Advertisement

Leading singularities and off-shell conformal integrals

  • James Drummond
  • Claude Duhr
  • Burkhard Eden
  • Paul HeslopEmail author
  • Jeffrey Pennington
  • Vladimir A. Smirnov
Open Access
Article

Abstract

The three-loop four-point function of stress-tensor multiplets in \( \mathcal{N}=4 \) super Yang-Mills theory contains two so far unknown, off-shell, conformal integrals, in addition to the known, ladder-type integrals. In this paper we evaluate the unknown integrals, thus obtaining the three-loop correlation function analytically. The integrals have the generic structure of rational functions multiplied by (multiple) polylogarithms. We use the idea of leading singularities to obtain the rational coefficients, the symbol — with an appropriate ansatz for its structure — as a means of characterising multiple polylogarithms, and the technique of asymptotic expansion of Feynman integrals to obtain the integrals in certain limits. The limiting behaviour uniquely fixes the symbols of the integrals, which we then lift to find the corresponding polylogarithmic functions. The final formulae are numerically confirmed. The techniques we develop can be applied more generally, and we illustrate this by analytically evaluating one of the integrals contributing to the same four-point function at four loops. This example shows a connection between the leading singularities and the entries of the symbol.

Keywords

Supersymmetric gauge theory Scattering Amplitudes Extended Supersymmetry 

References

  1. [1]
    M.T. Grisaru, M. Roček and W. Siegel, Zero three loop β-function in N = 4 super Yang-Mills theory, Phys. Rev. Lett. 45 (1980) 1063 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    W.E. Caswell and D. Zanon, Vanishing three loop β-function in N = 4 supersymmetric Yang-Mills theory, Phys. Lett. B 100 (1981) 152 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    P.S. Howe, K. Stelle and P. Townsend, The relaxed hypermultiplet: an unconstrained N = 2 superfield theory, Nucl. Phys. B 214 (1983) 519 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    S. Mandelstam, Light cone superspace and the vanishing of the β-function for the N = 4 model, in Proc. 21st Int. Conf. on High Energy Physics, P. Petiau and M. Proneuf eds., J. Phys. (France) 43 (1982) C-3 [INSPIRE].
  5. [5]
    S. Mandelstam, Light cone superspace and the ultraviolet finiteness of the N = 4 model, Nucl. Phys. B 213 (1983) 149 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    L. Brink, O. Lindgren and B.E. Nilsson, N = 4 Yang-Mills theory on the light cone, Nucl. Phys. B 212 (1983) 401 [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    L. Brink, O. Lindgren and B.E. Nilsson, The ultraviolet finiteness of the N = 4 Yang-Mills theory, Phys. Lett. B 123 (1983) 323 [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    P.S. Howe, K. Stelle and P. Townsend, Miraculous ultraviolet cancellations in supersymmetry made manifest, Nucl. Phys. B 236 (1984) 125 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    J. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    N. Beisert, V. Dippel and M. Staudacher, A novel long range spin chain and planar N = 4 super Yang-Mills, JHEP 07 (2004) 075 [hep-th/0405001] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    N. Beisert and M. Staudacher, Long-range PSU(2, 2|4) Bethe ansätze for gauge theory and strings, Nucl. Phys. B 727 (2005) 1 [hep-th/0504190] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    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) 1113] [hep-th/9711200] [INSPIRE].
  13. [13]
    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].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSzbMATHGoogle Scholar
  15. [15]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    G. Korchemsky, J. Drummond and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [INSPIRE].MathSciNetADSGoogle Scholar
  17. [17]
    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].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    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].MathSciNetADSGoogle Scholar
  20. [20]
    G. Korchemsky and A. Radyushkin, Loop space formalism and renormalization group for the infrared asymptotics of QCD, Phys. Lett. B 171 (1986) 459 [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    S. Ivanov, G. Korchemsky and A. Radyushkin, Infrared asymptotics of perturbative QCD: contour gauges, Yad. Fiz. 44 (1986) 230 [Sov. J. Nucl. Phys. 44 (1986) 145] [INSPIRE].
  22. [22]
    G. Korchemsky and A. Radyushkin, Infrared asymptotics of perturbative QCD. Quark and gluon propagators, Yad. Fiz. 45 (1987) 198 [Sov. J. Nucl. Phys. 45 (1987) 127] [INSPIRE].
  23. [23]
    G. Korchemsky and A. Radyushkin, Infrared asymptotics of perturbative QCD. Vertex functions, Yad. Fiz. 45 (1987) 1466 [Sov. J. Nucl. Phys. 45 (1987) 910] [INSPIRE].
  24. [24]
    G. Korchemsky and A. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    G. Korchemsky and G. Marchesini, Structure function for large x and renormalization of Wilson loop, Nucl. Phys. B 406 (1993) 225 [hep-ph/9210281] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    L.F. Alday, B. Eden, G.P. Korchemsky, J. Maldacena and E. Sokatchev, From correlation functions to Wilson loops, JHEP 09 (2011) 123 [arXiv:1007.3243] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    B. Eden, G.P. Korchemsky and E. Sokatchev, From correlation functions to scattering amplitudes, JHEP 12 (2011) 002 [arXiv:1007.3246] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    B. Eden, G.P. Korchemsky and E. Sokatchev, More on the duality correlators/amplitudes, Phys. Lett. B 709 (2012) 247 [arXiv:1009.2488] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton exchange and complete four point functions in the AdS/CFT correspondence, Nucl. Phys. B 562 (1999) 353 [hep-th/9903196] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    E. D’Hoker, S.D. Mathur, A. Matusis and L. Rastelli, The operator product expansion of N = 4 SYM and the 4 point functions of supergravity, Nucl. Phys. B 589 (2000) 38 [hep-th/9911222] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    G. Arutyunov and S. Frolov, Four point functions of lowest weight CPOs in N = 4 SYM 4 in supergravity approximation, Phys. Rev. D 62 (2000) 064016 [hep-th/0002170] [INSPIRE].MathSciNetADSGoogle Scholar
  32. [32]
    G. Arutyunov, F. Dolan, H. Osborn and E. Sokatchev, Correlation functions and massive Kaluza-Klein modes in the AdS/CFT correspondence, Nucl. Phys. B 665 (2003) 273 [hep-th/0212116] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Four point functions in N = 4 supersymmetric Yang-Mills theory at two loops, Nucl. Phys. B 557 (1999) 355 [hep-th/9811172] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Simplifications of four point functions in N = 4 supersymmetric Yang-Mills theory at two loops, Phys. Lett. B 466 (1999) 20 [hep-th/9906051] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    F. Gonzalez-Rey, I. Park and K. Schalm, A note on four point functions of conformal operators in N = 4 super Yang-Mills, Phys. Lett. B 448 (1999) 37 [hep-th/9811155] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    B. Eden, C. Schubert and E. Sokatchev, Three loop four point correlator in N = 4 SYM, Phys. Lett. B 482 (2000) 309 [hep-th/0003096] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, Anomalous dimensions in N = 4 SYM theory at order g 4, Nucl. Phys. B 584 (2000) 216 [hep-th/0003203] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky and E. Sokatchev, Unconstrained N = 2 matter, Yang-Mills and supergravity theories in harmonic superspace, Class. Quant. Grav. 1 (1984) 469 [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    G. Hartwell and P.S. Howe, (N,p,q) harmonic superspace, Int. J. Mod. Phys. A 10 (1995) 3901 [hep-th/9412147] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    B. Eden, A.C. Petkou, C. Schubert and E. Sokatchev, Partial nonrenormalization of the stress tensor four point function in N = 4 SYM and AdS/CFT, Nucl. Phys. B 607 (2001) 191 [hep-th/0009106] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    B. Eden, P.S. Howe, A. Pickering, E. Sokatchev and P.C. West, Four point functions in N = 2 superconformal field theories, Nucl. Phys. B 581 (2000) 523 [hep-th/0001138] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    F. Dolan and H. Osborn, Superconformal symmetry, correlation functions and the operator product expansion, Nucl. Phys. B 629 (2002) 3 [hep-th/0112251] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    P. Heslop and P. Howe, Four point functions in N = 4 SYM, JHEP 01 (2003) 043 [hep-th/0211252] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, Hidden symmetry of four-point correlation functions and amplitudes in N = 4 SYM, Nucl. Phys. B 862 (2012) 193 [arXiv:1108.3557] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N = 4 SYM, Nucl. Phys. B 862 (2012) 450 [arXiv:1201.5329] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    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
  47. [47]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    F. Cachazo, Sharpening the leading singularity, arXiv:0803.1988 [INSPIRE].
  49. [49]
    B. Eden, P. Heslop, G.P. Korchemsky, V.A. Smirnov and E. Sokatchev, Five-loop Konishi in N = 4 SYM, Nucl. Phys. B 862 (2012) 123 [arXiv:1202.5733] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    B. Eden, Three-loop universal structure constants in N = 4 SUSY Yang-Mills theory, arXiv:1207.3112 [INSPIRE].
  51. [51]
    V.A. Smirnov, Applied asymptotic expansions in momenta and masses, Springer Tracts Mod. Phys. 177 (2002) 1 [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    J. Drummond, J. Henn, V. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    N. Usyukina and A.I. Davydychev, An approach to the evaluation of three and four point ladder diagrams, Phys. Lett. B 298 (1993) 363 [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    N. Usyukina and A.I. Davydychev, Exact results for three and four point ladder diagrams with an arbitrary number of rungs, Phys. Lett. B 305 (1993) 136 [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    F.C.S. Brown, Single-valued multiple polylogarithms in one variable Comptes Rendus Math. 338 (2004) 527.CrossRefzbMATHGoogle Scholar
  56. [56]
    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].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    S. Caron-Huot, Loops in spacetime, in ECT*, Trento workshopScattering amplitudes: from QCD to maximally supersymmetric Yang-Mills theory and back, http://sites.google.com/site/trentoworkshop/program/, Trento Italy July 16-20 2012.
  58. [58]
    F. Chavez and C. Duhr, Three-mass triangle integrals and single-valued polylogarithms, JHEP 11 (2012) 114 [arXiv:1209.2722] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  59. [59]
    A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  60. [60]
    D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    E. Remiddi and J. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    J. Fleischer, A. Kotikov and O. Veretin, Analytic two loop results for selfenergy type and vertex type diagrams with one nonzero mass, Nucl. Phys. B 547 (1999) 343 [hep-ph/9808242] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    J. Drummond, Generalised ladders and single-valued polylogarithms, JHEP 02 (2013) 092 [arXiv:1207.3824] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  64. [64]
    M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    V.A. Smirnov, Analytic tools for Feynman integrals, Springer Tracts Mod. Phys. 250 (2012) 1 [INSPIRE].CrossRefGoogle Scholar
  66. [66]
    A. Pak and A. Smirnov, Geometric approach to asymptotic expansion of Feynman integrals, Eur. Phys. J. C 71 (2011) 1626 [arXiv:1011.4863] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    B. Jantzen, A.V. Smirnov and V.A. Smirnov, Expansion by regions: revealing potential and Glauber regions automatically, Eur. Phys. J. C 72 (2012) 2139 [arXiv:1206.0546] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    A. Smirnov, Algorithm FIREFeynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    S. Gorishnii, S. Larin, L. Surguladze and F. Tkachov, MINCER: program for multiloop calculations in quantum field theory for the Schoonschip system, Comput. Phys. Commun. 55 (1989) 381 [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    J. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14 (1999) 2037 [hep-ph/9806280] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  71. [71]
    S. Moch, P. Uwer and S. Weinzierl, Nested sums, expansion of transcendental functions and multiscale multiloop integrals, J. Math. Phys. 43 (2002) 3363 [hep-ph/0110083] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  72. [72]
    E.I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N = 4 super Yang-Mills, JHEP 11 (2005) 036 [hep-th/0506126] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  73. [73]
    C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, cs/0004015 [INSPIRE].
  74. [74]
    D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].ADSCrossRefGoogle Scholar
  75. [75]
    D. Maître, Extension of HPL to complex arguments, Comput. Phys. Commun. 183 (2012) 846 [hep-ph/0703052] [INSPIRE].ADSCrossRefGoogle Scholar
  76. [76]
    A. Smirnov and M. Tentyukov, Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA), Comput. Phys. Commun. 180 (2009) 735 [arXiv:0807.4129] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  77. [77]
    A. Smirnov, V. Smirnov and M. Tentyukov, FIESTA 2: parallelizeable multiloop numerical calculations, Comput. Phys. Commun. 182 (2011) 790 [arXiv:0912.0158] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  78. [78]
    O. Schnetz, Graphical functions and single-valued multiple polylogarithms, arXiv:1302.6445 [INSPIRE].
  79. [79]
    D.E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Alg. 58 (1979) 432.MathSciNetCrossRefzbMATHGoogle Scholar
  80. [80]
    A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math.AG/0103059.
  81. [81]
    F. Brown, On the decomposition of motivic multiple zeta values, arXiv:1102.1310 [INSPIRE].
  82. [82]
    C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  83. [83]
    P. Baikov and K. Chetyrkin, Four loop massless propagators: an algebraic evaluation of all master integrals, Nucl. Phys. B 837 (2010) 186 [arXiv:1004.1153] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  84. [84]
    R. Lee, A. Smirnov and V. Smirnov, Master integrals for four-loop massless propagators up to transcendentality weight twelve, Nucl. Phys. B 856 (2012) 95 [arXiv:1108.0732] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  85. [85]
    R. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
  86. [86]
    R. Lee, Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals, JHEP 07 (2008) 031 [arXiv:0804.3008] [INSPIRE].ADSCrossRefGoogle Scholar
  87. [87]
    A. Smirnov and V. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, arXiv:1302.5885 [INSPIRE].
  88. [88]
    S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].ADSCrossRefGoogle Scholar
  89. [89]
    N. Arkani-Hamed et al., Scattering amplitudes and the positive Grassmannian, arXiv:1212.5605 [INSPIRE].
  90. [90]
    A.E. Lipstein and L. Mason, From the holomorphic Wilson loop tod logloop-integrands for super-Yang-Mills amplitudes, JHEP 05 (2013) 106 [arXiv:1212.6228] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA 2013

Authors and Affiliations

  • James Drummond
    • 1
    • 2
  • Claude Duhr
    • 3
    • 4
  • Burkhard Eden
    • 5
  • Paul Heslop
    • 6
    Email author
  • Jeffrey Pennington
    • 7
  • Vladimir A. Smirnov
    • 5
    • 8
  1. 1.PH-TH, CERN, Case C01600Geneva 23Switzerland
  2. 2.LAPTH, CNRS et Université de SavoieAnnecy-le-Vieux CedexFrance
  3. 3.Institut für Theoretische PhysikETH ZürichZürichSwitzerland
  4. 4.Institute for Particle Physics Phenomenology, University of Durham, Science LaboratoriesDurhamU.K.
  5. 5.Institut für MathematikHumboldt-UniversitätBerlinGermany
  6. 6.Dept. of Mathematical Sciences, Durham University, Science LaboratoriesDurhamU.K.
  7. 7.SLAC National Accelerator LaboratoryStanford UniversityStanfordU.S.A.
  8. 8.Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear PhysicsMoscowRussia

Personalised recommendations