Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms

Abstract

We describe a family of finite, four-dimensional, L-loop Feynman integrals that involve weight-(L+1) hyperlogarithms integrated over (L-1)-dimensional elliptically fibered varieties we conjecture to be Calabi-Yau manifolds. At three loops, we identify the relevant K3 explicitly and we provide strong evidence that the four-loop integral involves a Calabi-Yau threefold. These integrals are necessary for the representation of amplitudes in many theories-from massless φ^{4} theory to integrable theories including maximally supersymmetric Yang-Mills theory in the planar limit-a fact we demonstrate.

From infinity to four dimensions: higher residue pairings and Feynman integrals

“…Finally, the theory of primitive forms, which gave rise to higher residue pairings, has been developed in order to generalize the classic theory of elliptic integrals to more general spaces [6,8]. We expect it to play a crucial role in recent developments connecting Feynman integrals to Calabi-Yau geometries [108][109][110][111][112][113][114][115]. but distinct phases.…”

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

“…In this work, we instead use sequences of residues to identify Calabi-Yau hypersurfaces in Feynman integrals, as done in ref. [67]. In particular, we begin with representations of (here non-marginal) Feynman integrals at L loops in terms of rational integrands involving only 2L integration variables (motivated by the conjectured bound of transcendental weight 2L at L loops in four dimensions).…”

“…In ref. [67], some of the authors provided evidence that the L-loop traintrack integral, depicted in figure 2, involves an integral over a Calabi-Yau (L−1)-fold. There, a manifestly dual-conformally invariant 2L-fold representation was given for this integral:…”

“…and (xy;zw) denotes the cross-ratio 6 Note that we have fixed a typo in f k from the published version of ref. [67].…”

“…In appendices A and B, we provide more background on the desingularization of hypersurfaces in weighted projective space and the computation of Hodge numbers and Euler characteristics. In appendix C we review loop-by-loop Feynman parametrization [67,[79][80][81] and derive a manifestly dual-conformal six-fold representation of the three-loop wheel integral, and in appendix D we derive a dual-conformal nine-fold representation of the four-loop wheel. In the latter case, we also describe several interesting kinematic limits and toy models.…”