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On the elimination of infinitesimal Gribov ambiguities in non-Abelian gauge theories

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Abstract

An alternative method to account for the Gribov ambiguities in gauge theories is presented. It is shown that, to eliminate Gribov ambiguities, at infinitesimal level, it is required to break the BRST symmetry in a soft manner. This can be done by introducing a suitable extra constraint that eliminates the infinitesimal Gribov copies. It is shown that the present approach is consistent with the well established known cases in the literature, i.e., the Landau and maximal Abelian gauges. The method is valid for gauges depending exclusively on the gauge field and is restricted to classical level. However, occasionally, we deal with quantum aspects of the technique, which are used to improve the results.

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Notes

  1. The Gribov operator is the Faddeev–Popov operator with the gauge condition employed.

  2. Global gauge invariance is also required. However, this symmetry can be broken in certain gauges, for instance, the maximal Abelian gauge.

  3. Although the method here developed can be applied to a class of gauges, the final result always depend on the original gauge constraint.

  4. We call attention to the fact that we are considering the general case where the gauge fixing is the same for all sector of the gauge group algebra. It turns out that the generalization to gauges where different sectors of the algebra have different gauge constraints is not difficult. The example of the maximal Abelian gauge will be discussed at Sect. 8.2.

  5. Since we are running out of geometrical fields which naturally arise in Y G , the only way to introduce extra fields with no reflection at the UV sector is to consider BRST doublets.

  6. Perhaps, a higher derivative term could be considered. We, however, opt to avoid the intricacies of dealing with such terms.

  7. By a simple algebraic analysis of (39), it is easy to find that this type of term would be present if, and only if, κ=2⇒n=2, z=4. At this restricted class of gauges resides the Landau gauge and the maximal Abelian gauge. However, that would exclude non-local gauges (which could be localizable by a suitable set of auxiliary fields) or other gauges such as, for instance, ∂ 2 ∂ μ A μ =0. In this example, these terms would be absent and the correspondent effects would probably appear from the higher derivative intricacies.

  8. The maximal Abelian gauge actually requires extra interacting terms. However, these terms originate from the fact that the gauge constraint is non-linear and demand quartic ghost interactions for renormalization purposes. However, this generalized maximal Abelian gauge is a more general gauge which does not respect our restriction of gauges depending exclusively on \(A_{\mu}^{A}\).

  9. In certain cases, e.g. the maximal Abelian gauge, the color invariance has to be treated in a different way. See Sect. 8.2

  10. See below the discussion within the alternative formulation.

  11. If the gauge fixing accepts the ghost mass term, even the Faddeev–Popov operator is altered. However, in this case, we cross the limits of this approach because, when restoring the BRST symmetry, this term will naturally generate a b-dependent term, modifying the gauge fixing itself. Thus, perhaps this term could only be considered through the LCO formalism.

  12. RFS is a level PQ-2 researcher under the program Produtividade em Pesquisa, 308845/2012-9.

References

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

    Article  MathSciNet  ADS  Google Scholar 

  2. L.D. Faddeev, V.N. Popov, Phys. Lett. B 25, 29 (1967)

    Article  ADS  Google Scholar 

  3. R.F. Sobreiro, S.P. Sorella, Introduction to the Gribov ambiguities in Euclidean Yang–Mills theories, in Lectures given at 13th Jorge Andre Swieca Summer School on Particle and Fields, Campos do Jordao, Brazil, 9–22 January, 2005. E-print: arXiv:hep-th/0504095

    Google Scholar 

  4. I.M. Singer, Commun. Math. Phys. 60, 7 (1978)

    Article  ADS  MATH  Google Scholar 

  5. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. 1 (Wiley, New York, 1963)

    MATH  Google Scholar 

  6. M. Daniel, C.M. Viallet, Rev. Mod. Phys. 52, 175 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  7. P. Cotta-Ramusino, C. Reina, J. Geom. Phys. 1(3), 121–155 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  8. G. Falqui, C. Reina, Commun. Math. Phys. 102, 503 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  9. M. Nakahara, Geometry, Topology and Physics. Graduate Student Series in Physics (Hilger, Bristol, 1990), 505 pp.

    Book  MATH  Google Scholar 

  10. R.A. Bertlmann, Anomalies in Quantum Field Theory. International Series of Monographs on Physics, vol. 91 (Clarendon, Oxford, 1996), 566 pp.

    MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  15. N. Maggiore, M. Schaden, Phys. Rev. D 50, 6616 (1994). arXiv:hep-th/9310111

    Article  ADS  Google Scholar 

  16. D. Dudal, R.F. Sobreiro, S.P. Sorella, H. Verschelde, Phys. Rev. D 72, 014016 (2005). arXiv:hep-th/0502183

    Article  ADS  Google Scholar 

  17. D. Dudal, J.A. Gracey, S.P. Sorella, N. Vandersickel, H. Verschelde, Phys. Rev. D 78, 065047 (2008). arXiv:0806.4348 [hep-th]

    Article  ADS  Google Scholar 

  18. D. Dudal, S.P. Sorella, N. Vandersickel, Phys. Rev. D 84, 065039 (2011). arXiv:1105.3371 [hep-th]

    Article  ADS  Google Scholar 

  19. M.A.L. Capri, V.E.R. Lemes, R.F. Sobreiro, S.P. Sorella, R. Thibes, Phys. Rev. D 72, 085021 (2005). hep-th/0507052

    Article  ADS  Google Scholar 

  20. M.A.L. Capri, V.E.R. Lemes, R.F. Sobreiro, S.P. Sorella, R. Thibes, Phys. Rev. D 74, 105007 (2006). hep-th/0609212

    Article  MathSciNet  ADS  Google Scholar 

  21. M.A.L. Capri, V.E.R. Lemes, R.F. Sobreiro, S.P. Sorella, R. Thibes, Phys. Rev. D 77, 105023 (2008). arXiv:0801.0566 [hep-th]

    Article  ADS  Google Scholar 

  22. M.A.L. Capri, A.J. Gomez, V.E.R. Lemes, R.F. Sobreiro, S.P. Sorella, Phys. Rev. D 79, 025019 (2009). arXiv:0811.2760 [hep-th]

    Article  ADS  Google Scholar 

  23. M.A.L. Capri, A.J. Gomez, M.S. Guimaraes, V.E.R. Lemes, S.P. Sorella, J. Phys. A 43, 245402 (2010). arXiv:1002.1659 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  24. I.L. Bogolubsky, E.M. Ilgenfritz, M. Muller-Preussker, A. Sternbeck, Proc. Sci. LAT2007, 290 (2007). arXiv:0710.1968 [hep-lat]

    Google Scholar 

  25. A. Cucchieri, T. Mendes, Proc. Sci. LAT2007, 297 (2007). arXiv:0710.0412 [hep-lat]

    Google Scholar 

  26. A. Sternbeck, L. von Smekal, D.B. Leinweber, A.G. Williams Proc. Sci. LAT2007, 340 (2007). arXiv:0710.1982 [hep-lat]

  27. A. Cucchieri, T. Mendes, Phys. Rev. D 81, 016005 (2010). arXiv:0904.4033 [hep-lat]

    Article  ADS  Google Scholar 

  28. L. Baulieu, S.P. Sorella, Phys. Lett. B 671, 481 (2009). arXiv:0808.1356 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  29. L. Baulieu, M.A.L. Capri, A.J. Gomez, V.E.R. Lemes, R.F. Sobreiro, S.P. Sorella, Eur. Phys. J. C 66, 451 (2010). arXiv:0901.3158 [hep-th]

    Article  ADS  Google Scholar 

  30. O. Piguet, S.P. Sorella, Lect. Notes Phys., M Monogr. 28, 1–134 (1995)

    Article  MathSciNet  Google Scholar 

  31. N. Vandersickel, arXiv:1104.1315 [hep-th]

  32. A. Cucchieri, D. Dudal, T. Mendes, N. Vandersickel, Phys. Rev. D 85, 094513 (2012). arXiv:1111.2327 [hep-lat]

    Article  ADS  Google Scholar 

  33. J. Serreau, M. Tissier, Phys. Lett. B 712, 97 (2012). arXiv:1202.3432 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  34. P. van Baal, Nucl. Phys. B 369, 259 (1992)

    Article  ADS  Google Scholar 

  35. A.A. Slavnov, J. High Energy Phys. 0808, 047 (2008). arXiv:0807.1795 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  36. A. Quadri, A.A. Slavnov, J. High Energy Phys. 1007, 087 (2010). arXiv:1002.2490 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  37. D. Zwanziger, Nucl. Phys. B 518, 237 (1998)

    Article  ADS  MATH  Google Scholar 

  38. M.A.L. Capri, A.J. Gomez, M.S. Guimaraes, V.E.R. Lemes, S.P. Sorella, D.G. Tedesco, Phys. Rev. D 82, 105019 (2010). arXiv:1009.4135 [hep-th]

    Article  ADS  Google Scholar 

  39. Work in progress

  40. D. Dudal, J.A. Gracey, V.E.R. Lemes, R.F. Sobreiro, S.P. Sorella, R. Thibes, H. Verschelde, J. High Energy Phys. 0507, 059 (2005). hep-th/0505037

    Article  MathSciNet  ADS  Google Scholar 

  41. M.A.L. Capri, R.F. Sobreiro, S.P. Sorella, Phys. Rev. D 73, 041701 (2006). hep-th/0512096

    Article  MathSciNet  ADS  Google Scholar 

  42. M.A.L. Capri, R.F. Sobreiro, S.P. Sorella, R. Thibes, Ann. Phys. 322, 1776 (2007). hep-th/0607117

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. G. Curci, R. Ferrari, Nuovo Cimento A 35, 1 (1976). Erratum, ibid. 47, 555 (1978)

    Article  ADS  Google Scholar 

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Acknowledgements

The authors are grateful to M.A.L. Capri, M.S. Guimarães, D. Dudal, S.A. Dias and L. Bonora for very useful discussions. The Conselho Nacional de Desenvolvimento Científico e TecnológicoFootnote 12 (CNPq-Brazil), The Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and the Pró-Reitoria de Pesquisa, Pós-Graduação e Inovação (PROPPI-UFF) are acknowledged for financial support.

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  1. Instituto de Física, Campus da Praia Vermelha, UFF—Universidade Federal Fluminense, Avenida General Milton Tavares de Souza s/n, 24210-346, Niterói, RJ, Brazil

    Antônio D. Pereira Jr. & Rodrigo F. Sobreiro

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  1. Antônio D. Pereira Jr.
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  2. Rodrigo F. Sobreiro
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Correspondence to Rodrigo F. Sobreiro.

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Pereira, A.D., Sobreiro, R.F. On the elimination of infinitesimal Gribov ambiguities in non-Abelian gauge theories. Eur. Phys. J. C 73, 2584 (2013). https://doi.org/10.1140/epjc/s10052-013-2584-6

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