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.
Highlights
Theoretical predictions for particle phenomenology strongly depend on our understanding of Feynman integrals
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 the previous section we have shown that these Yangian invariants have a fine structure; i.e., there are more elementary Yangian invariants whose linear combination is selected by imposing further symmetries of the considered Feynman integrals
Summary
Theoretical predictions for particle phenomenology strongly depend on our understanding of Feynman integrals. In the present paper we explore this connection between Feynman integrals and the theory of integrable models, which play a crucial role for developing analytical methods in all areas of physics These scalar fishnet integrals furnish some of the most important building blocks of quantum field theory at any loop order. Due to its interesting relation to the double box [9] as outlined below, we discuss the nine-variable hexagon integral: 2470-0010=2020=101(6)=066006(22) For this integral results are only known in a threeparticle on-shell case resulting in a function of six variables [10,11]. We discuss the analogous constraints for the double box and the related hexagon integral These constraints are formulated as systems of differential equations in the conformal cross ratios for the respective Feynman integrals. We close with an extended outlook pointing at various promising future directions
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