Elliptic Double-Box Integrals: Massless Scattering Amplitudes beyond Polylogarithms

Abstract

We derive an analytic representation of the ten-particle, two-loop double-box integral as an elliptic integral over weight-three polylogarithms. To obtain this form, we first derive a fourfold, rational (Feynman-)parametric representation for the integral, expressed directly in terms of dual-conformally invariant cross ratios; from this, the desired form is easily obtained. The essential features of this integral are illustrated by means of a simplified toy model, and we attach the relevant expressions for both integrals in ancillary files. We propose a normalization for such integrals that renders all of their polylogarithmic degenerations pure, and we discuss the need for a new "symbology" of mixed iterated elliptic and polylogarithmic integrals in order to bring them to a more canonical form.

Yangian bootstrap for conformal Feynman integrals

“…Below we will also demonstrate that the overall constant is fixed by the star-triangle integral (6) and takes the value c 4 = π 2 . This is in agreement with the results in the literature [17].…”

“…For the double box, the case with unit propagator powers and on-shell legs is known to be described by elliptic functions, whose theory in the context of Feynman integrals is still under construction [6][7][8]32]. It would be very interesting to apply the Yangian PDEs studied in this paper to an ansatz for this integral, once such an ansatz becomes available.…”

“…Note that using the star-triangle relation (6), this integral can be mapped (modulo an external propagator) to a two-loop integral with two connected three-point stars:…”

Embedding Feynman integral (Calabi-Yau) geometries in weighted projective space

“…The three-loop wheel integral allows for a three-parameter toy model similar to that of the elliptic double-box [80]. This toy model is defined by taking all six dual-momentum points defining the three-loop wheel integral to be light-like separated in sequence.…”

“…[67]). We do this by first deriving manifestly dual-conformal-invariant six-fold representations of these integrals using loop-byloop Feynman parametrization [67,[79][80][81]. As both integrals contribute to planar maximally supersymmetric Yang-Mills theory, we expect all six remaining integrations to be transcendental; therefore, each residue mimics a polylogarithmic integration.…”