Skip to main content
SpringerLink
Log in

Loop calculations in the three dimensional Gribov–Zwanziger Lagrangian

  • Regular Article - Theoretical Physics
  • Published:
The European Physical Journal C Aims and scope Submit manuscript

Abstract

The three dimensional Gribov–Zwanziger Lagrangian is analysed at one and two loops. Specifically, the two loop gap equation is evaluated and the Gribov mass is expressed in terms of the coupling constant. The one loop corrections to the propagators of all the fields are determined. It is shown that when the gap equation is satisfied the Faddeev–Popov ghost and both Bose and Grassmann localizing ghosts all enhance in the infrared limit at one loop. This verifies that the Kugo–Ojima confinement criterion holds to this order and we also show that both Grassmann ghosts are enhanced at two loops. For the Bose ghost we determine the full form of the propagator in the zero momentum limit for both the transverse and longitudinal pieces and confirm Zwanziger’s recent general analysis for the low energy behaviour. We provide an alternative but equivalent version of the horizon condition expressing it as the vacuum expectation value of an operator involving only the localizing Bose ghost field. The one loop static potential is also determined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Zwanziger, Nucl. Phys. B 321, 591 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  2. D. Zwanziger, Nucl. Phys. B 323, 513 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  3. D. Zwanziger, Nucl. Phys. B 378, 525 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  4. D. Zwanziger, Nucl. Phys. B 399, 477 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  5. V.N. Gribov, Nucl. Phys. B 139, 1 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  6. G. Dell’Antonio, D. Zwanziger, Nucl. Phys. B 326, 333 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  7. G. Dell’Antonio, D. Zwanziger, Commun. Math. Phys. 138, 291 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. D. Zwanziger, Nucl. Phys. B 364, 127 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  9. D. Zwanziger, Nucl. Phys. B 412, 657 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. D. Zwanziger, Phys. Rev. D 65, 094039 (2002)

    Article  ADS  Google Scholar 

  11. D. Zwanziger, Phys. Rev. D 69, 016002 (2004)

    Article  ADS  Google Scholar 

  12. T. Kugo, I. Ojima, Prog. Theor. Phys. Suppl. 66, 1 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  13. T. Kugo, I. Ojima, Prog. Theor. Phys. Suppl. 77, 1121 (1984)

    Google Scholar 

  14. T. Kugo, hep-th/9511033

  15. D. Dudal, S.P. Sorella, N. Vandersickel, H. Verschelde, Phys. Rev. D 79, 121701 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  16. N. Maggiore, M. Schaden, Phys. Rev. D 50, 6616 (1994)

    Article  ADS  Google Scholar 

  17. D. Dudal, R.F. Sobreiro, S.P. Sorella, H. Verschelde, Phys. Rev. D 72, 014016 (2005)

    Article  ADS  Google Scholar 

  18. D.J. Gross, F.J. Wilczek, Phys. Rev. Lett. 30, 1343 (1973)

    Article  ADS  Google Scholar 

  19. H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973)

    Article  ADS  Google Scholar 

  20. D.R.T. Jones, Nucl. Phys. B 75, 531 (1974)

    Article  ADS  Google Scholar 

  21. W.E. Caswell, Phys. Rev. Lett. 33, 244 (1974)

    Article  ADS  Google Scholar 

  22. O.V. Tarasov, A.A. Vladimirov, A.Yu. Zharkov, Phys. Lett. B 93, 429 (1980)

    Article  ADS  Google Scholar 

  23. J.A. Gracey, Phys. Lett. B 632, 282 (2006)

    Article  ADS  Google Scholar 

  24. J.A. Gracey, Phys. Lett. B 686, 319 (2010)

    Article  ADS  Google Scholar 

  25. J.A. Gracey, JHEP 05, 052 (2006)

    Article  ADS  Google Scholar 

  26. J.A. Gracey, JHEP 02, 078 (2010)

    Article  ADS  Google Scholar 

  27. D. Zwanziger, Phys. Rev. D 81, 125027 (2010)

    Article  ADS  Google Scholar 

  28. J.A. Gracey, JHEP 02, 009 (2010)

    Article  ADS  Google Scholar 

  29. A. Cucchieri, T. Mendes, PoS LAT2007 (2007), p. 297

  30. I.L. Bogolubsky, E.M. Ilgenfritz, M. Müller-Preussker, A. Sternbeck, PoS LAT2007 (2007), p. 290

  31. A. Cucchieri, T. Mendes, Phys. Rev. Lett. 100, 241601 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  32. A. Cucchieri, T. Mendes, Phys. Rev. D 78, 094503 (2008)

    Article  ADS  Google Scholar 

  33. O. Oliveira, P.J. Silva, Phys. Rev. D 79, 031501 (2009)

    Article  ADS  Google Scholar 

  34. Ph. Boucaud, J.P. Leroy, A.L. Yaounac, J. Micheli, O. Pène, J. Rodríguez-Quintero, JHEP 06, 099 (2008)

    Article  ADS  Google Scholar 

  35. A.C. Aguilar, D. Binosi, J. Papavassiliou, Phys. Rev. D 78, 025010 (2008)

    Article  ADS  Google Scholar 

  36. C.S. Fischer, A. Maas, J.M. Pawlowski, Ann. Phys. 324, 2408 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. D. Dudal, S.P. Sorella, N. Vandersickel, H. Verschelde, Phys. Rev. D 77, 071501 (2008)

    Article  ADS  Google Scholar 

  38. D. Dudal, J.A. Gracey, S.P. Sorella, N. Vandersickel, H. Verschelde, Phys. Rev. D 78, 065047 (2008)

    Article  ADS  Google Scholar 

  39. D. Zwanziger, arXiv:0904.2380 [hep-th]

  40. P. Nogueira, J. Comput. Phys. 105, 279 (1993)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  41. J.A.M. Vermaseren, math-ph/0010025

  42. A.K. Rajantie, Nucl. Phys. B 480, 729 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  43. A.K. Rajantie, Nucl. Phys. B 513, 761 (1998)

    Article  ADS  Google Scholar 

  44. W. Buchmüller, O. Philipsen, Nucl. Phys. B 443, 47 (1995)

    Article  ADS  Google Scholar 

  45. G. Alexanian, V.P. Nair, Phys. Lett. B 352, 435 (1995)

    Article  ADS  Google Scholar 

  46. R. Jackiw, S.Y. Pi, Phys. Lett. B 368, 131 (1996)

    Article  ADS  Google Scholar 

  47. J.M. Cornwall, Phys. Rev. D 57, 3694 (1998)

    Article  ADS  Google Scholar 

  48. F. Eberlein, Phys. Lett. B 439, 130 (1998)

    Article  ADS  Google Scholar 

  49. J.C. Taylor, Nucl. Phys. B 33, 436 (1971)

    Article  ADS  Google Scholar 

  50. M. Lüscher, K. Symanzik, P. Weisz, Nucl. Phys. B 173, 365 (1980)

    Article  ADS  Google Scholar 

  51. M. Lüscher, Nucl. Phys. B 180, 317 (1981)

    Article  ADS  Google Scholar 

  52. L. Susskind, Coarse grained quantum chromodynamics, in Les Houches 1976, Proceedings Weak and Electromagnetic Interactions at High Energy (North-Holland, Amsterdam, 1977), p. 207

    Google Scholar 

  53. W. Fischler, Nucl. Phys. B 129, 157 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  54. T. Appelquist, M. Dine, I.J. Muzinich, Phys. Rev. D 17, 2074 (1978)

    Article  ADS  Google Scholar 

  55. M. Peter, Phys. Rev. Lett. 78, 602 (1997)

    Article  ADS  Google Scholar 

  56. M. Peter, Nucl. Phys. B 501, 471 (1997)

    Article  ADS  Google Scholar 

  57. Y. Schröder, The static potential in QCD. DESY-THESIS-1999-021

  58. Y. Schröder, Phys. Lett. B 447, 321 (1999)

    Article  ADS  Google Scholar 

  59. A.V. Smirnov, V.A. Smirnov, M. Steinhauser, PoS RADCOR2007 (2007), p. 024

  60. A.V. Smirnov, V.A. Smirnov, M. Steinhauser, Nucl. Phys. Proc. Suppl. 183, 308 (2008)

    Article  ADS  Google Scholar 

  61. A.V. Smirnov, V.A. Smirnov, M. Steinhauser, Phys. Lett. B 668, 293 (2008)

    Article  ADS  Google Scholar 

  62. C. Anzai, Y. Kiyo, Y. Sumino, Phys. Rev. Lett. 104, 112003 (2010)

    Article  ADS  Google Scholar 

  63. A.V. Smirnov, V.A. Smirnov, M. Steinhauser, Phys. Rev. Lett. 104, 112002 (2010)

    Article  ADS  Google Scholar 

  64. A.V. Smirnov, V.A. Smirnov, M. Steinhauser, PoS RARCOR2009 (2010), p. 075

  65. Y. Schröder, The static potential in QCD3 at one loop, in Eotvos Conference in Science: Strong and Electroweak Matter (SEWM 97), Eger, Hungary (1997)

    Google Scholar 

  66. J.G.M. Gatheral, Phys. Lett. B 133, 90 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  67. J. Frenkel, J.C. Taylor, Nucl. Phys. B 246, 231 (1984)

    Article  ADS  Google Scholar 

  68. T. van Ritbergen, A.N. Schellekens, J.A.M. Vermaseren, Int. J. Mod. Phys. A 14, 41 (1999)

    Article  MATH  ADS  Google Scholar 

  69. S. Mandelstam, Phys. Rept. 67, 109 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  70. R. Anishetty, M. Baker, J.S. Ball, S.K. Kim, F. Zachariasen, Phys. Lett. B 86, 52 (1979)

    Article  ADS  Google Scholar 

  71. S. Mandelstam, Phys. Rev. D 20, 3223 (1979)

    Article  ADS  Google Scholar 

  72. G.B. West, Phys. Lett. B 115, 468 (1982)

    Article  ADS  Google Scholar 

  73. A. Hasenfratz, P. Hasenfratz, Phys. Lett. B 297, 166 (1992)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, P.O. Box 147, Liverpool, L69 3BX, UK

    J. A. Gracey

Authors
  1. J. A. Gracey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. A. Gracey.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gracey, J.A. Loop calculations in the three dimensional Gribov–Zwanziger Lagrangian. Eur. Phys. J. C 70, 451–477 (2010). https://doi.org/10.1140/epjc/s10052-010-1473-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjc/s10052-010-1473-5

Keywords

Search

Navigation