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We comment on the status of Steinmann-like constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar ${cal N}=4$ super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program. Based on physical discontinuities and Steinmann relations, we first summarize all possible double discontinuities (or first-two-entries) for (the symbol of) amplitudes and integrals in terms of dilogarithms, generalizing well-known results for $n=6,7$ to all multiplicities. As our main result, we find that extended-Steinmann relations hold for all finite integrals that we have checked, including various ladder integrals, generic double-pentagon integrals, as well as finite components of two-loop NMHV amplitudes for any $n$; with suitable normalization such as minimal subtraction, they hold for $n=8$ MHV amplitudes at three loops. We find interesting cancellation between contributions from rational and algebraic letters, and for the former we have also tested cluster-adjacency conditions using the so-called Sklyanin brackets. Finally, we propose a list of possible last-two-entries for $n$-point MHV amplitudes derived from $bar{Q}$ equations, which can be used to reduce the space of functions for higher-point MHV amplitudes.
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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 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.
Using the Gelfand-Kapranov-Zelevinsku{i} system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana amplitudes with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel $widehat Gamma$-class evaluation in the ambient spaces of the mirror, while the imaginary part of the amplitude in this regime is determined by the $widehat Gamma$-class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius $kappa$-constants, which determine the behaviour of the amplitudes, when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogenous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the amplitude as well as other master integrals with raised powers of the propagators in very short time to very high numerical precision for all values of the physical parameters. Using a recent $p$-adic analysis of the periods we determine the value of the maximal cut equal mass four-loop amplitude at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.
We study the four point function of the superconformal primary of the stress-tensor multiplet in four dimensional $mathcal{N}=4$ Super Yang Mills, at large t Hooft coupling and in a large $N$ expansion. This observable is holographically dual to four graviton amplitudes in type IIB supergravity on $AdS_5 times S^5$. We construct the most trascendental piece of the correlator at order $N^{-6}$ and compare it with the flat space limit of the corresponding two loops amplitude. This comparison allows us to conjecture structures of the correlator/amplitude which should be present at any loop order.
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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