Skip to main content
SpringerLink
Log in

Two-dimensional S-matrices from unitarity cuts

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

Using unitarity methods, we compute, for several massive two-dimensional models, the cut-constructible part of the one-loop 2 → 2 scattering S-matrices from the tree-level amplitudes. We apply our method to various integrable theories, finding evidence that for supersymmetric models the one-loop S-matrix is cut-constructible, while for models without supersymmetry (but with integrability) the missing rational terms are proportional to the tree-level S-matrix and therefore can be interpreted as a shift in the coupling. Finally, applying our procedure to the world-sheet theory for the light-cone gauge-fixed AdS5 × S 5 superstring we reproduce, at one-loop in the near-BMN expansion, the S-matrix known from integrability techniques.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Roiban, M. Spradlin and A. Volovich, Scattering amplitudes in gauge theories: progress and outlook, J. Phys. A 44 (2011) 450301.

    MathSciNet  Google Scholar 

  2. N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].

    MathSciNet  Google Scholar 

  3. G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  4. N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. T. Klose and K. Zarembo, Bethe ansatz in stringy σ-models, J. Stat. Mech. 0605 (2006) P05006 [hep-th/0603039] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  6. R. Roiban, A. Tirziu and A.A. Tseytlin, Asymptotic Bethe ansatz S-matrix and Landau-Lifshitz type effective 2D actions, J. Phys. A 39 (2006) 13129 [hep-th/0604199] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  7. T. Klose, T. McLoughlin, R. Roiban and K. Zarembo, Worldsheet scattering in AdS 5 × S 5, JHEP 03 (2007) 094 [hep-th/0611169] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. T. Klose and K. Zarembo, Reduced σ-model on AdS 5 × S 5 : one-loop scattering amplitudes, JHEP 02 (2007) 071 [hep-th/0701240] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. T. Klose, T. McLoughlin, J. Minahan and K. Zarembo, World-sheet scattering in AdS 5 × S 5 at two loops, JHEP 08 (2007) 051 [arXiv:0704.3891] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  10. A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized s matrices in two-dimensions as the exact solutions of certain relativistic quantum field models, Annals Phys. 120 (1979) 253 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. E. Ogievetsky, P. Wiegmann and N. Reshetikhin, The principal chiral field in two-dimensions on classical Lie algebras: the Bethe ansatz solution and factorized theory of scattering, Nucl. Phys. B 280 (1987) 45 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  12. P. Dorey, Exact S matrices, hep-th/9810026 [INSPIRE].

  13. I. Arefeva and V. Korepin, Scattering in two-dimensional model with lagrangian (1/γ)((∂ μ u)2 /2 + m 2 cos(u − 1)), Pisma Zh. Eksp. Teor. Fiz. 20 (1974) 680 [INSPIRE].

    Google Scholar 

  14. R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. Z. Bern and Y.-t. Huang, Basics of generalized unitarity, J. Phys. A 44 (2011) 454003 [arXiv:1103.1869] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  16. W. van Neerven, Dimensional regularization of mass and infrared singularities in two loop on-shell vertex functions, Nucl. Phys. B 268 (1986) 453 [INSPIRE].

    Article  ADS  Google Scholar 

  17. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].

    Article  ADS  Google Scholar 

  19. L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].

  20. R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge Univeristy Press, Cambridge U.K. (2002).

    Google Scholar 

  21. Z. Bern, J. Rozowsky and B. Yan, Two loop four gluon amplitudes in N = 4 super Yang-Mills, Phys. Lett. B 401 (1997) 273 [hep-ph/9702424] [INSPIRE].

    ADS  Google Scholar 

  22. Z. Bern and A. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [INSPIRE].

    Article  ADS  Google Scholar 

  23. T.J. Hollowood, J.L. Miramontes and Q.-H. Park, Massive integrable soliton theories, Nucl. Phys. B 445 (1995) 451 [hep-th/9412062] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. I. Bakas, Q.-H. Park and H.-J. Shin, Lagrangian formulation of symmetric space sine-Gordon models, Phys. Lett. B 372 (1996) 45 [hep-th/9512030] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  25. N. Dorey and T.J. Hollowood, Quantum scattering of charged solitons in the complex sine-Gordon model, Nucl. Phys. B 440 (1995) 215 [hep-th/9410140] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  26. B. Hoare and A. Tseytlin, On the perturbative S-matrix of generalized sine-Gordon models, JHEP 11 (2010) 111 [arXiv:1008.4914] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. H. de Vega and J. Maillet, Renormalization character and quantum S matrix for a classically integrable theory, Phys. Lett. B 101 (1981) 302 [INSPIRE].

    ADS  Google Scholar 

  28. T.J. Hollowood and J.L. Miramontes, Classical and quantum solitons in the symmetric space sine-Gordon theories, JHEP 04 (2011) 119 [arXiv:1012.0716] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. G. Felder, Elliptic quantum groups, hep-th/9412207 [INSPIRE].

  30. B. Hoare, T.J. Hollowood and J.L. Miramontes, Restoring unitarity in the q-deformed world-sheet S-matrix, arXiv:1303.1447 [INSPIRE].

  31. E. Witten, Nonabelian bosonization in two-dimensions, Commun. Math. Phys. 92 (1984) 455.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. V. Knizhnik and A. Zamolodchikov, Current algebra and Wess-Zumino model in two-dimensions, Nucl. Phys. B 247 (1984) 83 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. H. Leutwyler and M.A. Shifman, Perturbation theory in the Wess-Zumino-Novikov-Witten model, Int. J. Mod. Phys. A 7 (1992) 795 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  34. A.A. Tseytlin, Effective action of gauged WZW model and exact string solutions, Nucl. Phys. B 399 (1993) 601 [hep-th/9301015] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  35. A.A. Tseytlin, Conformal σ-models corresponding to gauged Wess-Zumino-Witten theories, Nucl. Phys. B 411 (1994) 509 [hep-th/9302083] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. B. de Wit, M.T. Grisaru and P. van Nieuwenhuizen, The WZNW model at two loops, Nucl. Phys. B 408 (1993) 299 [hep-th/9307027] [INSPIRE].

    Article  ADS  Google Scholar 

  37. K. Pohlmeyer, Integrable hamiltonian systems and interactions through quadratic constraints, Commun. Math. Phys. 46 (1976) 207.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. H. Eichenherr and K. Pohlmeyer, Lax pairs for certain generalizations of the sine-Gordon equation, Phys. Lett. B 89 (1979) 76 [INSPIRE].

    ADS  Google Scholar 

  39. A.A. Tseytlin, Spinning strings and AdS/CFT duality, hep-th/0311139 [INSPIRE].

  40. B. Barbashov and V. Nesterenko, Relativistic string model in a space-time of a constant curvature, Commun. Math. Phys. 78 (1981) 499.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. H. De Vega and N.G. Sanchez, Exact integrability of strings in D-Dimensional de Sitter space-time, Phys. Rev. D 47 (1993) 3394 [INSPIRE].

    ADS  Google Scholar 

  42. R. Metsaev and A.A. Tseytlin, Type IIB superstring action in AdS 5 × S 5 background, Nucl. Phys. B 533 (1998) 109 [hep-th/9805028] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  43. M. Grigoriev and A.A. Tseytlin, Pohlmeyer reduction of AdS 5 × S 5 superstring σ-model, Nucl. Phys. B 800 (2008) 450 [arXiv:0711.0155] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  44. A. Mikhailov and S. Schäfer-Nameki, Sine-Gordon-like action for the Superstring in AdS 5 × S 5, JHEP 05 (2008) 075 [arXiv:0711.0195] [INSPIRE].

    Article  ADS  Google Scholar 

  45. M. Grigoriev and A.A. Tseytlin, On reduced models for superstrings on AdS n × S n, Int. J. Mod. Phys. A 23 (2008) 2107 [arXiv:0806.2623] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  46. A. Babichenko, B. Stefanski Jr. and K. Zarembo, Integrability and the AdS 3 /CFT 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. D. Sorokin, A. Tseytlin, L. Wulff and K. Zarembo, Superstrings in AdS 2 × S 2 × T 6, J. Phys. A 44 (2011) 275401 [arXiv:1104.1793] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  48. R. Roiban and A.A. Tseytlin, UV finiteness of Pohlmeyer-reduced form of the AdS 5 × S 5 superstring theory, JHEP 04 (2009) 078 [arXiv:0902.2489] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  49. D.M. Schmidtt, Supersymmetry flows, semi-symmetric space sine-Gordon models and the Pohlmeyer reduction, JHEP 03 (2011) 021 [arXiv:1012.4713] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  50. T.J. Hollowood and J.L. Miramontes, The AdS 5 × S 5 semi-symmetric space sine-Gordon theory, JHEP 05 (2011) 136 [arXiv:1104.2429] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  51. M. Goykhman and E. Ivanov, Worldsheet Supersymmetry of Pohlmeyer-reduced AdS n × S n superstrings, JHEP 09 (2011) 078 [arXiv:1104.0706] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  52. D.M. Schmidtt, Integrability vs. supersymmetry: Poisson structures of the Pohlmeyer reduction, JHEP 11 (2011) 067 [arXiv:1106.4796] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  53. B. Hoare and A. Tseytlin, Tree-level S-matrix of Pohlmeyer reduced form of AdS 5 × S 5 superstring theory, JHEP 02 (2010) 094 [arXiv:0912.2958] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  54. B. Hoare and A. Tseytlin, Towards the quantum S-matrix of the Pohlmeyer reduced version of AdS 5 × S 5 superstring theory, Nucl. Phys. B 851 (2011) 161 [arXiv:1104.2423] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  55. K.-i. Kobayashi and T. Uematsu, S matrix of N = 2 supersymmetric sine-Gordon theory, Phys. Lett. B 275 (1992) 361 [hep-th/9110040] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  56. R. Shankar and E. Witten, The S matrix of the supersymmetric nonlinear σ-model, Phys. Rev. D 17 (1978) 2134 [INSPIRE].

    ADS  Google Scholar 

  57. I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  58. V. Kazakov, A. Marshakov, J. Minahan and K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP 05 (2004) 024 [hep-th/0402207] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  59. N. Berkovits, Quantum consistency of the superstring in AdS 5 × S 5 background, JHEP 03 (2005) 041 [hep-th/0411170] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  60. G. Arutyunov, S. Frolov and M. Staudacher, Bethe ansatz for quantum strings, JHEP 10 (2004) 016 [hep-th/0406256] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  61. R.A. Janik, The AdS 5 × S 5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  62. D. Volin, Minimal solution of the AdS/CFT crossing equation, J. Phys. A 42 (2009) 372001 [arXiv:0904.4929] [INSPIRE].

    MathSciNet  Google Scholar 

  63. N. Beisert, R. Hernandez and E. Lopez, A crossing-symmetric phase for AdS 5 × S 5 strings, JHEP 11 (2006) 070 [hep-th/0609044] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  64. N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].

    Article  Google Scholar 

  65. T. McLoughlin, Review of AdS/CFT Integrability, Chapter II.2: quantum strings in AdS 5 × S 5, Lett. Math. Phys. 99 (2012) 127 [arXiv:1012.3987] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  66. G. Arutyunov, S. Frolov, J. Plefka and M. Zamaklar, The off-shell symmetry algebra of the light-cone AdS 5 × S 5 superstring, J. Phys. A 40 (2007) 3583 [hep-th/0609157] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  67. G. Arutyunov, S. Frolov and M. Zamaklar, Finite-size effects from giant magnons, Nucl. Phys. B 778 (2007) 1 [hep-th/0606126] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  68. S. Frolov, J. Plefka and M. Zamaklar, The AdS 5 × S 5 superstring in light-cone gauge and its Bethe equations, J. Phys. A 39 (2006) 13037 [hep-th/0603008] [INSPIRE].

    MathSciNet  Google Scholar 

  69. R. Roiban, A. Tirziu and A.A. Tseytlin, Two-loop world-sheet corrections in AdS 5 × S 5 superstring, JHEP 07 (2007) 056 [arXiv:0704.3638] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  70. J.M. Maldacena and I. Swanson, Connecting giant magnons to the pp-wave: an interpolating limit of AdS 5 × S 5, Phys. Rev. D 76 (2007) 026002 [hep-th/0612079] [INSPIRE].

    ADS  Google Scholar 

  71. V. Giangreco Marotta Puletti, T. Klose and O. Ohlsson Sax, Factorized world-sheet scattering in near-flat AdS 5 × S 5, Nucl. Phys. B 792 (2008) 228 [arXiv:0707.2082] [INSPIRE].

    Article  Google Scholar 

  72. G. Arutyunov, S. Frolov and M. Zamaklar, The Zamolodchikov-Faddeev algebra for AdS 5 × S 5 superstring, JHEP 04 (2007) 002 [hep-th/0612229] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  73. N. Beisert, V. Dippel and M. Staudacher, A novel long range spin chain and planar N = 4 super Yang-Mills, JHEP 07 (2004) 075 [hep-th/0405001] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  74. G. Arutyunov and S. Frolov, On AdS 5 × S 5 string S-matrix, Phys. Lett. B 639 (2006) 378 [hep-th/0604043] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  75. C. Ahn and R.I. Nepomechie, Review of AdS/CFT integrability, chapter III.2: exact world-sheet S-matrix, Lett. Math. Phys. 99 (2012) 209 [arXiv:1012.3991] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  76. K. Zarembo, Strings on semisymmetric superspaces, JHEP 05 (2010) 002 [arXiv:1003.0465] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  77. A. Cagnazzo and K. Zarembo, B-field in AdS 3 /CFT 2 correspondence and integrability, JHEP 11 (2012)133 [Erratum ibid. 1304 (2013) 003] [arXiv:1209.4049] [INSPIRE].

  78. K. Zarembo, Worldsheet spectrum in AdS 4 /CFT 3 correspondence, JHEP 04 (2009) 135 [arXiv:0903.1747] [INSPIRE].

    Article  ADS  Google Scholar 

  79. C. Kalousios, C. Vergu and A. Volovich, Factorized tree-level scattering in AdS 4 × CP 3, JHEP 09 (2009) 049 [arXiv:0905.4702] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  80. N. Rughoonauth, P. Sundin and L. Wulff, Near BMN dynamics of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 07 (2012) 159 [arXiv:1204.4742] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  81. P. Sundin and L. Wulff, Worldsheet scattering in AdS 3 /CFT 2, JHEP 07 (2013) 007 [arXiv:1302.5349] [INSPIRE].

    Article  ADS  Google Scholar 

  82. B. Hoare and A. Tseytlin, On string theory on AdS 3 × S 3 × T 4 with mixed 3-form flux: tree-level S-matrix, Nucl. Phys. B 873 (2013) 682 [arXiv:1303.1037] [INSPIRE].

    Article  ADS  Google Scholar 

  83. T. Klose and T. McLoughlin, Worldsheet form factors in AdS/CFT, Phys. Rev. D 87 (2013) 026004 [arXiv:1208.2020] [INSPIRE].

    ADS  Google Scholar 

  84. O.T. Engelund, R.W. McKeown and R. Roiban, Generalized unitarity and the worldsheet S matrix in AdS n × S n × M 10−2n, arXiv:1304.4281 [INSPIRE].

Download references

Author information

Authors and Affiliations

  1. Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489, Berlin, Germany

    Lorenzo Bianchi, Valentina Forini & Ben Hoare

Authors
  1. Lorenzo Bianchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valentina Forini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ben Hoare
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ben Hoare.

Additional information

ArXiv ePrint: 1304.1798

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bianchi, L., Forini, V. & Hoare, B. Two-dimensional S-matrices from unitarity cuts. J. High Energ. Phys. 2013, 88 (2013). https://doi.org/10.1007/JHEP07(2013)088

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP07(2013)088

Keywords

Search

Navigation