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Generating all tree amplitudes in \( \mathcal{N} = 4\;{\text{SYM}} \) by Inverse Soft Limit

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Abstract

The idea of adding particles to construct amplitudes has been utilized in various ways in exploring the structure of scattering amplitudes. This idea is often called Inverse Soft Limit, namely it is the reverse mechanism of taking particles to be soft. We apply the Inverse Soft Limit to the tree-level amplitudes in \( \mathcal{N} = 4 \) super Yang-Mills theory, which allows us to generate full tree-level superamplitudes by adding “soft” particles in a certain way. With the help from Britto-Cachazo-Feng-Witten recursion relations, a systematic and concrete way of adding particles is determined recursively. The amplitudes constructed solely by adding particles not only have manifest Yangian symmetry, but also make the soft limit transparent. The method of generating amplitudes by Inverse Soft Limit can also be generalized for constructing form factors.

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References

  1. 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 

  2. 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 

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

    Article  MathSciNet  ADS  Google Scholar 

  4. 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 

  5. 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 

  6. N. Arkani-Hamed and J. Kaplan, On tree amplitudes in gauge theory and gravity, JHEP 04 (2008) 076 [arXiv:0801.2385] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  7. F. Cachazo, P. Svrček and E. Witten, MHV vertices and tree amplitudes in gauge theory, JHEP 09 (2004) 006 [hep-th/0403047] [INSPIRE].

    Article  ADS  Google Scholar 

  8. N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. A. Brandhuber, B.J. Spence and G. Travaglini, One-loop gauge theory amplitudes in N = 4 super Yang-Mills from MHV vertices, Nucl. Phys. B 706 (2005) 150 [hep-th/0407214] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  12. L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  13. J.M. Drummond, J.M. Henn, G.P. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  14. A. Brandhuber, P. Heslop and G. Travaglini, MHV amplitudes in N = 4 super Yang-Mills and Wilson loops, Nucl. Phys. B 794 (2008) 231 [arXiv:0707.1153] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. L.J. Mason and D. Skinner, The complete planar S-matrix of N = 4 SYM as a Wilson loop in twistor space, JHEP 12 (2010) 018 [arXiv:1009.2225] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  16. S. Caron-Huot, Notes on the scattering amplitude/Wilson loop duality, JHEP 07 (2011) 058 [arXiv:1010.1167] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  17. L.F. Alday, B. Eden, G.P. Korchemsky, J.M. Maldacena and E. Sokatchev, From correlation functions to Wilson loops, JHEP 09 (2011) 123 [arXiv:1007.3243] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. B. Eden, G.P. Korchemsky and E. Sokatchev, From correlation functions to scattering amplitudes, JHEP 12 (2011) 002 [arXiv:1007.3246] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. 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 

  20. M.B. Green, J.G. Russo and P. Vanhove, Ultraviolet properties of maximal supergravity, Phys. Rev. Lett. 98 (2007) 131602 [hep-th/0611273] [INSPIRE].

    Article  ADS  Google Scholar 

  21. S. Weinberg, Photons and gravitons in S-matrix theory: derivation of charge conservation and equality of gravitational and inertial mass, Phys. Rev. 135 (1964) B1049 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. 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 

  23. G.F. Sterman, Summation of large corrections to short distance hadronic cross-sections, Nucl. Phys. B 281 (1987) 310 [INSPIRE].

    Article  ADS  Google Scholar 

  24. S. Catani and L. Trentadue, Resummation of the QCD perturbative series for hard processes, Nucl. Phys. B 327 (1989) 323 [INSPIRE].

    Article  ADS  Google Scholar 

  25. J.C. Collins, Sudakov form-factors, Adv. Ser. Direct. High Energy Phys. 5 (1989) 573 [hep-ph/0312336] [INSPIRE].

    Google Scholar 

  26. 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 

  27. S. Catani and L. Trentadue, Comment on QCD exponentiation at large x, Nucl. Phys. B 353 (1991) 183 [INSPIRE].

    Article  ADS  Google Scholar 

  28. S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].

    Article  ADS  Google Scholar 

  29. G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [INSPIRE].

    Article  ADS  Google Scholar 

  30. C. Anastasiou, Z. Bern, L.J. Dixon and D.A. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91 (2003) 251602 [hep-th/0309040] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  31. Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  32. J. Golden and M. Spradlin, Collinear and soft limits of multi-loop integrands in N = 4 Yang-Mills, JHEP 05 (2012) 027 [arXiv:1203.1915] [INSPIRE].

    Article  ADS  Google Scholar 

  33. J.L. Bourjaily, A. Di Re, A. Shaikh, M. Spradlin and A. Volovich, The soft-collinear bootstrap: N = 4 Yang-Mills amplitudes at six and seven loops, JHEP 03 (2012) 032 [arXiv:1112.6432] [INSPIRE].

    Article  ADS  Google Scholar 

  34. N. Arkani-Hamed, Holography and the S-matrix, talk given at the International Workshop on Gauge and String Amplitudes, Durham U.K., 30 Mar-3 Apr 2009.

  35. N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, The S-matrix in twistor space, JHEP 03 (2010) 110 [arXiv:0903.2110] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. A.P. Hodges, Twistor diagrams for all tree amplitudes in gauge theory: a helicity-independent formalism, hep-th/0512336 [INSPIRE].

  37. 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 

  38. N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Local spacetime physics from the Grassmannian, JHEP 01 (2011) 108 [arXiv:0912.3249] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Unification of residues and Grassmannian dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. L. Dolan and P. Goddard, Gluon tree amplitudes in open twistor string theory, JHEP 12 (2009) 032 [arXiv:0909.0499] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  41. M. Spradlin and A. Volovich, From twistor string theory to recursion relations, Phys. Rev. D 80 (2009) 085022 [arXiv:0909.0229] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  42. D. Nandan, A. Volovich and C. Wen, A Grassmannian etude in NMHV minors, JHEP 07 (2010) 061 [arXiv:0912.3705] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  43. J.L. Bourjaily, J. Trnka, A. Volovich and C. Wen, The Grassmannian and the twistor string: connecting all trees in N = 4 SYM, JHEP 01 (2011) 038 [arXiv:1006.1899] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  44. L.J. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  45. M. Bullimore, L.J. Mason and D. Skinner, Twistor-strings, Grassmannians and leading singularities, JHEP 03 (2010) 070 [arXiv:0912.0539] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. M. Bullimore, Inverse soft factors and Grassmannian residues, JHEP 01 (2011) 055 [arXiv:1008.3110] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. D.C. Dunbar, J.H. Ettle and W.B. Perkins, The n-point MHV one-loop amplitude in N = 4 supergravity, Phys. Rev. Lett. 108 (2012) 061603 [arXiv:1111.1153] [INSPIRE].

    Article  ADS  Google Scholar 

  48. D.C. Dunbar, J.H. Ettle and W.B. Perkin, Constructing gravity amplitudes from real soft and collinear factorisation, arXiv:1203.0198 [INSPIRE].

  49. A.P. Hodges, New expressions for gravitational scattering amplitudes, arXiv:1108.2227 [INSPIRE].

  50. A.P. Hodges, A simple formula for gravitational MHV amplitudes, arXiv:1204.1930 [INSPIRE].

  51. D. Nguyen, M. Spradlin, A. Volovich and C. Wen, The tree formula for MHV graviton amplitudes, JHEP 07 (2010) 045 [arXiv:0907.2276] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  52. C. Boucher-Veronneau and A.J. Larkoski, Constructing amplitudes from their soft limits, JHEP 09 (2011) 130 [arXiv:1108.5385] [INSPIRE].

    Article  ADS  Google Scholar 

  53. W.L. van Neerven, Infrared behavior of on-shell form-factors in a N = 4 supersymmetric Yang-Mills field theory, Z. Phys. C 30 (1986) 595 [INSPIRE].

    ADS  Google Scholar 

  54. J.M. Drummond and J.M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  55. J.M. Maldacena and A. Zhiboedov, Form factors at strong coupling via a Y-system, JHEP 11 (2010) 104 [arXiv:1009.1139] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  56. A. Brandhuber, B. Spence, G. Travaglini and G. Yang, Form factors in N = 4 super Yang-Mills and periodic Wilson loops, JHEP 01 (2011) 134 [arXiv:1011.1899] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  57. A. Brandhuber, O. Gurdogan, R. Mooney, G. Travaglini and G. Yang, Harmony of super form factors, JHEP 10 (2011) 046 [arXiv:1107.5067] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  58. L.V. Bork, D.I. Kazakov and G.S. Vartanov, On MHV form factors in superspace for \( \mathcal{N} = 4\;SYM \) theory, JHEP 10 (2011) 133 [arXiv:1107.5551] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  59. A. Brandhuber, G. Travaglini and G. Yang, Analytic two-loop form factors in N = 4 SYM, JHEP 05 (2012) 082 [arXiv:1201.4170] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  60. L.V. Bork, On NMHV form factors in N = 4 SYM theory from generalized unitarity, arXiv:1203.2596 [INSPIRE].

  61. B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, The super-correlator/super-amplitude duality: part I, arXiv:1103.3714 [INSPIRE].

  62. B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, The super-correlator/super-amplitude duality: part II, arXiv:1103.4353 [INSPIRE].

  63. L.J. Dixon, E.W.N. Glover and V.V. Khoze, MHV rules for Higgs plus multi-gluon amplitudes, JHEP 12 (2004) 015 [hep-th/0411092] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  64. N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  66. J.M. Drummond, M. Spradlin, A. Volovich and C. Wen, Tree-level amplitudes in N = 8 supergravity, Phys. Rev. D 79 (2009) 105018 [arXiv:0901.2363] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  67. J.L. Bourjaily, Efficient tree-amplitudes in N = 4: automatic BCFW recursion in Mathematica, arXiv:1011.2447 [INSPIRE].

  68. 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 

  69. M. Spradlin, A. Volovich and C. Wen, Three applications of a bonus relation for gravity amplitudes, Phys. Lett. B 674 (2009) 69 [arXiv:0812.4767] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  70. S. He, D. Nandan and C. Wen, Note on bonus relations for N = 8 supergravity tree amplitudes, JHEP 02 (2011) 005 [arXiv:1011.4287] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  72. H. Elvang, Y.-T. Huang and C. Peng, On-shell superamplitudes in N < 4 SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Brown University, Box 1843, Providence, RI, 02912-1843, USA

    Dhritiman Nandan

  2. Centre for Research in String Theory, School of Physics and Astronomy, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK

    Congkao Wen

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  1. Dhritiman Nandan
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Nandan, D., Wen, C. Generating all tree amplitudes in \( \mathcal{N} = 4\;{\text{SYM}} \) by Inverse Soft Limit. J. High Energ. Phys. 2012, 40 (2012). https://doi.org/10.1007/JHEP08(2012)040

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