Yangian Bootstrap for Conformal Feynman Integrals

TL;DR

The paper develops a bootstrap program for conformal Feynman integrals using Yangian symmetry, applying it first to the D-dimensional four-point box and then to more intricate six-point configurations (double box and hexagon). For the box, Yangian invariance together with permutation symmetry fixes the integral to a specific Appell $F_4$-type structure in cross ratios, with the Bloch–Wigner function emerging as a special invariant in 4D and the overall constant fixed via a star-triangle coincidence limit. In the higher-point cases, the Yangian constraints translate into PDEs in multiple cross ratios, with the box solvable in full generality and the hexagon describable in terms of generalized Lauricella functions; however, the full linear combination matching the integral remains subtle due to convergence and region issues. The work also highlights a close relationship with Mellin–Barnes representations, where Yangian invariants correspond to sums of residues across compatible pole cones, offering a unified view of hypergeometric building blocks, region-wise analyticity, and potential recursive constants. Overall, the study demonstrates that integrability via the Yangian can powerfully constrain conformal Feynman integrals, providing algorithmic bootstrap procedures and revealing deep connections to classical integral techniques and multivariable hypergeometric functions, with clear paths for future extension to more complex diagrams and deformed propagator powers.

Abstract

We explore the idea to bootstrap Feynman integrals using integrability. In particular, we put the recently discovered Yangian symmetry of conformal Feynman integrals to work. As a prototypical example we demonstrate that the D-dimensional box integral with generic propagator powers is completely fixed by its symmetries to be a particular linear combination of Appell hypergeometric functions. In this context the Bloch-Wigner function arises as a special Yangian invariant in 4D. The bootstrap procedure for the box integral is naturally structured in algorithmic form. We then discuss the Yangian constraints for the six-point double box integral as well as for the related hexagon. For the latter we argue that the constraints are solved by a set of generalized Lauricella functions and we comment on complications in identifying the integral as a certain linear combination of these. Finally, we elaborate on the close relation to the Mellin-Barnes technique and argue that it generates Yangian invariants as sums of residues.

Figures (2)

• Figure 1: Regions I--III (green) as defined in (\ref{['eq:regionI']},\ref{['eq:regionII']},\ref{['eq:regionIII ']}). The striped (red) area indicates the region of Euclidean physical kinematics, while its complement in the above graph corresponds to Minkowski signature. The dashed boundary between the two regions is given by the line $4u = (1+u-v)^2$ or $z=\bar{z}$, respectively.
• Figure 2: Singularity structure of the Mellin--Barnes integrand (\ref{['eqn:MBintegrandboxint ']}). Colored lines correspond to poles of Gamma functions. The orange dot in the center of the plot marks the base point of the integration and can be moved inside the meshed (blue) triangle without changing the value of the integral. Red cones mark the regions in which residues need to be summed to obtain a valid series representation of the integral.