Yangian Symmetry for Bi-Scalar Loop Amplitudes

TL;DR

This work proves that planar amplitudes in the four-dimensional bi-scalar chiFT$_4$ exhibit an all-loop conformal Yangian symmetry. By encoding fishnet disc graphs into an inhomogeneous monodromy built from conformal Lax operators, the authors construct explicit level-one Yangian generators and establish invariance for both regular and irregular boundary graphs, including cuts that render on-shell amplitudes. The analysis connects dual conformal symmetry to a Yangian structure in momentum space and presents the first realization of the Yangian via Drinfeld’s framework, with a detailed expansion of the monodromy and cyclicity constraints. The results suggest powerful integrability tools for evaluating bi-scalar fishnet integrals and hint at deeper connections to the broader AdS/CFT integrability program and potential generalizations to related models.

Abstract

We establish an all-loop conformal Yangian symmetry for the full set of planar amplitudes in the recently proposed integrable bi-scalar field theory in four dimensions. This chiral theory is a particular double scaling limit of gamma-twisted weakly coupled N=4 SYM theory. Each amplitude with a certain order of scalar particles is given by a single fishnet Feynman graph of disc topology cut out of a regular square lattice. The Yangian can be realized by the action of a product of Lax operators with a specific sequence of inhomogeneity parameters on the boundary of the disc. Based on this observation, the Yangian generators of level one for generic bi-scalar amplitudes are explicitly constructed. Finally, we comment on the relation to the dual conformal symmetry of these scattering amplitudes.

Figures (16)

• Figure 1: Bi-scalar amplitude diagrams. (A): $M_1 = M_2 = 3$ (B): $M_1 =2, M_2 = 3$.
• Figure 2: A generic fishnet graph with regular boundary. It is drawn by solid lines. It depends on a number of variables $x_i^{\mu}$ which are coordinates of external legs. Each solid line of the fishnet graph represents a scalar propagator $x_{ij}^{-2}$. Integrations are over positions of vertices (denoted by filled blobs). The dual graph is drawn by dotted lines. The dual graph lives in the momentum representation with integrations over loop momenta. Its external momentum variables are defined as $p^\mu_i = x^\mu_i -x^\mu_{i+1}$. The dual graph does not necessarily correspond to an amplitude in the bi-scalar theory, since it could have interaction vertices of valency different from four. The external legs of the dual graph are amputated. The inflowing off-shell momenta are denoted by double lines.
• Figure 3: Correlator diagram and its dual (without external legs), which cannot be cut out of a simple sheet of regular square lattice. However, it can be cut out of a "double-sheet" regular square lattice with a branchpoint. More general graphs can be cut out of the lattices having various "conical" singularities, see footnote \ref{['ftnt3']}.
• Figure 4: Two-loop diagrams: double box topology in momentum variables (integration over momenta flowing in loops) and its dual double cross topology in region momentum variables (integration over position of the vertices -- filled blobs). (A): All inflowing momenta (depicted by double lines) are off shell, $p^2_{i} = x^2_{i\,i+1} \neq 0$ at $i = 1, \ldots, 6$. (B): Amplitude diagram for scattering of massless particles, i.e. the inflowing momenta (depicted by loosely dotted gray lines) are light-like $p^2_{i} = x^2_{i\,i+1} =0$ at $i = 1, \ldots, 10$. These constraints are imposed by means of delta functions $\delta(x_{i \, i+1}^2)$ depicted by dashed black lines.
• Figure 5: In this picture, a bi-scalar scattering amplitude of massless scalars is represented in the dual graph depicted by dotted lines. It has doubled (w.r.t. the previous Fig. \ref{['Fig2']}) external legs at the convex corners of the boundary, and no legs at the concave corners. The corresponding additional momentum variables at convex corners should satisfy the momentum conservation condition at each dual vertex. We obtain the admissible bi-scalar amplitude by identifying the points of the boundary of the original graph which end in the same square (i.e., the same site of the original lattice. Such points are surrounded by ellipses in the above figure. The masslessness is ensured by additional factors $\delta(p_j^2)$ multiplying the external legs of the dual graph.