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Graßmannian integrals in Minkowski signature, amplitudes, and integrability

  • Nils KanningEmail author
  • Matthias Staudacher
Open Access
Regular Article - Theoretical Physics
  • 51 Downloads

Abstract

We attempt to systematically derive tree-level scattering amplitudes in fourdimensional, planar, maximally supersymmetric Yang-Mills theory from integrability. We first review the connections between integrable spin chains, Yangian invariance, and the construction of such invariants in terms of Graßmannian contour integrals. Building upon these results, we equip a class of Graßmannian integrals for general symmetry algebras with unitary integration contours. These contours emerge naturally by paying special attention to the proper reality conditions of the algebras. Specializing to \( \mathfrak{p}\mathfrak{s}\mathfrak{u}\left(2,2\Big|4\right) \) and thus to maximal superconformal symmetry in Minkowski space, we find in a number of examples expressions similar to, but subtly different from the perturbative physical scattering amplitudes. Our results suggest a subtle breaking of Yangian invariance for the latter, with curious implications for their construction from integrability.

Keywords

Lattice Integrable Models Matrix Models Scattering Amplitudes Supersymmetric Gauge Theory 

Notes

Open Access

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References

  1. [1]
    N. Kanning, On the integrable structure of super Yang-Mills scattering amplitudes, Ph.D. Thesis, Humboldt University, Berlin Germany (2016) [arXiv:1811.06324] [INSPIRE].
  2. [2]
    L.J. Dixon, A brief introduction to modern amplitude methods, in Proceedings of 2012 European School of High-Energy Physics (ESHEP 2012), La Pommeraye France (2012), pg. 31 [arXiv:1310.5353] [INSPIRE].
  3. [3]
    J.M. Henn and J.C. Plefka, Scattering Amplitudes in Gauge Theories, Lect. Notes Phys. 883 (2014) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    H. Elvang and Y.-t. Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge University Press, Cambridge U.K. (2015).Google Scholar
  5. [5]
    R. Roiban, Review of AdS/CFT Integrability, Chapter V.1: Scattering Amplitudesa Brief Introduction, Lett. Math. Phys. 99 (2012) 455 [arXiv:1012.4001] [INSPIRE].
  6. [6]
    J.M. Drummond, Review of AdS/CFT Integrability, Chapter V.2: Dual Superconformal Symmetry, Lett. Math. Phys. 99 (2012) 481 [arXiv:1012.4002] [INSPIRE].
  7. [7]
    R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
  8. [8]
    B.L. van der Waerden, Spinoranalyse, Gesell. Wiss. Göttingen Nachr. Math-Phys. Kl. 1929 (1928) 100, http://eudml.org/doc/59283.
  9. [9]
    H. Weyl, Electron and Gravitation. 1 (in German), Z. Phys. 56 (1929) 330 [INSPIRE].
  10. [10]
    J.M. Drummond and J.M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [INSPIRE].
  11. [11]
    S.J. Parke and T.R. Taylor, An Amplitude for n Gluon Scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].
  12. [12]
    V.P. Nair, A Current Algebra for Some Gauge Theory Amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].
  13. [13]
    V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985) 254 [INSPIRE].Google Scholar
  14. [14]
    M.L. Nazarov, Quantum Berezinian and the Classical Capelli Identity, Lett. Math. Phys. 21 (1991) 123.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    F. Loebbert, Lectures on Yangian Symmetry, J. Phys. A 49 (2016) 323002 [arXiv:1606.02947] [INSPIRE].
  16. [16]
    L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Relativistic gravitation and gravitational radiation. Proceedings of School of Physics, Les Houches France (1995), pg. 149 [hep-th/9605187] [INSPIRE].
  17. [17]
    R. Frassek, N. Kanning, Y. Ko and M. Staudacher, Bethe Ansatz for Yangian Invariants: Towards Super Yang-Mills Scattering Amplitudes, Nucl. Phys. B 883 (2014) 373 [arXiv:1312.1693] [INSPIRE].
  18. [18]
    D. Chicherin, S. Derkachov and R. Kirschner, Yang-Baxter operators and scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 881 (2014) 467 [arXiv:1309.5748] [INSPIRE].
  19. [19]
    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].
  20. [20]
    N. Beisert, On Yangian Symmetry in Planar N = 4 SYM, in Quantum chromodynamics and beyond: Gribov-80 memorial volume. Proceedings of Memorial Workshop devoted to the 80th birthday of V.N. Gribov, Trieste Italy (2010), pg. 175 [arXiv:1004.5423] [INSPIRE].
  21. [21]
    L. Ferro, J. Plefka and M. Staudacher, Yangian Symmetry in Maximally Supersymmetric Yang-Mills Theory, in Space-Time-Matter: Analytic and Geometric Structures, de Gruyter, Berlin Germany (2018), pg. 288.Google Scholar
  22. [22]
    T. Bargheer, N. Beisert, W. Galleas, F. Loebbert and T. McLoughlin, Exacting N = 4 Superconformal Symmetry, JHEP 11 (2009) 056 [arXiv:0905.3738] [INSPIRE].
  23. [23]
    A. Sever and P. Vieira, Symmetries of the N = 4 SYM S-matrix, arXiv:0908.2437 [INSPIRE].
  24. [24]
    T. Bargheer, N. Beisert and F. Loebbert, Exact Superconformal and Yangian Symmetry of Scattering Amplitudes, J. Phys. A 44 (2011) 454012 [arXiv:1104.0700] [INSPIRE].
  25. [25]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    L.J. Mason and D. Skinner, Dual Superconformal Invariance, Momentum Twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Scattering Amplitudes and the Positive Grassmannian, [arXiv:1212.5605] [INSPIRE].
  28. [28]
    N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press, Cambridge U.K. (2016).Google Scholar
  29. [29]
    J.M. Drummond and L. Ferro, Yangians, Grassmannians and T-duality, JHEP 07 (2010) 027 [arXiv:1001.3348] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    J.M. Drummond and L. Ferro, The Yangian origin of the Grassmannian integral, JHEP 12 (2010) 010 [arXiv:1002.4622] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    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].
  32. [32]
    D. Nandan, A. Volovich and C. Wen, A Grassmannian Etude in NMHV Minors, JHEP 07 (2010) 061 [arXiv:0912.3705] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Unification of Residues and Grassmannian Dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    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].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    L. Ferro, T. Lukowski, C. Meneghelli, J. Plefka and M. Staudacher, Spectral Parameters for Scattering Amplitudes in N = 4 Super Yang-Mills Theory, JHEP 01 (2014) 094 [arXiv:1308.3494] [INSPIRE].
  36. [36]
    N. Beisert, J. Broedel and M. Rosso, On Yangian-invariant regularization of deformed on-shell diagrams in \( \mathcal{N}=4 \) super-Yang-Mills theory, J. Phys. A 47 (2014) 365402 [arXiv:1401.7274] [INSPIRE].
  37. [37]
    J. Broedel, M. de Leeuw and M. Rosso, A dictionary between R-operators, on-shell graphs and Yangian algebras, JHEP 06 (2014) 170 [arXiv:1403.3670] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    J. Broedel, M. de Leeuw and M. Rosso, Deformed one-loop amplitudes in \( \mathcal{N}=4 \) super-Yang-Mills theory, JHEP 11 (2014) 091 [arXiv:1406.4024] [INSPIRE].
  39. [39]
    L. Ferro, T. Lukowski and M. Staudacher, \( \mathcal{N}=4 \) scattering amplitudes and the deformed Graßmannian, Nucl. Phys. B 889 (2014) 192 [arXiv:1407.6736] [INSPIRE].
  40. [40]
    T. Bargheer, Y.-t. Huang, F. Loebbert and M. Yamazaki, Integrable Amplitude Deformations for N = 4 Super Yang-Mills and ABJM Theory, Phys. Rev. D 91 (2015) 026004 [arXiv:1407.4449] [INSPIRE].
  41. [41]
    D. Chicherin and R. Kirschner, Yangian symmetric correlators, Nucl. Phys. B 877 (2013) 484 [arXiv:1306.0711] [INSPIRE].
  42. [42]
    N. Kanning, T. Lukowski and M. Staudacher, A shortcut to general tree-level scattering amplitudes in \( \mathcal{N}=4 \) SYM via integrability, Fortsch. Phys. 62 (2014) 556 [arXiv:1403.3382] [INSPIRE].
  43. [43]
    B.I. Zwiebel, From Scattering Amplitudes to the Dilatation Generator in N = 4 SYM, J. Phys. A 45 (2012) 115401 [arXiv:1111.0083] [INSPIRE].
  44. [44]
    N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
  45. [45]
    N. Beisert and M. Staudacher, The N = 4 SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439 [hep-th/0307042] [INSPIRE].
  46. [46]
    N. Beisert, The complete one loop dilatation operator of N = 4 superYang-Mills theory, Nucl. Phys. B 676 (2004) 3 [hep-th/0307015] [INSPIRE].
  47. [47]
    N. Beisert, The Analytic Bethe Ansatz for a Chain with Centrally Extended su(2|2) Symmetry, J. Stat. Mech. 0701 (2007) P01017 [nlin/0610017].
  48. [48]
    S. Salaff, A Nonzero Determinant Related to Schurs Matrix, Trans. Am. Math. Soc. 127 (1967) 349.Google Scholar
  49. [49]
    D.T. Stoyanov and I.T. Todorov, Majorana representations of the Lorentz group and infinite component fields, J. Math. Phys. 9 (1968) 2146 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  50. [50]
    G. Mack and I. Todorov, Irreducibility of the ladder representations of u(2,2) when restricted to the Poincaré subgroup, J. Math. Phys. 10 (1969) 2078 [INSPIRE].
  51. [51]
    I.T. Todorov, Discrete Series of Hermitian Representations of the Lie Algebra of u(p, q), Preprint IC-66-71 (1966).Google Scholar
  52. [52]
    I. Bars and M. Günaydin, Unitary Representations of Noncompact Supergroups, Commun. Math. Phys. 91 (1983) 31 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    N. Arkani-Hamed, F. Cachazo and C. Cheung, The Grassmannian Origin of Dual Superconformal Invariance, JHEP 03 (2010) 036 [arXiv:0909.0483] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Generalized unitarity for N = 4 super-amplitudes, Nucl. Phys. B 869(2013) 452 [arXiv:0808.0491] [INSPIRE].
  55. [55]
    J.M. Drummond, J. 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].
  56. [56]
    H. Dorn, H. Münkler and C. Spielvogel, Conformal geometry of null hexagons for Wilson loops and scattering amplitudes, Phys. Part. Nucl. 45 (2014) 692 [arXiv:1211.5537] [INSPIRE].CrossRefGoogle Scholar
  57. [57]
    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    N. Kanning and M. Staudacher, work in progress.Google Scholar
  59. [59]
    K.I. Gross and R.A. Kunze, Fourier Bessel Transforms and Holomorphic Discrete Series, in Conference on Harmonic Analysis, College Park Maryland U.S.A. (1971), pg. 79.Google Scholar
  60. [60]
    G. Post, Properties of Massless Relativistic Fields Under the Conformal Group, J. Math. Phys. 17 (1976) 24 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    G.P. Korchemsky and E. Sokatchev, Twistor transform of all tree amplitudes in N = 4 SYM theory, Nucl. Phys. B 829 (2010) 478 [arXiv:0907.4107] [INSPIRE].
  62. [62]
    L. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, American Mathematical Society, Providence U.S.A. (1963).Google Scholar
  63. [63]
    J.L. Bourjaily, Efficient Tree-Amplitudes in N = 4: Automatic BCFW Recursion in Mathematica, arXiv:1011.2447 [INSPIRE].
  64. [64]
    J.L. Bourjaily, Positroids, Plabic Graphs and Scattering Amplitudes in Mathematica,arXiv:1212.6974 [INSPIRE].
  65. [65]
    N. Kanning and M. Staudacher, unpublished.Google Scholar
  66. [66]
    B.V. Shabat, Introduction to Complex Analysis: Part II, Functions of Several Variables, American Mathematical Society, Providence U.S.A. (1992).Google Scholar
  67. [67]
    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].
  68. [68]
    M. Günaydin and D. Volin, The complete unitary dual of non-compact Lie superalgebra su(p, q|m) via the generalised oscillator formalism and non-compact Young diagrams, arXiv:1712.01811 [INSPIRE].
  69. [69]
    N. Kanning, Y. Ko and M. Staudacher, Graßmannian integrals as matrix models for non-compact Yangian invariants, Nucl. Phys. B 894 (2015) 407 [arXiv:1412.8476] [INSPIRE].
  70. [70]
    H. Leutwyler and A.V. Smilga, Spectrum of Dirac operator and role of winding number in QCD, Phys. Rev. D 46 (1992) 5607 [INSPIRE].
  71. [71]
    D.J. Gross and E. Witten, Possible Third Order Phase Transition in the Large N Lattice Gauge Theory, Phys. Rev. D 21 (1980) 446 [INSPIRE].
  72. [72]
    E. Brézin and D.J. Gross, The External Field Problem in the Large N Limit of QCD, Phys. Lett. 97B (1980) 120 [INSPIRE].
  73. [73]
    I. Bars and F. Green, Complete Integration of U (N) Lattice Gauge Theory in a Large N Limit, Phys. Rev. D 20 (1979) 3311 [INSPIRE].
  74. [74]
    B. Schlittgen and T. Wettig, Generalizations of some integrals over the unitary group, J. Phys. A 36 (2003) 3195 [math-ph/0209030] [INSPIRE].
  75. [75]
    A.B. Balantekin, Character expansions, Itzykson-Zuber integrals and the QCD partition function, Phys. Rev. D 62 (2000) 085017 [hep-th/0007161] [INSPIRE].
  76. [76]
    B. Eden and M. Staudacher, Integrability and transcendentality, J. Stat. Mech. 0611 (2006) P11014 [hep-th/0603157] [INSPIRE].
  77. [77]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
  78. [78]
    A. Mironov, A. Morozov and G.W. Semenoff, Unitary matrix integrals in the framework of generalized Kontsevich model. 1. Brezin-Gross-Witten model, Int. J. Mod. Phys. A 11 (1996) 5031 [hep-th/9404005] [INSPIRE].
  79. [79]
    A.Y. Orlov, New Solvable Matrix Integrals, Int. J. Mod. Phys. A 19 (2004) 276 [nlin/0209063].
  80. [80]
    T. Miwa, M. Jimbo and E. Date, Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, Cambridge University Press, Cambridge U.K. (2000).Google Scholar
  81. [81]
    V. Bargmann, On a Hilbert Space of Analytic Functions and an Associated Integral Transform Part I, Commun. Pure Appl. Math. 14 (1961) 187.Google Scholar
  82. [82]
    L.A. Takhtajan, Quantum Mechanics for Mathematicians, American Mathematical Society, Providence U.S.A. (2008).Google Scholar
  83. [83]
    J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, Oxford U.K. (2005).Google Scholar
  84. [84]
    H. Boos, F. Göhmann, A. Klümper, K.S. Nirov and A.V. Razumov, Exercises with the universal R-matrix, J. Phys. A 43 (2010) 415208 [arXiv:1004.5342] [INSPIRE].
  85. [85]
    B.S. Shastry, Exact Integrability of the One-Dimensional Hubbard Model, Phys. Rev. Lett. 56 (1986) 2453 [INSPIRE].
  86. [86]
    C. Gómez, M. Ruiz-Altaba and G. Sierra, Quantum Groups in Two-Dimensional Physics, Cambridge University Press, Cambridge U.K. (2005).Google Scholar
  87. [87]
    P.D. Miller, Applied Asymptotic Analysis, American Mathematical Society, Providence U.S.A. (2006).Google Scholar
  88. [88]
    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].
  89. [89]
    Z. Bern et al., The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev. D 78 (2008) 045007 [arXiv:0803.1465] [INSPIRE].
  90. [90]
    A. Hurwitz, Über die Erzeugung der Invarianten durch Integration, Gesell. Wiss. Göttingen Nachr. Math-Phys. Kl. 1897 (1897) 71, http://eudml.org/doc/58378.
  91. [91]
    R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, John Wiley & Sons, New York U.S.A. (1974).Google Scholar

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Authors and Affiliations

  1. 1.Institut für Mathematik und Institut für PhysikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Arnold Sommerfeld Center for Theoretical PhysicsLudwig-Maximilians-UniversitätMünchenGermany

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