Application of the operator product expansion to the short distance behavior of nuclear potentials

  • Sinya Aoki
  • Janos Balog
  • Peter Weisz
Open Access


We investigate the short distance behavior of nucleon-nucleon (NN) potentials defined through Bethe-Salpeter wave functions, by perturbatively calculating anomalous dimensions of 6-quark operators in QCD. Thanks to the asymptotic freedom of QCD, 1-loop computations give certain exact results for the potentials in the zero distance limit. In particular the functional form of the S-state central NN potential at short distance r is predicted to be a little weaker than r −2. On the other hand, due to the intriguing character of the anomalous dimension spectrum, perturbative considerations alone can not determine whether this potential is repulsive or attractive at short distances. A crude estimation suggests that the force at short distance is repulsive, as found numerically in lattice QCD. A similar behavior is found for the tensor potential.


Asymptotic Freedom Renormalization Group QCD Lattice QCD 


  1. [1]
    N. Ishii, S. Aoki and T. Hatsuda, The Nuclear Force from Lattice QCD, Phys. Rev. Lett. 99 (2007) 022001 [nucl-th/0611096] [SPIRES].CrossRefADSGoogle Scholar
  2. [2]
    S. Aoki, T. Hatsuda and N. Ishii, Nuclear Force from Monte Carlo Simulations of Lattice Quantum Chromodynamics, Comput. Sci. Dis. 1 (2008) 015009 [arXiv:0805.2462] [SPIRES].CrossRefGoogle Scholar
  3. [3]
    S. Aoki, T. Hatsuda and N. Ishii, Theoretical Foundation of the Nuclear Force in QCD and its applications to Central and Tensor Forces in Quenched Lattice QCD Simulations, Prog. Theor. Phys. 123 (2010) 89 [arXiv:0909.5585] [SPIRES].MATHCrossRefADSGoogle Scholar
  4. [4]
    N. Ishii, S. Aoki and T. Hatsuda, Nuclear forces from quenched and NF=2+1 full lattice QCD using the PACS-CS gauge configurations, PoS(Lattice 2008)155.
  5. [5]
    K. Hashimoto, T. Sakai and S. Sugimoto, Nuclear Force from String Theory, Prog. Theor. Phys. 122 (2009) 427 [arXiv:0901.4449] [SPIRES].MATHCrossRefADSGoogle Scholar
  6. [6]
    S. Aoki, J. Balog and P. Weisz, Bethe–Salpeter wave functions in integrable models, Prog. Theor. Phys. 121 (2009) 1003 [arXiv:0805.3098] [SPIRES].MATHCrossRefADSGoogle Scholar
  7. [7]
    S. Aoki, J. Balog and P. Weisz, Repulsive core of the NN potential and operator product expansion, PoS(LAT2009)132.
  8. [8]
    J. Collins, Renormalization, Cambridge University Press, Cambridge U.K. (1984).MATHCrossRefGoogle Scholar
  9. [9]
    M.E. Peskin, Anomalous Dimensions Of Three Quark Operators, Phys. Lett. B 88 (1979) 128 [SPIRES].ADSGoogle Scholar
  10. [10]
    C. Itzykson and J.B. Zuber, Quantum Field Theory, International Series In Pure and Applied Physics, McGraw-Hill, New York U.S.A. (1980).Google Scholar
  11. [11]
    G. Altarelli and L. Maiani, Octet Enhancement of Nonleptonic Weak Interactions in Asymptotically Free Gauge Theories, Phys. Lett. B 52 (1974) 351 [SPIRES].ADSGoogle Scholar
  12. [12]
    M.K. Gaillard and B.W. Lee, Delta I = 1/2 Rule for Nonleptonic Decays in Asymptotically Free Field Theories, Phys. Rev. Lett. 33 (1974) 108 [SPIRES].CrossRefADSGoogle Scholar
  13. [13]
    J.A. Bagger, K.T. Matchev and R.-J. Zhang, QCD corrections to flavor-changing neutral currents in the supersymmetric standard model, Phys. Lett. B 412 (1997) 77 [hep-ph/9707225] [SPIRES].ADSGoogle Scholar
  14. [14]
    M. Jamin and M. Kremer, Anomalous Dimensions Of Spin 0 Four Quark Operators Without Derivatives, Nucl. Phys. B 277 (1986) 349 [SPIRES].CrossRefADSGoogle Scholar
  15. [15]
    M.K. Gaillard, B.W. Lee and J.L. Rosner, Search for Charm, Rev. Mod. Phys. 47 (1975) 277 [SPIRES].CrossRefADSGoogle Scholar
  16. [16]
    K. Sasaki and N. Ishizuka, I = 2 Two-Pion Wave Function and Scattering Phase Shift, Phys. Rev. D 78 (2008) 014511 [arXiv:0804.2941] [SPIRES].ADSGoogle Scholar

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© The Author(s) 2010

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Graduate School of Pure and Applied SciencesUniversity of TsukubaTsukuba, IbarakiJapan
  2. 2.Research Institute for Particle and Nuclear PhysicsBudapestHungary
  3. 3.Max-Planck-Institut für PhysikMünchenGermany

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