Download PDF

Non-linear gauge transformations in D = 10 SYM theory and the BCJ duality

  • Seungjin Lee
  • Carlos R. Mafra
  • Oliver Schlotterer
Open Access
Regular Article - Theoretical Physics
First Online:
Received:
Accepted:

Abstract

Recent progress on scattering amplitudes in super Yang-Mills and super-string theory benefitted from the use of multiparticle superfields. They universally capture tree-level subdiagrams, and their generating series solve the non-linear equations of ten-dimensional super Yang-Mills. We provide simplified recursions for multiparticle superfields and relate them to earlier representations through non-linear gauge transformations of their generating series. Moreover, we discuss the gauge transformations which enforce their Lie symmetries as suggested by the Bern-Carrasco-Johansson duality between color and kine-matics. Another gauge transformation due to Harnad and Shnider is shown to streamline the theta-expansion of multiparticle superfields, bypassing the need to use their recursion relations beyond the lowest components. The findings of this work tremendously simplify the component extraction from kinematic factors in pure spinor superspace.

Keywords

Superspaces Superstrings and Heterotic Strings 
Download to read the full article text

Notes

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.

References

  1. [1]
    E. Witten, Twistor-like transform in ten-dimensions, Nucl. Phys. B 266 (1986) 245 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N -point superstring disk amplitude I. Pure spinor computation, Nucl. Phys. B 873 (2013) 419 [arXiv:1106.2645] [INSPIRE].
  4. [4]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N -point superstring disk amplitude II. Amplitude and hypergeometric function structure, Nucl. Phys. B 873 (2013) 461 [arXiv:1106.2646] [INSPIRE].
  5. [5]
    C.R. Mafra and O. Schlotterer, The structure of n-point one-loop open superstring amplitudes, JHEP 08 (2014) 099 [arXiv:1203.6215] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    C.R. Mafra and O. Schlotterer, Multiparticle SYM equations of motion and pure spinor BRST blocks, JHEP 07 (2014) 153 [arXiv:1404.4986] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    C.R. Mafra and O. Schlotterer, Solution to the nonlinear field equations of ten dimensional supersymmetric Yang-Mills theory, Phys. Rev. D 92 (2015) 066001 [arXiv:1501.05562] [INSPIRE].ADSGoogle Scholar
  8. [8]
    Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    Z. Bern, T. Dennen, Y.-t. Huang and M. Kiermaier, Gravity as the square of gauge theory, Phys. Rev. D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE].ADSGoogle Scholar
  10. [10]
    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
  11. [11]
    J.J.M. Carrasco, Gauge and gravity amplitude relations, arXiv:1506.00974 [INSPIRE].
  12. [12]
    C. Reutenauer, Free Lie algebras, London Mathematical Society Monographs, London Mathematical Society, U.K. (1993).Google Scholar
  13. [13]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    C.R. Mafra and O. Schlotterer, Towards one-loop SYM amplitudes from the pure spinor BRST cohomology, Fortsch. Phys. 63 (2015) 105 [arXiv:1410.0668] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    C.R. Mafra and O. Schlotterer, Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace, JHEP 10 (2015) 124 [arXiv:1505.02746] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    J.P. Harnad and S. Shnider, Constraints and field equations for ten-dimensional super Yang-Mills theory, Commun. Math. Phys. 106 (1986) 183 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    G. Policastro and D. Tsimpis, R 4 , purified, Class. Quant. Grav. 23 (2006) 4753 [hep-th/0603165] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    C.R. Mafra, PSS: a form program to evaluate pure spinor superspace expressions, arXiv:1007.4999 [INSPIRE].
  19. [19]
    C.R. Mafra and O. Schlotterer, Berends-Giele recursions and the BCJ duality in superspace and components, arXiv:1510.08846 [INSPIRE].
  20. [20]
    W. Siegel, Superfields in higher dimensional space-time, Phys. Lett. B 80 (1979) 220 [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    N. Berkovits, ICTP lectures on covariant quantization of the superstring, hep-th/0209059 [INSPIRE].
  22. [22]
    F.A. Berends and W.T. Giele, Recursive calculations for processes with n-gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    C.R. Mafra, Towards field theory amplitudes from the cohomology of pure spinor superspace, JHEP 11 (2010) 096 [arXiv:1007.3639] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    C.R. Mafra, O. Schlotterer, S. Stieberger and D. Tsimpis, A recursive method for SYM n-point tree amplitudes, Phys. Rev. D 83 (2011) 126012 [arXiv:1012.3981] [INSPIRE].ADSGoogle Scholar
  25. [25]
    M.B. Green, C.R. Mafra and O. Schlotterer, Multiparticle one-loop amplitudes and S-duality in closed superstring theory, JHEP 10 (2013) 188 [arXiv:1307.3534] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    J. Bjornsson, Multi-loop amplitudes in maximally supersymmetric pure spinor field theory, JHEP 01 (2011) 002 [arXiv:1009.5906] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    J. Bjornsson and M.B. Green, 5 loops in 24/5 dimensions, JHEP 08 (2010) 132 [arXiv:1004.2692] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    F.A. Berends and W.T. Giele, Multiple soft gluon radiation in parton processes, Nucl. Phys. B 313 (1989) 595 [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    R. Ree, Lie elements and an algebra associated with shuffles, Ann. Math. 68 (1958) 210.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    A.A. Rosly and K.G. Selivanov, On amplitudes in selfdual sector of Yang-Mills theory, Phys. Lett. B 399 (1997) 135 [hep-th/9611101] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    A.A. Rosly and K.G. Selivanov, Gravitational SD perturbiner, hep-th/9710196 [INSPIRE].
  32. [32]
    K.G. Selivanov, Postclassicism in tree amplitudes, hep-th/9905128 [INSPIRE].
  33. [33]
    K.G. Selivanov, On tree form-factors in (supersymmetric) Yang-Mills theory, Commun. Math. Phys. 208 (2000) 671 [hep-th/9809046] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    W.A. Bardeen, Selfdual Yang-Mills theory, integrability and multiparton amplitudes, Prog. Theor. Phys. Suppl. 123 (1996) 1 [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    C.R. Mafra and O. Schlotterer, Cohomology foundations of one-loop amplitudes in pure spinor superspace, arXiv:1408.3605 [INSPIRE].
  36. [36]
    N. Berkovits and C.R. Mafra, Some superstring amplitude computations with the non-minimal pure spinor formalism, JHEP 11 (2006) 079 [hep-th/0607187] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    C.R. Mafra and O. Schlotterer, PSS: from pure spinor superspace to components, http://www.damtp.cam.ac.uk/user/crm66/SYM/pss.html.
  38. [38]
    N. Berkovits, Covariant quantization of the superparticle using pure spinors, JHEP 09 (2001) 016 [hep-th/0105050] [INSPIRE].
  39. [39]
    N. Berkovits, Multiloop amplitudes and vanishing theorems using the pure spinor formalism for the superstring, JHEP 09 (2004) 047 [hep-th/0406055] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    N. Berkovits, Super-Poincaré covariant two-loop superstring amplitudes, JHEP 01 (2006) 005 [hep-th/0503197] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    N. Berkovits and C.R. Mafra, Equivalence of two-loop superstring amplitudes in the pure spinor and RNS formalisms, Phys. Rev. Lett. 96 (2006) 011602 [hep-th/0509234] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Seungjin Lee
    • 1
  • Carlos R. Mafra
    • 2
    • 3
  • Oliver Schlotterer
    • 1
  1. 1.Max-Planck-Institut für Gravitationsphysik Albert-Einstein-InstitutPotsdamGermany
  2. 2.Institute for Advanced StudySchool of Natural SciencesPrincetonU.S.A.
  3. 3.DAMTP, University of CambridgeCambridgeU.K.

Personalised recommendations