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$AdS_3 \times S^3 \times M^4$ string S-matrices from unitarity cuts

Lorenzo Bianchi, Ben Hoare

TL;DR

The paper develops and applies a one-loop unitarity-cut framework to two-dimensional integrable theories describing string worldsheet dynamics in AdS_3 × S^3 × M^4 backgrounds. By extending previous work to handle multiple masses and external-leg corrections, it demonstrates that the loop S-matrix is built from tree-level data and finite bubble integrals, with logarithmic terms confined to universal dressing phases and rational terms fixed by integrability. For RR backgrounds on T^4 and on S^3 × S^1, the results reproduce the exact S-matrix expansions and semiclassical phases, supporting the conjecture that integrable S-matrices are cut-constructible up to coupling shifts. In the mixed RR/NSNS flux case, the authors conjecture the one-loop dressing phases and show consistency with known limits, highlighting the framework's potential to illuminate deformations of all-loop phases. Overall, the work validates unitarity-based constructions as a robust tool for extracting one-loop S-matrix data in diverse AdS_3 string backgrounds and sets the stage for further explorations of massless modes and higher-loop corrections.

Abstract

Continuing the program initiated in arXiv:1304.1798 we investigate unitarity methods applied to two-dimensional integrable field theories. The one-loop computation is generalized to encompass theories with different masses in the asymptotic spectrum and external leg corrections. Additionally, the prescription for working with potentially singular cuts is modified to cope with an ambiguity that was not encountered before. The resulting methods are then applied to three light-cone gauge string theories; i) $AdS_3 \times S^3 \times T^4$ supported by RR flux, ii) $AdS_3 \times S^3 \times S^3 \times S^1$ supported by RR flux and iii) $AdS_3 \times S^3 \times T^4$ supported by a mix of RR and NSNS fluxes. In the first case we find agreement with the exact result following from symmetry considerations and in the second case with one-loop semiclassical computations. This agreement crucially includes the rational terms and hence supports the conjecture that S-matrices of integrable field theories are cut-constructible, up to a possible shift in the coupling. In the final case, under the assumption that our methods continue to give all rational terms, we give a conjecture for the one-loop phases.

$AdS_3 \times S^3 \times M^4$ string S-matrices from unitarity cuts

TL;DR

The paper develops and applies a one-loop unitarity-cut framework to two-dimensional integrable theories describing string worldsheet dynamics in AdS_3 × S^3 × M^4 backgrounds. By extending previous work to handle multiple masses and external-leg corrections, it demonstrates that the loop S-matrix is built from tree-level data and finite bubble integrals, with logarithmic terms confined to universal dressing phases and rational terms fixed by integrability. For RR backgrounds on T^4 and on S^3 × S^1, the results reproduce the exact S-matrix expansions and semiclassical phases, supporting the conjecture that integrable S-matrices are cut-constructible up to coupling shifts. In the mixed RR/NSNS flux case, the authors conjecture the one-loop dressing phases and show consistency with known limits, highlighting the framework's potential to illuminate deformations of all-loop phases. Overall, the work validates unitarity-based constructions as a robust tool for extracting one-loop S-matrix data in diverse AdS_3 string backgrounds and sets the stage for further explorations of massless modes and higher-loop corrections.

Abstract

Continuing the program initiated in arXiv:1304.1798 we investigate unitarity methods applied to two-dimensional integrable field theories. The one-loop computation is generalized to encompass theories with different masses in the asymptotic spectrum and external leg corrections. Additionally, the prescription for working with potentially singular cuts is modified to cope with an ambiguity that was not encountered before. The resulting methods are then applied to three light-cone gauge string theories; i) supported by RR flux, ii) supported by RR flux and iii) supported by a mix of RR and NSNS fluxes. In the first case we find agreement with the exact result following from symmetry considerations and in the second case with one-loop semiclassical computations. This agreement crucially includes the rational terms and hence supports the conjecture that S-matrices of integrable field theories are cut-constructible, up to a possible shift in the coupling. In the final case, under the assumption that our methods continue to give all rational terms, we give a conjecture for the one-loop phases.

Paper Structure

This paper contains 24 sections, 115 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. General principles
  3. Theories with a single mass
  4. Theories with multiple masses
  5. External leg corrections
  6. Structure of the result
  7. Massive sector for AdS3 x S3 x M4 supported RR flux
  8. Exact and tree-level S-matrices
  9. Massive sector for AdS3 x S3 x T4
  10. Massive sector for AdS3 x S3 x S3 x S1
  11. A general tree-level S-matrix for the AdS3 x S3 x M4 theories
  12. Result from unitarity techniques
  13. Coefficients of the logarithms
  14. Rational terms from the s-channel
  15. The t-channel contribution and the dressing phases
  16. ...and 9 more sections

Figures (3)

  • Figure 1: Diagrams representing s-, t- and u-channel cuts contributing to the four-point one-loop amplitude.
  • Figure 2: Diagrams contributing to external leg corrections at one-loop.
  • Figure 3: Cut of a two-point function obtained by fusing two form factors. The double line indicates an off-shell state.