Yangian bootstrap for conformal Feynman integrals

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. *

Massive Conformal Symmetry and Integrability for Feynman Integrals

“…If this constraint is not satisfied, we still have level-one momentum symmetryP μ , which corresponds to a massive generalization of momentum space conformal symmetry that we introduce below. In the massless limit this matches the observations of [5][6][7] on the integrability of massless Feynman integrals in D ¼ 3, 4, and 6 spacetime dimensions.…”

“…In addition to limiting the number of variables, this new symmetry strongly constrains the functional form of the integrals. While these symmetry properties naturally extend the observations on the integrability of massless Feynman integrals [5][6][7], to the knowledge of the authors this is the first occurence of quantum integrability in massive quantum field theory in D > 2 spacetime dimensions.…”

“…Yangian bootstrap.-Let us now employ the above nonlocal symmetry to bootstrap Feynman integrals with massive propagators. This discussion extends the algorithm of [7] to the massive case. We start with some examples containing conformal vertices obeying the condition P j a j ¼ D. Notably, after solving these integrals, we can obtain an infinite class of integrals by acting with r mass derivatives ∂ m k , which yields an integral with propagator weightsã j with P jãj ¼ D þ r. Conformal, two points, two masses, one loop: As a simple starting example consider the two-point integral (see, e.g., [10,13,14] [15]:…”

“…This may be possible along the lines of the massless case [5,6], but it could be tricky to find a proof that applies directly to all cases considered in this Letter (e.g., to generic spacetime dimensions D). Second, the applicability of Yangian symmetry to massive Feynman integrals and generic (double) n-gons allows us to refine the algorithm of [7] on many more examples whose complexity interpolates between the simplest massless cases with 2 and 9 variables, respectively. This may finally open the door to computing, e.g., the massless hexagon or double box integral.…”

Massive fishnets

“…While non-unitarity may be considered a significant drawback, it is compensated by a one-to-one correspondence between correlation functions (or scattering amplitudes) and Feynman integrals. This correspondence allows to translate the symmetries of correlators directly to the fundamental building blocks of generic quantum field theories: N = 4 SYM → γ-deformation → fishnet theory → Feynman graphs (1.1) Hence, the integrability features of the AdS/CFT duality find their way to phenomenologically interesting quantities and can be used to fix certain Feynman integrals completely [41].…”