相关论文: Symbol Alphabets from Tensor Diagrams
Symbol alphabets of n-particle amplitudes in N=4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4,n) as well as certain algebraic functions of cluster variables. The first paper arXiv:2007.00646 in this series…
Symbol alphabets of n-particle amplitudes in N=4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4,n) as well as certain algebraic functions of cluster variables. In this paper we solve the C Z = 0 matrix…
The full 245-letter symbol alphabet for all planar massless two-loop six-point Feynman integrals was recently determined in arXiv:2412.19884 and arXiv:2501.01847. In a parallel mathematical development, it was shown in arXiv:2408.14956 that…
Symbol alphabets of n-particle amplitudes in N=4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4,n) as well as certain algebraic functions of cluster variables. In this paper we suggest an algorithm for…
It is widely expected that NMHV amplitudes in planar, maximally supersymmetric Yang-Mills theory require symbol letters that are not rationally expressible in terms of momentum-twistor (or cluster) variables starting at two loops for eight…
Motivated by reformulating Yangian invariants in planar ${\cal N}=4$ SYM directly as $d\log$ forms on momentum-twistor space, we propose a purely algebraic problem of determining the arguments of the $d\log$'s, which we call "letters", for…
There is a remarkable well-known connection between the G$(4,n)$ cluster algebra and $n$-particle amplitudes in $\mathcal{N}=4$ SYM theory. For $n \ge 8$ two long-standing open questions have been to find a mathematically natural way to…
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…
We further exploit the relation between tropical Grassmannians and $\operatorname{Gr}(4,n)$ cluster algebras in order to make and refine predictions for the singularities of scattering amplitudes in $\mathcal{N}=4$ planar super Yang-Mills…
We derive analytic results for the symbol of certain two-loop Feynman integrals relevant for seven- and eight-point two-loop scattering amplitudes in planar $\mathcal{N}=4$ super-Yang--Mills theory. We use a bootstrap inspired strategy,…
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)…
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…
We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $\mathcal{N}=4$ super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry…
We propose a geometrical approach to generate symbol letters of amplitudes/integrals in planar $\mathcal{N}=4$ Super Yang-Mills theory, known as {\it Schubert problems}. Beginning with one-loop integrals, we find that intersections of lines…
We conjecture a new set of analytic relations for scattering amplitudes in planar N=4 super Yang-Mills theory. They generalise the Steinmann relations and are expressed in terms of the cluster algebras associated to Gr(4,n). In terms of the…
Scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory exhibit singularities which reflect various aspects of the cluster algebras associated to the Grassmannians ${\rm Gr}(4,n)$ and their tropical counterparts. Here we…
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 $D_2\simeq A_1^2$, we…
Multi-loop scattering amplitudes in N=4 Yang-Mills theory possess cluster algebra structure. In order to develop a computational framework which exploits this connection, we show how to construct bases of Goncharov polylogarithm functions,…
We provide evidence through two loops, that rational letters of polylogarithmic Feynman integrals are captured by the Landau equations, when the latter are recast as a polynomial of the kinematic variables of the integral, known as the…
We study the symbol and the alphabet for two-loop NMHV amplitudes in planar ${\cal N}=4$ super-Yang-Mills from the $\bar{Q}$ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop…