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Quantum mechanics of null polygonal Wilson loops

A. V. Belitsky, S. E. Derkachov, A. N Manashov

TL;DR

The paper addresses the spectral problem for multiparticle GKP excitations that govern the OPE dynamics of null polygonal Wilson loops in planar $\mathcal{N}=4$ SYM. It maps the problem to an open $SL(2,\mathbb{R})$ spin chain and solves it using the Baxter $\mathbb{Q}$-operator together with Sklyanin's Separation of Variables, enabling explicit construction of eigenfunctions and eigenvalues. The authors derive integral and recursive representations for the $N$-particle eigenfunctions, establish their orthogonality, and formulate square and hexagon transitions; crucially, hexagon transitions factorize into a product of one-particle kernels, confirming conjectured structure. These results provide a nonperturbative framework for the OPE of polygonal Wilson loops and lay the groundwork for extensions to inhomogeneous and supersymmetric settings, with direct relevance to nonperturbative amplitudes in $\mathcal{N}=4$ SYM.

Abstract

Scattering amplitudes in maximally supersymmetric gauge theory are dual to super-Wilson loops on null polygonal contours. The operator product expansion for the latter revealed that their dynamics is governed by the evolution of multiparticle GKP excitations. They were shown to emerge from the spectral problem of an underlying open spin chain. In this work we solve this model with the help of the Baxter Q-operator and Sklyanin's Separation of Variables methods. We provide an explicit construction for eigenfunctions and eigenvalues of GKP excitations. We demonstrate how the former define the so-called multiparticle hexagon transitions in super-Wison loops and prove their factorized form suggested earlier.

Quantum mechanics of null polygonal Wilson loops

TL;DR

The paper addresses the spectral problem for multiparticle GKP excitations that govern the OPE dynamics of null polygonal Wilson loops in planar SYM. It maps the problem to an open spin chain and solves it using the Baxter -operator together with Sklyanin's Separation of Variables, enabling explicit construction of eigenfunctions and eigenvalues. The authors derive integral and recursive representations for the -particle eigenfunctions, establish their orthogonality, and formulate square and hexagon transitions; crucially, hexagon transitions factorize into a product of one-particle kernels, confirming conjectured structure. These results provide a nonperturbative framework for the OPE of polygonal Wilson loops and lay the groundwork for extensions to inhomogeneous and supersymmetric settings, with direct relevance to nonperturbative amplitudes in SYM.

Abstract

Scattering amplitudes in maximally supersymmetric gauge theory are dual to super-Wilson loops on null polygonal contours. The operator product expansion for the latter revealed that their dynamics is governed by the evolution of multiparticle GKP excitations. They were shown to emerge from the spectral problem of an underlying open spin chain. In this work we solve this model with the help of the Baxter Q-operator and Sklyanin's Separation of Variables methods. We provide an explicit construction for eigenfunctions and eigenvalues of GKP excitations. We demonstrate how the former define the so-called multiparticle hexagon transitions in super-Wison loops and prove their factorized form suggested earlier.

Paper Structure

This paper contains 35 sections, 252 equations, 15 figures.

Table of Contents

  1. Motivation
  2. Soft-collinear expansion of polygons
  3. Light-cone operators
  4. Open spin chain
  5. Scalar product
  6. Integrals of motion
  7. Factorized $R$-matrices and Hamiltonians
  8. Intertwiners
  9. Hamiltonians
  10. Baxter operators and Baxter equations
  11. Baxter operator $\mathbb{Q}^+$
  12. Baxter operator $\mathbb{Q}^-$
  13. Baxter equations
  14. Hamiltonians from Baxter operators
  15. Eigenfunctions
  16. ...and 20 more sections

Figures (15)

  • Figure 1: Single (a,b) and two-particle (c) contributions to OPE of the octagon. The one-loop graph in (a) given by the scalar propagator exchanged between the cusps produces a components of the tree NHMV amplitude. The graph in panel (b) displays one of the perturbative corrections due to the Hamiltonian acting on the light-cone operator. In (c) we show the two-particle contribution that produces subleading effects in the OPE.
  • Figure 2: The diagrammatic representation of the kernel of the Baxter operator $\mathbb{Q}^+$, Eq. (\ref{['Q+kernel']}).
  • Figure 3: The diagrammatic representation of the kernel of the Baxter operator $\mathbb{Q}^-$.
  • Figure 4: A subtree (shown in solid lines) in $N$-site pyramid (solid and dashed lines) originating from the vertex $V_{n-1}^{(1)}$ ending on sites from $z_1$ to $z_n$.
  • Figure 5: Diagrammatic representation for $N$-particle eigenstate (\ref{['Deigenfunction']}). Here $\alpha_n^\pm \equiv s \pm i u_n$ stands for the power of the propagator (see Appendix \ref{['FeynmanAppendix']} for conventions). Each vertex corresponds to a coordinate in the pyramid that is integrated with the measure (\ref{['MeasureSL2']}). The top leftmost vertex is located at $w = 0$.
  • ...and 10 more figures