Double soft theorems in gauge and string theories

  • Anastasia Volovich
  • Congkao Wen
  • Michael Zlotnikov
Open Access
Regular Article - Theoretical Physics


We investigate the tree-level S-matrix in gauge theories and open superstring theory with several soft particles. We show that scattering amplitudes with two or three soft gluons of non-identical helicities behave universally in the limit, with multi-soft factors which are not the product of individual soft gluon factors. The results are obtained from the BCFW recursion relations in four dimensions, and further extended to arbitrary dimensions using the CHY formula. We also find new soft theorems for double soft limits of scalars and fermions in \( \mathcal{N}=4 \) and pure \( \mathcal{N}=2 \) SYM. Finally, we show that the double-soft-scalar theorems can be extended to open superstring theory without receiving any α′ corrections.


Scattering Amplitudes Supersymmetric gauge theory Extended Supersymmetry 


  1. [1]
    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516.MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    T.H. Burnett and N.M. Kroll, Extension of the low soft photon theorem, Phys. Rev. Lett. 20 (1968) 86 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    D.J. Gross and R. Jackiw, Low-Energy Theorem for Graviton Scattering, Phys. Rev. 166 (1968) 1287 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    R. Jackiw, Low-Energy Theorems for Massless Bosons: Photons and Gravitons, Phys. Rev. 168 (1968) 1623 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    J.S. Bell and R. Van Royen, On the low-burnett-kroll theorem for soft-photon emission, Nuovo Cim. A 60 (1969) 62 [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    V. Del Duca, High-energy Bremsstrahlung Theorems for Soft Photons, Nucl. Phys. B 345 (1990) 369 [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
  9. [9]
    E. Casali, Soft sub-leading divergences in Yang-Mills amplitudes, JHEP 08 (2014) 077 [arXiv:1404.5551] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    A.J. Larkoski, Conformal Invariance of the Subleading Soft Theorem in Gauge Theory, Phys. Rev. D 90 (2014) 087701 [arXiv:1405.2346] [INSPIRE].ADSGoogle Scholar
  11. [11]
    F. Cachazo and E.Y. Yuan, Are Soft Theorems Renormalized?, arXiv:1405.3413 [INSPIRE].
  12. [12]
    J. Broedel, M. de Leeuw, J. Plefka and M. Rosso, Constraining subleading soft gluon and graviton theorems, Phys. Rev. D 90 (2014) 065024 [arXiv:1406.6574] [INSPIRE].ADSGoogle Scholar
  13. [13]
    J. Broedel, M. de Leeuw, J. Plefka and M. Rosso, Local contributions to factorized soft graviton theorems at loop level, Phys. Lett. B 746 (2015) 293 [arXiv:1411.2230] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    B.U.W. Schwab, Subleading Soft Factor for String Disk Amplitudes, JHEP 08 (2014) 062 [arXiv:1406.4172] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    Z. Bern, S. Davies, P. Di Vecchia and J. Nohle, Low-Energy Behavior of Gluons and Gravitons from Gauge Invariance, Phys. Rev. D 90 (2014) 084035 [arXiv:1406.6987] [INSPIRE].ADSGoogle Scholar
  16. [16]
    C.D. White, Diagrammatic insights into next-to-soft corrections, Phys. Lett. B 737 (2014) 216 [arXiv:1406.7184] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].ADSGoogle Scholar
  18. [18]
    Y.-J. Du, B. Feng, C.-H. Fu and Y. Wang, Note on Soft Graviton theorem by KLT Relation, JHEP 11 (2014) 090 [arXiv:1408.4179] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    H. Lüo, P. Mastrolia and W.J. Torres Bobadilla, Subleading soft behavior of QCD amplitudes, Phys. Rev. D 91 (2015) 065018 [arXiv:1411.1669] [INSPIRE].ADSGoogle Scholar
  20. [20]
    B.U.W. Schwab, A Note on Soft Factors for Closed String Scattering, JHEP 03 (2015) 140 [arXiv:1411.6661] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  21. [21]
    A. Sabio Vera and M.A. Vazquez-Mozo, The Double Copy Structure of Soft Gravitons, JHEP 03 (2015) 070 [arXiv:1412.3699] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  22. [22]
    A.J. Larkoski, D. Neill and I.W. Stewart, Soft Theorems from Effective Field Theory, JHEP 06 (2015) 077 [arXiv:1412.3108] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    M. Campiglia and A. Laddha, New symmetries for the Gravitational S-matrix, JHEP 04 (2015) 076 [arXiv:1502.02318] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    P. Di Vecchia, R. Marotta and M. Mojaza, Soft theorem for the graviton, dilaton and the Kalb-Ramond field in the bosonic string, JHEP 05 (2015) 137 [arXiv:1502.05258] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    T. He, P. Mitra and A. Strominger, 2D Kac-Moody Symmetry of 4D Yang-Mills Theory, arXiv:1503.02663 [INSPIRE].
  26. [26]
    S. Pasterski, A. Strominger and A. Zhiboedov, New Gravitational Memories, arXiv:1502.06120 [INSPIRE].
  27. [27]
    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Higher-Dimensional Supertranslations and Weinbergs Soft Graviton Theorem, arXiv:1502.07644 [INSPIRE].
  28. [28]
    D. Kapec, V. Lysov and A. Strominger, Asymptotic Symmetries of Massless QED in Even Dimensions, arXiv:1412.2763 [INSPIRE].
  29. [29]
    V. Lysov, S. Pasterski and A. Strominger, Lows Subleading Soft Theorem as a Symmetry of QED, Phys. Rev. Lett. 113 (2014) 111601 [arXiv:1407.3814] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    T. He, P. Mitra, A.P. Porfyriadis and A. Strominger, New Symmetries of Massless QED, JHEP 10 (2014) 112 [arXiv:1407.3789] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S-matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinbergs soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG 2010) 010 [arXiv:1102.4632] [INSPIRE].
  38. [38]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    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
  40. [40]
    B.U.W. Schwab and A. Volovich, Subleading Soft Theorem in Arbitrary Dimensions from Scattering Equations, Phys. Rev. Lett. 113 (2014) 101601 [arXiv:1404.7749] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    N. Afkhami-Jeddi, Soft Graviton Theorem in Arbitrary Dimensions, arXiv:1405.3533 [INSPIRE].
  42. [42]
    M. Zlotnikov, Sub-sub-leading soft-graviton theorem in arbitrary dimension, JHEP 10 (2014) 148 [arXiv:1407.5936] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    C. Kalousios and F. Rojas, Next to subleading soft-graviton theorem in arbitrary dimensions, JHEP 01 (2015) 107 [arXiv:1407.5982] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    T. Adamo, E. Casali and D. Skinner, Perturbative gravity at null infinity, Class. Quant. Grav. 31 (2014) 225008 [arXiv:1405.5122] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    Y. Geyer, A.E. Lipstein and L. Mason, Ambitwistor strings at null infinity and (subleading) soft limits, Class. Quant. Grav. 32 (2015) 055003 [arXiv:1406.1462] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    A.E. Lipstein, Soft Theorems from Conformal Field Theory, JHEP 06 (2015) 166 [arXiv:1504.01364] [INSPIRE].CrossRefGoogle Scholar
  47. [47]
    T. Adamo and E. Casali, Perturbative gauge theory at null infinity, Phys. Rev. D 91 (2015) 125022 [arXiv:1504.02304] [INSPIRE].ADSGoogle Scholar
  48. [48]
    Z. Bern, S. Davies and J. Nohle, On Loop Corrections to Subleading Soft Behavior of Gluons and Gravitons, Phys. Rev. D 90 (2014) 085015 [arXiv:1405.1015] [INSPIRE].ADSGoogle Scholar
  49. [49]
    S. He, Y.-t. Huang and C. Wen, Loop Corrections to Soft Theorems in Gauge Theories and Gravity, JHEP 12 (2014) 115 [arXiv:1405.1410] [INSPIRE].
  50. [50]
    F. Cachazo and E.Y. Yuan, Are Soft Theorems Renormalized?, arXiv:1405.3413 [INSPIRE].
  51. [51]
    M. Bianchi, S. He, Y.-t. Huang and C. Wen, More on Soft Theorems: Trees, Loops and Strings, arXiv:1406.5155 [INSPIRE].
  52. [52]
    S. Weinberg, Pion scattering lengths, Phys. Rev. Lett. 17 (1966) 616 [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    R. Jackiw and L. Soloviev, Low-energy-theorem approach to single-particle singularities in the presence of massless bosons, Phys. Rev. 173 (1968) 1485 [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  55. [55]
    S.L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial vector current, Phys. Rev. 137 (1965) B1022.ADSCrossRefGoogle Scholar
  56. [56]
    W.-M. Chen, Y.-t. Huang and C. Wen, From U(1) to E8: soft theorems in supergravity amplitudes, JHEP 03 (2015) 150 [arXiv:1412.1811] [INSPIRE].
  57. [57]
    W.-M. Chen, Y.-t. Huang and C. Wen, New fermionic soft theorems, Phys. Rev. Lett. 115 (2015) 021603 [arXiv:1412.1809] [INSPIRE].
  58. [58]
    F. Cachazo, S. He and E.Y. Yuan, New Double Soft Emission Theorems, arXiv:1503.04816 [INSPIRE].
  59. [59]
    T. He, P. Mitra and A. Strominger, 2D Kac-Moody Symmetry of 4D Yang-Mills Theory, arXiv:1503.02663 [INSPIRE].
  60. [60]
    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].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    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].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    T. Klose, T. McLoughlin, D. Nandan, J. Plefka and G. Travaglini, Double-Soft Limits of Gluons and Gravitons, arXiv:1504.05558 [INSPIRE].
  63. [63]
    C. Boucher-Veronneau and A.J. Larkoski, Constructing Amplitudes from Their Soft Limits, JHEP 09 (2011) 130 [arXiv:1108.5385] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    D. Nandan and C. Wen, Generating All Tree Amplitudes in N = 4 SYM by Inverse Soft Limit, JHEP 08 (2012) 040 [arXiv:1204.4841] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  65. [65]
    V.P. Nair, A Current Algebra for Some Gauge Theory Amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    A. Brandhuber, P. Heslop and G. Travaglini, A Note on dual superconformal symmetry of the N = 4 super Yang-Mills S-matrix, Phys. Rev. D 78 (2008) 125005 [arXiv:0807.4097] [INSPIRE].MathSciNetADSGoogle Scholar
  67. [67]
    H. Elvang and M. Kiermaier, Stringy KLT relations, global symmetries and E_7(7) violation, JHEP 10 (2010) 108 [arXiv:1007.4813] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  68. [68]
    N. Beisert, H. Elvang, D.Z. Freedman, M. Kiermaier, A. Morales and S. Stieberger, E7(7) constraints on counterterms in N = 8 supergravity, Phys. Lett. B 694 (2010) 265 [arXiv:1009.1643] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  69. [69]
    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].MathSciNetADSCrossRefGoogle Scholar
  70. [70]
    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].MathSciNetADSCrossRefGoogle Scholar
  71. [71]
    H. Elvang, Y.-t. Huang and C. Peng, On-shell superamplitudes in N < 4 SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Anastasia Volovich
    • 1
  • Congkao Wen
    • 2
  • Michael Zlotnikov
    • 1
  1. 1.Brown University Department of PhysicsProvidenceU.S.A.
  2. 2.I.N.F.N. Sezione di Roma “Tor Vergata”RomaItaly

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