Poles in the S-matrix of relativistic Chern-Simons matter theories from quantum mechanics

  • Yogesh Dandekar
  • Mangesh Mandlik
  • Shiraz Minwalla
Open Access
Regular Article - Theoretical Physics


An all orders formula for the S-matrix for 2 → 2 scattering in large N Chern-Simons theory coupled to a fundamental scalar has recently been conjectured. We find a scaling limit of the theory in which the pole in this S-matrix is near threshold. We argue that the theory must be well described by non-relativistic quantum mechanics in this limit, and determine the relevant Schroedinger equation. We demonstrate that the S-matrix obtained from this Schroedinger equation agrees perfectly with this scaling limit of the relativistic S-matrix; in particular the pole structures match exactly. We view this matching as a nontrivial consistency check of the conjectured field theory S-matrix.


Scattering Amplitudes Chern-Simons Theories Anyons 1/N Expansion 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959) 485 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    F. Wilczek, Fractional statistics and anyon superconductivity, World Scientific, Singapore (1990) [INSPIRE].CrossRefMATHGoogle Scholar
  4. [4]
    R. Jackiw, Dynamical symmetry of the magnetic vortex, Annals Phys. 201 (1990) 83 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    D. Bak, R. Jackiw and S.-Y. Pi, Non-Abelian Chern-Simons particles and their quantization, Phys. Rev. D 49 (1994) 6778 [hep-th/9402057] [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    D. Bak and O. Bergman, Perturbative analysis of non-Abelian Aharonov-Bohm scattering, Phys. Rev. D 51 (1995) 1994 [hep-th/9403104] [INSPIRE].ADSGoogle Scholar
  7. [7]
    G. Amelino-Camelia and D. Bak, Schrödinger selfadjoint extension and quantum field theory, Phys. Lett. B 343 (1995) 231 [hep-th/9406213] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    S. Park and D. Bak, Exact 4-point scattering amplitude of the superconformal Schrödinger Chern-Simons theory, J. Korean Phys. Soc. 60 (2012) 714 [arXiv:1203.0068] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    S. Jain et al., Unitarity, crossing symmetry and duality of the S-matrix in large-N Chern-Simons theories with fundamental matter, arXiv:1404.6373 [INSPIRE].
  10. [10]
    O. Aharony, G. Gur-Ari and R. Yacoby, D = 3 bosonic vector models coupled to Chern-Simons gauge theories, JHEP 03 (2012) 037 [arXiv:1110.4382] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    S. Giombi et al., Chern-Simons theory with vector fermion matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  13. [13]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, Class. Quant. Grav. 30 (2013) 104003 [arXiv:1204.3882] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    C.-M. Chang, S. Minwalla, T. Sharma and X. Yin, ABJ triality: from higher spin fields to strings, J. Phys. A 46 (2013) 214009 [arXiv:1207.4485] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  15. [15]
    O. Aharony, G. Gur-Ari and R. Yacoby, Correlation functions of large-N Chern-Simons-matter theories and bosonization in three dimensions, JHEP 12 (2012) 028 [arXiv:1207.4593] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    S. Jain, S.P. Trivedi, S.R. Wadia and S. Yokoyama, Supersymmetric Chern-Simons theories with vector matter, JHEP 10 (2012) 194 [arXiv:1207.4750] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    S. Yokoyama, Chern-Simons-Fermion vector model with chemical potential, JHEP 01 (2013) 052 [arXiv:1210.4109] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    G. Gur-Ari and R. Yacoby, Correlators of large-N fermionic Chern-Simons vector models, JHEP 02 (2013) 150 [arXiv:1211.1866] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    O. Aharony, S. Giombi, G. Gur-Ari, J. Maldacena and R. Yacoby, The thermal free energy in large-N Chern-Simons-matter theories, JHEP 03 (2013) 121 [arXiv:1211.4843] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    S. Jain et al., Phases of large-N vector Chern-Simons theories on S 2 × S 1, JHEP 09 (2013) 009 [arXiv:1301.6169] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    T. Takimi, Duality and higher temperature phases of large-N Chern-Simons matter theories on S 2 × S 1, JHEP 07 (2013) 177 [arXiv:1304.3725] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    S. Jain, S. Minwalla and S. Yokoyama, Chern-Simons duality with a fundamental boson and fermion, JHEP 11 (2013) 037 [arXiv:1305.7235] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    Y. Frishman and J. Sonnenschein, Breaking conformal invariancelarge-N Chern-Simons theory coupled to massive fundamental fermions, JHEP 12 (2013) 091 [arXiv:1306.6465] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    S. Yokoyama, A note on large-N thermal free energy in supersymmetric Chern-Simons vector models, JHEP 01 (2014) 148 [arXiv:1310.0902] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  25. [25]
    S.-J. Kim and C.-K. Lee, Quantum description of anyons: role of contact terms, Phys. Rev. D 55 (1997) 2227 [hep-th/9606054] [INSPIRE].ADSGoogle Scholar
  26. [26]
    S.N.M. Ruijsenaars, The Aharonov-Bohm effect and scattering theory, Annals Phys. 146 (1983) 1 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    R. Jackiw, Delta function potentials in two-dimensional and three-dimensional quantum mechanics, (1991) [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Yogesh Dandekar
    • 1
  • Mangesh Mandlik
    • 1
  • Shiraz Minwalla
    • 1
  1. 1.Department of Theoretical PhysicsTata Institute of Fundamental ResearchMumbaiIndia

Personalised recommendations