Skip to main content
SpringerLink
Log in

One-loop four-point amplitudes in pure and matter-coupled \( \mathcal{N}\leq 4 \) supergravity

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We construct all supergravity theories that can be obtained through factorized orbifold projections of \( \mathcal{N}=8 \) supergravity, exposing their double-copy structure, and calculate their one-loop four-point scattering amplitudes. We observe a unified structure in both matter and gravity amplitudes, and demonstrate that the four-graviton amplitudes are insensitive to the precise nature of the matter couplings. We show that these amplitudes are identical for the two different realizations of \( \mathcal{N}=4 \) supergravity with two vector multiplets, and argue that this feature extends to all multiplicities and loop orders as well as to higher dimensions. We also construct a selected set of supergravities obtained through a non-factorized orbifold action. Furthermore we calculate one-loop four-point amplitudes for all pure super-Yang-Mills theories with less-than-maximal supersymmetry using the duality between color and kinematics, finding here a unified expression that holds for all four gluon amplitudes in these theories. We recover the related amplitudes of factorized \( \mathcal{N}\leq 4 \) supergravities employing the double-copy construction. We observe a requirement that the four-point loop-level amplitudes have non-local integrand representations, exhibiting a mild non-locality in the form of inverse powers of the three external Mandelstam invariants. These are the first loop-level color-kinematic-satisfying representations in reduced supersymmetry theories.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. Beisert, C. Kristjansen and M. Staudacher, The Dilatation operator of conformal \( \mathcal{N}=4 \) super Yang-Mills theory, Nucl. Phys. B 664 (2003) 131 [hep-th/0303060] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. N. Beisert and M. Staudacher, The \( \mathcal{N}=4 \) SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439 [hep-th/0307042] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].

    MathSciNet  Google Scholar 

  4. 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].

    Article  MathSciNet  ADS  Google Scholar 

  5. 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].

    Article  MathSciNet  ADS  Google Scholar 

  6. N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Hodges and J. Trnka, A note on polytopes for scattering amplitudes, JHEP 04 (2012) 081 [arXiv:1012.6030] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  7. L. Dolan, C.R. Nappi and E. Witten, Yangian symmetry in D = 4 superconformal Yang-Mills theory, hep-th/0401243 [INSPIRE].

  8. J. Drummond, J. Henn, V. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in \( \mathcal{N}=4 \) super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  10. J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in \( \mathcal{N}=4 \) super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  12. Z. Bern, J. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  13. Z. Bern et al., Three-Loop Superfiniteness of \( \mathcal{N}=8 \) Supergravity, Phys. Rev. Lett. 98 (2007) 161303 [hep-th/0702112] [INSPIRE].

    Article  ADS  Google Scholar 

  14. Z. Bern, J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of \( \mathcal{N}=8 \) Supergravity, Phys. Rev. D 78 (2008) 105019 [arXiv:0808.4112] [INSPIRE].

    ADS  Google Scholar 

  15. Z. Bern, J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The Ultraviolet Behavior of \( \mathcal{N}=8 \) Supergravity at Four Loops, Phys. Rev. Lett. 103 (2009) 081301 [arXiv:0905.2326] [INSPIRE].

    Article  ADS  Google Scholar 

  16. Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Amplitudes and Ultraviolet Behavior of \( \mathcal{N}=8 \) Supergravity, Fortsch. Phys. 59 (2011) 561 [arXiv:1103.1848] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Z. Bern, L.J. Dixon and R. Roiban, Is \( \mathcal{N}=8 \) supergravity ultraviolet finite?, Phys. Lett. B 644 (2007) 265 [hep-th/0611086] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  18. R. Kallosh, \( \mathcal{N}=8 \) Supergravity on the Light Cone, Phys. Rev. D 80 (2009) 105022 [arXiv:0903.4630] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  19. M.B. Green, J.G. Russo and P. Vanhove, String theory dualities and supergravity divergences, JHEP 06 (2010) 075 [arXiv:1002.3805] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  20. M.B. Green, J.G. Russo and P. Vanhove, Automorphic properties of low energy string amplitudes in various dimensions, Phys. Rev. D 81 (2010) 086008 [arXiv:1001.2535] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  21. G. Bossard and H. Nicolai, Counterterms vs. dualities, JHEP 08 (2011) 074 [arXiv:1105.1273] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. G. Chalmers, On the finiteness of \( \mathcal{N}=8 \) quantum supergravity, hep-th/0008162 [INSPIRE].

  23. M.B. Green, J.G. Russo and P. Vanhove, Non-renormalisation conditions in type-II string theory and maximal supergravity, JHEP 02 (2007) 099 [hep-th/0610299] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. R. Kallosh, E 7(7) symmetry and finiteness of \( \mathcal{N}=8 \) supergravity, JHEP 03 (2012) 083 [arXiv:1103.4115] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  25. R. Kallosh, \( \mathcal{N}=8 \) counterterms and E 7(7) current conservation, JHEP 06 (2011) 073 [arXiv:1104.5480] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  26. M.B. Green, H. Ooguri and J.H. Schwarz, Nondecoupling of Maximal Supergravity from the Superstring, Phys. Rev. Lett. 99 (2007) 041601 [arXiv:0704.0777] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on orbifolds, Nucl. Phys. B 261 (1985) 678 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on orbifolds. 2., Nucl. Phys. B 274 (1986) 285 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The conformal field theory of orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  30. S. Kachru and E. Silverstein, 4 − D conformal theories and strings on orbifolds, Phys. Rev. Lett. 80 (1998) 4855 [hep-th/9802183] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. A.E. Lawrence, N. Nekrasov and C. Vafa, On conformal field theories in four-dimensions, Nucl. Phys. B 533 (1998) 199 [hep-th/9803015] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. M. Bershadsky and A. Johansen, Large-N limit of orbifold field theories, Nucl. Phys. B 536 (1998) 141 [hep-th/9803249] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. D.C. Dunbar, J.H. Ettle and W.B. Perkins, Perturbative expansion of \( \mathcal{N}<8 \) Supergravity, Phys. Rev. D 83 (2011) 065015 [arXiv:1011.5378] [INSPIRE].

    ADS  Google Scholar 

  34. Z. Bern, C. Boucher-Veronneau and H. Johansson, \( \mathcal{N}\geq 4 \) Supergravity Amplitudes from Gauge Theory at One Loop, Phys. Rev. D 84 (2011) 105035 [arXiv:1107.1935] [INSPIRE].

    ADS  Google Scholar 

  35. Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, Absence of Three-Loop Four-Point Divergences in \( \mathcal{N}=4 \) Supergravity, Phys. Rev. Lett. 108 (2012) 201301 [arXiv:1202.3423] [INSPIRE].

    Article  ADS  Google Scholar 

  36. Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, Ultraviolet Cancellations in Half-Maximal Supergravity as a Consequence of the Double-Copy Structure, Phys. Rev. D 86 (2012) 105014 [arXiv:1209.2472] [INSPIRE].

    ADS  Google Scholar 

  37. P. Tourkine and P. Vanhove, An R 4 non-renormalisation theorem in \( \mathcal{N}=4 \) supergravity, Class. Quant. Grav. 29 (2012) 115006 [arXiv:1202.3692] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  38. R. Kallosh, On Absence of 3-loop Divergence in \( \mathcal{N}=4 \) Supergravity, Phys. Rev. D 85 (2012) 081702 [arXiv:1202.4690] [INSPIRE].

    ADS  Google Scholar 

  39. S. Ferrara, R. Kallosh and A. Van Proeyen, Conjecture on Hidden Superconformal Symmetry of \( \mathcal{N}=4 \) Supergravity, Phys. Rev. D 87 (2013) 025004 [arXiv:1209.0418] [INSPIRE].

    ADS  Google Scholar 

  40. G. Bossard, P. Howe and K. Stelle, Anomalies and divergences in \( \mathcal{N}=4 \) supergravity, Phys. Lett. B 719 (2013) 424 [arXiv:1212.0841] [INSPIRE].

    ADS  Google Scholar 

  41. P. Tourkine and P. Vanhove, One-loop four-graviton amplitudes in \( \mathcal{N}=4 \) supergravity models, arXiv:1208.1255 [INSPIRE].

  42. M. Fischler, Finiteness calculations for O(4) through O(8) extended supergravity and O(4) supergravity coupled to selfdual O(4) matter, Phys. Rev. D 20 (1979) 396 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  43. E. Fradkin and A.A. Tseytlin, One loop infinities in dimensionally reduced supergravities, Phys. Lett. B 137 (1984) 357 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  44. 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].

    Article  MathSciNet  ADS  Google Scholar 

  45. T. Bargheer, S. He and T. McLoughlin, New Relations for Three-Dimensional Supersymmetric Scattering Amplitudes, Phys. Rev. Lett. 108 (2012) 231601 [arXiv:1203.0562] [INSPIRE].

    Article  ADS  Google Scholar 

  46. Y.-t. Huang and H. Johansson, Equivalent D = 3 Supergravity Amplitudes from Double Copies of Three-Algebra and Two-Algebra Gauge Theories, arXiv:1210.2255 [INSPIRE].

  47. P.H. Damgaard, R. Huang, T. Sondergaard and Y. Zhang, The complete KLT-map between gravity and gauge theories, JHEP 08 (2012) 101 [arXiv:1206.1577] [INSPIRE].

    Article  ADS  Google Scholar 

  48. 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].

    Article  MathSciNet  ADS  Google Scholar 

  49. Z. Bern, L.J. Dixon and D.A. Kosower, Progress in one loop QCD computations, Ann. Rev. Nucl. Part. Sci. 46 (1996) 109 [hep-ph/9602280] [INSPIRE].

    Article  ADS  Google Scholar 

  50. 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].

    Article  ADS  Google Scholar 

  51. H. Kawai, D. Lewellen and S. Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  52. Z. Bern, L.J. Dixon, M. Perelstein and J. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  53. M. Günaydin, G. Sierra and P. Townsend, Exceptional Supergravity Theories and the MAGIC Square, Phys. Lett. B 133 (1983) 72 [INSPIRE].

    ADS  Google Scholar 

  54. A. Sen and C. Vafa, Dual pairs of type-II string compactification, Nucl. Phys. B 455 (1995) 165 [hep-th/9508064] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  55. M. Günaydin, Lectures on Spectrum Generating Symmetries and U-duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace, in Springer Proceedings in Physics. Vol. 134: Proceedings of the School on Attractor Mechanism 2007, Springer-Verlag, Berlin Germany (2010) [arXiv:0908.0374] [INSPIRE].

  56. E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  57. M. Günaydin, G. Sierra and P. Townsend, The geometry of \( \mathcal{N}=2 \) Maxwell-Einstein supergravity and Jordan algebras, Nucl. Phys. B 242 (1984) 244 [INSPIRE].

    Article  ADS  Google Scholar 

  58. A. Dabholkar and J.A. Harvey, String islands, JHEP 02 (1999) 006 [hep-th/9809122] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  59. S. Mizoguchi, On asymmetric orbifolds and the D = 5 no-modulus supergravity, Phys. Lett. B 523 (2001) 351 [hep-th/0109193] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  60. R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in \( \mathcal{N}=4 \) super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  61. H. Elvang, D.Z. Freedman and M. Kiermaier, Recursion relations, generating functions and unitarity sums in \( \mathcal{N}=4 \) SYM theory, JHEP 04 (2009) 009 [arXiv:0808.1720] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  62. Z. Bern, J. Carrasco, H. Ita, H. Johansson and R. Roiban, On the structure of supersymmetric sums in multi-loop unitarity cuts, Phys. Rev. D 80 (2009) 065029 [arXiv:0903.5348] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  63. N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  64. M. Bianchi, H. Elvang and D.Z. Freedman, Generating tree amplitudes in \( \mathcal{N}=4 \) SYM and \( \mathcal{N}=8 \) SG, JHEP 09 (2008) 063 [arXiv:0805.0757] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  65. S. Lal and S. Raju, The next-to-simplest quantum field theories, Phys. Rev. D 81 (2010) 105002 [arXiv:0910.0930] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  66. H. Elvang, Y.-t. Huang and C. Peng, On-shell superamplitudes in \( \mathcal{N}<4 \) SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  67. T. Dennen, Y.-t. Huang and W. Siegel, Supertwistor space for 6D maximal super Yang-Mills, JHEP 04 (2010) 127 [arXiv:0910.2688] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  68. M.B. Green, J.H. Schwarz and L. Brink, \( \mathcal{N}=4 \) Yang-Mills and \( \mathcal{N}=8 \) Supergravity as Limits of String Theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].

    Article  ADS  Google Scholar 

  69. D.C. Dunbar and P.S. Norridge, Calculation of graviton scattering amplitudes using string based methods, Nucl. Phys. B 433 (1995) 181 [hep-th/9408014] [INSPIRE].

    Article  ADS  Google Scholar 

  70. D.C. Dunbar, B. Julia, D. Seminara and M. Trigiante, Counterterms in type-I supergravities, JHEP 01 (2000) 046 [hep-th/9911158] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  71. C. Cheung and D. O’Connell, Amplitudes and spinor-helicity in six dimensions, JHEP 07 (2009) 075 [arXiv:0902.0981] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  72. Z. Bern, J.J. Carrasco, T. Dennen, Y.-t. Huang and H. Ita, Generalized unitarity and six-dimensional helicity, Phys. Rev. D 83 (2011) 085022 [arXiv:1010.0494] [INSPIRE].

    ADS  Google Scholar 

  73. A. Brandhuber, D. Korres, D. Koschade and G. Travaglini, One-loop Amplitudes in six-dimensional (1,1) theories from generalised unitarity, JHEP 02 (2011) 077 [arXiv:1010.1515] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  74. Y.-t. Huang, Non-chiral S-matrix of \( \mathcal{N}=4 \) Super Yang-Mills, arXiv:1104.2021 [INSPIRE].

  75. Z. Bern, L.J. Dixon, D. Dunbar, M. Perelstein and J. 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].

    Article  ADS  Google Scholar 

  76. S.H. Henry Tye and Y. Zhang, Dual identities inside the gluon and the graviton scattering amplitudes, JHEP 06 (2010) 071 [Erratum ibid. 1104 (2011) 114] [arXiv:1003.1732] [INSPIRE].

  77. N. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, Monodromy and Jacobi-like relations for color-ordered amplitudes, JHEP 06 (2010) 003 [arXiv:1003.2403] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  78. J. Broedel and J.J.M. Carrasco, Virtuous trees at five and six points for Yang-Mills and gravity, Phys. Rev. D 84 (2011) 085009 [arXiv:1107.4802] [INSPIRE].

    ADS  Google Scholar 

  79. 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].

    ADS  Google Scholar 

  80. R. Monteiro and D. O’Connell, The kinematic algebra from the self-dual sector, JHEP 07 (2011) 007 [arXiv:1105.2565] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  81. M. Kiermaier, Gravity as square of gauge theory, presented at Amplitudes 2010, London U.K. (2010), http://www.strings.ph.qmul.ac.uk/∼theory/Amplitudes2010.

  82. N. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, The momentum kernel of gauge and gravity theories, JHEP 01 (2011) 001 [arXiv:1010.3933] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  83. C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  84. Z. Bern and T. Dennen, A Color Dual Form for Gauge-Theory Amplitudes, Phys. Rev. Lett. 107 (2011) 081601 [arXiv:1103.0312] [INSPIRE].

    Article  ADS  Google Scholar 

  85. Z. Bern, J. Carrasco, L. 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].

    ADS  Google Scholar 

  86. J.J. Carrasco and H. Johansson, Five-Point Amplitudes in \( \mathcal{N}=4 \) super-Yang-Mills Theory and \( \mathcal{N}=8 \) Supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].

    ADS  Google Scholar 

  87. Z. Bern, C. Boucher-Veronneau and H. Johansson, \( \mathcal{N}\geq 4 \) Supergravity Amplitudes from Gauge Theory at One Loop, Phys. Rev. D 84 (2011) 105035 [arXiv:1107.1935] [INSPIRE].

    ADS  Google Scholar 

  88. D.C. Dunbar, J.H. Ettle and W.B. Perkins, Perturbative expansion of \( \mathcal{N}<8 \) Supergravity, Phys. Rev. D 83 (2011) 065015 [arXiv:1011.5378] [INSPIRE].

    ADS  Google Scholar 

  89. C. Boucher-Veronneau and L. Dixon, \( \mathcal{N}\geq 4 \) supergravity amplitudes from gauge theory at two loops, JHEP 12 (2011) 046 [arXiv:1110.1132] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  90. S. Oxburgh and C. White, BCJ duality and the double copy in the soft limit, JHEP 02 (2013) 127 [arXiv:1210.1110] [INSPIRE].

    Article  ADS  Google Scholar 

  91. J. Broedel and L.J. Dixon, Color-kinematics duality and double-copy construction for amplitudes from higher-dimension operators, JHEP 10 (2012) 091 [arXiv:1208.0876] [INSPIRE].

    Article  ADS  Google Scholar 

  92. S.G. Naculich, H. Nastase and H.J. Schnitzer, Linear relations between \( \mathcal{N}\geq 4 \) supergravity and subleading-color SYM amplitudes, JHEP 01 (2012) 041 [arXiv:1111.1675] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  93. J.J.M. Carrasco and H. Johansson, Generic multiloop methods and application to \( \mathcal{N}=4 \) super-Yang-Mills, J. Phys. A 44 (2011) 454004 [arXiv:1103.3298] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  94. Z. Bern and A. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [INSPIRE].

    Article  ADS  Google Scholar 

  95. J. Broedel and J.J.M. Carrasco, Virtuous Trees at Five and Six Points for Yang-Mills and Gravity, Phys. Rev. D 84 (2011) 085009 [arXiv:1107.4802] [INSPIRE].

    ADS  Google Scholar 

  96. T. Bargheer, S. He and T. McLoughlin, New Relations for Three-Dimensional Supersymmetric Scattering Amplitudes, Phys. Rev. Lett. 108 (2012) 231601 [arXiv:1203.0562] [INSPIRE].

    Article  ADS  Google Scholar 

  97. Y.-t. Huang and H. Johansson, Equivalent D = 3 Supergravity Amplitudes from Double Copies of Three-Algebra and Two-Algebra Gauge Theories, arXiv:1210.2255 [INSPIRE].

Download references

Author information

Authors and Affiliations

  1. Stanford Institute for Theoretical Physics and Department of Physics, Stanford University, Stanford, CA, 94305, U.S.A.

    John Joseph M. Carrasco

  2. Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA, 16802, U.S.A.

    Marco Chiodaroli, Murat Günaydin & Radu Roiban

Authors
  1. John Joseph M. Carrasco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Chiodaroli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Murat Günaydin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Radu Roiban
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Chiodaroli.

Additional information

ArXiv ePrint: 1212.1146

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carrasco, J.J.M., Chiodaroli, M., Günaydin, M. et al. One-loop four-point amplitudes in pure and matter-coupled \( \mathcal{N}\leq 4 \) supergravity. J. High Energ. Phys. 2013, 56 (2013). https://doi.org/10.1007/JHEP03(2013)056

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP03(2013)056

Keywords

Search

Navigation