The analytic structure and the transcendental weight of the BFKL ladder at NLL accuracy

  • Vittorio Del Duca
  • Claude Duhr
  • Robin Marzucca
  • Bram VerbeekEmail author
Open Access
Regular Article - Theoretical Physics


We study some analytic properties of the BFKL ladder at next-to-leading logarithmic accuracy (NLLA). We use a procedure by Chirilli and Kovchegov to construct the NLO eigenfunctions, and we show that the BFKL ladder can be evaluated order by order in the coupling in terms of certain generalised single-valued multiple polylogarithms recently introduced by Schnetz. We develop techniques to evaluate the BFKL ladder at any loop order, and we present explicit results up to five loops. Using the freedom in defining the matter content of the NLO BFKL eigenvalue, we obtain conditions for the BFKL ladder in momentum space at NLLA to have maximal transcendental weight. We observe that, unlike in moment space, the result in momentum space in \( \mathcal{N} \) = 4 SYM is not identical to the maximal weight part of QCD, and moreover that there is no gauge theory with this property. We classify the theories for which the BFKL ladder at NLLA has maximal weight in terms of their field content, and we find that these theories are highly constrained: there are precisely four classes of theories with this property involving only fundamental and adjoint matter, all of which have a vanishing one-loop beta function and a matter content that fits into supersymmetric multiplets. Our findings indicate that theories which have maximal weight are highly constrained and point to the possibility that there is a connection between maximal transcendental weight and superconformal symmetry.


Perturbative QCD Scattering Amplitudes Supersymmetric Gauge Theory 


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.


  1. [1]
    V.S. Fadin, E.A. Kuraev and L.N. Lipatov, On the Pomeranchuk Singularity in Asymptotically Free Theories, Phys. Lett. 60B (1975) 50 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Multi-Reggeon Processes in the Yang-Mills Theory, Sov. Phys. JETP 44 (1976) 443 [INSPIRE].ADSGoogle Scholar
  3. [3]
    E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk Singularity in Nonabelian Gauge Theories, Sov. Phys. JETP 45 (1977) 199 [Zh. Eksp. Teor. Fiz. 72 (1977) 377] [INSPIRE].
  4. [4]
    I.I. Balitsky and L.N. Lipatov, The Pomeranchuk Singularity in Quantum Chromodynamics, Sov. J. Nucl. Phys. 28 (1978) 822 [INSPIRE].Google Scholar
  5. [5]
    V.S. Fadin and L.N. Lipatov, BFKL Pomeron in the next-to-leading approximation, Phys. Lett. B 429 (1998) 127 [hep-ph/9802290] [INSPIRE].
  6. [6]
    M. Ciafaloni and G. Camici, Energy scale(s) and next-to-leading BFKL equation, Phys. Lett. B 430 (1998) 349 [hep-ph/9803389] [INSPIRE].
  7. [7]
    G.A. Chirilli and Y.V. Kovchegov, Solution of the NLO BFKL Equation and a Strategy for Solving the All-Order BFKL Equation, JHEP 06 (2013) 055 [arXiv:1305.1924] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    G.A. Chirilli and Y.V. Kovchegov, γγ Cross Section at NLO and Properties of the BFKL Evolution at Higher Orders, JHEP 05 (2014) 099 [Erratum ibid. 1508 (2015) 075] [arXiv:1403.3384] [INSPIRE].
  9. [9]
    V. Del Duca, L.J. Dixon, C. Duhr and J. Pennington, The BFKL equation, Mueller-Navelet jets and single-valued harmonic polylogarithms, JHEP 02 (2014) 086 [arXiv:1309.6647] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    F.C.S. Brown, Single-valued multiple polylogarithms in one variable, C. R. Acad. Sci. Paris Ser. I 338 (2004) 527.CrossRefzbMATHGoogle Scholar
  11. [11]
    O. Schnetz, Numbers and Functions in Quantum Field Theory, arXiv:1606.08598 [INSPIRE].
  12. [12]
    A.V. Kotikov and L.N. Lipatov, NLO corrections to the BFKL equation in QCD and in supersymmetric gauge theories, Nucl. Phys. B 582 (2000) 19 [hep-ph/0004008] [INSPIRE].
  13. [13]
    A.V. Kotikov and L.N. Lipatov, DGLAP and BFKL equations in the N = 4 supersymmetric gauge theory, Nucl. Phys. B 661 (2003) 19 [Erratum ibid. B 685 (2004) 405] [hep-ph/0208220] [INSPIRE].
  14. [14]
    A.V. Kotikov, L.N. Lipatov and V.N. Velizhanin, Anomalous dimensions of Wilson operators in N = 4 SYM theory, Phys. Lett. B 557 (2003) 114 [hep-ph/0301021] [INSPIRE].
  15. [15]
    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].
  16. [16]
    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].
  17. [17]
    L.N. Lipatov, The Bare Pomeron in Quantum Chromodynamics, Sov. Phys. JETP 63 (1986) 904 [Zh. Eksp. Teor. Fiz. 90 (1986) 1536] [INSPIRE].
  18. [18]
    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
  19. [19]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
  20. [20]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076].MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059 [INSPIRE].
  22. [22]
    F.C.S. Brown, Single-valued hyperlogarithms and unipotent differential equations,∼brown/RHpaper5.pdf.
  23. [23]
    F. Brown, Single-valued Motivic Periods and Multiple Zeta Values, SIGMA 2 (2014) e25 [arXiv:1309.5309] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  24. [24]
    F.C.S. Brown, Notes on motivic periods, arXiv:1512.06410.
  25. [25]
    V. Del Duca et al., Multi-Regge kinematics and the moduli space of Riemann spheres with marked points, JHEP 08 (2016) 152 [arXiv:1606.08807] [INSPIRE].ADSMathSciNetCrossRefGoogle 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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    J.A.M. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14 (1999) 2037 [hep-ph/9806280] [INSPIRE].
  28. [28]
    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].
  29. [29]
    S. Weinzierl, Expansion around half integer values, binomial sums and inverse binomial sums, J. Math. Phys. 45 (2004) 2656 [hep-ph/0402131] [INSPIRE].
  30. [30]
    O. Schnetz, Graphical functions and single-valued multiple polylogarithms, Commun. Num. Theor. Phys. 08 (2014) 589 [arXiv:1302.6445] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    A. Gadde, E. Pomoni and L. Rastelli, The Veneziano Limit of N = 2 Superconformal QCD: Towards the String Dual of N = 2 SU(N (c)) SYM with N (f ) = 2N (c), arXiv:0912.4918 [INSPIRE].
  32. [32]
    N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to five gluon amplitudes, Phys. Rev. Lett. 70 (1993) 2677 [hep-ph/9302280] [INSPIRE].
  34. [34]
    Z. Bern and A.G. Morgan, Supersymmetry relations between contributions to one loop gauge boson amplitudes, Phys. Rev. D 49 (1994) 6155 [hep-ph/9312218] [INSPIRE].
  35. [35]
    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].
  36. [36]
    L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].
  37. [37]
    S. Catani and M. Grazzini, The soft gluon current at one loop order, Nucl. Phys. B 591 (2000) 435 [hep-ph/0007142] [INSPIRE].
  38. [38]
    C. Duhr and T. Gehrmann, The two-loop soft current in dimensional regularization, Phys. Lett. B 727 (2013) 452 [arXiv:1309.4393] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  39. [39]
    Y. Li and H.X. Zhu, Single soft gluon emission at two loops, JHEP 11 (2013) 080 [arXiv:1309.4391] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    A. Grozin, J.M. Henn, G.P. Korchemsky and P. Marquard, Three Loop Cusp Anomalous Dimension in QCD, Phys. Rev. Lett. 114 (2015) 062006 [arXiv:1409.0023] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    A. Grozin, J.M. Henn, G.P. Korchemsky and P. Marquard, The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions, JHEP 01 (2016) 140 [arXiv:1510.07803] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    W.E. Caswell, Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order, Phys. Rev. Lett. 33 (1974) 244 [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    D.R.T. Jones, Two Loop Diagrams in Yang-Mills Theory, Nucl. Phys. B 75 (1974) 531 [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    O.V. Tarasov, A.A. Vladimirov and A. Yu. Zharkov, The Gell-Mann-Low Function of QCD in the Three Loop Approximation, Phys. Lett. 93B (1980) 429 [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    S.A. Larin and J.A.M. Vermaseren, The Three loop QCD β-function and anomalous dimensions, Phys. Lett. B 303 (1993) 334 [hep-ph/9302208] [INSPIRE].
  47. [47]
    T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The Four loop β-function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390] [INSPIRE].
  48. [48]
    M. Czakon, The Four-loop QCD β-function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261] [INSPIRE].
  49. [49]
    F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, The five-loop β-function of Yang-Mills theory with fermions, JHEP 02 (2017) 090 [arXiv:1701.01404] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    R. Andree and D. Young, Wilson Loops in N = 2 Superconformal Yang-Mills Theory, JHEP 09 (2010) 095 [arXiv:1007.4923] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    M. Leoni, A. Mauri and A. Santambrogio, Four-point amplitudes in \( \mathcal{N} \) = 2 SCQCD, JHEP 09 (2014) 017 [Erratum ibid. 1502 (2015) 022] [arXiv:1406.7283] [INSPIRE].
  52. [52]
    M. Leoni, A. Mauri and A. Santambrogio, On the amplitude/Wilson loop duality in N = 2 SCQCD, Phys. Lett. B 747 (2015) 325 [arXiv:1502.07614] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  53. [53]
    L.N. Lipatov, Reggeization of the Vector Meson and the Vacuum Singularity in Nonabelian Gauge Theories, Sov. J. Nucl. Phys. 23 (1976) 338 [Yad. Fiz. 23 (1976) 642] [INSPIRE].
  54. [54]
    I.I. Balitsky, L.N. Lipatov and V.S. Fadin, Regge processes in nonabelian gauge theories (in Russian).Google Scholar
  55. [55]
    V.S. Fadin, R. Fiore, M.G. Kozlov and A.V. Reznichenko, Proof of the multi-Regge form of QCD amplitudes with gluon exchanges in the NLA, Phys. Lett. B 639 (2006) 74 [hep-ph/0602006] [INSPIRE].
  56. [56]
    M.G. Kozlov, A.V. Reznichenko and V.S. Fadin, Check of the gluon-reggeization condition in the next-to-leading order: Quark part, Phys. Atom. Nucl. 74 (2011) 758 [Yad. Fiz. 74 (2011) 784] [INSPIRE].
  57. [57]
    M.G. Kozlov, A.V. Reznichenko and V.S. Fadin, Check of the gluon-Reggeization condition in the next-to-leading order: Gluon part, Phys. Atom. Nucl. 75 (2012) 493 [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    M.G. Kozlov, A.V. Reznichenko and V.S. Fadin, Impact factor for gluon production in multi-Regge kinematics in the next-to-leading order, Phys. Atom. Nucl. 75 (2012) 850 [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    V.S. Fadin, M.G. Kozlov and A.V. Reznichenko, Gluon Reggeization in Yang-Mills Theories, Phys. Rev. D 92 (2015) 085044 [arXiv:1507.00823] [INSPIRE].ADSMathSciNetGoogle Scholar
  60. [60]
    V. Del Duca and E.W.N. Glover, The High-energy limit of QCD at two loops, JHEP 10 (2001) 035 [hep-ph/0109028] [INSPIRE].
  61. [61]
    V. Del Duca, C. Duhr, E. Gardi, L. Magnea and C.D. White, An infrared approach to Reggeization, Phys. Rev. D 85 (2012) 071104 [arXiv:1108.5947] [INSPIRE].ADSGoogle Scholar
  62. [62]
    V. Del Duca, C. Duhr, E. Gardi, L. Magnea and C.D. White, The Infrared structure of gauge theory amplitudes in the high-energy limit, JHEP 12 (2011) 021 [arXiv:1109.3581] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    V. Del Duca, G. Falcioni, L. Magnea and L. Vernazza, High-energy QCD amplitudes at two loops and beyond, Phys. Lett. B 732 (2014) 233 [arXiv:1311.0304] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  64. [64]
    V. Del Duca, G. Falcioni, L. Magnea and L. Vernazza, Analyzing high-energy factorization beyond next-to-leading logarithmic accuracy, JHEP 02 (2015) 029 [arXiv:1409.8330] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    J.M. Henn and B. Mistlberger, Four-Gluon Scattering at Three Loops, Infrared Structure and the Regge Limit, Phys. Rev. Lett. 117 (2016) 171601 [arXiv:1608.00850] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    V.S. Fadin, Particularities of the NNLLA BFKL, AIP Conf. Proc. 1819 (2017) 060003 [arXiv:1612.04481] [INSPIRE].CrossRefGoogle Scholar
  67. [67]
    S. Caron-Huot, E. Gardi and L. Vernazza, Two-parton scattering in the high-energy limit, JHEP 06 (2017) 016 [arXiv:1701.05241] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsETH ZürichZürichSwitzerland
  2. 2.Theoretical Physics DepartmentCERNGeneva 23Switzerland
  3. 3.Center for Cosmology, Particle Physics and Phenomenology (CP3)Université catholique de LouvainLouvain-La-NeuveBelgium

Personalised recommendations