Cluster algebras for Feynman integrals

We initiate the study of cluster algebras in Feynman integrals in dimensional regularization. We provide evidence that four-point Feynman integrals with one off-shell leg are described by a $C_{2}$ cluster algebra, and we find cluster adjacency relations that restrict the allowed function space. By embedding $C_{2}$ inside the $A_3$ cluster algebra, we identify these adjacencies with the extended Steinmann relations for six-particle massless scattering. The cluster algebra connection we find restricts the functions space for vector boson or Higgs plus jet amplitudes, and for form factors recently considered in $mathcal{N}=4$ super Yang-Mills. We explain general procedures for studying relationships between alphabets of generalized polylogarithmic functions and cluster algebras, and use them to provide various identifications of one-loop alphabets with cluster algebras. In particular, we show how one can obtain one-loop alphabets for five-particle scattering from a recently discussed dual conformal eight-particle alphabet related to the $G(4,8)$ cluster algebra.

A Novel Algorithm for Nested Summation and Hypergeometric Expansions

We consider a class of sums over products of Z-sums whose arguments differ by a symbolic integer. Such sums appear, for instance, in the expansion of Gauss hypergeometric functions around integer indices that depend on a symbolic parameter. We present a telescopic algorithm for efficiently converting these sums into generalized polylogarithms, Z-sums, and cyclotomic harmonic sums for generic values of this parameter. This algorithm is illustrated by computing the double pentaladder integrals through ten loops, and a family of massive self-energy diagrams through $O(epsilon^6)$ in dimensional regularization. We also outline the general telescopic strategy of this algorithm, which we anticipate can be applied to other classes of sums.

Evaluating the 6-point Remainder Function Near the Collinear Limit

The simplicity of maximally supersymmetric Yang-Mills theory makes it an ideal theoretical laboratory for developing computational tools, which eventually find their way to QCD applications. In this contribution, we continue the investigation of a recent proposal by Basso, Sever and Vieira, for the nonperturbative description of its planar scattering amplitudes, as an expansion around collinear kinematics. The method of arXiv:1310.5735, for computing the integrals the latter proposal predicts for the leading term in the expansion of the 6-point remainder function, is extended to one of the subleading terms. In particular, we focus on the contribution of the 2-gluon bound state in the dual flux tube picture, proving its general form at any order in the coupling, and providing explicit expressions up to 6 loops. These are included in the ancillary file accompanying the version of this article on the arXiv.

Hexagon Wilson Loop OPE and Harmonic Polylogarithms

A recent, integrability-based conjecture in the framework of the Wilson loop OPE for N=4 SYM theory, predicts the leading OPE contribution for the hexagon MHV remainder function and NMHV ratio function to all loops, in integral form. We prove that these integrals evaluate to a particular basis of harmonic polylogarithms, at any order in the weak coupling expansion. The proof constitutes an algorithm for the direct computation of the integrals, which we employ in order to obtain the full (N)MHV OPE contribution in question up to 6 loops, and certain parts of it up to 12 loops. We attach computer-readable files with our results, as well as an algorithm implementation which may be readily used to generate higher-loop corrections. The feasibility of obtaining the explicit kinematical dependence of the first term in the OPE in principle at arbitrary loop order, offers promise for the suitability of this approach as a non-perturbative description of Wilson loops/scattering amplitudes.