Journal of High Energy Physics

, 2019:190 | Cite as

Transcendental structure of multiloop massless correlators and anomalous dimensions

  • P.A. Baikov
  • K.G. ChetyrkinEmail author
Open Access
Regular Article - Theoretical Physics


We give a short account of recent advances in our understanding of the π- dependent terms in massless (Euclidean) 2-point functions as well as in generic anomalous dimensions (ADs) and β-functions. We extend the considerations of [1] by two more loops, that is for the case of 6- and 7-loop correlators and 7- and 8-loop renormalization group (RG) functions. Our predictions for the (π-dependent terms) of the 7-loop RG functions for the case of the O(n) 𝜙4 theory are in full agreement with the recent results from [2]. All available 7- and 8-loop results for QCD and the scalar O(n) ϕ4 theory obtained within the large Nf approach to the quantum field theory (see, e.g. [3]) are also in full agreement with our results.


Perturbative QCD Renormalization Group 


Open Access

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  1. [1]
    P.A. Baikov and K.G. Chetyrkin, The structure of generic anomalous dimensions and no-π theorem for massless propagators, JHEP06 (2018) 141 [arXiv:1804.10088] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    O. Schnetz, Numbers and Functions in Quantum Field Theory, Phys. Rev.D 97 (2018) 085018 [arXiv:1606.08598] [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    J.A. Gracey, Large Nf quantum field theory, Int. J. Mod. Phys.A 33 (2019) 1830032 [arXiv:1812.05368] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  4. [4]
    S.G. Gorishnii, A.L. Kataev and S.A. Larin, The O(\( {\alpha}_s^3 \))-corrections to σ tot (e +e → hadrons) and Γ(τ → ντ + hadrons) in QCD, Phys. Lett.B 259 (1991) 144 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    R. Ayoub, Euler and the zeta function, Am. Math. Mon.81 (1974) 1067.MathSciNetCrossRefGoogle Scholar
  6. [6]
    P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Adler Function, Bjorken Sum Rule, and the Crewther Relation to Order \( {\alpha}_s^4 \)in a General Gauge Theory, Phys. Rev. Lett.104 (2010) 132004 [arXiv:1001.3606] [INSPIRE].
  7. [7]
    P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Five-loop fermion anomalous dimension for a general gauge group from four-loop massless propagators, JHEP04 (2017) 119 [arXiv:1702.01458] [INSPIRE].
  8. [8]
    K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, New Approach to Evaluation of Multiloop Feynman Integrals: The Gegenbauer Polynomial x Space Technique, Nucl. Phys.B 174 (1980) 345 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    M. Jamin and R. Miravitllas, Absence of even-integer ζ -function values in Euclidean physical quantities in QCD, Phys. Lett.B 779 (2018) 452 [arXiv:1711.00787] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    J. Davies and A. Vogt, Absence of π 2terms in physical anomalous dimensions in DIS: Verification and resulting predictions, Phys. Lett.B 776 (2018) 189 [arXiv:1711.05267] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    K.G. Chetyrkin, G. Falcioni, F. Herzog and J.A.M. Vermaseren, Five-loop renormalisation of QCD in covariant gauges, JHEP10 (2017) 179 [arXiv:1709.08541] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    B. Ruijl, F. Herzog, T. Ueda, J.A.M. Vermaseren and A. Vogt, The R -operation and five-loop calculations, PoS(RADCOR2017)011 [arXiv:1801.06084] [INSPIRE].
  13. [13]
    F. Herzog, S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, Five-loop contributions to low-N non-singlet anomalous dimensions in QCD, Phys. Lett.B 790 (2019) 436 [arXiv:1812.11818] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Five-Loop Running of the QCD coupling constant, Phys. Rev. Lett.118 (2017) 082002 [arXiv:1606.08659] [INSPIRE].
  15. [15]
    F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, The five-loop β-function of Yang-Mills theory with fermions, JHEP02 (2017) 090 [arXiv:1701.01404] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    T. Luthe, A. Maier, P. Marquard and Y. Schröder, The five-loop Beta function for a general gauge group and anomalous dimensions beyond Feynman gauge, JHEP10 (2017) 166 [arXiv:1709.07718] [INSPIRE].
  17. [17]
    P.A. Baikov and K.G. Chetyrkin, Four Loop Massless Propagators: An Algebraic Evaluation of All Master Integrals, Nucl. Phys.B 837 (2010) 186 [arXiv:1004.1153] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    R.N. Lee, A.V. Smirnov and V.A. Smirnov, Master Integrals for Four-Loop Massless Propagators up to Transcendentality Weight Twelve, Nucl. Phys.B 856 (2012) 95 [arXiv:1108.0732] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    E. Panzer, On the analytic computation of massless propagators in dimensional regularization, Nucl. Phys.B 874 (2013) 567 [arXiv:1305.2161] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Georgoudis, V. Goncalves, E. Panzer and R. Pereira, Five-loop massless propagator integrals, arXiv:1802.00803 [INSPIRE].
  21. [21]
    F. Brown, Feynman amplitudes, coaction principle, and cosmic Galois group, Commun. Num. Theor. Phys.11 (2017) 453 [arXiv:1512.06409] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  22. [22]
    E. Panzer and O. Schnetz, The Galois coaction on 𝜙4periods, Commun. Num. Theor. Phys.11 (2017) 657 [arXiv:1603.04289] [INSPIRE].CrossRefGoogle Scholar
  23. [23]
    D.J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via feynman diagrams up to 9 loops, Phys. Lett.B 393 (1997) 403 [hep-th/9609128] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    D. Broadhurst, Multiple Zeta Values and Modular Forms in Quantum Field Theory, Springer Vienna, Vienna, Austria (2013), pp. 33–73.zbMATHGoogle Scholar
  25. [25]
    P.A. Baikov and K.G. Chetyrkin, No-π Theorem for Euclidean Massless Correlators, PoS(LL2018)008 [arXiv:1808.00237] [INSPIRE].
  26. [26]
    D.J. Broadhurst, Dimensionally continued multiloop gauge theory, hep-th/9909185 [INSPIRE].
  27. [27]
    J. Blumlein, D.J. Broadhurst and J.A.M. Vermaseren, The Multiple Zeta Value Data Mine, Comput. Phys. Commun.181 (2010) 582 [arXiv:0907.2557] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    D.J. Broadhurst and D. Kreimer, Knots and numbers in ϕ 4theory to 7 loops and beyond, Int. J. Mod. Phys.C 6 (1995) 519 [hep-ph/9504352] [INSPIRE].
  29. [29]
    E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. thesis, Humboldt University, 2015, [arXiv:1506.07243] [INSPIRE].
  30. [30]
    J.A. Gracey, The QCD Beta function at 𝒪 (1/Nf), Phys. Lett.B 373 (1996) 178 [hep-ph/9602214] [INSPIRE].
  31. [31]
    M. Ciuchini, S.E. Derkachov, J.A. Gracey and A.N. Manashov, Quark mass anomalous dimension at 𝒪 \( \left(1/{N}_f^2\right) \)in QCD, Phys. Lett.B 458 (1999) 117 [hep-ph/9903410] [INSPIRE].
  32. [32]
    M. Ciuchini, S.E. Derkachov, J.A. Gracey and A.N. Manashov, Computation of quark mass anomalous dimension at 𝒪 \( \left(1/{N}_f^2\right) \)in quantum chromodynamics, Nucl. Phys.B 579 (2000) 56 [hep-ph/9912221] [INSPIRE].
  33. [33]
    A.N. Vasiliev, Yu.M. Pismak and Yu.R. Khonkonen, Simple Method of Calculating the Critical Indices in the 1/N Expansion, Theor. Math. Phys.46 (1981) 104 [INSPIRE].CrossRefGoogle Scholar
  34. [34]
    A.N. Vasiliev, Yu.M. Pismak and Yu.R. Khonkonen, 1/N Expansion: Calculation of the Exponents η and ν in the Order 1/N 2for Arbitrary Number of Dimensions, Theor. Math. Phys.47 (1981) 465 [INSPIRE].CrossRefGoogle Scholar
  35. [35]
    A.N. Vasiliev, Yu.M. Pismak and Yu.R. Khonkonen, 1/N Expansion: Calculation Of the Exponent η in the Order 1/N 3by the Conformal Bootstrap Method, Theor. Math. Phys.50 (1982) 127 [INSPIRE].CrossRefGoogle Scholar
  36. [36]
    D.J. Broadhurst, J.A. Gracey and D. Kreimer, Beyond the triangle and uniqueness relations: Nonzeta counterterms at large N from positive knots, Z. Phys.C 75 (1997) 559 [hep-th/9607174] [INSPIRE].Google Scholar
  37. [37]
    A.V. Kotikov and S. Teber, On the Landau-Khalatnikov-Fradkin transformation and the mystery of even ζ -values in Euclidean massless correlators, arXiv:1906.10930 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Skobeltsyn Institute of Nuclear PhysicsLomonosov Moscow State UniversityMoscowRussian Federation
  2. 2.Institut für Theoretische TeilchenphysikKarlsruhe Institute of Technology (KIT)KarlsruheGermany
  3. 3.II Institut für Theoretische PhysikUniversität HamburgHamburgGermany

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