example of a conformal scalar fishnet Feynman graph in four dimensions. Filled blobs denote loop integrations, white blobs represent external points $x_k$.
• Figure 2: The box integral in momentum (black) and dual (green) coordinate space. It will be convenient to distinguish (a) off-shell and (b) on-shell external momenta. Note the relabelling of coordinates with respect to (\ref{['eq:boxint ']}).
• Figure 3: (a): Monodromy encircling a sample fishnet graph representing the left hand side of (\ref{['eq:moneigen ']}). (b): Intermediate step of the proof of Yangian symmetry.
• Figure 4: Rules employed to prove the Yangian invariance. (a): The intertwining relation (\ref{['eq:inter ']}). (b) and (c): Pulling the monodromy contour through an integration vertex, cf. (\ref{['eq:rule2 ']}).
• Figure 5: Double-box integral with massless external legs. Dashed black lines represent delta functions $\delta(x_{i,i+1}^2)$ forcing the momenta (dotted lines) on shell.