Hypergeometry from $\mathrm{\widehat P}$-Symmetry: Feynman Integrals in One and Two Dimensions

TL;DR

The paper demonstrates that track and triangle-track Feynman integrals in $1$ and $2$ dimensions are fully fixed by nonlocal $\widehat{P}$-symmetries of Yangian type, enabling explicit bootstrap of all track graphs up to six external points and four loops, as well as the full one-loop family. It links these results to Aomoto–Gelfand hypergeometric functions and a spectral-transform method, and shows how the 1D results lift to 2D via a diagonal double-copy structure, including conformal double-box integrals. The work provides detailed 1D constructions for many topologies (from triangles to hexagons) and recasts them in hypergeometric languages (Appell, Horn, Lauricella), while also deriving comb-channel conformal partial waves and exploring the AG framework. Altogether, the study offers a comprehensive, symmetry–driven bootstrap of low-dimensional Feynman integrals and a practical bridge to higher dimensions and conformal theories, with potential applications to integrability-inspired computational tools. The results suggest a robust path to extend these methods to massive propagators and more general Witten-diagram configurations, enriching both mathematical and physical perspectives on Feynman integrals.

Abstract

Feynman integrals with generic propagator powers in one and two spacetime dimensions are investigated from various perspectives. In particular, we argue that the class of track integrals at any loop order is fixed by the recently found $\mathrm{\widehat P}$-symmetries of Yangian type. All track integrals up to six external points (and four loops) are bootstrapped explicitly as well as the full family of one-loop integrals at any multiplicity. Moreover, the triangle tracks at generic loop order, which constitute the most generic family of track-type integrals, are bootstrapped in this way. The results are compared to the direct evaluation via a `spectral transform' from the integrability toolbox that turns out to be particularly efficient for position-space tree integrals in lower dimensions. We prove that all $\mathrm{\widehat P}$-symmetries of these integrals can be derived from the framework of Aomoto--Gelfand hypergeometric functions, which applies to integrals in one and two dimensions. Finally, we also demonstrate the method's applicability to conformal integrals by deriving the complete results for all comb-channel conformal partial waves as well as the conformal double-box integral. We explicitly go through all examples of the above integrals in 1D and then provide a straightforward recipe for how to read off their 2D counterparts.

Figures (1)

• Figure 1: Spectral transform of a single propagator. We introduce auxiliary spacetime points $3$ and $4$ as well as a spectral parameter $u$ into the propagator powers.