String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry

  • Marcus Berg
  • Igor BuchbergerEmail author
  • Oliver Schlotterer
Open Access
Regular Article - Theoretical Physics


We compute one-loop amplitudes in six-dimensional Yang-Mills theory with half-maximal supersymmetry from first principles: imposing gauge invariance and locality on an ansatz made from string-theory inspired kinematic building blocks yields unique expressions for the 3- and 4-point amplitudes. We check that the results are reproduced in the field-theory limit α → 0 of string amplitudes in K3 orbifolds, using simplifications made in a companion string-theory paper [1].


Scattering Amplitudes Superstrings and Heterotic Strings Supersymmetric Gauge Theory Supersymmetric Effective Theories 


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]
    M. Berg, I. Buchberger and O. Schlotterer, From maximal to minimal supersymmetry in string loop amplitudes, JHEP 04 (2017) 163 [arXiv:1603.05262] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    M. Bianchi and A.V. Santini, String predictions for near future colliders from one-loop scattering amplitudes around D-brane worlds, JHEP 12 (2006) 010 [hep-th/0607224] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M. Bianchi and D. Consoli, Simplifying one-loop amplitudes in superstring theory, JHEP 01 (2016) 043 [arXiv:1508.00421] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    P. Tourkine and P. Vanhove, One-loop four-graviton amplitudes in \( \mathcal{N}=4 \) supergravity models, Phys. Rev. D 87 (2013) 045001 [arXiv:1208.1255] [INSPIRE].
  5. [5]
    A. Ochirov and P. Tourkine, BCJ duality and double copy in the closed string sector, JHEP 05 (2014) 136 [arXiv:1312.1326] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    J.A. Minahan, One loop amplitudes on orbifolds and the renormalization of coupling constants, Nucl. Phys. B 298 (1988) 36 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    E. Witten, Twistor-like transform in ten-dimensions, Nucl. Phys. B 266 (1986) 245 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    P.S. Howe, Pure spinors lines in superspace and ten-dimensional supersymmetric theories, Phys. Lett. B 258 (1991) 141 [Addendum ibid. B 259 (1991) 511] [INSPIRE].
  9. [9]
    P.S. Howe, Pure spinors, function superspaces and supergravity theories in ten-dimensions and eleven-dimensions, Phys. Lett. B 273 (1991) 90 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  11. [11]
    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
  12. [12]
    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
  13. [13]
    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].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    C.R. Mafra and O. Schlotterer, One-loop superstring six-point amplitudes and anomalies in pure spinor superspace, JHEP 04 (2016) 148 [arXiv:1603.04790] [INSPIRE].ADSGoogle Scholar
  15. [15]
    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].
  16. [16]
    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
  17. [17]
    J.J.M. Carrasco, M. Chiodaroli, M. Günaydin and R. Roiban, One-loop four-point amplitudes in pure and matter-coupled N ≤ 4 supergravity, JHEP 03 (2013) 056 [arXiv:1212.1146] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    H. Johansson and A. Ochirov, Pure gravities via color-kinematics duality for fundamental matter, JHEP 11 (2015) 046 [arXiv:1407.4772] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    F.A. Berends and W.T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    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
  21. [21]
    C.R. Mafra and O. Schlotterer, Berends-Giele recursions and the BCJ duality in superspace and components, JHEP 03 (2016) 097 [arXiv:1510.08846] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    S. Lee, C.R. Mafra and O. Schlotterer, Non-linear gauge transformations in D = 10 SYM theory and the BCJ duality, JHEP 03 (2016) 090 [arXiv:1510.08843] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    F.A. Berends, W.T. Giele and H. Kuijf, Exact and approximate expressions for multi-gluon scattering, Nucl. Phys. B 333 (1990) 120 [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
  26. [26]
    J. Polchinski, String theory. Volume 2: superstring theory and beyond, Cambridge University Press, Cambridge U.K. (1998).Google Scholar
  27. [27]
    M.B. Green, J.H. Schwarz and E. Witten, Superstring theory. Volume 2: loop amplitudes, anomalies and phenomenology, Cambridge University Press, Cambridge U.K. (1987).Google Scholar
  28. [28]
    W.-M. Chen, Y.-t. Huang and D.A. McGady, Anomalies without an action, arXiv:1402.7062 [INSPIRE].
  29. [29]
    C.R. Mafra and O. Schlotterer, Cohomology foundations of one-loop amplitudes in pure spinor superspace, arXiv:1408.3605 [INSPIRE].
  30. [30]
    L.E. Ibanez and A.M. Uranga, String theory and particle physics: an introduction to string phenomenology, Cambridge University Press, Cambridge U.K. (2012).Google Scholar
  31. [31]
    H. Elvang and M. Kiermaier, Stringy KLT relations, global symmetries and E 7(7) violation, JHEP 10 (2010) 108 [arXiv:1007.4813] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  32. [32]
    Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Simplifying multiloop integrands and ultraviolet divergences of gauge theory and gravity amplitudes, Phys. Rev. D 85 (2012) 105014 [arXiv:1201.5366] [INSPIRE].ADSGoogle Scholar
  33. [33]
    J. Polchinski, String theory. Volume 1: an introduction to the bosonic string, Cambridge University Press, Cambridge U.K. (1998).Google Scholar
  34. [34]
    A. Sen, One loop mass renormalization of unstable particles in superstring theory, JHEP 11 (2016) 050 [arXiv:1607.06500] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  35. [35]
    Z. Bern and A.G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [INSPIRE].
  36. [36]
    P. Anastasopoulos, M. Bianchi, E. Dudas and E. Kiritsis, Anomalies, anomalous U(1)’s and generalized Chern-Simons terms, JHEP 11 (2006) 057 [hep-th/0605225] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    J.J.M. Carrasco, Gauge and gravity amplitude relations, arXiv:1506.00974 [INSPIRE].
  38. [38]
    Z. Bern, S. Davies, T. Dennen, A.V. Smirnov and V.A. Smirnov, Ultraviolet properties of N = 4 supergravity at four loops, Phys. Rev. Lett. 111(2013)231302 [arXiv:1309.2498] [INSPIRE].
  39. [39]
    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].
  40. [40]
    R.H. Boels, R.S. Isermann, R. Monteiro and D. O’Connell, Colour-kinematics duality for one-loop rational amplitudes, JHEP 04 (2013) 107 [arXiv:1301.4165] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    Z. Bern, S. Davies, T. Dennen, Y.-t. Huang and J. Nohle, Color-kinematics duality for pure Yang-Mills and gravity at one and two loops, Phys. Rev. D 92 (2015) 045041 [arXiv:1303.6605] [INSPIRE].ADSMathSciNetGoogle Scholar
  42. [42]
    J. Nohle, Color-kinematics duality in one-loop four-gluon amplitudes with matter, Phys. Rev. D 90 (2014) 025020 [arXiv:1309.7416] [INSPIRE].ADSGoogle Scholar
  43. [43]
    M. Chiodaroli, Q. Jin and R. Roiban, Color/kinematics duality for general abelian orbifolds of N = 4 super Yang-Mills theory, JHEP 01 (2014) 152 [arXiv:1311.3600] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    G. Mogull and D. O’Connell, Overcoming obstacles to colour-kinematics duality at two loops, JHEP 12 (2015) 135 [arXiv:1511.06652] [INSPIRE].ADSMathSciNetGoogle Scholar
  45. [45]
    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
  46. [46]
    Z. Bern, C. Boucher-Veronneau and H. Johansson, N ≥ 4 supergravity amplitudes from gauge theory at one loop, Phys. Rev. D 84 (2011) 105035 [arXiv:1107.1935] [INSPIRE].ADSGoogle Scholar
  47. [47]
    A. Gregori, E. Kiritsis, C. Kounnas, N.A. Obers, P.M. Petropoulos and B. Pioline, R 2 corrections and nonperturbative dualities of N = 4 string ground states, Nucl. Phys. B 510 (1998) 423 [hep-th/9708062] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  48. [48]
    S. He and O. Schlotterer, New relations for gauge-theory and gravity amplitudes at loop level, Phys. Rev. Lett. 118 (2017) 161601 [arXiv:1612.00417] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    Z. Bern, J.J. Carrasco, W.-M. Chen, H. Johansson and R. Roiban, Gravity amplitudes as generalized double copies of gauge-theory amplitudes, Phys. Rev. Lett. 118 (2017) 181602 [arXiv:1701.02519] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    H. Johansson and Y.T. Huang, private communication.Google Scholar
  51. [51]
    S. He, O. Schlotterer and Y. Zhang, New BCJ representations for one-loop amplitudes in gauge theories and gravity, arXiv:1706.00640 [INSPIRE].
  52. [52]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
  53. [53]
    M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, Supergravity and the S matrix, Phys. Rev. D 15 (1977) 996.ADSGoogle Scholar
  54. [54]
    M.T. Grisaru and H.N. Pendleton, Some properties of scattering amplitudes in supersymmetric theories, Nucl. Phys. B 124 (1977) 81 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    S.J. Parke and T.R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    L.A. Barreiro and R. Medina, Revisiting the S-matrix approach to the open superstring low energy effective lagrangian, JHEP 10 (2012) 108 [arXiv:1208.6066] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    L.A. Barreiro and R. Medina, RNS derivation of N-point disk amplitudes from the revisited S-matrix approach, Nucl. Phys. B 886 (2014) 870 [arXiv:1310.5942] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    R.H. Boels and R. Medina, Graviton and gluon scattering from first principles, Phys. Rev. Lett. 118 (2017) 061602 [arXiv:1607.08246] [INSPIRE].CrossRefGoogle Scholar
  59. [59]
    R. Blumenhagen, D. Lüst and S. Theisen, Basic concepts of string theory, Springer, Germany (2013).Google Scholar
  60. [60]
    C. Angelantonj and A. Sagnotti, Open strings, Phys. Rept. 371 (2002) 1 [hep-th/0204089] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    E.G. Gimon and C.V. Johnson, K3 orientifolds, Nucl. Phys. B 477 (1996) 715 [hep-th/9604129] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 supergravity as limits of string theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    J. Broedel, C.R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112 [arXiv:1412.5535] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  64. [64]
    Z. Bern and D.A. Kosower, The computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  65. [65]
    C. Schubert, Perturbative quantum field theory in the string inspired formalism, Phys. Rept. 355 (2001) 73 [hep-th/0101036] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    N.E.J. Bjerrum-Bohr and P. Vanhove, Explicit cancellation of triangles in one-loop gravity amplitudes, JHEP 04 (2008) 065 [arXiv:0802.0868] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    P. Tourkine, Tropical amplitudes, Annales Henri Poincaré 18 (2017) 2199 [arXiv:1309.3551] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    N. Berkovits, Covariant quantization of the Green-Schwarz superstring in a Calabi-Yau background, Nucl. Phys. B 431 (1994) 258 [hep-th/9404162] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    N. Berkovits, C. Vafa and E. Witten, Conformal field theory of AdS background with Ramond-Ramond flux, JHEP 03 (1999) 018 [hep-th/9902098] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    C.R. Mafra, Towards field theory amplitudes from the cohomology of pure spinor superspace, JHEP 11 (2010) 096 [arXiv:1007.3639] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    C.R. Mafra, O. Schlotterer, S. Stieberger and D. Tsimpis, Six open string disk amplitude in pure spinor superspace, Nucl. Phys. B 846 (2011) 359 [arXiv:1011.0994] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    E.G. Gimon and J. Polchinski, Consistency conditions for orientifolds and d manifolds, Phys. Rev. D 54 (1996) 1667 [hep-th/9601038] [INSPIRE].ADSMathSciNetGoogle Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Marcus Berg
    • 1
  • Igor Buchberger
    • 1
    Email author
  • Oliver Schlotterer
    • 2
  1. 1.Department of PhysicsKarlstad UniversityKarlstadSweden
  2. 2.Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-InstitutPotsdamGermany

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