Advertisement

Scattering equations, supergravity integrands, and pure spinors

  • Tim Adamo
  • Eduardo Casali
Open Access
Regular Article - Theoretical Physics

Abstract

The tree-level S-matrix of type II supergravity can be computed in scattering equation form by correlators in a worldsheet theory analogous to a chiral, infinite tension limit of the pure spinor formalism. By defining a non-minimal version of this theory, we give a prescription for computing correlators on higher genus worldsheets which manifest spacetime supersymmetry. These correlators are conjectured to provide the loop integrands of supergravity scattering amplitudes, supported on the scattering equations. We give nontrivial evidence in support of this conjecture at genus one and two with four external states. Throughout, we find a close correspondence with the pure spinor formalism of superstring theory, particularly regarding regulators and zero-mode counting.

Keywords

Scattering Amplitudes Superstrings and Heterotic Strings Supergravity Models 

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]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, arXiv:1412.3479 [INSPIRE].
  4. [4]
    L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    T. Adamo, E. Casali and D. Skinner, A Worldsheet Theory for Supergravity, JHEP 02 (2015) 116 [arXiv:1409.5656] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    T. Adamo, E. Casali and D. Skinner, Ambitwistor strings and the scattering equations at one loop, JHEP 04 (2014) 104 [arXiv:1312.3828] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    E. Casali and P. Tourkine, Infrared behaviour of the one-loop scattering equations and supergravity integrands, JHEP 04 (2015) 013 [arXiv:1412.3787] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  8. [8]
    N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  9. [9]
    N. Berkovits, ICTP lectures on covariant quantization of the superstring, hep-th/0209059 [INSPIRE].
  10. [10]
    O.A. Bedoya and N. Berkovits, GGI Lectures on the Pure Spinor Formalism of the Superstring, arXiv:0910.2254 [INSPIRE].
  11. [11]
    H. Gomez and C.R. Mafra, The closed-string 3-loop amplitude and S-duality, JHEP 10 (2013) 217 [arXiv:1308.6567] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    N. Berkovits, Infinite Tension Limit of the Pure Spinor Superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    H. Gomez and E.Y. Yuan, N-point tree-level scattering amplitude in the new Berkovitsstring, JHEP 04 (2014) 046 [arXiv:1312.5485] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    N. Berkovits, Pure spinor formalism as an N = 2 topological string, JHEP 10 (2005) 089 [hep-th/0509120] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    N. Berkovits and N. Nekrasov, Multiloop superstring amplitudes from non-minimal pure spinor formalism, JHEP 12 (2006) 029 [hep-th/0609012] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    N. Berkovits, Covariant quantization of the superparticle using pure spinors, JHEP 09 (2001) 016 [hep-th/0105050] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    J. Bjornsson, Multi-loop amplitudes in maximally supersymmetric pure spinor field theory, JHEP 01 (2011) 002 [arXiv:1009.5906] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    N. Berkovits, Twistor Origin of the Superstring, JHEP 03 (2015) 122 [arXiv:1409.2510] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  19. [19]
    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
  20. [20]
    N. Berkovits, Relating the RNS and pure spinor formalisms for the superstring, JHEP 08 (2001) 026 [hep-th/0104247] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    C.R. Mafra, Four-point one-loop amplitude computation in the pure spinor formalism, JHEP 01 (2006) 075 [hep-th/0512052] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    N. Berkovits, Super-Poincaré covariant two-loop superstring amplitudes, JHEP 01 (2006) 005 [hep-th/0503197] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    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
  24. [24]
    C.R. Mafra, Superstring Scattering Amplitudes with the Pure Spinor Formalism, arXiv:0902.1552 [INSPIRE].
  25. [25]
    J.P. Harnad and S. Shnider, Constraints and field equations for ten-dimensional super Yang-Mills theory, Commun. Math. Phys. 106 (1986) 183 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    E. Witten, Twistor-Like Transform in Ten-Dimensions, Nucl. Phys. B 266 (1986) 245 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    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
  28. [28]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    T. Kugo and I. Ojima, Local Covariant Operator Formalism of Nonabelian Gauge Theories and Quark Confinement Problem, Prog. Theor. Phys. Suppl. 66 (1979) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    G.N. Rybkin, State space in BRST quantization and Kugo-Ojima quartets, Int. J. Mod. Phys. A 6 (1991) 1675 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    J. Bjornsson and M.B. Green, 5 loops in 24/5 dimensions, JHEP 08 (2010) 132 [arXiv:1004.2692] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    I. Oda and M. Tonin, Y-formalism and b ghost in the non-minimal pure spinor formalism of superstrings, Nucl. Phys. B 779 (2007) 63 [arXiv:0704.1219] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    N. Berkovits and S.A. Cherkis, Higher-dimensional twistor transforms using pure spinors, JHEP 12 (2004) 049 [hep-th/0409243] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    Y. Aisaka and N. Berkovits, Pure Spinor Vertex Operators in Siegel Gauge and Loop Amplitude Regularization, JHEP 07 (2009) 062 [arXiv:0903.3443] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    P.A. Grassi and P. Vanhove, Higher-loop amplitudes in the non-minimal pure spinor formalism, JHEP 05 (2009) 089 [arXiv:0903.3903] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    J.D. Fay, Theta Functions on Riemann Surfaces, Lect. Notes Math. 352 (1973) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    E. D’Hoker and D.H. Phong, The Geometry of String Perturbation Theory, Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    J. Hoogeveen and K. Skenderis, BRST quantization of the pure spinor superstring, JHEP 11 (2007) 081 [arXiv:0710.2598] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    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
  41. [41]
    H. Gomez and C.R. Mafra, The Overall Coefficient of the Two-loop Superstring Amplitude Using Pure Spinors, JHEP 05 (2010) 017 [arXiv:1003.0678] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    E. D’Hoker and D.H. Phong, Two loop superstrings. 1. Main formulas, Phys. Lett. B 529 (2002) 241 [hep-th/0110247] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    E. D’Hoker and D.H. Phong, Two-loop superstrings VI: Non-renormalization theorems and the 4-point function, Nucl. Phys. B 715 (2005) 3 [hep-th/0501197] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  44. [44]
    C.R. Mafra, Pure Spinor Superspace Identities for Massless Four-point Kinematic Factors, JHEP 04 (2008) 093 [arXiv:0801.0580] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    E. D’Hoker, M. Gutperle and D.H. Phong, Two-loop superstrings and S-duality, Nucl. Phys. B 722 (2005) 81 [hep-th/0503180] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    E.P. Verlinde and H.L. Verlinde, Chiral Bosonization, Determinants and the String Partition Function, Nucl. Phys. B 288 (1987) 357 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    J. Polchinski, Factorization of Bosonic String Amplitudes, Nucl. Phys. B 307 (1988) 61 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B 530 (1998) 401 [hep-th/9802162] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    T. Adamo, Worldsheet factorization for twistor-strings, JHEP 04 (2014) 080 [arXiv:1310.8602] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    C.R. Mafra, Towards Field Theory Amplitudes From the Cohomology of Pure Spinor Superspace, JHEP 11 (2010) 096 [arXiv:1007.3639] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    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
  52. [52]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    C.R. Mafra and O. Schlotterer, A solution to the non-linear equations of D = 10 super Yang-Mills theory, arXiv:1501.05562 [INSPIRE].
  54. [54]
    C.R. Mafra and O. Schlotterer, Cohomology foundations of one-loop amplitudes in pure spinor superspace, arXiv:1408.3605 [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics & Theoretical PhysicsUniversity of CambridgeCambridgeUnited Kingdom

Personalised recommendations