Conformal Methods for Massless Feynman Integrals and Large NfMethods

  • John A. Gracey
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)


We review the large N method of calculating high order information on the renormalization group functions in a quantum field theory which is based on conformal integration methods. As an example these techniques are applied to a typical graph contributing to the β-function of O(N) ϕ4 theory at O(1∕N 2). The possible future directions for the large N methods are discussed in light of the development of more recent techniques such as the Laporta algorithm.



It is with pleasure that I thank the organizers of the meeting for permission to include this article in these proceedings. It is based on a talk presented at Quantum Field Theory, Periods and Polylogarithms III, Humboldt University, Berlin in June 2012 which was also in honour of Dr D.J. Broadhurst’s 65th birthday. The Axodraw package, [24], was used to draw the figures in the article.


  1. 1.
    Vasil’ev, A.N., Pismak, Y.M., Honkonen, J.R.: Simple method of calculating the critical indices in the 1∕N expansion. Theor. Math. Phys. 46, 104–113 (1981)CrossRefGoogle Scholar
  2. 2.
    Vasil’ev, A.N., Pismak, Y.M., Honkonen, J.R.: 1∕N expansion: calculation of the exponents η and ν in the order 1∕N 2 for arbitrary number of dimensions. Theor. Math. Phys. 47, 465–475 (1981)CrossRefGoogle Scholar
  3. 3.
    d’Eramo, M., Peliti, L., Parisi, G.: Theoretical predictions for critical exponents at the λ-point of Bose liquids. Lett. Nuovo Cim. 2, 878–880 (1971)CrossRefGoogle Scholar
  4. 4.
    Ussyukina, N.I.: Calculation of multiloop diagrams in high orders of perturbation theory. Phys. Lett. B267, 382–388 (1991)ADSCrossRefGoogle Scholar
  5. 5.
    Gracey, J.A.: On the evaluation of massless Feynman diagrams by the method of uniqueness JAG. Phys. Lett. B277, 469–473 (1992)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Broadhurst, D.J.: Exploiting the 1,440-fold symmetry of the master two-loop diagram. Z. Phys. C32, 249–253 (1986)ADSGoogle Scholar
  7. 7.
    Barfoot, D.T., Broadhurst, D.J.: Z 2 × S 6 symmetry of the two-loop diagram. Z. Phys. C41, 81–85 (1988)MathSciNetGoogle Scholar
  8. 8.
    Broadhurst, D.J., Kreimer, D.: Knots and numbers in ϕ4 theory to 7 loops and beyond. Int. J. Mod. Phys. C6, 519–524 (1995)MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Bierenbaum, I., Weinzierl, S.: The massless two loop two point function. Eur. Phys. J. C32, 67–78 (2003)ADSCrossRefGoogle Scholar
  10. 10.
    Chetyrkin, K.G., Tkachov, F.V.: Integration by parts: the algorithm to calculate β-functions in 4 loops. Nucl. Phys. B192, 159–204 (1981)ADSCrossRefGoogle Scholar
  11. 11.
    Laporta, S.: High precision calculation of multiloop Feynman integrals by difference equations. Int. J. Mod. Phys. A15, 5087–5159 (2000)MathSciNetADSGoogle Scholar
  12. 12.
    Gracey, J.A.: Computation of critical exponent η at O(1∕N f 2) in quantum electrodynamics in arbitrary dimensions. Nucl. Phys. B414, 614–648 (1994)ADSCrossRefGoogle Scholar
  13. 13.
    Vasil’ev, A.N., Pismak, Y.M., Honkonen, J.R.: 1∕N expansion: calculation of the exponent η in the order 1∕N 3 by the conformal bootstrap method. Theor. Math. Phys. 50, 127–134 (1982)CrossRefGoogle Scholar
  14. 14.
    Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena. International Series of Monographs on Physics. Clarendon Press, Oxford (2002)CrossRefGoogle Scholar
  15. 15.
    Broadhurst, D.J., Gracey, J.A., Kreimer, D.: Beyond the triangle and uniqueness relations: non-zetas counterterms at large N from positive knots. Z. Phys. C75, 559–574 (1997)MathSciNetGoogle Scholar
  16. 16.
    Ussyukina, N.I., Davydychev, A.I.: New results for two-loop off-shell three-point diagrams. Phys. Lett. B332, 159–167 (1994)ADSCrossRefGoogle Scholar
  17. 17.
    Broadhurst, D.J.: Evaluation of a class of Feynman diagrams for all numbers of loops and dimensions. Phys. Lett. B164, 356–360 (1985)ADSCrossRefGoogle Scholar
  18. 18.
    Gracey, J.A.: Quark, gluon and ghost anomalous dimensions at O(1∕N f) in quantum chromodynamics. Phys. Lett. B318, 177–183 (1993)ADSCrossRefGoogle Scholar
  19. 19.
    Gracey, J.A.: The QCD β-function at O(1∕N f). Phys. Lett. B373, 178–184 (1996)ADSCrossRefGoogle Scholar
  20. 20.
    Ciuchini, M., Derkachov, S.E., Gracey, J.A., Manashov, A.N.: Computation of quark mass anomalous dimension at O(1∕N f 2) in quantum chromodynamics. Nucl. Phys. B579, 56–100 (2000)ADSCrossRefGoogle Scholar
  21. 21.
    Gracey, J.A.: Anomalous dimension of non-singlet Wilson operators at O(1∕N f) in deep inelastic scattering. Phys. Lett. B322, 141–146 (1994)ADSCrossRefGoogle Scholar
  22. 22.
    Bennett, J.F., Gracey, J.A.: Determination of the anomalous dimension of gluonic operators in deep inelastic scattering at O(1∕N f). Nucl. Phys. B517, 241–268 (1998)ADSCrossRefGoogle Scholar
  23. 23.
    Hasenfratz, A., Hasenfratz, P.: The equivalence of the SU(N) Yang-Mills theory with a purely fermionic model. Phys. Lett. B297, 166–170 (1992)ADSCrossRefGoogle Scholar
  24. 24.
    Vermaseren, J.A.M.: Axodraw. Comput. Phys. Commun. 83, 45–58 (1994)ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Theoretical Physics Division, Department of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

Personalised recommendations