Yangian Bootstrap for Conformal Feynman Integrals

TL;DR

This paper demonstrates how Yangian symmetry can be used to uniquely determine conformal Feynman integrals, revealing their structure as hypergeometric functions and connecting to Mellin-Barnes techniques.

Contribution

It introduces a bootstrap approach leveraging Yangian symmetry to fix conformal Feynman integrals, providing explicit solutions and algorithmic procedures.

Findings

The D-dimensional box integral is fixed by symmetry to a combination of Appell functions.

Yangian invariants include the Bloch-Wigner function in 4D.

Constraints for six-point integrals relate to Lauricella functions.

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.