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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.
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We propose that the symbol alphabet for classes of planar, dual-conformal-invariant Feynman integrals can be obtained as truncated cluster algebras purely from their kinematics, which correspond to boundaries of (compactifications of) $G_+(4,n)/T$ for the $n$-particle massless kinematics. For one-, two-, three-mass-easy hexagon kinematics with $n=7,8,9$, we find finite cluster algebras $D_4$, $D_5$ and $D_6$ respectively, in accordance with previous result on alphabets of these integrals. As the main example, we consider hexagon kinematics with two massive corners on opposite sides and find a truncated affine $D_4$ cluster algebra whose polytopal realization is a co-dimension 4 boundary of that of $G_+(4,8)/T$ with 39 facets; the normal vectors for 38 of them correspond to g-vectors and the remaining one gives a limit ray, which yields an alphabet of $38$ rational letters and $5$ algebraic ones with the unique four-mass-box square root. We construct the space of integrable symbols with this alphabet and physical first-entry conditions, whose dimension can be reduced using conditions from a truncated version of cluster adjacency. Already at weight $4$, by imposing last-entry conditions inspired by the $n=8$ double-pentagon integral, we are able to uniquely determine an integrable symbol that gives the algebraic part of the most generic double-pentagon integral. Finally, we locate in the space the $n=8$ double-pentagon ladder integrals up to four loops using differential equations derived from Wilson-loop $dlog$ forms, and we find a remarkable pattern about the appearance of algebraic letters.
We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and (seven-point) double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $D_2simeq A_1^2$, we show that penta-box ladder has an alphabet of $D_3simeq A_3$ and provide strong evidence that the alphabet of double-penta ladder can be identified with a $D_4$ cluster algebra. We relate the symbol letters to the ${bf u}$ variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop ${rm d}log$ representation, which allows us to predict higher-loop alphabet recursively; by applying such recursions to six-dimensional hexagon integrals, we also find $D_5$ and $D_6$ cluster functions for the two-mass-easy and three-mass-easy case, respectively.
Canonical Feynman integrals are of great interest in the study of scattering amplitudes at the multi-loop level. We propose to construct $dlog$-form integrals of the hypergeometric type, treat them as a representation of Feynman integrals, and project them into master integrals using intersection theory. This provides a constructive way to build canonical master integrals whose differential equations can be solved easily. We use our method to investigate both the maximally cut integrals and the uncut ones at one and two loops, and demonstrate its applicability in problems with multiple scales.
We study Feynman integrals and scattering amplitudes in ${cal N}=4$ super-Yang-Mills by exploiting the duality with null polygonal Wilson loops. Certain Feynman integrals, including one-loop and two-loop chiral pentagons, are given by Feynman diagrams of a supersymmetric Wilson loop, where one can perform loop integrations and be left with simple integrals along edges. As the main application, we compute analytically for the first time, the symbol of the generic ($ngeq 12$) double pentagon, which gives two-loop MHV amplitudes and components of NMHV amplitudes to all multiplicities. We represent the double pentagon as a two-fold $mathrm{d} log$ integral of a one-loop hexagon, and the non-trivial part of the integration lies at rationalizing square roots contained in the latter. We obtain a remarkably compact algebraic words which contain $6$ algebraic letters for each of the $16$ square roots, and they all nicely cancel in combinations for MHV amplitudes and NMHV components which are free of square roots. In addition to $96$ algebraic letters, the alphabet consists of $152$ dual conformal invariant combinations of rational letters.
We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.
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