Scattering equations, twistor-string formulas and double-soft limits in four dimensions

Open Access
Regular Article - Theoretical Physics

Abstract

We study scattering equations and formulas for tree amplitudes of various theories in four dimensions, in terms of spinor helicity variables and on-shell superspace for supersymmetric theories. As originally obtained in Witten’s twistor string theory and other twistor-string models, the equations can take either polynomial or rational forms, and we clarify the simple relation between them. We present new, four-dimensional formulas for all tree amplitudes in the non-linear sigma model, a special Galileon theory and the maximally supersymmetric completion of the Dirac-Born-Infeld theory. Furthermore, we apply the formulas to study various double-soft theorems in these theories, including the emissions of a pair of soft photons, fermions and scalars for super-amplitudes in super-DBI theory.

Keywords

Scattering Amplitudes Effective field theories Spontaneous Symmetry Breaking 

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, Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    K. Hinterbichler and A. Joyce, Hidden symmetry of the Galileon, Phys. Rev. D 92 (2015) 023503 [arXiv:1501.07600] [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    G.R. Dvali, G. Gabadadze and M. Porrati, 4 − D gravity on a brane in 5 − D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    K. Kampf and J. Novotny, Unification of Galileon Dualities, JHEP 10 (2014) 006 [arXiv:1403.6813] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    F. Cachazo, S. He and E.Y. Yuan, Scattering in Three Dimensions from Rational Maps, JHEP 10 (2013) 141 [arXiv:1306.2962] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].ADSGoogle Scholar
  11. [11]
    D. Fairlie and D. Roberts, Dual Models without Tachyons — a New Approach, unpublished Durham preprint PRINT-72-2440 (1972).Google Scholar
  12. [12]
    D. Roberts, Mathematical Structure of Dual Amplitudes, Ph.D. Thesis, Durham University, Durham U.K. (1972), http://etheses.dur.ac.uk/8662/1/8662 5593.PDF.
  13. [13]
    D.B. Fairlie, A Coding of Real Null Four-Momenta into World-Sheet Co-ordinates, Adv. Math. Phys. 2009 (2009) 284689 [arXiv:0805.2263] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    D.J. Gross and P.F. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B 303 (1988) 407 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    E. Witten, Parity invariance for strings in twistor space, Adv. Theor. Math. Phys. 8 (2004) 779 [hep-th/0403199] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    Y. Makeenko and P. Olesen, The QCD scattering amplitude from area behaved Wilson loops, Phys. Lett. B 709 (2012) 285 [arXiv:1111.5606] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    F. Cachazo, Fundamental BCJ Relation in N = 4 SYM From The Connected Formulation, arXiv:1206.5970 [INSPIRE].
  18. [18]
    N. Berkovits, Infinite Tension Limit of the Pure Spinor Superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    K. Ohmori, Worldsheet Geometries of Ambitwistor String, JHEP 06 (2015) 075 [arXiv:1504.02675] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    E. Casali, Y. Geyer, L. Mason, R. Monteiro and K.A. Roehrig, New Ambitwistor String Theories, JHEP 11 (2015) 038 [arXiv:1506.08771] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    R. Roiban, M. Spradlin and A. Volovich, On the tree level S matrix of Yang-Mills theory, Phys. Rev. D 70 (2004) 026009 [hep-th/0403190] [INSPIRE].ADSMathSciNetGoogle Scholar
  23. [23]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    F. Cachazo and Y. Geyer, A ’Twistor String’ Inspired Formula For Tree-Level Scattering Amplitudes in N = 8 SUGRA, arXiv:1206.6511 [INSPIRE].
  25. [25]
    F. Cachazo and D. Skinner, Gravity from Rational Curves in Twistor Space, Phys. Rev. Lett. 110 (2013) 161301 [arXiv:1207.0741] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    F. Cachazo, L. Mason and D. Skinner, Gravity in Twistor Space and its Grassmannian Formulation, SIGMA 10 (2014) 051 [arXiv:1207.4712] [INSPIRE].MathSciNetMATHGoogle Scholar
  27. [27]
    D. Skinner, Twistor Strings for N = 8 Supergravity, arXiv:1301.0868 [INSPIRE].
  28. [28]
    Y. Geyer, A.E. Lipstein and L.J. Mason, Ambitwistor Strings in Four Dimensions, Phys. Rev. Lett. 113 (2014) 081602 [arXiv:1404.6219] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    F. Cachazo and G. Zhang, Minimal Basis in Four Dimensions and Scalar Blocks, arXiv:1601.06305 [INSPIRE].
  30. [30]
    T. Adamo, E. Casali, K.A. Roehrig and D. Skinner, On tree amplitudes of supersymmetric Einstein-Yang-Mills theory, JHEP 12 (2015) 177 [arXiv:1507.02207] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    A.A. Tseytlin, Born-Infeld action, supersymmetry and string theory, hep-th/9908105 [INSPIRE].
  32. [32]
    E. Bergshoeff, F. Coomans, R. Kallosh, C.S. Shahbazi and A. Van Proeyen, Dirac-Born-Infeld-Volkov-Akulov and Deformation of Supersymmetry, JHEP 08 (2013) 100 [arXiv:1303.5662] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    D.V. Volkov and V.P. Akulov, Possible universal neutrino interaction, JETP Lett. 16 (1972) 438 [INSPIRE].ADSMATHGoogle Scholar
  34. [34]
    F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
  35. [35]
    S.L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial vector current, Phys. Rev. 137 (1965) B1022.ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    S. Weinberg, Pion scattering lengths, Phys. Rev. Lett. 17 (1966) 616 [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    F. Cachazo, S. He and E.Y. Yuan, New Double Soft Emission Theorems, Phys. Rev. D 92 (2015) 065030 [arXiv:1503.04816] [INSPIRE].ADSMathSciNetGoogle Scholar
  39. [39]
    W.-M. Chen, Y.-t. Huang and C. Wen, New Fermionic Soft Theorems for Supergravity Amplitudes, Phys. Rev. Lett. 115 (2015) 021603 [arXiv:1412.1809] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    F. Cachazo, Resultants and Gravity Amplitudes, arXiv:1301.3970 [INSPIRE].
  43. [43]
    N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Unification of Residues and Grassmannian Dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, The S-matrix in Twistor Space, JHEP 03 (2010) 110 [arXiv:0903.2110] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    S. He, A Link Representation for Gravity Amplitudes, JHEP 10 (2013) 139 [arXiv:1207.4064] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    V.P. Nair, A Current Algebra for Some Gauge Theory Amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    R. Boels, K.J. Larsen, N.A. Obers and M. Vonk, MHV, CSW and BCFW: Field theory structures in string theory amplitudes, JHEP 11 (2008) 015 [arXiv:0808.2598] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    H. Luo and C. Wen, Recursion relations from soft theorems, JHEP 03 (2016) 088 [arXiv:1512.06801] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    F. Cachazo, P. Cha and S. Mizera, Extensions of Theories from Soft Limits, arXiv:1604.03893 [INSPIRE].
  51. [51]
    W.-M. Chen, Y.-t. Huang and C. Wen, Exact coefficients for higher dimensional operators with sixteen supersymmetries, JHEP 09 (2015) 098 [arXiv:1505.07093] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    T. Klose, T. McLoughlin, D. Nandan, J. Plefka and G. Travaglini, Double-Soft Limits of Gluons and Gravitons, JHEP 07 (2015) 135 [arXiv:1504.05558] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    A. Volovich, C. Wen and M. Zlotnikov, Double Soft Theorems in Gauge and String Theories, JHEP 07 (2015) 095 [arXiv:1504.05559] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    P. Di Vecchia, R. Marotta and M. Mojaza, Double-soft behavior for scalars and gluons from string theory, JHEP 12 (2015) 150 [arXiv:1507.00938] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    K. Kampf, J. Novotny and J. Trnka, Tree-level Amplitudes in the Nonlinear σ-model, JHEP 05 (2013) 032 [arXiv:1304.3048] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    Y.-J. Du and H. Lüo, On single and double soft behaviors in NLSM, JHEP 08 (2015) 058 [arXiv:1505.04411] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    I. Low, Double Soft Theorems and Shift Symmetry in Nonlinear σ-models, Phys. Rev. D 93 (2016) 045032 [arXiv:1512.01232] [INSPIRE].ADSGoogle Scholar
  58. [58]
    W.-M. Chen, Y.-t. Huang and C. Wen, From U(1) to E 8 : soft theorems in supergravity amplitudes, JHEP 03 (2015) 150 [arXiv:1412.1811] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  59. [59]
    G. Georgiou, Multi-soft theorems in Gauge Theory from MHV Diagrams, JHEP 08 (2015) 128 [arXiv:1505.08130] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  60. [60]
    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
  61. [61]
    Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop Integrands for Scattering Amplitudes from the Riemann Sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    F. Cachazo, S. He and E.Y. Yuan, One-Loop Corrections from Higher Dimensional Tree Amplitudes, arXiv:1512.05001 [INSPIRE].
  64. [64]
    A. Lipstein and V. Schomerus, Towards a Worldsheet Description of N = 8 Supergravity, arXiv:1507.02936 [INSPIRE].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of SciencesBeijingP.R. China
  2. 2.Center for Cosmology, Particle Physics and Phenomenology (CP3), Université catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.Department of PhysicsRenmin University of ChinaBeijingP.R. China
  4. 4.Institute of High Energy Physics and Theoretical Physics Center for Science FacilitiesChinese Academy of SciencesBeijingP.R. China
  5. 5.Center for High Energy PhysicsPeking UniversityBeijingP.R. China

Personalised recommendations