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Six-point gluon scattering amplitudes from Z_4-symmetric integrable model

Yasuyuki Hatsuda, Katsushi Ito, Kazuhiro Sakai, Yuji Satoh

TL;DR

This work connects six-point gluon scattering amplitudes in strong-coupling ${\cal N}=4$ SYM to the ${\mathbb Z}_4$-symmetric integrable model by solving its TBA/ Y-system. It derives analytic and numerical results for the remainder function ${R}$ via a perturbative expansion around the CFT point with a chemical potential and a large-mass expansion, separating contributions from the free energy and from BDS-related terms. The authors obtain the small-$|Z|$ (CFT) behavior of the remainder through a perturbative free energy and a controlled expansion of Y-functions, and they derive the leading large-$|Z|$ corrections, showing ${R}\to {\pi^2}/{12}$ with exponentially suppressed corrections. The results corroborate numerical calculations and illustrate a concrete realization of the AdS/CFT correspondence where a two-dimensional integrable model underpins the strong-coupling amplitude, with implications for general $n$-point extensions and cross-connection to weak-coupling results.

Abstract

We study six-point gluon scattering amplitudes in N=4 super Yang-Mills theory at strong coupling by investigating the thermodynamic Bethe ansatz equations of the underlying Z_4-symmetric integrable model both analytically and numerically. By the conformal field theory (CFT) perturbation, we compute the free energy part of the remainder function with generic chemical potential near the CFT/small mass limit. Combining this with the expansion of the Y-functions, we obtain the remainder function near the small mass limit up to a function of the chemical potential, which can be evaluated numerically. We also find the leading corrections to the remainder function near the large mass limit. We confirm that these results are in good agreement with numerical computations.

Six-point gluon scattering amplitudes from Z_4-symmetric integrable model

TL;DR

This work connects six-point gluon scattering amplitudes in strong-coupling SYM to the -symmetric integrable model by solving its TBA/ Y-system. It derives analytic and numerical results for the remainder function via a perturbative expansion around the CFT point with a chemical potential and a large-mass expansion, separating contributions from the free energy and from BDS-related terms. The authors obtain the small- (CFT) behavior of the remainder through a perturbative free energy and a controlled expansion of Y-functions, and they derive the leading large- corrections, showing with exponentially suppressed corrections. The results corroborate numerical calculations and illustrate a concrete realization of the AdS/CFT correspondence where a two-dimensional integrable model underpins the strong-coupling amplitude, with implications for general -point extensions and cross-connection to weak-coupling results.

Abstract

We study six-point gluon scattering amplitudes in N=4 super Yang-Mills theory at strong coupling by investigating the thermodynamic Bethe ansatz equations of the underlying Z_4-symmetric integrable model both analytically and numerically. By the conformal field theory (CFT) perturbation, we compute the free energy part of the remainder function with generic chemical potential near the CFT/small mass limit. Combining this with the expansion of the Y-functions, we obtain the remainder function near the small mass limit up to a function of the chemical potential, which can be evaluated numerically. We also find the leading corrections to the remainder function near the large mass limit. We confirm that these results are in good agreement with numerical computations.

Paper Structure

This paper contains 12 sections, 87 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Review of TBA system for six-point amplitudes
  3. Classical string solutions with a null polygonal boundary
  4. Hitchin system
  5. Auxiliary linear problem and small solutions
  6. Y-system
  7. TBA equations
  8. Area and remainder function
  9. Free energy around CFT point
  10. Remainder function around CFT point
  11. Remainder function in large mass region
  12. Conclusions

Figures (11)

  • Figure 1: (a) The $|Z|$-dependence of the function $1/U_1$ for different values of $\phi$ at $\varphi=-{\pi/48}$. (b) $1/U_2$. (c) $1/U_3$.
  • Figure 2: (a) 3D plots of free energy $A_{\hbox{\scriptsize free}}$ as a function of $|Z|$ and $\phi$ ($0\le |Z|\le 5,\ 0\le \phi\le 3\pi/2$) at $\varphi=-\pi/48$. (b) $|Z|$-$A_{\hbox{\scriptsize free}}$ graphs for various values of $\phi$ at $\varphi=-\pi/48$.
  • Figure 3: (a) 3D plots of numerical data of remainder function $R$ as a function of $|Z|$ and $\phi$ at $\varphi=-{\pi/48}$. (b) $|Z|$-$R$ graphs for various values of $\phi$ at $\varphi=-{\pi/48}$.
  • Figure 4: (a) Plots of $(|Z|,A_{\hbox{\scriptsize free}})$ for various values of $\phi$ at $\varphi=-{\pi/48}$. (b) Plots of $(|Z|,A_{\hbox{\scriptsize free}})$ for $0\leq |Z|\leq 0.2$ at $\phi=\pi/4,\ \varphi=-\pi/48$.
  • Figure 5: Plots of $A_{\hbox{\scriptsize free}}$ for $0\leq \phi\leq \pi$ at $|Z|=0.1$ and $\varphi=-{\pi/48}$. The dashed line represents the graph calculated by the formula (\ref{['eq:free316']}).
  • ...and 6 more figures