, Volume 78, Issue 6, pp 927–946 | Cite as

Large-N c quantum chromodynamics and harmonic sums

  • EDUARDO DE RAFAELEmail author


In the large-N c limit of QCD, two-point functions of local operators become harmonic sums. I review some properties which follow from this fact and which are relevant for phenomenological applications. This has led us to consider a class of analytic number theory functions as toy models of large-N c QCD which also is discussed.


Large-Nc QCD harmonic sums Riemann zeros quantum field theory 


11.10.−z 11.10.Cd 11.10.Ef 11.10.Gh 11.10.Hi 11.15.Bt 11.55.Hx 12.38.Cy 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G t’ Hooft, Nucl. Phys. B75, 461 (1974)ADSCrossRefGoogle Scholar
  2. [2]
    C Vafa and E Witten, Nucl. Phys. B72, 461 (1974)Google Scholar
  3. [3]
    G ’t Hooft, NATO Adv . Study Inst. Ser. B Phys. 59, 135 (1980)Google Scholar
  4. [4]
    S Coleman and E Witten, Phys. Rev . Lett. 45, 100 (1980)MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    C Vafa and E Witten, Nucl. Phys. B234, 173 (1984)MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    M Knecht, La chromodynamique quantique à basse énergie, cours donné à la 27eme session de l’Ecole d’Eté de Gif, La Chromodynamique Quantique sous toutes ses couleurs, LPC Clermont-Ferrand, 18–22 Sept. 1995, A.-M. Lutz ed., IN2P3Google Scholar
  7. [7]
    E Witten, Nucl. Phys. B160, 57 (1979)MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    S Weinberg, Physica A96, 327 (1984)ADSGoogle Scholar
  9. [9]
    J Wess and B Zumino, Phys. Lett. B37, 95 (1971)MathSciNetADSGoogle Scholar
  10. [10]
    E Witten, Nucl. Phys. B223, 422 (1983)MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    J Gasser and H Leutwyler, Nucl. Phys. B250, 465 (1985)ADSCrossRefGoogle Scholar
  12. [12]
    G Ecker, J Gasser, A Pich and E de Rafael, Nucl. Phys. B321, 311 (1989)ADSCrossRefGoogle Scholar
  13. [13]
    T Hambye, S Peris and E de Rafael, J. High Energy Phys. 027, 0305 (2003)Google Scholar
  14. [14]
    M Knecht and E de Rafael, Phys. Lett. B424, 335 (1998)ADSGoogle Scholar
  15. [15]
    T Das, G S Guralnik, V S Mathur, F E Low and J E Young, Phys. Rev . Lett. 18, 759 (1967)ADSCrossRefGoogle Scholar
  16. [16]
    M Knecht, S Peris and E de Rafael, Phys. Lett. B443, 255 (1998)ADSGoogle Scholar
  17. [17]
    M A Shifman, A I Vainshtein and V I Zakharov, Nucl. Phys. B147, 385, 447 (1979)ADSCrossRefGoogle Scholar
  18. [18]
    E Witten, Phys. Rev. Lett. 51, 2351 (1983)MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    J Comellas, J I Latorre and J Tarón, Phys. Lett. B360, 109 (1995)ADSGoogle Scholar
  20. [20]
    A Manohar and G Georgi, Nucl. Phys. B233, 232 (1984)Google Scholar
  21. [21]
    D Espriu, E de Rafael and J Tarón, Nucl. Phys. B345, 22 (1990)ADSCrossRefGoogle Scholar
  22. [22]
    S Weinberg, Phys. Rev. Lett. 105, 261601 (2010)ADSCrossRefGoogle Scholar
  23. [23]
    E de Rafael, Phys. Lett. B703, 60 (2011)ADSGoogle Scholar
  24. [24]
    Y Nambu and G Jona-Lasinio, Phys. Rev. 122, 345 (1961)ADSCrossRefGoogle Scholar
  25. [25]
    J Bijnens, Ch Bruno and E de Rafael, Nucl. Phys. B390, 501 (1993)ADSCrossRefGoogle Scholar
  26. [26]
    G Ecker, J Gasser, H Leutwyler, A Pich and E de Rafael, Phys. Lett. B321, 425 (1989)ADSGoogle Scholar
  27. [27]
    V Cirigliano, G Ecker, H Neufeld and A Pich, J. High Energy Phys. 012, 0306 (2003)Google Scholar
  28. [28]
    E de Rafael, Nucl. Phys. (Proc. Suppl.) B119, 71 (2003)ADSCrossRefGoogle Scholar
  29. [29]
    S Peris and E de Rafael, Phys. Lett. B490, 213 (2000)ADSGoogle Scholar
  30. [30]
    P Masjuan and S Peris, J. High Energy Phys. 0705, 040 (2007)ADSCrossRefGoogle Scholar
  31. [31]
    ALEPH Collaboration: R Barate et al, Z. Phys. C76, 15 (1997); ibid, Eur. Phys. J. C4, 409 (1998)Google Scholar
  32. [32]
    Ph Flajolet, X Gourdon and Ph Dumas, Theor. Comp. Sci. 144, 3 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    S Friot, D Greynat and E de Rafael, Phys. Lett. B628, 73 (2006)ADSGoogle Scholar
  34. [34]
    J-Ph Aguilar, D Greynat and E de Rafael, Phys. Rev . D77, 093010 (2008)ADSGoogle Scholar
  35. [35]
    D Greynat and S Peris, Phys. Rev . D82, 034030 (2010)ADSGoogle Scholar
  36. [36]
    B Blok, M A Shifman and D X Zhang, Phys. Rev. D57, 2691 (1998); Erratum, ibid. D59, 019901 (1999)Google Scholar
  37. [37]
    M Golterman, S Peris, B Phily and E de Rafael, J. High Energy Phys. 0201, 024 (2002)ADSCrossRefGoogle Scholar
  38. [38]
    O Catà, M Golterman and S Peris, J. High Energy Phys. 0508, 076 (2005)ADSCrossRefGoogle Scholar
  39. [39]
    Tom M Apostol, Introduction to analytic number theory, Ch.12 (Springer-Verlag, 1976)Google Scholar
  40. [40]
    J F Donoghue and E  Golowich, Phys. Lett. 478, 172 (2000)Google Scholar
  41. [41]
    M Gonzalez-Alonso, A Pich and J Prades, Phys. Rev. D81, 074007 (2010)ADSGoogle Scholar
  42. [42]
    K Maltman et al, arXiv:1110.5562v1 [hep-ph]Google Scholar
  43. [43]
    Julian Havil, GAMMA, exploring Euler’s constant (Princeton University Press, 2003)Google Scholar
  44. [44]
    Jeffrey Stopple, A primer of analytic number theoryFrom Pythagoras to Riemann (Cambridge University Press, 2003)Google Scholar

Copyright information

© Indian Academy of Sciences 2012

Authors and Affiliations

  1. 1.Centre de Physique ThéoriqueUnité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix Marseille 1Marseille Cedex 9France

Personalised recommendations