Star integrals, convolutions and simplices

  • Dhritiman Nandan
  • Miguel F. Paulos
  • Marcus Spradlin
  • Anastasia Volovich
Open Access
Article

Abstract

We explore single and multi-loop conformal integrals, such as the ones appearing in dual conformal theories in flat space. Using Mellin amplitudes, a large class of higher loop integrals can be written as simple integro-differential operators on star integrals: one-loop n-gon integrals in n dimensions. These are known to be given by volumes of hyperbolic simplices. We explicitly compute the five-dimensional pentagon integral in full generality using Schläfli’s formula. Then, as a first step to understanding higher loops, we use spline technology to construct explicitly the 6d hexagon and 8d octagon integrals in two-dimensional kinematics. The fully massive hexagon and octagon integrals are then related to the double box and triple box integrals respectively. We comment on the classes of functions needed to express these integrals in general kinematics, involving elliptic functions and beyond.

Keywords

Supersymmetric gauge theory Scattering Amplitudes AdS-CFT Correspondence 

References

  1. [1]
    L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills theories, Nucl. Phys. B 121 (1977) 77 [INSPIRE].MathSciNetADSCrossRefGoogle 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]
    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
  4. [4]
    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].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    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].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    J. Drummond and L. Ferro, Yangians, grassmannians and T-duality, JHEP 07 (2010) 027 [arXiv:1001.3348] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    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].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    J. Drummond and L. Ferro, The yangian origin of the grassmannian integral, JHEP 12 (2010) 010 [arXiv:1002.4622] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    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].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    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].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    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].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    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
  13. [13]
    Z. Bern, J. Carrasco, H. Johansson and D. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [INSPIRE].MathSciNetADSGoogle Scholar
  14. [14]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    N. Arkani-Hamed, F. Cachazo and C. Cheung, The grassmannian origin of dual superconformal invariance, JHEP 03 (2010) 036 [arXiv:0909.0483] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov et al., Scattering amplitudes and the positive grassmannian, arXiv:1212.5605 [INSPIRE].
  17. [17]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    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
  19. [19]
    J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    S. Caron-Huot, Notes on the scattering amplitude/Wilson loop duality, JHEP 07 (2011) 058 [arXiv:1010.1167] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    L. 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].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    R. Roiban, M. Spradlin and A. Volovich, Special issue: scattering amplitudes in gauge theories: progress and outlook, J. Phys. A 44 (2011) n. 45.MathSciNetGoogle Scholar
  23. [23]
    S. Caron-Huot, Loops and trees, JHEP 05 (2011) 080 [arXiv:1007.3224] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    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].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    R.H. Boels, On BCFW shifts of integrands and integrals, JHEP 11 (2010) 113 [arXiv:1008.3101] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    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
  27. [27]
    Z. Bern, L. Dixon, D. Kosower, R. Roiban, M. Spradlin 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
  28. [28]
    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
  29. [29]
    V. Del Duca, C. Duhr and V.A. Smirnov, An analytic result for the two-loop hexagon Wilson loop in N = 4 SYM, JHEP 03 (2010) 099 [arXiv:0911.5332] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    V. Del Duca, C. Duhr and V.A. Smirnov, The two-loop hexagon Wilson loop in N = 4 SYM, JHEP 05 (2010) 084 [arXiv:1003.1702] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    A.B. Goncharov, Polylogarithms and motivic Galois groups, Proc. Symp. Pure Math. 55 (1994) 43.MathSciNetCrossRefGoogle Scholar
  32. [32]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497.MathSciNetMATHGoogle Scholar
  33. [33]
    A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math J. 128 (2005) 209 [arXiv:math/0208144].MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    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
  35. [35]
    D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    A. Brandhuber, G. Travaglini and G. Yang, Analytic two-loop form factors in N = 4 SYM, JHEP 05 (2012) 082 [arXiv:1201.4170] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    L.J. Dixon, C. Duhr and J. Pennington, Single-valued harmonic polylogarithms and the multi-Regge limit, JHEP 10 (2012) 074 [arXiv:1207.0186] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    J. Drummond, Generalised ladders and single-valued polylogarithms, JHEP 02 (2013) 092 [arXiv:1207.3824] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    J. Pennington, The six-point remainder function to all loop orders in the multi-Regge limit, JHEP 01 (2013) 059 [arXiv:1209.5357] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    G. Mack, D-independent representation of conformal field theories in D dimensions via transformation to auxiliary dual resonance models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
  46. [46]
    G. Mack, D-dimensional conformal field theories with anomalous dimensions as dual resonance models, Bulg. J. Phys. 36 (2009) 214 [arXiv:0909.1024] [INSPIRE].MathSciNetMATHGoogle Scholar
  47. [47]
    J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    A.L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A natural language for AdS/CFT correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    M.F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074 [arXiv:1107.1504] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    A.L. Fitzpatrick and J. Kaplan, Analyticity and the holographic S-matrix, JHEP 10 (2012) 127 [arXiv:1111.6972] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    D. Nandan, A. Volovich and C. Wen, On Feynman rules for Mellin amplitudes in AdS/CFT, JHEP 05 (2012) 129 [arXiv:1112.0305] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    A.L. Fitzpatrick and J. Kaplan, Unitarity and the holographic S-matrix, JHEP 10 (2012) 032 [arXiv:1112.4845] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    A.L. Fitzpatrick and J. Kaplan, AdS field theory from conformal field theory, JHEP 02 (2013) 054 [arXiv:1208.0337] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    M.F. Paulos, M. Spradlin and A. Volovich, Mellin amplitudes for dual conformal integrals, JHEP 08 (2012) 072 [arXiv:1203.6362] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  55. [55]
    J.M. Drummond, J.M. Henn and J. Trnka, New differential equations for on-shell loop integrals, JHEP 04 (2011) 083 [arXiv:1010.3679] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    L.J. Dixon, J.M. Drummond and J.M. Henn, The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N = 4 SYM, JHEP 06 (2011) 100 [arXiv:1104.2787] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    L. Ferro, Differential equations for multi-loop integrals and two-dimensional kinematics, JHEP 04 (2013) 160 [arXiv:1204.1031] [INSPIRE].CrossRefGoogle Scholar
  58. [58]
    A.I. Davydychev, Some exact results for N point massive Feynman integrals, J. Math. Phys. 32 (1991) 1052 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  59. [59]
    A.I. Davydychev and R. Delbourgo, A geometrical angle on Feynman integrals, J. Math. Phys. 39 (1998) 4299 [hep-th/9709216] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  60. [60]
    L. Mason and D. Skinner, Amplitudes at weak coupling as polytopes in AdS 5, J. Phys. A 44 (2011) 135401 [arXiv:1004.3498] [INSPIRE].MathSciNetADSGoogle Scholar
  61. [61]
    O. Schnetz, The geometry of one-loop amplitudes, arXiv:1010.5334 [INSPIRE].
  62. [62]
    M.F. Paulos, Loops, polytopes and splines, arXiv:1210.0578 [INSPIRE].
  63. [63]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Hodges and J. Trnka, A note on polytopes for scattering amplitudes, JHEP 04 (2012) 081 [arXiv:1012.6030] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  64. [64]
    A. Goncharov, Volumes of hyperbolic manifolds and mixed Tate motives, alg-geom/9601021.
  65. [65]
    M. Spradlin and A. Volovich, Symbols of one-loop integrals from mixed Tate motives, JHEP 11 (2011) 084 [arXiv:1105.2024] [INSPIRE].Google Scholar
  66. [66]
    V. Del Duca, C. Duhr and V.A. Smirnov, The massless hexagon integral in d = 6 dimensions, Phys. Lett. B 703 (2011) 363 [arXiv:1104.2781] [INSPIRE].ADSGoogle Scholar
  67. [67]
    V. Del Duca, C. Duhr and V.A. Smirnov, The one-loop one-mass hexagon integral in d = 6 dimensions, JHEP 07 (2011) 064 [arXiv:1105.1333] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    V. Del Duca, L.J. Dixon, J.M. Drummond, C. Duhr, J.M. Henn et al., The one-loop six-dimensional hexagon integral with three massive corners, Phys. Rev. D 84 (2011) 045017 [arXiv:1105.2011] [INSPIRE].ADSGoogle Scholar
  69. [69]
    S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    P.A.M. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429.MathSciNetCrossRefGoogle Scholar
  72. [72]
    S. Weinberg, Six-dimensional methods for four-dimensional conformal field theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].ADSGoogle Scholar
  73. [73]
    Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  74. [74]
    P. Heslop and V.V. Khoze, Analytic results for MHV Wilson loops, JHEP 11 (2010) 035 [arXiv:1007.1805] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  75. [75]
    L.F. Alday, Some analytic results for two-loop scattering amplitudes, JHEP 07 (2011) 080 [arXiv:1009.1110] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  76. [76]
    P. Heslop and V.V. Khoze, Wilson loops @ 3-loops in special kinematics, JHEP 11 (2011) 152 [arXiv:1109.0058] [INSPIRE].ADSCrossRefGoogle Scholar
  77. [77]
    T. Goddard, P. Heslop and V.V. Khoze, Uplifting amplitudes in special kinematics, JHEP 10 (2012) 041 [arXiv:1205.3448] [INSPIRE].ADSCrossRefGoogle Scholar
  78. [78]
    S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B 704 (2005) 349 [hep-ph/0406160] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  79. [79]
    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, arXiv:0905.1473 [INSPIRE].

Copyright information

© SISSA 2013

Authors and Affiliations

  • Dhritiman Nandan
    • 1
  • Miguel F. Paulos
    • 1
    • 2
  • Marcus Spradlin
    • 1
    • 2
  • Anastasia Volovich
    • 1
    • 2
  1. 1.Department of PhysicsBrown UniversityProvidenceU.S.A
  2. 2.Theory Division, Physics DepartmentCERNGeneva 23Switzerland

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