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Three-point functions in planar \( \mathcal{N} = {4} \) super Yang-Mills theory for scalar operators up to length five at the one-loop order

  • George Georgiou
  • Valeria Gili
  • André Großardt
  • Jan Plefka
Article

Abstract

We report on a systematic perturbative study of three-point functions in planar SU(N) \( \mathcal{N} = {4} \) super Yang-Mills theory at the one-loop level involving scalar field operators up to length five. For this we have computed a sample of 40 structure constants involving primary operators of up to and including length five which are built entirely from scalar fields. A combinatorial dressing technique has been developed to promote tree-level correlators to one-loop level. In addition we have resolved the mixing up to the order \( g_{\text{YM}}^2 \) level of the operators involved, which amounts to mixings with bi-fermions, with bi-derivative insertions as well as self-mixing contributions in the scalar sector. This work supersedes a preprint by two of the authors from 2010 which had neglected the mixing contributions.

Keywords

Supersymmetric gauge theory Extended Supersymmetry AdS-CFT Correspondence 

References

  1. [1]
    J. Minahan and K. Zarembo, The Bethe ansatz for \( \mathcal{N} = {4} \) super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    N. Beisert, C. Kristjansen and M. Staudacher, The dilatation operator of conformal \( \mathcal{N} = {4} \) super Yang-Mills theory, Nucl. Phys. B 664 (2003) 131 [hep-th/0303060] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].MathSciNetADSGoogle Scholar
  4. [4]
    N. Beisert and M. Staudacher, The \( \mathcal{N} = {4} \) SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439 [hep-th/0307042] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1133] [hep-th/9711200] [INSPIRE].
  6. [6]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSzbMATHGoogle Scholar
  7. [7]
    S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].MathSciNetADSGoogle Scholar
  8. [8]
    L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills theories, Nucl. Phys. B 121 (1977) 77 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    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
  10. [10]
    N. Beisert, The SU(2|3) dynamic spin chain, Nucl. Phys. B 682 (2004) 487 [hep-th/0310252] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    M. Staudacher, The factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].MathSciNetGoogle Scholar
  13. [13]
    A.A. Tseytlin, Semiclassical strings in AdS 5 × S 5 and scalar operators in N = 4 SYM theory, Comptes Rendus Physique 5 (2004) 1049 [hep-th/0407218] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    A. Belitsky, V. Braun, A. Gorsky and G. Korchemsky, Integrability in QCD and beyond, Int. J. Mod. Phys. A 19 (2004) 4715 [hep-th/0407232] [INSPIRE].MathSciNetADSGoogle Scholar
  15. [15]
    K. Zarembo, Semiclassical Bethe Ansatz and AdS/CFT, Comptes Rendus Physique 5 (2004) 1081 [hep-th/0411191] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    J. Plefka, Spinning strings and integrable spin chains in the AdS/CFT correspondence, Living Rev. Rel. 8 (2005) 9 [hep-th/0507136] [INSPIRE].Google Scholar
  17. [17]
    J. Minahan, A brief introduction to the Bethe ansatz in \( \mathcal{N} = {4} \) super-Yang-Mills, J. Phys. A 39 (2006) 12657 [INSPIRE].MathSciNetGoogle Scholar
  18. [18]
    G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].MathSciNetADSGoogle Scholar
  19. [19]
    N. Beisert, Higher-loop integrability in \( \mathcal{N} = {4} \) gauge theory, Comptes Rendus Physique 5 (2004) 1039 [hep-th/0409147] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    N. Beisert, The dilatation operator of \( \mathcal{N} = {4} \) super Yang-Mills theory and integrability, Phys. Rept. 405 (2005) 1 [hep-th/0407277] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    J. Ambjørn, R.A. Janik and C. Kristjansen, Wrapping interactions and a new source of corrections to the spin-chain/string duality, Nucl. Phys. B 736 (2006) 288 [hep-th/0510171] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    Z. Bajnok, A. Hegedus, R.A. Janik and T. Lukowski, Five loop Konishi from AdS/CFT, Nucl. Phys. B 827 (2010) 426 [arXiv:0906.4062] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    F. Fiamberti, A. Santambrogio and C. Sieg, Five-loop anomalous dimension at critical wrapping order in N = 4 SYM, JHEP 03 (2010) 103 [arXiv:0908.0234] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    N. Gromov, V. Kazakov, A. Kozak and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills Theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. [26]
    D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe Ansatz for planar AdS/CFT: a proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].MathSciNetGoogle Scholar
  27. [27]
    G. Arutyunov and S. Frolov, Thermodynamic Bethe Ansatz for the AdS 5 × S 5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of planar \( \mathcal{N} = {4} \) supersymmetric Yang-Mills theory: Konishi dimension at any coupling, Phys. Rev. Lett. 104 (2010) 211601 [arXiv:0906.4240] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    S. Frolov, Konishi operator at intermediate coupling, J. Phys. A 44 (2011) 065401 [arXiv:1006.5032] [INSPIRE].ADSGoogle Scholar
  30. [30]
    J. Erickson, G. Semenoff and K. Zarembo, Wilson loops in \( \mathcal{N} = {4} \) supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    N. Drukker and D.J. Gross, An exact prediction of \( \mathcal{N} = {4} \) SUSYM theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. [32]
    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].MathSciNetADSGoogle Scholar
  33. [33]
    J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in \( \mathcal{N} = {4} \) super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    L.F. Alday and R. Roiban, Scattering amplitudes, Wilson loops and the string/gauge theory correspondence, Phys. Rept. 468 (2008) 153 [arXiv:0807.1889] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    J. Henn, Duality between Wilson loops and gluon amplitudes, Fortsch. Phys. 57 (2009) 729 [arXiv:0903.0522] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. [37]
    B. Eden, P.S. Howe and P.C. West, Nilpotent invariants in N = 4 SYM, Phys. Lett. B 463 (1999) 19 [hep-th/9905085] [INSPIRE].MathSciNetADSGoogle Scholar
  38. [38]
    G. Arutyunov, B. Eden and E. Sokatchev, On nonrenormalization and OPE in superconformal field theories, Nucl. Phys. B 619 (2001) 359 [hep-th/0105254] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    P. Heslop and P.S. Howe, OPEs and three-point correlators of protected operators in N = 4 SYM, Nucl. Phys. B 626 (2002) 265 [hep-th/0107212] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large-N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  41. [41]
    A. Basu, M.B. Green and S. Sethi, Some systematics of the coupling constant dependence of N = 4 Yang-Mills, JHEP 09 (2004) 045 [hep-th/0406231] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, Properties of the Konishi multiplet in N = 4 SYM theory, JHEP 05 (2001) 042 [hep-th/0104016] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    N. Beisert, C. Kristjansen, J. Plefka, G. Semenoff and M. Staudacher, BMN correlators and operator mixing in \( \mathcal{N} = {4} \) super Yang-Mills theory, Nucl. Phys. B 650 (2003) 125 [hep-th/0208178] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    R. Roiban and A. Volovich, Yang-Mills correlation functions from integrable spin chains, JHEP 09 (2004) 032 [hep-th/0407140] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    K. Okuyama and L.-S. Tseng, Three-point functions in N = 4 SYM theory at one-loop, JHEP 08 (2004) 055 [hep-th/0404190] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    L.F. Alday, J.R. David, E. Gava and K. Narain, Structure constants of planar N = 4 Yang-Mills at one loop, JHEP 09 (2005) 070 [hep-th/0502186] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  47. [47]
    L.F. Alday, J.R. David, E. Gava and K. Narain, Towards a string bit formulation of N = 4 super Yang-Mills, JHEP 04 (2006) 014 [hep-th/0510264] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    G. Georgiou, V.L. Gili and R. Russo, Operator mixing and three-point functions in N = 4 SYM, JHEP 10 (2009) 009 [arXiv:0907.1567] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    G. Arutyunov, S. Frolov and A. Petkou, Perturbative and instanton corrections to the OPE of CPOs in N = 4 SYM(4), Nucl. Phys. B 602 (2001) 238 [Erratum ibid. B 609 (2001) 540] [hep-th/0010137] [INSPIRE].
  50. [50]
    C.-S. Chu, V.V. Khoze and G. Travaglini, Three point functions in N = 4 Yang-Mills theory and pp waves, JHEP 06 (2002) 011 [hep-th/0206005] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    G. Georgiou, V.L. Gili and R. Russo, Operator Mixing and the AdS/CFT correspondence, JHEP 01 (2009) 082 [arXiv:0810.0499] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    P.-Y. Casteill, R. Janik, A. Jarosz and C. Kristjansen, Quasilocality of joining/splitting strings from coherent states, JHEP 12 (2007) 069 [arXiv:0710.4166] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    T. Yoneya, Holography in the large J limit of AdS/CFT correspondence and its applications, Prog. Theor. Phys. Suppl. 164 (2007) 82 [hep-th/0607046] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    S. Dobashi and T. Yoneya, Resolving the holography in the plane-wave limit of AdS/CFT correspondence, Nucl. Phys. B 711 (2005) 3 [hep-th/0406225] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  55. [55]
    A. Tsuji, Holography of Wilson loop correlator and spinning strings, Prog. Theor. Phys. 117 (2007) 557 [hep-th/0606030] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  56. [56]
    R.A. Janik, P. Surowka and A. Wereszczynski, On correlation functions of operators dual to classical spinning string states, JHEP 05 (2010) 030 [arXiv:1002.4613] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    E. Buchbinder and A. Tseytlin, On semiclassical approximation for correlators of closed string vertex operators in AdS/CFT, JHEP 08 (2010) 057 [arXiv:1005.4516] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  58. [58]
    K. Zarembo, Holographic three-point functions of semiclassical states, JHEP 09 (2010) 030 [arXiv:1008.1059] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  59. [59]
    M.S. Costa, R. Monteiro, J.E. Santos and D. Zoakos, On three-point correlation functions in the gauge/gravity duality, JHEP 11 (2010) 141 [arXiv:1008.1070] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  60. [60]
    R. Roiban and A. Tseytlin, On semiclassical computation of 3-point functions of closed string vertex operators in AdS 5 × S 5, Phys. Rev. D 82 (2010) 106011 [arXiv:1008.4921] [INSPIRE].ADSGoogle Scholar
  61. [61]
    S. Ryang, Correlators of vertex operators for circular strings with winding numbers in AdS 5 × S 5, JHEP 01 (2011) 092 [arXiv:1011.3573] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    T. Klose and T. McLoughlin, A light-cone approach to three-point functions in AdS 5 × S 5, arXiv:1106.0495 [INSPIRE].
  63. [63]
    S. Ryang, Extremal correlator of three vertex operators for circular winding strings in AdS 5 × S 5, JHEP 11 (2011) 026 [arXiv:1109.3242] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  64. [64]
    R. Hernandez, Three-point correlation functions from semiclassical circular strings, J. Phys. A 44 (2011) 085403 [arXiv:1011.0408] [INSPIRE].ADSGoogle Scholar
  65. [65]
    J. Russo and A. Tseytlin, Large spin expansion of semiclassical 3-point correlators in AdS 5 × S 5, JHEP 02 (2011) 029 [arXiv:1012.2760] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  66. [66]
    G. Georgiou, Two and three-point correlators of operators dual to folded string solutions at strong coupling, JHEP 02 (2011) 046 [arXiv:1011.5181] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  67. [67]
    C. Park and B.-H. Lee, Correlation functions of magnon and spike, Phys. Rev. D 83 (2011) 126004 [arXiv:1012.3293] [INSPIRE].ADSGoogle Scholar
  68. [68]
    D. Bak, B. Chen and J.-B. Wu, Holographic correlation functions for open strings and branes, JHEP 06 (2011) 014 [arXiv:1103.2024] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  69. [69]
    A. Bissi, C. Kristjansen, D. Young and K. Zoubos, Holographic three-point functions of giant gravitons, JHEP 06 (2011) 085 [arXiv:1103.4079] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  70. [70]
    R. Hernandez, Three-point correlators for giant magnons, JHEP 05 (2011) 123 [arXiv:1104.1160] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    C. Ahn and P. Bozhilov, Three-point Correlation functions of Giant magnons with finite size, Phys. Lett. B 702 (2011) 286 [arXiv:1105.3084] [INSPIRE].MathSciNetADSGoogle Scholar
  72. [72]
    D. Arnaudov, R. Rashkov and T. Vetsov, Three and four-point correlators of operators dual to folded string solutions in AdS 5 × S 5, Int. J. Mod. Phys. A 26 (2011) 3403 [arXiv:1103.6145] [INSPIRE].MathSciNetADSGoogle Scholar
  73. [73]
    C. Ahn and P. Bozhilov, Three-point correlation function of giant magnons in the Lunin-Maldacena background, Phys. Rev. D 84 (2011) 126011 [arXiv:1106.5656] [INSPIRE].ADSGoogle Scholar
  74. [74]
    J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability, JHEP 09 (2011) 028 [arXiv:1012.2475] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  75. [75]
    J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability II. Weak/strong coupling match, JHEP 09 (2011) 029 [arXiv:1104.5501] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  76. [76]
    N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability III. Classical tunneling, arXiv:1111.2349 [INSPIRE].
  77. [77]
    O. Foda, N = 4 SYM structure constants as determinants, JHEP 03 (2012) 096 [arXiv:1111.4663] [INSPIRE].ADSGoogle Scholar
  78. [78]
    A. Bissi, T. Harmark and M. Orselli, Holographic 3-point function at one loop, JHEP 02 (2012) 133 [arXiv:1112.5075] [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    G. Georgiou, SL(2) sector: weak/strong coupling agreement of three-point correlators, JHEP 09 (2011) 132 [arXiv:1107.1850] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  80. [80]
    R.A. Janik and A. Wereszczynski, Correlation functions of three heavy operators: the AdS contribution, JHEP 12 (2011) 095 [arXiv:1109.6262] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    E. Buchbinder and A. Tseytlin, Semiclassical correlators of three states with large S 5 charges in string theory in AdS 5 × S 5, Phys. Rev. D 85 (2012) 026001 [arXiv:1110.5621] [INSPIRE].ADSGoogle Scholar
  82. [82]
    S. Gubser, I. Klebanov and A.M. Polyakov, A semiclassical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  83. [83]
    Y. Kazama and S. Komatsu, On holographic three point functions for GKP strings from integrability, JHEP 01 (2012) 110 [arXiv:1110.3949] [INSPIRE].ADSCrossRefGoogle Scholar
  84. [84]
    N. Drukker and J. Plefka, The structure of n-point functions of chiral primary operators in N = 4 super Yang-Mills at one-loop, JHEP 04 (2009) 001 [arXiv:0812.3341] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  85. [85]
    E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Extremal correlators in the AdS/CFT correspondence, hep-th/9908160 [INSPIRE].
  86. [86]
    G. Georgiou, V. Gili and J. Plefka, The two-loop dilatation operator of N = 4 super Yang-Mills theory in the SO(6) sector, JHEP 12 (2011) 075 [arXiv:1106.0724] [INSPIRE].ADSCrossRefGoogle Scholar
  87. [87]
    A. Grossardt and J. Plefka, One-loop spectroscopy of scalar three-point functions in planar N = 4 super Yang-Mills theory, arXiv:1007.2356 [INSPIRE].
  88. [88]
    G. Georgiou and G. Travaglini, Fermion BMN operators, the dilatation operator of N = 4 SYM and pp wave string interactions, JHEP 04 (2004) 001 [hep-th/0403188] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  89. [89]
    A. Grossardt, Three-Point Functions in Superconformal Field Theories, MSc Thesis, Institute of Physics, Humboldt-University Berlin, Berlin Germany (2010), http://qft.physik.hu-berlin.de.

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • George Georgiou
    • 1
  • Valeria Gili
    • 2
  • André Großardt
    • 3
    • 4
    • 5
  • Jan Plefka
    • 5
  1. 1.Demokritos National Research Center, Institute of Nuclear PhysicsAthensGreece
  2. 2.Centre for Research in String Theory, School of PhysicsQueen Mary University of LondonLondonUnited Kingdom
  3. 3.ZARM BremenBremenGermany
  4. 4.Institute of Theoretical PhysicsUniversity of HannoverHannoverGermany
  5. 5.Institut für PhysikHumboldt-Universität zu BerlinBerlinGermany

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