Advertisement

Journal of High Energy Physics

, 2019:107 | Cite as

The self-dual classical double copy, and the Eguchi-Hanson instanton

  • David S. Berman
  • Erick Chacón
  • Andrés Luna
  • Chris D. WhiteEmail author
Open Access
Regular Article - Theoretical Physics
  • 4 Downloads

Abstract

The double copy is a map from non-abelian gauge theories to gravity, that has been demonstrated both for scattering amplitudes and exact classical solutions. In this study, we reconsider the double copy for exact solutions that are self-dual in either the gauge or gravity theory. In this case, one may formulate a general double copy in terms of a certain differential operator, which generates the gauge and gravity solutions from a harmonic function residing in a biadjoint scalar theory. As an illustration, we examine the single copy of the well-known Eguchi-Hanson instanton in gravity. The gauge field thus obtained represents an abelian-like object whose field is dipole-like at large distances, and which has no magnetic or electric charge.

Keywords

Scattering Amplitudes Solitons Monopoles and Instantons 

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]
    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].
  2. [2]
    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].
  3. [3]
    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].
  4. [4]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal basis for gauge theory amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    S. Stieberger, Open & closed vs. pure open string disk amplitudes, arXiv:0907.2211 [INSPIRE].
  6. [6]
    N.E.J. 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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    B. Feng, R. Huang and Y. Jia, Gauge amplitude identities by on-shell recursion relation in S-matrix program, Phys. Lett. B 695 (2011) 350 [arXiv:1004.3417] [INSPIRE].
  8. [8]
    S.H. Henry Tye and Y. Zhang, Dual identities inside the gluon and the graviton scattering amplitudes, JHEP 06 (2010) 071 [Erratum ibid. 04 (2011) 114] [arXiv:1003.1732] [INSPIRE].
  9. [9]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    R. Monteiro and D. O’Connell, The kinematic algebra from the self-dual sector, JHEP 07 (2011) 007 [arXiv:1105.2565] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard, R. Monteiro and D. O’Connell, Algebras for amplitudes, JHEP 06 (2012) 061 [arXiv:1203.0944] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    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].
  13. [13]
    Z. Bern et al., 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].
  14. [14]
    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].
  15. [15]
    Z. Bern, J.S. Rozowsky and B. Yan, Two loop four gluon amplitudes in N = 4 super-Yang-Mills, Phys. Lett. B 401 (1997) 273 [hep-ph/9702424] [INSPIRE].
  16. [16]
    J.J. Carrasco and H. Johansson, Five-point amplitudes in N = 4 Super-Yang-Mills theory and N = 8 supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].
  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].
  18. [18]
    C.R. Mafra and O. Schlotterer, The structure of n-point one-loop open superstring amplitudes, JHEP 08 (2014) 099 [arXiv:1203.6215] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    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
  20. [20]
    N.E.J. Bjerrum-Bohr, T. Dennen, R. Monteiro and D. O’Connell, Integrand oxidation and one-loop colour-dual numerators in N = 4 gauge theory, JHEP 07 (2013) 092 [arXiv:1303.2913] [INSPIRE].
  21. [21]
    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].
  22. [22]
    Z. Bern, S. Davies and T. Dennen, The ultraviolet structure of half-maximal supergravity with matter multiplets at two and three loops, Phys. Rev. D 88 (2013) 065007 [arXiv:1305.4876] [INSPIRE].
  23. [23]
    J. Nohle, Color-kinematics duality in one-loop four-gluon amplitudes with matter, Phys. Rev. D 90 (2014) 025020 [arXiv:1309.7416] [INSPIRE].
  24. [24]
    Z. Bern et al., Ultraviolet properties of N = 4 supergravity at four loops, Phys. Rev. Lett. 111 (2013) 231302 [arXiv:1309.2498] [INSPIRE].
  25. [25]
    S.G. Naculich, H. Nastase and H.J. Schnitzer, All-loop infrared-divergent behavior of most-subleading-color gauge-theory amplitudes, JHEP 04 (2013) 114 [arXiv:1301.2234] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Y.-J. Du, B. Feng and C.-H. Fu, Dual-color decompositions at one-loop level in Yang-Mills theory, JHEP 06 (2014) 157 [arXiv:1402.6805] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    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
  28. [28]
    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].
  29. [29]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    S. He, R. Monteiro and O. Schlotterer, String-inspired BCJ numerators for one-loop MHV amplitudes, JHEP 01 (2016) 171 [arXiv:1507.06288] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Z. Bern, S. Davies and J. Nohle, Double-copy constructions and unitarity cuts, Phys. Rev. D 93 (2016) 105015 [arXiv:1510.03448] [INSPIRE].
  32. [32]
    G. Mogull and D. O’Connell, Overcoming obstacles to colour-kinematics duality at two loops, JHEP 12 (2015) 135 [arXiv:1511.06652] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  33. [33]
    M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Spontaneously broken Yang-Mills-Einstein supergravities as double copies, JHEP 06 (2017) 064 [arXiv:1511.01740] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    Z. Bern et al., Five-loop four-point integrand of N = 8 supergravity as a generalized double copy, Phys. Rev. D 96 (2017) 126012 [arXiv:1708.06807] [INSPIRE].
  35. [35]
    H. Johansson and A. Ochirov, Color-kinematics duality for QCD amplitudes, JHEP 01 (2016) 170 [arXiv:1507.00332] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    S. Oxburgh and C.D. White, BCJ duality and the double copy in the soft limit, JHEP 02 (2013) 127 [arXiv:1210.1110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    C.D. White, Factorization properties of soft graviton amplitudes, JHEP 05 (2011) 060 [arXiv:1103.2981] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    S. Melville, S.G. Naculich, H.J. Schnitzer and C.D. White, Wilson line approach to gravity in the high energy limit, Phys. Rev. D 89 (2014) 025009 [arXiv:1306.6019] [INSPIRE].
  39. [39]
    A. Luna, S. Melville, S.G. Naculich and C.D. White, Next-to-soft corrections to high energy scattering in QCD and gravity, JHEP 01 (2017) 052 [arXiv:1611.02172] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    R. Saotome and R. Akhoury, Relationship Between Gravity and Gauge Scattering in the High Energy Limit, JHEP 01 (2013) 123 [arXiv:1210.8111] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    A. Sabio Vera, E. Serna Campillo and M.A. Vazquez-Mozo, Color-kinematics duality and the Regge limit of inelastic amplitudes, JHEP 04 (2013) 086 [arXiv:1212.5103] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  42. [42]
    H. Johansson et al., Color-kinematics duality in multi-Regge kinematics and dimensional reduction, JHEP 10 (2013) 215 [arXiv:1307.3106] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    H. Johansson, A. Sabio Vera, E. Serna Campillo and M.A. Vazquez-Mozo, Color-kinematics duality and dimensional reduction for graviton emission in Regge limit, arXiv:1310.1680 [INSPIRE].
  44. [44]
    R. Monteiro, D. O’Connell and C.D. White, Black holes and the double copy, JHEP 12 (2014) 056 [arXiv:1410.0239] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    A. Luna, R. Monteiro, D. O’Connell and C.D. White, The classical double copy for Taub-NUT spacetime, Phys. Lett. B 750 (2015) 272 [arXiv:1507.01869] [INSPIRE].
  46. [46]
    A. Luna et al., The double copy: Bremsstrahlung and accelerating black holes, JHEP 06 (2016) 023 [arXiv:1603.05737] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    A.K. Ridgway and M.B. Wise, Static spherically symmetric Kerr-Schild metrics and implications for the classical double copy, Phys. Rev. D 94 (2016) 044023 [arXiv:1512.02243] [INSPIRE].
  48. [48]
    A. Anastasiou et al., Yang-Mills origin of gravitational symmetries, Phys. Rev. Lett. 113 (2014) 231606 [arXiv:1408.4434] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    L. Borsten and M.J. Duff, Gravity as the square of Yang-Mills?, Phys. Scripta 90 (2015) 108012 [arXiv:1602.08267] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    A. Anastasiou et al., Twin supergravities from Yang-Mills theory squared, Phys. Rev. D 96 (2017) 026013 [arXiv:1610.07192] [INSPIRE].
  51. [51]
    A. Anastasiou et al., Are all supergravity theories Yang-Mills squared?, Nucl. Phys. B 934 (2018) 606 [arXiv:1707.03234] [INSPIRE].
  52. [52]
    G.L. Cardoso, S. Nagy and S. Nampuri, A double copy for \( \mathcal{N}=2 \) supergravity: a linearised tale told on-shell, JHEP 10 (2016) 127 [arXiv:1609.05022] [INSPIRE].
  53. [53]
    L. Borsten, D = 6, \( \mathcal{N}=\left(2,0\right) \) and \( \mathcal{N}=\left(4,0\right) \) theories, Phys. Rev. D 97 (2018) 066014 [arXiv:1708.02573] [INSPIRE].
  54. [54]
    A. Anastasiou et al., The mile high magic pyramid, arXiv:1711.08476 [INSPIRE].
  55. [55]
    A. Anastasiou et al., Gravity as gauge theory squared: a ghost story, Phys. Rev. Lett. 121 (2018) 211601 [arXiv:1807.02486] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    G. Lopes Cardoso, G. Inverso, S. Nagy and S. Nampuri, Comments on the double copy construction for gravitational theories, PoS(CORFU2017)177 [arXiv:1803.07670] [INSPIRE].
  57. [57]
    W.D. Goldberger and A.K. Ridgway, Radiation and the classical double copy for color charges, Phys. Rev. D 95 (2017) 125010 [arXiv:1611.03493] [INSPIRE].
  58. [58]
    W.D. Goldberger, S.G. Prabhu and J.O. Thompson, Classical gluon and graviton radiation from the bi-adjoint scalar double copy, Phys. Rev. D 96 (2017) 065009 [arXiv:1705.09263] [INSPIRE].
  59. [59]
    W.D. Goldberger and A.K. Ridgway, Bound states and the classical double copy, Phys. Rev. D 97 (2018) 085019 [arXiv:1711.09493] [INSPIRE].
  60. [60]
    W.D. Goldberger, J. Li and S.G. Prabhu, Spinning particles, axion radiation and the classical double copy, Phys. Rev. D 97 (2018) 105018 [arXiv:1712.09250] [INSPIRE].
  61. [61]
    A. Luna et al., Perturbative spacetimes from Yang-Mills theory, JHEP 04 (2017) 069 [arXiv:1611.07508] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    A. Luna, I. Nicholson, D. O’Connell and C.D. White, Inelastic black hole scattering from charged scalar amplitudes, JHEP 03 (2018) 044 [arXiv:1711.03901] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    C.-H. Shen, Gravitational radiation from color-kinematics duality, JHEP 11 (2018) 162 [arXiv:1806.07388] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  64. [64]
    M. Levi, Effective field theories of post-newtonian gravity: a comprehensive review, arXiv:1807.01699 [INSPIRE].
  65. [65]
    J. Plefka, J. Steinhoff and W. Wormsbecher, Effective action of dilaton gravity as the classical double copy of Yang-Mills theory, arXiv:1807.09859 [INSPIRE].
  66. [66]
    C. Cheung, I.Z. Rothstein and M.P. Solon, From scattering amplitudes to classical potentials in the post-Minkowskian expansion, Phys. Rev. Lett. 121 (2018) 251101 [arXiv:1808.02489] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    M. Carrillo González, R. Penco and M. Trodden, Radiation of scalar modes and the classical double copy, JHEP 11 (2018) 065 [arXiv:1809.04611] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  68. [68]
    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
  69. [69]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  70. [70]
    L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    T. Adamo, E. Casali, L. Mason and S. Nekovar, Scattering on plane waves and the double copy, Class. Quant. Grav. 35 (2018) 015004 [arXiv:1706.08925] [INSPIRE].
  72. [72]
    N. Bahjat-Abbas, A. Luna and C.D. White, The Kerr-Schild double copy in curved spacetime, JHEP 12 (2017) 004 [arXiv:1710.01953] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    M. Carrillo-González, R. Penco and M. Trodden, The classical double copy in maximally symmetric spacetimes, JHEP 04 (2018) 028 [arXiv:1711.01296] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    K. Lee, Kerr-Schild double field theory and classical double copy, JHEP 10 (2018) 027 [arXiv:1807.08443] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  75. [75]
    R. Monteiro, I. Nicholson and D. O’Connell, Spinor-helicity and the algebraic classification of higher-dimensional spacetimes, arXiv:1809.03906 [INSPIRE].
  76. [76]
    R. Monteiro and D. O’Connell, The kinematic algebras from the scattering equations, JHEP 03 (2014) 110 [arXiv:1311.1151] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    K.P. Tod, Self-dual Kerr-Schild metrics and null Maxwell fields, J. Math. Phys. 23 (1982) 1147.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  78. [78]
    T. Eguchi and A.J. Hanson, Asymptotically flat selfdual solutions to euclidean gravity, Phys. Lett. B 74 (1978) 249.Google Scholar
  79. [79]
    T. Eguchi and A.J. Hanson, Selfdual solutions to Euclidean gravity, Annals Phys. 120 (1979) 82 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  80. [80]
    T. Eguchi and A.J. Hanson, Gravitational instantons, Gen. Rel. Grav. 11 (1979) 315 [INSPIRE].
  81. [81]
    A.A. Belavin et al., Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975) 85 [INSPIRE].
  82. [82]
    T. Ortin, Gravity and strings, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2015).Google Scholar
  83. [83]
    S. Lee, R. Roychowdhury and H.S. Yang, Test of emergent gravity, Phys. Rev. D 88 (2013) 086007 [arXiv:1211.0207] [INSPIRE].
  84. [84]
    J. Lee and H.S. Yang, Quantized Kähler geometry and quantum gravity, J. Korean Phys. Soc. 72 (2018) 1421 [arXiv:1804.09171] [INSPIRE].ADSCrossRefGoogle Scholar
  85. [85]
    A.H. Taub, Empty space-times admitting a three parameter group of motions, Ann. Math. 53 (1951) 472.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  86. [86]
    E. Newman, L. Tamburino and T. Unti, Empty-space generalization of the Schwarzschild metric, J. Math. Phys. 4 (1963) 915.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  87. [87]
    A. Parkes, A cubic action for selfdual Yang-Mills, Phys. Lett. B 286 (1992) 265 [hep-th/9203074] [INSPIRE].
  88. [88]
    G. ’t Hooft, A physical interpretation of gravitational instantons, Nucl. Phys. B 315 (1989) 517 [INSPIRE].
  89. [89]
    T. Eguchi, P.B. Gilkey and A.J. Hanson, Gravitation, gauge theories and differential geometry, Phys. Rept. 66 (1980) 213 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  90. [90]
    G.W. Gibbons and S.W. Hawking, Classification of gravitational instanton symmetries, Commun. Math. Phys. 66 (1979) 291 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  91. [91]
    G. Burnett-Stuart, Sparling-Tod metric = Eguchi-Hanson, Twistor Newsletter 9 (1979) 6.Google Scholar
  92. [92]
    P.A.M. Dirac, Quantised singularities in the electromagnetic field,, Proc. Roy. Soc. Lond. A 133 (1931) 60 [INSPIRE].
  93. [93]
    C.D. White, Exact solutions for the biadjoint scalar field, Phys. Lett. B 763 (2016) 365 [arXiv:1606.04724] [INSPIRE].
  94. [94]
    P.-J. De Smet and C.D. White, Extended solutions for the biadjoint scalar field, Phys. Lett. B 775 (2017) 163 [arXiv:1708.01103] [INSPIRE].
  95. [95]
    G. ’t Hooft, Computation of the quantum effects due to a four-dimensional pseudoparticle, Phys. Rev. D 14 (1976) 3432 [Erratum ibid. D 18 (1978) 2199] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • David S. Berman
    • 1
  • Erick Chacón
    • 2
  • Andrés Luna
    • 3
  • Chris D. White
    • 1
    Email author
  1. 1.Centre for Research in String Theory, School of Physics and AstronomyQueen Mary University of LondonLondonU.K.
  2. 2.Departamento de FísicaCentro de Investigación y de Estudios Avanzados del Instituto Politécnico NacionalMéxico D.F.México
  3. 3.Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and AstronomyUniversity of CaliforniaLos AngelesU.S.A.

Personalised recommendations