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Cluster Polylogarithms for Scattering Amplitudes

John Golden, Miguel F. Paulos, Marcus Spradlin, Anastasia Volovich

TL;DR

The paper introduces cluster polylogarithm functions as a framework to express two-loop scattering amplitudes in ${ m N}=4$ SYM through cluster algebras. It shows that all weight-4 cluster functions arise from a single $A_2$ building block, and, by imposing locality on generalized Stasheff polytopes, assembles these into a unique $A_3$ function that encodes the complete cluster structure of two-loop $n$-particle MHV amplitudes for all $n$, with $n=7$ treated explicitly. The authors construct and analyze the $A_2$ and $A_3$ functions, discuss their locality and Poisson properties, and demonstrate how the $A_3$ function yields a compact representation of the most intricate part of $R_7^{(2)}$ in terms of ${ m Gr}(4,7)$ coordinates, including a discussion of remaining analytic ambiguities. The work provides a concrete, cluster-algebraic basis for higher-loop, higher-point MHV amplitudes and lays groundwork for analytic completions and future extensions to more complex amplitudes. It also motivates potential reformulations of SYM amplitudes that make the underlying cluster structure more explicit and computationally tractable.

Abstract

Motivated by the cluster structure of two-loop scattering amplitudes in N=4 Yang-Mills theory we define "cluster polylogarithm functions". We find that all such functions of weight 4 are made up of a single simple building block associated to the A_2 cluster algebra. Adding the requirement of locality on generalized Stasheff polytopes, we find that these A_2 building blocks arrange themselves to form a unique function associated to the A_3 cluster algebra. This A_3 function manifests all of the cluster algebraic structure of the two-loop n-particle MHV amplitudes for all n, and we use it to provide an explicit representation for the most complicated part of the n=7 amplitude as an example.

Cluster Polylogarithms for Scattering Amplitudes

TL;DR

The paper introduces cluster polylogarithm functions as a framework to express two-loop scattering amplitudes in SYM through cluster algebras. It shows that all weight-4 cluster functions arise from a single building block, and, by imposing locality on generalized Stasheff polytopes, assembles these into a unique function that encodes the complete cluster structure of two-loop -particle MHV amplitudes for all , with treated explicitly. The authors construct and analyze the and functions, discuss their locality and Poisson properties, and demonstrate how the function yields a compact representation of the most intricate part of in terms of coordinates, including a discussion of remaining analytic ambiguities. The work provides a concrete, cluster-algebraic basis for higher-loop, higher-point MHV amplitudes and lays groundwork for analytic completions and future extensions to more complex amplitudes. It also motivates potential reformulations of SYM amplitudes that make the underlying cluster structure more explicit and computationally tractable.

Abstract

Motivated by the cluster structure of two-loop scattering amplitudes in N=4 Yang-Mills theory we define "cluster polylogarithm functions". We find that all such functions of weight 4 are made up of a single simple building block associated to the A_2 cluster algebra. Adding the requirement of locality on generalized Stasheff polytopes, we find that these A_2 building blocks arrange themselves to form a unique function associated to the A_3 cluster algebra. This A_3 function manifests all of the cluster algebraic structure of the two-loop n-particle MHV amplitudes for all n, and we use it to provide an explicit representation for the most complicated part of the n=7 amplitude as an example.

Paper Structure

This paper contains 13 sections, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cluster polylogarithm functions
  3. Polylogarithm functions, symbols, and the coproduct
  4. Integrability
  5. Cluster $\mathcal{A}$- and $\mathcal{X}$-coordinates
  6. Cluster polylogarithm functions
  7. The $A_2$ function
  8. The $A_3$ function
  9. Cluster polylogarithms for $\mathop{\mathrm{Gr}}\nolimits(4,7)$ and the amplitude $R_7^{(2)}$
  10. Conclusion
  11. Functional representatives
  12. The $A_2$ function
  13. The $A_3$ function

Figures (2)

  • Figure 1: The $A_2$ cluster algebra: to each oriented edge is associated a cluster $\mathcal{X}$-variable (reversing an arrow requires inverting the associated variable), and to each vertex is associated the pair of variables (called the cluster) associated to the edges emanating from that vertex. Moving from one cluster to an adjacent one along some edge is accomplished by mutating on the variable associated to that edge.
  • Figure 2: The Stasheff polytope for the $A_3$ cluster algebra. The caption of fig. (\ref{['fig:A2']}) applies, except that here a cluster of three $\mathcal{X}$-coordinates is associated to each vertex. The three quadrilateral faces are shaded blue to distinguish them visually from the six pentagonal faces. The interior of this polytope can be identified with the blow-up of the positive domain in $\mathop{\mathrm{Conf}}\nolimits_6(\mathbb{P}^3)$, see for example Brown.