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A bstract We propose a geometrical approach to generate symbol letters of amplitudes/integrals in planar N = 4 Super Yang-Mills theory, known as Schubert problems . Beginning with one-loop integrals, we find that intersections of lines in momentum twistor space are always ordered on a given line, once the external kinematics Z is in the positive region G + (4, n ). Remarkably, cross-ratios of these ordered intersections on a line, which are guaranteed to be positive now, nicely coincide with symbol letters of corresponding Feynman integrals, whose positivity is then concluded directly from such geometrical configurations. In particular, we reproduce from this approach the 18 multiplicative independent algebraic letters for n = 8 amplitudes up to three loops. Finally, we generalize the discussion to two-loop Schubert problems and, again from ordered points on a line, generate a new kind of algebraic letters which mix two distinct square roots together. They have been found recently in the alphabet of two-loop double-box integral with n ≥ 9, and they are expected to appear in amplitudes at k + ℓ ≥ 4.

A bstract We present a systematic method for computing cosmological amplitudes, including in-in correlators and wavefunction coefficients. Specialising to cases with conformally-coupled external scalars and massive scalar exchanges, we introduce a decomposition into massive family trees, which capture the nested time structure common to these observables. We then evaluate these building blocks using the Method of Brackets (MoB), a multivariate extension of Ramanujan’s master theorem that operates directly on the integrand, translating integrals into discrete summations via a compact set of algebraic rules. This yields infinite series representations valid across the full space of external momenta and internal energies. We also develop Feynman-like diagrammatic rules that map interaction graphs to summand structures, enabling efficient and scalable computation. The resulting expressions make time evolution manifest, smoothly interpolate to the conformal limit, and are well-suited for both numerical evaluation and analytic analysis of massive field effects in cosmology.

A bstract 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 d log forms, and we find a remarkable pattern about the appearance of algebraic letters.

A bstract We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is D 2 ≃ A 1 2 , we show that penta-box ladder has an alphabet of D 3  ≃  A 3 and provide strong evidence that the alphabet of seven-point double-penta ladders can be identified with a D 4 cluster algebra. We relate the symbol letters to the 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 d log representation, which allows us to predict higher-loop alphabet recursively; by applying it to certain eight-point and nine-point double-penta ladders, we also find D 5 and D 6 cluster functions respectively.

A bstract We take the first step in generalizing the so-called “Schubert analysis”, originally proposed in twistor space for four-dimensional kinematics, to the study of symbol letters and more detailed information on canonical differential equations for Feynman integral families in general dimensions with general masses. The basic idea is to work in embedding space and compute possible cross-ratios built from (Lorentz products of) maximal cut solutions for all integrals in the family. We demonstrate the power of the method using the most general one-loop integrals, as well as various two-loop planar integral families (such as sunrise, double-triangle and double-box) in general dimensions. Not only can we obtain all symbol letters as cross-ratios from maximal-cut solutions, but we also reproduce entries in the canonical differential equations satisfied by a basis of d log integrals.

A bstract Following [ 1 ], we derive a recursion relation by applying a one-parameter deformation of kinematic variables for tree-level scattering amplitudes in bi-adjoint ϕ 3 theory. The recursion relies on properties of the amplitude that can be made manifest in the underlying kinematic associahedron, and it provides triangulations for the latter. Furthermore, we solve the recursion relation and present all-multiplicity results for the amplitude: by reformulating the associahedron in terms of its vertices, it is given explicitly as a sum of “volume” of simplicies for any triangulation, which is an analogy of BCFW representation/triangulation of amplituhedron for N = 4 SYM.

A bstract We revisit the conjectural method called Schubert analysis for generating the alphabet of symbol letters for Feynman integrals, which was based on geometries of intersecting lines associated with corresponding cut diagrams. We explain the effectiveness of this somewhat mysterious method by relating such geometries to the corresponding Landau singularities, which also amounts to “uplifting” Landau singularities of a Feynman integral to its symbol letters. We illustrate this Landau-based Schubert analysis using various multi-loop Feynman integrals in four dimensions and present an automated Mathematica notebook for it. We then apply the method to a simplified problem of studying alphabets of physical quantities such as scattering amplitudes and form factors in planar N = 4 super-Yang-Mills. By focusing on a small set of Landau diagrams (as opposed to all relevant Feynman integrals), we show how this method nicely produces the two-loop alphabet of n -point MHV amplitudes and that of the n = 4 MHV form factors. A byproduct of our analysis is an explicit representation of any symbol alphabet obtained this way as the union of various type- A cluster algebras.

A bstract 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 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 MHV amplitudes up to 9 points derived from Q ¯ equations, which can be used to reduce the space of functions for higher-point MHV amplitudes.

A bstract Multi-loop Feynman integrals are key objects for the high-order correction computations in high energy phenomenology. These integrals with multiple scales may have complicated symbol structures, and we show that twistor geometries of closely related dual conformal integrals shed light on their alphabet and symbol structures. In this paper, first, as a cutting-edge example, we derive the two-loop four-external-mass Feynman integrals with uniform transcendental (UT) weights, based on the latest developments on UT integrals. Then we find that all the symbol letters of these integrals can be explained non-trivially by studying the so-called Schubert problem of certain dual conformal integrals with a point at infinity. Certain properties of the symbol such as first two entries and extended Steinmann relations are also studied from analogous properties of dual conformal integrals.

A bstract Recently considerable efforts have been devoted to computing cosmological correlators and the corresponding wavefunction coefficients, as well as understanding their analytical structures. In this note, we revisit the computation of these “cosmological amplitudes” associated with any tree or loop graph for conformal scalars with time-dependent interactions in the power-law FRW universe, directly in terms of iterated time integrals. We start by decomposing any such cosmological amplitude (for loop graph, the “integrand” prior to loop integrations) as a linear combination of basic time integrals , one for each directed graph . We derive remarkably simple first-order differential equations involving such time integrals with edges “contracted” one at a time, which can be solved recursively and the solution takes the form of Euler-Mellin integrals/generalized hypergeometric functions. By combining such equations, we then derive a complete system of differential equations for all time integrals needed for a given graph. Our method works for any graph: for a tree graph with n nodes, this system can be transformed into the canonical differential equations of size 4 n −1 equivalent to the graphic rules derived recently , and we also derive the system of differential equations for loop integrands e.g. of all-loop two-site graphs and one-loop n -gon graphs. Finally, we show how the differential equations truncate for the de Sitter (dS) case (in a way similar to differential equations for Feynman integrals truncate for integer dimensions), which immediately yields the complete symbol for the dS amplitude with interesting structures e.g. for n -site chains and n -gon cases.