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BCJ duality and the double copy in the soft limit

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Abstract

We examine the structure of infrared singularities in QCD and quantum General Relativity, from the point of view of the recently conjectured double copy property which relates scattering amplitudes in non-Abelian gauge theories with gravitational counterparts. We show that IR divergences in both theories are consistent with the double copy procedure, to all orders in perturbation theory, thus providing all loop-level evidence for the conjecture. We further comment on the relevance, or otherwise, to the so-called dipole formula, a conjecture for the complete structure of IR singularities in QCD.

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References

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

    Article  MathSciNet  ADS  Google Scholar 

  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 [ar**v:1004.0476] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. Z. Bern, T. Dennen, Y.-t. Huang and M. Kiermaier, Gravity as the Square of Gauge Theory, Phys. Rev. D 82 (2010) 065003 [ar**v:1004.0693] [INSPIRE].

    ADS  Google Scholar 

  4. 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 [ar**v:1201.5366] [INSPIRE].

    ADS  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

  6. 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] [ar**v:1003.1732] [INSPIRE].

    Article  ADS  Google Scholar 

  7. N. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes, JHEP 06 (2010) 003 [ar**v:1003.2403] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. 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 [ar**v:1107.4802] [INSPIRE].

    ADS  Google Scholar 

  9. R. Monteiro and D. O’Connell, The Kinematic Algebra From the Self-Dual Sector, JHEP 07 (2011) 007 [ar**v:1105.2565] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  10. 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 

  11. 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 

  12. 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].

    Article  ADS  Google Scholar 

  13. Z. Bern, J. 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].

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  15. Z. Bern, J. Carrasco and H. Johansson, Progress on Ultraviolet Finiteness of Supergravity, ar**v:0902.3765 [INSPIRE].

  16. C. Boucher-Veronneau and L. Dixon, N ≥ 4 Supergravity Amplitudes from Gauge Theory at Two Loops, JHEP 12 (2011) 046 [ar**v:1110.1132] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  17. S.G. Naculich, H. Nastase and H.J. Schnitzer, Linear relations between N ≥ 4 supergravity and subleading-color SYM amplitudes, JHEP 01 (2012) 041 [ar**v:1111.1675] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. S.G. Naculich, H. Nastase and H.J. Schnitzer, Subleading-color contributions to gluon-gluon scattering in N = 4 SYM theory and relations to N = 8 supergravity, JHEP 11 (2008) 018 [ar**v:0809.0376] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  20. J. Grammer, G. and D. Yennie, Improved treatment for the infrared divergence problem in quantum electrodynamics, Phys. Rev. D 8 (1973) 4332 [INSPIRE].

  21. A.H. Mueller, On the asymptotic behavior of the Sudakov form-factor, Phys. Rev. D 20 (1979) 2037 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  22. J.C. Collins, Algorithm to compute corrections to the Sudakov form-factor, Phys. Rev. D 22 (1980) 1478 [INSPIRE].

    ADS  Google Scholar 

  23. A. Sen, Asymptotic Behavior of the Sudakov Form-Factor in QCD, Phys. Rev. D 24 (1981) 3281 [INSPIRE].

    ADS  Google Scholar 

  24. A. Sen, Asymptotic Behavior of the Wide Angle On-Shell Quark Scattering Amplitudes in Nonabelian Gauge Theories, Phys. Rev. D 28 (1983) 860 [INSPIRE].

    ADS  Google Scholar 

  25. J. Gatheral, Exponentiation of eikonal cross-sections in nonabelian gauge theories, Phys. Lett. B 133 (1983) 90 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  26. J. Frenkel and J. Taylor, Nonabelian eikonal exponentiation, Nucl. Phys. B 246 (1984) 231 [INSPIRE].

    Article  ADS  Google Scholar 

  27. L. Magnea and G.F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev. D 42 (1990) 4222 [INSPIRE].

    ADS  Google Scholar 

  28. A. Mitov, G. Sterman and I. Sung, Diagrammatic Exponentiation for Products of Wilson Lines, Phys. Rev. D 82 (2010) 096010 [ar**v:1008.0099] [INSPIRE].

    ADS  Google Scholar 

  29. E. Gardi, E. Laenen, G. Stavenga and C.D. White, Webs in multiparton scattering using the replica trick, JHEP 11 (2010) 155 [ar**v:1008.0098] [INSPIRE].

    Article  ADS  Google Scholar 

  30. L.J. Dixon, L. Magnea and G.F. Sterman, Universal structure of subleading infrared poles in gauge theory amplitudes, JHEP 08 (2008) 022 [ar**v:0805.3515] [INSPIRE].

    Article  ADS  Google Scholar 

  31. T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [ar**v:0901.0722] [INSPIRE].

    Article  ADS  Google Scholar 

  32. T. Becher and M. Neubert, On the Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP 06 (2009) 081 [ar**v:0903.1126] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [ar**v:0901.1091] [INSPIRE].

    Article  ADS  Google Scholar 

  34. E. Gardi and L. Magnea, Infrared singularities in QCD amplitudes, Nuovo Cim. C32N5-6 (2009) 137 [ar**v:0908.3273] [INSPIRE].

    Google Scholar 

  35. L.J. Dixon, E. Gardi and L. Magnea, On soft singularities at three loops and beyond, JHEP 02 (2010) 081 [ar**v:0910.3653] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. V. Del Duca, C. Duhr, E. Gardi, L. Magnea and C.D. White, An infrared approach to Reggeization, Phys. Rev. D 85 (2012) 071104 [ar**v:1108.5947] [INSPIRE].

    ADS  Google Scholar 

  37. V. Del Duca, C. Duhr, E. Gardi, L. Magnea and C.D. White, The Infrared structure of gauge theory amplitudes in the high-energy limit, JHEP 12 (2011) 021 [ar**v:1109.3581] [INSPIRE].

    Article  ADS  Google Scholar 

  38. L. Vernazza, Analysis of the Anomalous-Dimension Matrix of n-jet Operators at 4 Loops, PoS(EPS-HEP2011)284 [ar**v:1112.3375] [INSPIRE].

  39. V. Ahrens, M. Neubert and L. Vernazza, Structure of Infrared Singularities of Gauge-Theory Amplitudes at Three and Four Loops, JHEP 09 (2012) 138 [ar**v:1208.4847] [INSPIRE].

    Article  ADS  Google Scholar 

  40. N. Kidonakis, Two-loop soft anomalous dimensions and NNLL resummation for heavy quark production, Phys. Rev. Lett. 102 (2009) 232003 [ar**v:0903.2561] [INSPIRE].

    Article  ADS  Google Scholar 

  41. A. Mitov, G.F. Sterman and I. Sung, The Massive Soft Anomalous Dimension Matrix at Two Loops, Phys. Rev. D 79 (2009) 094015 [ar**v:0903.3241] [INSPIRE].

    ADS  Google Scholar 

  42. T. Becher and M. Neubert, Infrared singularities of QCD amplitudes with massive partons, Phys. Rev. D 79 (2009) 125004 [Erratum ibid. D 80 (2009) 109901] [ar**v:0904.1021] [INSPIRE].

    ADS  Google Scholar 

  43. M. Beneke, P. Falgari and C. Schwinn, Soft radiation in heavy-particle pair production: All-order colour structure and two-loop anomalous dimension, Nucl. Phys. B 828 (2010) 69 [ar**v:0907.1443] [INSPIRE].

    Article  ADS  Google Scholar 

  44. M. Czakon, A. Mitov and G.F. Sterman, Threshold Resummation for Top-Pair Hadroproduction to Next-to-Next-to-Leading Log, Phys. Rev. D 80 (2009) 074017 [ar**v:0907.1790] [INSPIRE].

    ADS  Google Scholar 

  45. A. Ferroglia, M. Neubert, B.D. Pecjak and L.L. Yang, Two-loop divergences of scattering amplitudes with massive partons, Phys. Rev. Lett. 103 (2009) 201601 [ar**v:0907.4791] [INSPIRE].

    Article  ADS  Google Scholar 

  46. A. Ferroglia, M. Neubert, B.D. Pecjak and L.L. Yang, Two-loop divergences of massive scattering amplitudes in non-abelian gauge theories, JHEP 11 (2009) 062 [ar**v:0908.3676] [INSPIRE].

    Article  ADS  Google Scholar 

  47. J.-y. Chiu, A. Fuhrer, R. Kelley and A.V. Manohar, Factorization Structure of Gauge Theory Amplitudes and Application to Hard Scattering Processes at the LHC, Phys. Rev. D 80 (2009) 094013 [ar**v:0909.0012] [INSPIRE].

    ADS  Google Scholar 

  48. A. Mitov, G.F. Sterman and I. Sung, Computation of the Soft Anomalous Dimension Matrix in Coordinate Space, Phys. Rev. D 82 (2010) 034020 [ar**v:1005.4646] [INSPIRE].

    ADS  Google Scholar 

  49. S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516.

    Article  MathSciNet  ADS  Google Scholar 

  50. S.G. Naculich and H.J. Schnitzer, Eikonal methods applied to gravitational scattering amplitudes, JHEP 05 (2011) 087 [ar**v:1101.1524] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  51. C.D. White, Factorization Properties of Soft Graviton Amplitudes, JHEP 05 (2011) 060 [ar**v:1103.2981] [INSPIRE].

    Article  ADS  Google Scholar 

  52. R. Akhoury, R. Saotome and G. Sterman, Collinear and Soft Divergences in Perturbative Quantum Gravity, Phys. Rev. D 84 (2011) 104040 [ar**v:1109.0270] [INSPIRE].

    ADS  Google Scholar 

  53. Z. Bern and Y.-t. Huang, Basics of Generalized Unitarity, J. Phys. A 44 (2011) 454003 [ar**v:1103.1869] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  54. S. Catani and M. Seymour, The Dipole formalism for the calculation of QCD jet cross-sections at next-to-leading order, Phys. Lett. B 378 (1996) 287 [hep-ph/9602277] [INSPIRE].

    Article  ADS  Google Scholar 

  55. S. Catani and M. Seymour, A General algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291 [Erratum ibid. B 510 (1998) 503-504] [hep-ph/9605323] [INSPIRE].

    Article  ADS  Google Scholar 

  56. S.M. Aybat, L.J. Dixon and G.F. Sterman, The Two-loop anomalous dimension matrix for soft gluon exchange, Phys. Rev. Lett. 97 (2006) 072001 [hep-ph/0606254] [INSPIRE].

    Article  ADS  Google Scholar 

  57. S.M. Aybat, L.J. Dixon and G.F. Sterman, The Two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [INSPIRE].

    ADS  Google Scholar 

  58. E. Gardi and C.D. White, General properties of multiparton webs: Proofs from combinatorics, JHEP 03 (2011) 079 [ar**v:1102.0756] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  59. E. Gardi, J.M. Smillie and C.D. White, On the renormalization of multiparton webs, JHEP 09 (2011) 114 [ar**v:1108.1357] [INSPIRE].

    Article  ADS  Google Scholar 

  60. Y.-T. Chien, M.D. Schwartz, D. Simmons-Duffin and I.W. Stewart, Jet Physics from Static Charges in AdS, Phys. Rev. D 85 (2012) 045010 [ar**v:1109.6010] [INSPIRE].

    ADS  Google Scholar 

  61. M. Beneke and G. Kirilin, Soft-collinear gravity, JHEP 09 (2012) 066 [ar**v:1207.4926] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  62. E. Laenen, G. Stavenga and C.D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP 03 (2009) 054 [ar**v:0811.2067] [INSPIRE].

    Article  ADS  Google Scholar 

  63. E. Laenen, L. Magnea, G. Stavenga and C.D. White, Next-to-eikonal corrections to soft gluon radiation: a diagrammatic approach, JHEP 01 (2011) 141 [ar**v:1010.1860] [INSPIRE].

    Article  ADS  Google Scholar 

  64. D. Miller and C. White, The Gravitational cusp anomalous dimension from AdS space, Phys. Rev. D 85 (2012) 104034 [ar**v:1201.2358] [INSPIRE].

    ADS  Google Scholar 

  65. H.W. Hamber, Discrete and continuum quantum gravity, ar**v:0704.2895 [INSPIRE].

  66. J.J.M. Carrasco, M. Chiodaroli, M. Günaydin and R. Roiban, One-loop four-point amplitudes in pure and matter-coupled N ≥ 4 supergravity, ar**v:1212.1146 [INSPIRE].

  67. C. Boucher-Veronneau and A.J. Larkoski, Constructing Amplitudes from Their Soft Limits, JHEP 09 (2011) 130 [ar**v:1108.5385] [INSPIRE].

    Article  ADS  Google Scholar 

  68. D.C. Dunbar, J.H. Ettle and W.B. Perkins, Constructing Gravity Amplitudes from Real Soft and Collinear Factorisation, Phys. Rev. D 86 (2012) 026009 [ar**v:1203.0198] [INSPIRE].

    ADS  Google Scholar 

  69. B.S. DeWitt, Quantum Theory of Gravity. 3. Applications of the Covariant Theory, Phys. Rev. 162 (1967) 1239 [INSPIRE].

    Article  ADS  Google Scholar 

  70. R. Saotome and R. Akhoury, Relationship Between Gravity and Gauge Scattering in the High Energy Limit, JHEP 01 (2013) 123 [ar**v:1210.8111] [INSPIRE].

    Article  ADS  Google Scholar 

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Oxburgh, S., White, C.D. BCJ duality and the double copy in the soft limit. J. High Energ. Phys. 2013, 127 (2013). https://doi.org/10.1007/JHEP02(2013)127

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