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On the Singularities of the Magnon S-matrix

Nick Dorey, Diego M. Hofman, Juan Maldacena

TL;DR

This work clarifies the analytic structure of the magnon S-matrix in the planar N=4 SYM / AdS5×S5 setting, showing that a broad set of near-real poles are double poles arising from the exchange of pairs of BPS magnons and that their locations are fixed by the BPS dispersion, in agreement with the BES dressing phase. It demonstrates that these poles do not signal new bound states; instead, certain non-BPS classical solutions decompose into BPS magnons, resolving previous ambiguities about bound states. The authors connect the pole structure to both quantum (Coleman–Thun type) diagrams and semiclassical/giant-magnon limits, and provide an integral representation and pinch-analysis for the BES phase to locate these poles precisely. Collectively, the results reinforce the picture that the asymptotic spectrum consists of BPS boundstates and clarify how pole structure constrains the exact S-matrix in this integrable gauge/string duality. Implications include a physical basis for the BES proposal and a framework for understanding dressing-factor singularities within the spin-chain / worldsheet correspondence.

Abstract

We investigate the analytic structure of the magnon S-matrix in the spin-chain description of planar ${\cal N}=4$ SUSY Yang-Mills/$AdS_{5}\times S^{5}$ strings. Semiclassical analysis suggests that the exact S-matrix must have a large family of poles near the real axis in momentum space. In this article we show that these are double poles corresponding to the exchange of pairs of BPS magnons. Their locations in the complex plane are uniquely fixed by the known dispersion relation for the BPS particles. The locations precisely agree with the recent conjecture for the $S$ matrix by Beisert, Hernandez, Lopez, Eden and Staudacher (hep-th/0609044 and hep-th/0610251). These poles do not signal the presence of new bound states. In fact, a certain non-BPS localized classical solution, which was thought to give rise to new bound states, can actually decay into a pair of BPS magnons.

On the Singularities of the Magnon S-matrix

TL;DR

This work clarifies the analytic structure of the magnon S-matrix in the planar N=4 SYM / AdS5×S5 setting, showing that a broad set of near-real poles are double poles arising from the exchange of pairs of BPS magnons and that their locations are fixed by the BPS dispersion, in agreement with the BES dressing phase. It demonstrates that these poles do not signal new bound states; instead, certain non-BPS classical solutions decompose into BPS magnons, resolving previous ambiguities about bound states. The authors connect the pole structure to both quantum (Coleman–Thun type) diagrams and semiclassical/giant-magnon limits, and provide an integral representation and pinch-analysis for the BES phase to locate these poles precisely. Collectively, the results reinforce the picture that the asymptotic spectrum consists of BPS boundstates and clarify how pole structure constrains the exact S-matrix in this integrable gauge/string duality. Implications include a physical basis for the BES proposal and a framework for understanding dressing-factor singularities within the spin-chain / worldsheet correspondence.

Abstract

We investigate the analytic structure of the magnon S-matrix in the spin-chain description of planar SUSY Yang-Mills/ strings. Semiclassical analysis suggests that the exact S-matrix must have a large family of poles near the real axis in momentum space. In this article we show that these are double poles corresponding to the exchange of pairs of BPS magnons. Their locations in the complex plane are uniquely fixed by the known dispersion relation for the BPS particles. The locations precisely agree with the recent conjecture for the matrix by Beisert, Hernandez, Lopez, Eden and Staudacher (hep-th/0609044 and hep-th/0610251). These poles do not signal the presence of new bound states. In fact, a certain non-BPS localized classical solution, which was thought to give rise to new bound states, can actually decay into a pair of BPS magnons.

Paper Structure

This paper contains 18 sections, 67 equations, 14 figures.

Table of Contents

  1. Introduction
  2. S-matrix Singularities
  3. Poles from physical processes
  4. Simple poles
  5. Double poles
  6. Classical, semiclassical and approximate results
  7. Localized classical solutions and the absence of non-BPS bound states
  8. The semiclassical limit of the quantum theory
  9. Scattering of giant magnons and a non-relativistic limit
  10. A toy model for double poles
  11. Poles in the Beisert-Eden-Hernandez-Lopez-Staudacher S-matrix
  12. Integral expression for the dressing factor
  13. Poles of the dressing factor in the giant magnon region
  14. When do we pinch the contour?
  15. Conclusions
  16. ...and 3 more sections

Figures (14)

  • Figure 1: Diagrams associated to singularities of the S-matrix.
  • Figure 2: Three-point vertices.
  • Figure 3: Formation of a bound state in the s-channel
  • Figure 4: Crossing transformation of vertex.
  • Figure 5: t-channel contribution
  • ...and 9 more figures