Advertisement

Spectral parameters for scattering amplitudes in \( \mathcal{N} \) =4 super Yang-Mills theory

  • Livia Ferro
  • Tomasz Lukowski
  • Carlo Meneghelli
  • Jan Plefka
  • Matthias Staudacher
Open Access
Article

Abstract

Planar \( \mathcal{N} \) = 4 Super Yang-Mills theory appears to be a quantum integrable four-dimensional conformal theory. This has been used to find equations believed to describe its exact spectrum of anomalous dimensions. Integrability seemingly also extends to the planar space-time scattering amplitudes of the \( \mathcal{N} \) = 4 model, which show strong signs of Yangian invariance. However, in contradistinction to the spectral problem, this has not yet led to equations determining the exact amplitudes. We propose that the missing element is the spectral parameter, ubiquitous in integrable models. We show that it may indeed be included into recent on-shell approaches to scattering amplitude integrands, providing a natural deformation of the latter. Under some constraints, Yangian symmetry is preserved. Finally we speculate that the spectral parameter might also be the regulator of choice for controlling the infrared divergences appearing when integrating the integrands in exactly four dimensions.

Keywords

Scattering Amplitudes Integrable Field Theories AdS-CFT Correspondence 

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]
    L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills theories, Nucl. Phys. B 121 (1977) 77 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    F. Gliozzi, J. Scherk and D.I. Olive, Supersymmetry, supergravity theories and the dual spinor model, Nucl. Phys. B 122 (1977) 253 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  4. [4]
    L. Ferro, T. Lukowski, C. Meneghelli, J. Plefka and M. Staudacher, Harmonic R-matrices for scattering amplitudes and spectral regularization, Phys. Rev. Lett. 110 (2013) 121602 [arXiv:1212.0850] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [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].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    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].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    A. Brandhuber, P. Heslop and G. Travaglini, A note on dual superconformal symmetry of the N =4 super Yang-Mills S-matrix, Phys. Rev. D 78 (2008) 125005 [arXiv:0807.4097] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    H. Elvang, D.Z. Freedman and M. Kiermaier, Recursion relations, generating functions and unitarity sums in N = 4 SYM theory, JHEP 04 (2009) 009 [arXiv:0808.1720] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    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].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    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].ADSCrossRefGoogle Scholar
  12. [12]
    J. Drummond and J. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    L.J. Dixon, Scattering amplitudes: the most perfect microscopic structures in the universe, J. Phys. A 44 (2011) 454001 [arXiv:1105.0771] [INSPIRE].ADSGoogle Scholar
  14. [14]
    R. Roiban, Review of AdS/CFT integrability, chapter V.1: scattering amplitudesa brief introduction, Lett. Math. Phys. 99 (2012) 455 [arXiv:1012.4001] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    J. Drummond, Review of AdS/CFT integrability, chapter V.2: dual superconformal symmetry, Lett. Math. Phys. 99 (2012) 481 [arXiv:1012.4002] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    L.F. Alday, Review of AdS/CFT integrability, chapter V.3: scattering amplitudes at strong coupling, Lett. Math. Phys. 99 (2012) 507 [arXiv:1012.4003] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    J. Drummond, J. Henn, G. 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].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    J. Drummond, G. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    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].ADSCrossRefMathSciNetGoogle Scholar
  21. [21]
    J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    N. Berkovits and J. Maldacena, Fermionic T-duality, dual superconformal symmetry and the amplitude/Wilson loop connection, JHEP 09 (2008) 062 [arXiv:0807.3196] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    N. Beisert, R. Ricci, A.A. Tseytlin and M. Wolf, Dual superconformal symmetry from AdS 5× S 5 superstring integrability, Phys. Rev. D 78 (2008) 126004 [arXiv:0807.3228] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 01 (2007) P01021 [hep-th/0610251] [INSPIRE].Google Scholar
  25. [25]
    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].ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    T. Bargheer, N. Beisert, W. Galleas, F. Loebbert and T. McLoughlin, Exacting N = 4 superconformal symmetry, JHEP 11 (2009) 056 [arXiv:0905.3738] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  27. [27]
    A. Sever and P. Vieira, Symmetries of the N = 4 SYM S-matrix, arXiv:0908.2437 [INSPIRE].
  28. [28]
    N. Beisert, J. Henn, T. McLoughlin and J. Plefka, One-loop superconformal and Yangian symmetries of scattering amplitudes in N = 4 super Yang-Mills, JHEP 04 (2010) 085 [arXiv:1002.1733] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  29. [29]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    L. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  31. [31]
    J. Drummond and L. Ferro, Yangians, Grassmannians and T-duality, JHEP 07 (2010) 027 [arXiv:1001.3348] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  32. [32]
    J. Drummond and L. Ferro, The Yangian origin of the Grassmannian integral, JHEP 12 (2010) 010 [arXiv:1002.4622] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  33. [33]
    G. Korchemsky and E. Sokatchev, Superconformal invariants for scattering amplitudes in N =4 SYM theory, Nucl. Phys. B 839 (2010) 377 [arXiv:1002.4625] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  34. [34]
    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].ADSCrossRefMathSciNetGoogle Scholar
  35. [35]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  36. [36]
    N. Arkani-Hamed et al., Scattering amplitudes and the positive Grassmannian, arXiv:1212.5605 [INSPIRE].
  37. [37]
    B.I. Zwiebel, From scattering amplitudes to the dilatation generator in N = 4 SYM, J. Phys. A 45 (2012) 115401 [arXiv:1111.0083] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    L. Faddeev, How algebraic Bethe ansatz works for integrable model, in Relativistic gravitation and gravitational radiation, J.-A. Marck and J.-P. Lasota eds., Cambridge University Press, Cambridge U.K. (1997) [hep-th/9605187] [INSPIRE].Google Scholar
  39. [39]
    L.F. Alday, J.M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of N =4 super Yang-Mills, JHEP 01 (2010) 077 [arXiv:0908.0684] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  40. [40]
    J.M. Henn, S.G. Naculich, H.J. Schnitzer and M. Spradlin, Higgs-regularized three-loop four-gluon amplitude in N = 4 SYM: exponentiation and Regge limits, JHEP 04 (2010) 038 [arXiv:1001.1358] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  41. [41]
    J.M. Henn, S.G. Naculich, H.J. Schnitzer and M. Spradlin, More loops and legs in Higgs-regulated N = 4 SYM amplitudes, JHEP 08 (2010) 002 [arXiv:1004.5381] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  42. [42]
    A.E. Lipstein and L. Mason, From dlogs to dilogs; the super Yang-Mills MHV amplitude revisited, arXiv:1307.1443 [INSPIRE].
  43. [43]
    J.L. Bourjaily, S. Caron-Huot and J. Trnka, Dual-conformal regularization of infrared loop divergences and the chiral box expansion, arXiv:1303.4734 [INSPIRE].
  44. [44]
    A.P. Hodges, A twistor approach to the regularization of divergences, Proc. Roy. Soc. Lond. A 397 (1985) 341 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  45. [45]
    A.P. Hodges, Twistor diagrams for all tree amplitudes in gauge theory: a helicity-independent formalism, hep-th/0512336 [INSPIRE].
  46. [46]
    P. Kulish, N.Y. Reshetikhin and E. Sklyanin, Yang-Baxter equation and representation theory. 1, Lett. Math. Phys. 5 (1981) 393 [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  47. [47]
    V. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985) 254 [Dokl. Akad. Nauk Ser. Fiz. 283 (1985) 1060] [INSPIRE].
  48. [48]
    V. Drinfel’d, Quantum groups, J. Sov. Math. 41 (1988) 898 [Zap. Nauchn. Semin. 155 (1986) 18] [INSPIRE].
  49. [49]
    V. Drinfeld, A new realization of Yangians and quantized affine algebras, Sov. Math. Dokl. 36 (1988) 212 [INSPIRE].MathSciNetGoogle Scholar
  50. [50]
    N. Beisert and M. Staudacher, The N = 4 SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439 [hep-th/0307042] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  51. [51]
    N. Beisert, The complete one loop dilatation operator of N = 4 super Yang-Mills theory, Nucl. Phys. B 676 (2004) 3 [hep-th/0307015] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  52. [52]
    V.V. Bazhanov, R. Frassek, T. Lukowski, C. Meneghelli and M. Staudacher, Baxter Q-operators and representations of Yangians, Nucl. Phys. B 850 (2011) 148 [arXiv:1010.3699] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  53. [53]
    M. Günaydin and N. Marcus, The spectrum of the S 5 compactification of the chiral N = 2, D = 10 supergravity and the unitary supermultiplets of U(2,2/4), Class. Quant. Grav. 2 (1985) L11 [INSPIRE].CrossRefzbMATHGoogle Scholar
  54. [54]
    R. Frassek, N. Kanning, Y. Ko and M. Staudacher, Bethe ansatz for Yangian invariants: towards super Yang-Mills scattering amplitudes, arXiv:1312.1693 [INSPIRE].
  55. [55]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  56. [56]
    R.H. Boels, On BCFW shifts of integrands and integrals, JHEP 11 (2010) 113 [arXiv:1008.3101] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  57. [57]
    A. Postnikov, Total positivity, Grassmannians and networks, math.CO/0609764 [INSPIRE].
  58. [58]
    S. Fomin and A. Zelevinsky, Cluster algebras I: foundations, math.RT/0104151.
  59. [59]
    L. Dolan, C.R. Nappi and E. Witten, Yangian symmetry in D = 4 superconformal Yang-Mills theory, in Quantum theory and symmetries, P.C. Argyres et al. eds., World Scientific, Singapore (2004) [hep-th/0401243] [INSPIRE].Google Scholar
  60. [60]
    G. ’t Hooft and M. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  61. [61]
    C. Bollini and J. Giambiagi, Dimensional renormalization: the number of dimensions as a regularizing parameter, Nuovo Cim. B 12 (1972) 20 [INSPIRE].Google Scholar
  62. [62]
    J. Ashmore, A method of gauge invariant regularization, Lett. Nuovo Cim. 4 (1972) 289 [INSPIRE].CrossRefGoogle Scholar
  63. [63]
    G. Cicuta and E. Montaldi, Analytic renormalization via continuous space dimension, Lett. Nuovo Cim. 4 (1972) 329 [INSPIRE].CrossRefGoogle Scholar
  64. [64]
    R.K. Ellis and J. Sexton, QCD radiative corrections to parton parton scattering, Nucl. Phys. B 269 (1986) 445 [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    Z. Bern and D.A. Kosower, The computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  66. [66]
    Z. Bern, A. De Freitas, L.J. Dixon and H. Wong, Supersymmetric regularization, two loop QCD amplitudes and coupling shifts, Phys. Rev. D 66 (2002) 085002 [hep-ph/0202271] [INSPIRE].ADSGoogle Scholar
  67. [67]
    W. Siegel, Supersymmetric dimensional regularization via dimensional reduction, Phys. Lett. B 84 (1979) 193 [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    C.G. Bollini, J.J. Giambiagi and A. Gonzales Dominguez, Analytic regularization and the divergences of quantum field theories, Nuovo Cim. 31 (1964) 550.CrossRefGoogle Scholar
  69. [69]
    E.R. Speer, Generalized Feynman amplitudes, J. Math. Phys. 9 (1968) 1404.ADSCrossRefGoogle Scholar
  70. [70]
    L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for scattering amplitudes, J. Phys. A 43 (2010) 485401 [arXiv:1002.2459] [INSPIRE].MathSciNetGoogle Scholar
  71. [71]
    A. Sever, P. Vieira and T. Wang, From polygon Wilson loops to spin chains and back, JHEP 12 (2012) 065 [arXiv:1208.0841] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  72. [72]
    B. Basso, A. Sever and P. Vieira, Space-time S-matrix and flux-tube S-matrix at finite coupling, Phys. Rev. Lett. 111 (2013) 091602 [arXiv:1303.1396] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    B. Basso, A. Sever and P. Vieira, Space-time S-matrix and flux tube S-matrix II. Extracting and matching data, arXiv:1306.2058 [INSPIRE].
  74. [74]
    B. Keller, Cluster algebras, quiver representations and triangulated categories, arXiv:0807.1960.
  75. [75]
    J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic amplitudes and cluster coordinates, arXiv:1305.1617 [INSPIRE].

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Livia Ferro
    • 1
  • Tomasz Lukowski
    • 2
  • Carlo Meneghelli
    • 3
    • 4
  • Jan Plefka
    • 1
  • Matthias Staudacher
    • 2
    • 5
  1. 1.Institut für PhysikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Institut für Mathematik und Institut für PhysikHumboldt-Universität zu Berlin, IRIS AdlershofBerlinGermany
  3. 3.Fachbereich MathematikUniversität HamburgHamburgGermany
  4. 4.Theory Group, DESYHamburgGermany
  5. 5.Max-Planck Institut für Gravitationsphysik, Albert-Einstein-InstitutPotsdamGermany

Personalised recommendations