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Journal of High Energy Physics

, Volume 2015, Issue 12, pp 1–22 | Cite as

Spin-2 form factors at three loop in QCD

  • Taushif Ahmed
  • Goutam Das
  • Prakash Mathews
  • Narayan RanaEmail author
  • V. Ravindran
Open Access
Regular Article - Theoretical Physics

Abstract

Spin-2 fields are often candidates in physics beyond the Standard Model namely the models with extra-dimensions where spin-2 Kaluza-Klein gravitons couple to the fields of the Standard Model. Also, in the context of Higgs searches, spin-2 fields have been studied as an alternative to the scalar Higgs boson. In this article, we present the complete three loop QCD radiative corrections to the spin-2 quark-antiquark and spin-2 gluon-gluon form factors in SU(N) gauge theory with n f light flavors. These form factors contribute to both quark-antiquark and gluon-gluon initiated processes involving spin-2 particle in the hadronic reactions at the LHC. We have studied the structure of infrared singularities in these form factors up to three loop level using Sudakov integro-differential equation and found that the anomalous dimensions originating from soft and collinear regions of the loop integrals coincide with those of the electroweak vector boson and Higgs form factors confirming the universality of the infrared singularities in QCD amplitudes.

Keywords

NLO Computations 

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]
    ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE].
  2. [2]
    CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
  3. [3]
    J. Ellis, V. Sanz and T. You, Prima Facie Evidence against Spin-Two Higgs Impostors, Phys. Lett. B 726 (2013) 244 [arXiv:1211.3068] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, The Hierarchy problem and new dimensions at a millimeter, Phys. Lett. B 429 (1998) 263 [hep-ph/9803315] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, New dimensions at a millimeter to a Fermi and superstrings at a TeV, Phys. Lett. B 436 (1998) 257 [hep-ph/9804398] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  6. [6]
    N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity, Phys. Rev. D 59 (1999) 086004 [hep-ph/9807344] [INSPIRE].ADSGoogle Scholar
  7. [7]
    L. Randall and R. Sundrum, A Large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370 [hep-ph/9905221] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    R. Fok, C. Guimaraes, R. Lewis and V. Sanz, It is a Graviton! or maybe not, JHEP 12 (2012) 062 [arXiv:1203.2917] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    R. Hamberg, W.L. van Neerven and T. Matsuura, A Complete calculation of the order α s2 correction to the Drell-Yan K factor, Nucl. Phys. B 359 (1991) 343 [Erratum ibid. B 644 (2002) 403] [INSPIRE].
  10. [10]
    T. Ahmed, M. Mahakhud, N. Rana and V. Ravindran, Drell-Yan Production at Threshold to Third Order in QCD, Phys. Rev. Lett. 113 (2014) 112002 [arXiv:1404.0366] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, Threshold resummation at N 3 LL accuracy and soft-virtual cross sections at N 3 LO, Nucl. Phys. B 888 (2014) 75 [arXiv:1405.4827] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  12. [12]
    R.V. Harlander and W.B. Kilgore, Next-to-next-to-leading order Higgs production at hadron colliders, Phys. Rev. Lett. 88 (2002) 201801 [hep-ph/0201206] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    V. Ravindran, J. Smith and W.L. van Neerven, NNLO corrections to the total cross-section for Higgs boson production in hadron hadron collisions, Nucl. Phys. B 665 (2003) 325 [hep-ph/0302135] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    C. Anastasiou et al., Higgs boson gluon-fusion production at threshold in N 3 LO QCD, Phys. Lett. B 737 (2014) 325 [arXiv:1403.4616] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    Y. Li, A. von Manteuffel, R.M. Schabinger and H.X. Zhu, N 3 LO Higgs boson and Drell-Yan production at threshold: The one-loop two-emission contribution, Phys. Rev. D 90 (2014) 053006 [arXiv:1404.5839] [INSPIRE].ADSGoogle Scholar
  17. [17]
    D. de Florian, J. Mazzitelli, S. Moch and A. Vogt, Approximate N 3 LO Higgs-boson production cross section using physical-kernel constraints, JHEP 10 (2014) 176 [arXiv:1408.6277] [INSPIRE].ADSGoogle Scholar
  18. [18]
    C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, Higgs Boson Gluon-Fusion Production in QCD at Three Loops, Phys. Rev. Lett. 114 (2015) 212001 [arXiv:1503.06056] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    R.V. Harlander and W.B. Kilgore, Higgs boson production in bottom quark fusion at next-to-next-to leading order, Phys. Rev. D 68 (2003) 013001 [hep-ph/0304035] [INSPIRE].ADSGoogle Scholar
  20. [20]
    T. Ahmed, N. Rana and V. Ravindran, Higgs boson production through bb annihilation at threshold in N 3 LO QCD, JHEP 10 (2014) 139 [arXiv:1408.0787] [INSPIRE].ADSGoogle Scholar
  21. [21]
    O. Brein, A. Djouadi and R. Harlander, NNLO QCD corrections to the Higgs-strahlung processes at hadron colliders, Phys. Lett. B 579 (2004) 149 [hep-ph/0307206] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    M.C. Kumar, M.K. Mandal and V. Ravindran, Associated production of Higgs boson with vector boson at threshold N 3 LO in QCD, JHEP 03 (2015) 037 [arXiv:1412.3357] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    D. de Florian, M. Mahakhud, P. Mathews, J. Mazzitelli and V. Ravindran, Quark and gluon spin-2 form factors to two-loops in QCD, JHEP 02 (2014) 035 [arXiv:1312.6528] [INSPIRE].CrossRefGoogle Scholar
  24. [24]
    S. Moch, J.A.M. Vermaseren and A. Vogt, Three-loop results for quark and gluon form-factors, Phys. Lett. B 625 (2005) 245 [hep-ph/0508055] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    S. Moch, J.A.M. Vermaseren and A. Vogt, The Quark form-factor at higher orders, JHEP 08 (2005) 049 [hep-ph/0507039] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Quark and gluon form factors to three loops, Phys. Rev. Lett. 102 (2009) 212002 [arXiv:0902.3519] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, Calculation of the quark and gluon form factors to three loops in QCD, JHEP 06 (2010) 094 [arXiv:1004.3653] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  28. [28]
    T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, The quark and gluon form factors to three loops in QCD through to O(epsˆ2), JHEP 11 (2010) 102 [arXiv:1010.4478] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  29. [29]
    T. Gehrmann and D. Kara, The Hbb form factor to three loops in QCD, JHEP 09 (2014) 174 [arXiv:1407.8114] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    ATLAS collaboration, Search for high-mass dilepton resonances in pp collisions at \( \sqrt{s}=8 \) TeV with the ATLAS detector, Phys. Rev. D 90 (2014) 052005 [arXiv:1405.4123] [INSPIRE].
  31. [31]
    CMS collaboration, Search for heavy narrow dilepton resonances in pp collisions at \( \sqrt{s}=7 \) TeV and \( \sqrt{s}=8 \) TeV, Phys. Lett. B 720 (2013) 63 [arXiv:1212.6175] [INSPIRE].
  32. [32]
    ATLAS collaboration, Search for high-mass diphoton resonances in pp collisions at \( \sqrt{s}=8 \) TeV with the ATLAS detector, Phys. Rev. D 92 (2015) 032004 [arXiv:1504.05511] [INSPIRE].
  33. [33]
    J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., The automated computation of tree-level and next-to-leading order differential cross sections and their matching to parton shower simulations, JHEP 07 (2014) 079 [arXiv:1405.0301] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    P. Mathews, V. Ravindran, K. Sridhar and W.L. van Neerven, Next-to-leading order QCD corrections to the Drell-Yan cross section in models of TeV-scale gravity, Nucl. Phys. B 713 (2005) 333 [hep-ph/0411018] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    P. Mathews and V. Ravindran, Angular distribution of Drell-Yan process at hadron colliders to NLO-QCD in models of TeV scale gravity, Nucl. Phys. B 753 (2006) 1 [hep-ph/0507250] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    M.C. Kumar, P. Mathews and V. Ravindran, PDF and scale uncertainties of various DY distributions in ADD and RS models at hadron colliders, Eur. Phys. J. C 49 (2007) 599 [hep-ph/0604135] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    R. Frederix, M.K. Mandal, P. Mathews, V. Ravindran and S. Seth, Drell-Yan, ZZ, W + W production in SM & ADD model to NLO+PS accuracy at the LHC, Eur. Phys. J. C 74 (2014) 2745 [arXiv:1307.7013] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    M.C. Kumar, P. Mathews, V. Ravindran and A. Tripathi, Direct photon pair production at the LHC to order α s in TeV scale gravity models, Nucl. Phys. B 818 (2009) 28 [arXiv:0902.4894] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  39. [39]
    R. Frederix, M.K. Mandal, P. Mathews, V. Ravindran, S. Seth, P. Torrielli et al., Diphoton production in the ADD model to NLO+parton shower accuracy at the LHC, JHEP 12 (2012) 102 [arXiv:1209.6527] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    N. Agarwal, V. Ravindran, V.K. Tiwari and A. Tripathi, Next-to-leading order QCD corrections to the Z boson pair production at the LHC in Randall Sundrum model, Phys. Lett. B 686 (2010) 244 [arXiv:0910.1551] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    N. Agarwal, V. Ravindran, V.K. Tiwari and A. Tripathi, W + W production in Large extra dimension model at next-to-leading order in QCD at the LHC, Phys. Rev. D 82 (2010) 036001 [arXiv:1003.5450] [INSPIRE].ADSGoogle Scholar
  42. [42]
    G. Das, P. Mathews, V. Ravindran and S. Seth, RS resonance in di-final state production at the LHC to NLO+PS accuracy, JHEP 10 (2014) 188 [arXiv:1408.3970] [INSPIRE].ADSGoogle Scholar
  43. [43]
    D. de Florian, M. Mahakhud, P. Mathews, J. Mazzitelli and V. Ravindran, Next-to-Next-to-Leading Order QCD Corrections in Models of TeV-Scale Gravity, JHEP 04 (2014) 028 [arXiv:1312.7173] [INSPIRE].CrossRefGoogle Scholar
  44. [44]
    T. Ahmed, M. Mahakhud, P. Mathews, N. Rana and V. Ravindran, Two-Loop QCD Correction to massive spin-2 resonance3 gluons, JHEP 05 (2014) 107 [arXiv:1404.0028] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    S. Catani, The Singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  47. [47]
    T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [arXiv:0901.0722] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    N.K. Nielsen, The Energy Momentum Tensor in a Nonabelian Quark Gluon Theory, Nucl. Phys. B 120 (1977) 212 [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    M.F. Zoller and K.G. Chetyrkin, OPE of the energy-momentum tensor correlator in massless QCD, JHEP 12 (2012) 119 [arXiv:1209.1516] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    P. Mathews, V. Ravindran and K. Sridhar, NLO - QCD corrections to e + e hadrons in models of TeV-scale gravity, JHEP 08 (2004) 048 [hep-ph/0405292] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
  54. [54]
    J.A.M. Vermaseren, The FORM project, Nucl. Phys. Proc. Suppl. 183 (2008) 19 [arXiv:0806.4080] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    A. von Manteuffel and C. Studerus, Reduze 2 - Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].
  56. [56]
    F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. 100B (1981) 65.ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations, JHEP 07 (2004) 046 [hep-ph/0404258] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    A.V. Smirnov, Algorithm FIRE - Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  61. [61]
    C. Studerus, Reduze-Feynman Integral Reduction in C++, Comput. Phys. Commun. 181 (2010) 1293 [arXiv:0912.2546] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
  63. [63]
    R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].CrossRefGoogle Scholar
  64. [64]
    R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP 11 (2013) 165 [arXiv:1308.6676] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    S.G. Gorishnii, S.A. Larin, L.R. Surguladze and F.V. Tkachov, Mincer: Program for Multiloop Calculations in Quantum Field Theory for the Schoonschip System, Comput. Phys. Commun. 55 (1989) 381 [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    S. Bekavac, Calculation of massless Feynman integrals using harmonic sums, Comput. Phys. Commun. 175 (2006) 180 [hep-ph/0505174] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  67. [67]
    T. Gehrmann, T. Huber and D. Maître, Two-loop quark and gluon form-factors in dimensional regularisation, Phys. Lett. B 622 (2005) 295 [hep-ph/0507061] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    T. Gehrmann, G. Heinrich, T. Huber and C. Studerus, Master integrals for massless three-loop form-factors: One-loop and two-loop insertions, Phys. Lett. B 640 (2006) 252 [hep-ph/0607185] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    G. Heinrich, T. Huber and D. Maître, Master integrals for fermionic contributions to massless three-loop form-factors, Phys. Lett. B 662 (2008) 344 [arXiv:0711.3590] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    G. Heinrich, T. Huber, D.A. Kosower and V.A. Smirnov, Nine-Propagator Master Integrals for Massless Three-Loop Form Factors, Phys. Lett. B 678 (2009) 359 [arXiv:0902.3512] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    R.N. Lee, A.V. Smirnov and V.A. Smirnov, Analytic Results for Massless Three-Loop Form Factors, JHEP 04 (2010) 020 [arXiv:1001.2887] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    V. Ravindran, J. Smith and W.L. van Neerven, Two-loop corrections to Higgs boson production, Nucl. Phys. B 704 (2005) 332 [hep-ph/0408315] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  73. [73]
    S.M. Aybat, L.J. Dixon and G.F. Sterman, The Two-loop anomalous dimension matrix for soft gluon exchange, Phys. Rev. Lett. 97 (2006) 072001 [hep-ph/0606254] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  74. [74]
    S.M. Aybat, L.J. Dixon and G.F. Sterman, The Two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [INSPIRE].ADSGoogle Scholar
  75. [75]
    V.V. Sudakov, Vertex parts at very high-energies in quantum electrodynamics, Sov. Phys. JETP 3 (1956) 65 [INSPIRE].MathSciNetzbMATHGoogle Scholar
  76. [76]
    A.H. Mueller, On the Asymptotic Behavior of the Sudakov Form-factor, Phys. Rev. D 20 (1979) 2037 [INSPIRE].ADSGoogle Scholar
  77. [77]
    J.C. Collins, Algorithm to Compute Corrections to the Sudakov Form-factor, Phys. Rev. D 22 (1980) 1478 [INSPIRE].ADSGoogle Scholar
  78. [78]
    A. Sen, Asymptotic Behavior of the Sudakov Form-Factor in QCD, Phys. Rev. D 24 (1981) 3281 [INSPIRE].ADSGoogle Scholar
  79. [79]
    V. Ravindran, On Sudakov and soft resummations in QCD, Nucl. Phys. B 746 (2006) 58 [hep-ph/0512249] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  80. [80]
    V. Ravindran, Higher-order threshold effects to inclusive processes in QCD, Nucl. Phys. B 752 (2006) 173 [hep-ph/0603041] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    S. Moch, J.A.M. Vermaseren and A. Vogt, The Three loop splitting functions in QCD: The Nonsinglet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  82. [82]
    A. Vogt, S. Moch and J.A.M. Vermaseren, The Three-loop splitting functions in QCD: The Singlet case, Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  83. [83]
    T. Gehrmann, J.M. Henn and T. Huber, The three-loop form factor in N = 4 super Yang-Mills, JHEP 03 (2012) 101 [arXiv:1112.4524] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  84. [84]
    A.V. Kotikov and L.N. Lipatov, On the highest transcendentality in N = 4 SUSY, Nucl. Phys. B 769 (2007) 217 [hep-th/0611204] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  85. [85]
    R.N. Lee and V.A. Smirnov, Analytic ϵ-expansions of Master Integrals Corresponding to Massless Three-Loop Form Factors and Three-Loop g-2 up to Four-Loop Transcendentality Weight, JHEP 02 (2011) 102 [arXiv:1010.1334] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  86. [86]
    A.V. Kotikov, L.N. Lipatov, A.I. Onishchenko and V.N. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett. B 595 (2004) 521 [Erratum ibid. B 632 (2006) 754] [hep-th/0404092] [INSPIRE].
  87. [87]
    A.V. Kotikov and L.N. Lipatov, DGLAP and BFKL evolution equations in the N = 4 supersymmetric gauge theory, hep-ph/0112346 [INSPIRE].

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Taushif Ahmed
    • 1
  • Goutam Das
    • 2
  • Prakash Mathews
    • 2
  • Narayan Rana
    • 1
    Email author
  • V. Ravindran
    • 1
  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.Saha Institute of Nuclear PhysicsKolkataIndia

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