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The European Physical Journal A

, Volume 35, Issue 2, pp 189–205 | Cite as

Power law running of the effective gluon mass

  • A. C. Aguilar
  • J. Papavassiliou
Regular Article - Theoretical Physics

Abstract.

The dynamically generated effective gluon mass is known to depend non-trivially on the momentum, decreasing sufficiently fast in the deep ultraviolet, in order for the renormalizability of QCD to be preserved. General arguments based on the analogy with the constituent quark masses, as well as explicit calculations using the operator-product expansion, suggest that the gluon mass falls off as the inverse square of the momentum, relating it to the gauge-invariant gluon condensate of dimension four. In this article we demonstrate that the power law running of the effective gluon mass is indeed dynamically realized at the level of the non-perturbative Schwinger-Dyson equation. We study a gauge-invariant non-linear integral equation involving the gluon self-energy, and establish the conditions necessary for the existence of infrared finite solutions, described in terms of a momentum-dependent gluon mass. Assuming a simplified form for the gluon propagator, we derive a secondary integral equation that controls the running of the mass in the deep ultraviolet. Depending on the values chosen for certain parameters entering into the Ansatz for the fully dressed three-gluon vertex, this latter equation yields either logarithmic solutions, familiar from previous linear studies, or a new type of solutions, displaying power law running. In addition, it furnishes a non-trivial integral constraint, which restricts significantly (but does not determine fully) the running of the mass in the intermediate and infrared regimes. The numerical analysis presented is in complete agreement with the analytic results obtained, showing clearly the appearance of the two types of momentum dependence, well-separated in the relevant space of parameters. Several technical improvements, various open issues, and possible future directions, are briefly discussed.

PACS.

12.38.Lg Other nonperturbative calculations 12.38.Aw General properties of QCD (dynamics, confinement etc.) 

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References

  1. 1.
    J.M. Cornwall, Nucl. Phys. B 157, 392 (1979).CrossRefADSGoogle Scholar
  2. 2.
    J.M. Cornwall, Phys. Rev. D 26, 1453 (1982).CrossRefADSGoogle Scholar
  3. 3.
    C.W. Bernard, Nucl. Phys. B 219, 341 (1983)CrossRefADSGoogle Scholar
  4. 4.
    K.I. Kondo, Phys. Lett. B 514, 335 (2001)MATHCrossRefADSGoogle Scholar
  5. 5.
    J.C.R. Bloch, Few Body Syst. 33, 111 (2003).CrossRefADSGoogle Scholar
  6. 6.
    A.C. Aguilar, A.A. Natale, JHEP 0408, 057 (2004).CrossRefADSGoogle Scholar
  7. 7.
    D. Dudal, J.A. Gracey, V.E.R. Lemes, M.S. Sarandy, R.F. Sobreiro, S.P. Sorella, H. Verschelde, Phys. Rev. D 70, 114038 (2004)CrossRefADSGoogle Scholar
  8. 8.
    A.C. Aguilar, J. Papavassiliou, JHEP 0612, 012 (2006).CrossRefADSGoogle Scholar
  9. 9.
    G. Parisi, R. Petronzio, Phys. Lett. B 94, 51 (1980).CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    A.C. Mattingly, P.M. Stevenson, Phys. Rev. Lett. 69, 1320 (1992)CrossRefADSGoogle Scholar
  11. 11.
    A. Mihara, A.A. Natale, Phys. Lett. B 482, 378 (2000)CrossRefADSGoogle Scholar
  12. 12.
    In addition, the non-perturbative behavior of QCD Green's functions found in lattice simulations may be described in terms of effectively massive gluon propagators, see, for example, C. Alexandrou, P. de Forcrand, E. Follana, Phys. Rev. D 63, 094504 (2001)CrossRefADSGoogle Scholar
  13. 13.
    J.S. Schwinger, Phys. Rev. 125, 397 (1962)MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147, 448Google Scholar
  15. 15.
    J.M. Cornwall, J. Papavassiliou, Phys. Rev. D 40, 3474 (1989).CrossRefADSGoogle Scholar
  16. 16.
    D. Binosi, J. Papavassiliou, Phys. Rev. D 66, 111901 (2002)CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    L.F. Abbott, Nucl. Phys. B 185, 189 (1981).CrossRefADSGoogle Scholar
  18. 18.
    D. Binosi, J. Papavassiliou, JHEP 0703, 041 (2007).CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    J. Papavassiliou, J.M. Cornwall, Phys. Rev. D 44, 1285 (1991).CrossRefADSGoogle Scholar
  20. 20.
    A.M. Badalian, V.L. Morgunov, Phys. Rev. D 60, 116008 (1999).CrossRefADSGoogle Scholar
  21. 21.
    A.C. Aguilar, A.A. Natale, P.S. Rodrigues da Silva, Phys. Rev. Lett. 90, 152001 (2003)CrossRefADSGoogle Scholar
  22. 22.
    S.J. Brodsky, S. Menke, C. Merino, J. Rathsman, Phys. Rev. D 67, 055008 (2003)CrossRefADSGoogle Scholar
  23. 23.
    The freezing of the QCD coupling has also been advocated in various different approaches, e.g., A.C. Mattingly, P.M. Stevenson, Phys. Rev. D 49, 437 (1994)CrossRefGoogle Scholar
  24. 24.
    See, for example, K.D. Lane, Phys. Rev. D 10, 2605 (1974)CrossRefGoogle Scholar
  25. 25.
    J.M. Cornwall, W.S. Hou, Phys. Rev. D 34, 585 (1986).CrossRefADSGoogle Scholar
  26. 26.
    M. Lavelle, Phys. Rev. D 44, 26 (1991).CrossRefADSGoogle Scholar
  27. 27.
    It is important to notice that the conventional gluon self-energy contains in addition unphysical condensates involving ghost operators, see, M.J. Lavelle, M. Schaden, Phys. Lett. B 208, 297 (1988)CrossRefGoogle Scholar
  28. 28.
    The full SDE for the BFM gluon self-energy was first derived in R.B. Sohn, Nucl. Phys. B 273, 468 (1986)CrossRefADSGoogle Scholar
  29. 29.
    D. Binosi, J. Papavassiliou, Phys. Rev. D 66, 025024 (2002)CrossRefADSMathSciNetGoogle Scholar
  30. 30.
    P. Gambino, P.A. Grassi, Phys. Rev. D 62, 076002 (2000)CrossRefADSGoogle Scholar
  31. 31.
    Note in passing that this type of generalized Feynman gauge cannot be obtained through an appropriate choice of the (constant) gauge-fixing parameter $\xi$. Instead, it is reminiscent of the so-called ``stagnant gauge'', presented in C.H. Llewellyn Smith, Nucl. Phys. B 165, 423 (1980)CrossRefGoogle Scholar
  32. 32.
    A. Salam, Phys. Rev. 130, 1287 (1963)CrossRefADSMathSciNetGoogle Scholar
  33. 33.
    R. Jackiw, K. Johnson, Phys. Rev. D 8, 2386 (1973)CrossRefADSGoogle Scholar
  34. 34.
    J.S. Ball, T.W. Chiu, Phys. Rev. D 22, 2550 (1980)CrossRefADSGoogle Scholar
  35. 35.
    M. Binger, S.J. Brodsky, Phys. Rev. D 74, 054016 (2006).CrossRefADSGoogle Scholar
  36. 36.
    D. Binosi, J. Papavassiliou, Nucl. Phys. Proc. Suppl. 121, 281 (2003).MATHCrossRefADSGoogle Scholar
  37. 37.
    J.E. King, Phys. Rev. D 27, 1821 (1983)CrossRefADSGoogle Scholar
  38. 38.
    The mass scale $\mu_2$ is associated with the quark condensate $\langle{\bar{\psi}}\psi \rangle$ of dimension three, while $\mu_1$ with $M_0$, a bare quark mass that breaks chiral symmetry explicitly.Google Scholar
  39. 39.
    See, for example, I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Fifth Edition (Academic Press, London, 1994).Google Scholar
  40. 40.
    F.R. Graziani, Z. Phys. C 33, 397 (1987).CrossRefADSGoogle Scholar
  41. 41.
    I.I. Kogan, A. Kovner, Phys. Rev. D 52, 3719 (1995). CrossRefADSGoogle Scholar
  42. 42.
    E.V. Gorbar, A.A. Natale, Phys. Rev. D 61, 054012 (2000).CrossRefADSGoogle Scholar
  43. 43.
    In addition to $\langle G^2 \rangle$, another quantity that may be relevant to these considerations is the gauge-invariant non-local condensate of dimension two, usually denoted by $\langle A^2_{\min} \rangle$, obtained through the minimization of $\int \upd^4 x (A_{\mu})^2$ over all gauge transformations Gubarev:2000eu,Gracey:2007ki, or variants of it involving also ghost condensates Kondo:2001nq. $\langle A^2_{\min} \rangle$ should not to be confused with $\langle 0| : A_{\mu}^{a} A^{\mu}_{a} : |0 \rangle$, the local gauge-variant condensate of dimension twoGoogle Scholar
  44. 44.
    F.V. Gubarev, L. Stodolsky, V.I. Zakharov, Phys. Rev. Lett. 86, 2220 (2001)CrossRefADSGoogle Scholar
  45. 45.
    J.A. Gracey, arXiv:0706.1440 [hep-th] and references therein.Google Scholar
  46. 46.
    J.M. Cornwall, R. Jackiw, E. Tomboulis, Phys. Rev. D 10, 2428 (1974).MATHCrossRefADSGoogle Scholar
  47. 47.
    J.M. Cornwall, Physica A 158, 97 (1989).CrossRefADSGoogle Scholar
  48. 48.
    D. Atkinson, J.C.R. Bloch, Phys. Rev. D 58, 094036 (1998)CrossRefADSGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departamento de Fısica Teórica and IFIC, Centro MixtoUniversidad de Valencia - CSICBurjassot, ValenciaSpain

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