Cluster algebras for Feynman integrals

Abstract

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 C2 cluster algebra, and we find cluster adjacency relations that restrict the allowed function space. By embedding C2 inside the A3 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 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.

Truncated cluster algebras and Feynman integrals with algebraic letters

“…For example, currently we do not know any positroid cell for two-mass-hard hexagon kinematics. In [49], the alphabet of the latter was conjectured to be a subset of octagon alphabet that are annihilated by first-order differential operators encoding the kinematics. This seems to be a general method when we know the alphabet of G + (4, n)/T , and it would be interesting to study the relation of such subsets to our truncated cluster algebras.…”

“…We have looked at higher-dimensional cases, e.g. for one-mass heptagon kinematics with n = 8, our method gives a co-dimension 2 boundary of G + (4, 8)/T which has 100 + 1 facets, where we have 100 g-vectors and 1 limit ray (the subset from differential operators of [49] is smaller). Since the computation for G + (4, n)/T cluster algebra becomes very difficult beyond n = 8 (there are recent results for n = 9 using a subset of all Plücker coordinates [30]), it is crucial to develop both methods for studying higher-point DCI integrals.…”

“…Last but not least, cluster algebra structures have been observed for Feynman integrals that are not DCI including those with IR divergence; their alphabet can sometimes be obtained from that in DCI case, e.g. the pentagon alphabet with one-massive leg can be obtained from sending a dual point to infinity in the two-mass-hard hexagon kinematics [49]. It would be extremely interesting to find possible truncated cluster algebras for these more general integrals, which should again be directly related to their canonical differential equations.…”

“…the same A 3 and E 6 control n = 6, 7 multi-loop integrals in N = 4 SYM [19,45]. Very recently, cluster algebra structures have been discovered for Feynman integrals beyond those in planar N = 4 SYM [49]: there is strong evidence for a C 2 cluster algebra and adjacency for four-point Feynman integrals with an off-shell leg, and various cluster-algebra alphabets have been found for one-loop integrals, and general five-particle alphabet which play an important role in recent two-loop computations [50][51][52]. Apart from connection to cluster algebras, knowledge of alphabet (and further information) can be used for bootstrapping Feynman integrals [46,53] (see also [54]).…”

Cluster Superalgebras and Stringy Integrals

“…The first two of these four connections are currently confined (see however [12]) to the realm of planar maximally supersymmetric Yang-Mills theory and are tied, in particular, to the rich mathematical structure of the Grassmannian Gr(k, n). It is natural to wonder whether super versions of these connections could be described in terms of cluster super algebras associated to the super Grassmanian Gr(k|l, m|n) (the space of k|l planes in C m|n ).…”