Four loop scattering in the Nambu-Goto theory

  • Peter Conkey
  • Sergei Dubovsky
Open Access
Regular Article - Theoretical Physics


We initiate the study of multiloop scattering amplitudes in the Nambu-Goto theory on the worldsheet of a non-critical string. We start with a brute force calculation of two loop four particle scattering. Somewhat surprisingly, even though non-trivial UV counterterms are present at this order, on-shell amplitudes remain polynomial in the momenta of colliding particles. We show that this can be understood as a consequence of existence of certain close by (semi)integrable models. Furthermore, these arguments can be extended to obtain the answer for three and four loop scattering, bypassing the brute force calculation. The resulting amplitudes develop non-polynomial (logarithmic) dependence on the momenta starting at three loops.


Bosonic Strings Integrable Field Theories Long strings 


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]
    L.J. Dixon, Scattering amplitudes: the most perfect microscopic structures in the universe, J. Phys. A44 (2011) 454001 [arXiv:1105.0771].ADSMathSciNetMATHGoogle Scholar
  2. [2]
    H. Elvang and Y.-t. Huang, Scattering amplitudes, arXiv:1308.1697 [INSPIRE].
  3. [3]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].
  4. [4]
    N. Arkani-Hamed et al., Scattering amplitudes and the positive Grassmannian, arXiv:1212.5605.
  5. [5]
    Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    Z. Bern, S. Davies and T. Dennen, Enhanced ultraviolet cancellations in \( \mathcal{N} \) = 5 supergravity at four loops, Phys. Rev. D 90 (2014) 105011 [arXiv:1409.3089] [INSPIRE].ADSGoogle Scholar
  7. [7]
    C.F. Berger et al., An automated implementation of on-shell methods for one-loop amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [INSPIRE].ADSGoogle Scholar
  8. [8]
    S. Dubovsky, R. Flauger and V. Gorbenko, Effective string theory revisited, JHEP 09 (2012) 044 [arXiv:1203.1054] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    S. Dubovsky, R. Flauger and V. Gorbenko, Solving the simplest theory of quantum gravity, JHEP 09 (2012) 133 [arXiv:1205.6805] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    S. Dubovsky and V. Gorbenko, Towards a theory of the QCD string, JHEP 02 (2016) 022 [arXiv:1511.01908] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Quantum dynamics of a massless relativistic string, Nucl. Phys. B 56 (1973) 109 [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    A. Athenodorou, B. Bringoltz and M. Teper, Closed flux tubes and their string description in D = 3+1 SU(N) gauge theories, JHEP 02 (2011) 030 [arXiv:1007.4720] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    A. Athenodorou, B. Bringoltz and M. Teper, Closed flux tubes and their string description in D = 2 + 1 SU(N) gauge theories, JHEP 05 (2011) 042 [arXiv:1103.5854] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    A. Athenodorou and M. Teper, Closed flux tubes in higher representations and their string description in D = 2 + 1 SU(N ) gauge theories, JHEP 06 (2013) 053 [arXiv:1303.5946] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    A. Athenodorou and M. Teper, Closed flux tubes in D = 2 + 1 SU(N ) gauge theories: dynamics and effective string description, arXiv:1602.07634 [INSPIRE].
  16. [16]
    S. Dubovsky, R. Flauger and V. Gorbenko, Evidence from lattice data for a new particle on the worldsheet of the QCD flux tube, Phys. Rev. Lett. 111 (2013) 062006 [arXiv:1301.2325] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    S. Dubovsky, R. Flauger and V. Gorbenko, Flux tube spectra from approximate integrability at low energies, J. Exp. Theor. Phys. 120 (2015) 399 [arXiv:1404.0037] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    Z. Bern, C. Cheung, H.-H. Chi, S. Davies, L. Dixon and J. Nohle, Evanescent effects can alter ultraviolet divergences in quantum gravity without physical consequences, Phys. Rev. Lett. 115 (2015) 211301 [arXiv:1507.06118] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    S. Dubovsky, V. Gorbenko and M. Mirbabayi, Natural tuning: towards a proof of concept, JHEP 09 (2013) 045 [arXiv:1305.6939] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    J. Polchinski and A. Strominger, Effective string theory, Phys. Rev. Lett. 67 (1991) 1681 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    A. Alloul, N.D. Christensen, C. Degrande, C. Duhr and B. Fuks, FeynRules 2.0 — A complete toolbox for tree-level phenomenology, Comput. Phys. Commun. 185 (2014) 2250 [arXiv:1310.1921] [INSPIRE].
  22. [22]
    T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260] [INSPIRE].
  23. [23]
    R. Mertig, M. Böhm and A. Denner, FEYN CALC: computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    P. Cooper, S. Dubovsky, V. Gorbenko, A. Mohsen and S. Storace, Looking for integrability on the worldsheet of confining strings, JHEP 04 (2015) 127 [arXiv:1411.0703] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Center for Cosmology and Particle Physics, Department of PhysicsNew York UniversityNew YorkU.S.A.

Personalised recommendations