Skip to main content
SpringerLink
Log in

Two- and three-point functions in two-dimensional Landau-gauge Yang-Mills theory: continuum results

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We investigate the Dyson-Schwinger equations for the gluon and ghost propagators and the ghost-gluon vertex of Landau-gauge gluodynamics in two dimensions. While this simplifies some aspects of the calculations as compared to three and four dimensions, new complications arise due to a mixing of different momentum regimes. As a result, the solutions for the propagators are more sensitive to changes in the three-point functions and the ansätze used for them at the leading order in a vertex expansion. Here, we therefore go beyond this common truncation by including the ghost-gluon vertex self-consistently for the first time, while using a model for the three-gluon vertex which reproduces the known infrared asymptotics and the zeros at intermediate momenta as observed on the lattice. A separate computation of the three-gluon vertex from the results is used to confirm the stability of this behavior a posteriori. We also present further arguments for the absence of the decoupling solution in two dimensions. Finally, we show how in general the infrared exponent κ of the scaling solutions in two, three and four dimensions can be changed by allowing an angle dependence and thus an essential singularity of the ghost-gluon vertex in the infrared.

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. L. von Smekal, A. Hauck and R. Alkofer, A solution to coupled Dyson-Schwinger equations for gluons and ghosts in Landau gauge, Annals Phys. 267 (1998) 1 [Erratum ibid. 269 (1998) 182] [hep-ph/9707327] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  2. L. von Smekal, R. Alkofer and A. Hauck, The infrared behavior of gluon and ghost propagators in Landau gauge QCD, Phys. Rev. Lett. 79 (1997) 3591 [hep-ph/9705242] [INSPIRE].

    Article  ADS  Google Scholar 

  3. D. Zwanziger, Nonperturbative Landau gauge and infrared critical exponents in QCD, Phys. Rev. D 65 (2002) 094039 [hep-th/0109224] [INSPIRE].

    ADS  Google Scholar 

  4. C. Lerche and L. von Smekal, On the infrared exponent for gluon and ghost propagation in Landau gauge QCD, Phys. Rev. D 65 (2002) 125006 [hep-ph/0202194] [INSPIRE].

    ADS  Google Scholar 

  5. D. Zwanziger, Time independent stochastic quantization, DS equations and infrared critical exponents in QCD, Phys. Rev. D 67 (2003) 105001 [hep-th/0206053] [INSPIRE].

    ADS  Google Scholar 

  6. C. Fischer and R. Alkofer, Infrared exponents and running coupling of SU(N ) Yang-Mills theories, Phys. Lett. B 536 (2002) 177 [hep-ph/0202202] [INSPIRE].

    ADS  Google Scholar 

  7. J.M. Pawlowski, D.F. Litim, S. Nedelko and L. von Smekal, Infrared behavior and fixed points in Landau gauge QCD, Phys. Rev. Lett. 93 (2004) 152002 [hep-th/0312324] [INSPIRE].

    Article  ADS  Google Scholar 

  8. D. Zwanziger, Nonperturbative Faddeev-Popov formula and infrared limit of QCD, Phys. Rev. D 69 (2004) 016002 [hep-ph/0303028] [INSPIRE].

    ADS  Google Scholar 

  9. R. Alkofer, C.S. Fischer and F.J. Llanes-Estrada, Vertex functions and infrared fixed point in Landau gauge SU(N ) Yang-Mills theory, Phys. Lett. B 611 (2005) 279 [Erratum ibid. 670 (2009)460-461] [hep-th/0412330] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  10. P. Silva and O. Oliveira, Gribov copies, lattice QCD and the gluon propagator, Nucl. Phys. B 690 (2004)177 [hep-lat/0403026] [INSPIRE].

    Article  ADS  Google Scholar 

  11. I. Bogolubsky, E. Ilgenfritz, M. Muller-Preussker and A. Sternbeck, The Landau gauge gluon and ghost propagators in 4d SU(3) gluodynamics in large lattice volumes, PoS LAT2007 (2007)290 [arXiv:0710.1968] [INSPIRE].

  12. O. Oliveira and P. Silva, Infrared gluon and ghost propagators exponents from lattice QCD, Eur. Phys. J. C 62 (2009) 525 [arXiv:0705.0964] [INSPIRE].

    Article  ADS  Google Scholar 

  13. A. Cucchieri and T. Mendes, What’s up with IR gluon and ghost propagators in Landau gauge? a puzzling answer from huge lattices, PoS LAT2007 (2007) 297 [arXiv:0710.0412] [INSPIRE].

  14. D. Dudal, S. Sorella, N. Vandersickel and H. Verschelde, New features of the gluon and ghost propagator in the infrared region from the Gribov-Zwanziger approach, Phys. Rev. D 77 (2008)071501 [arXiv:0711.4496] [INSPIRE].

    ADS  Google Scholar 

  15. D. Dudal, J.A. Gracey, S.P. Sorella, N. Vandersickel and H. Verschelde, A refinement of the Gribov-Zwanziger approach in the Landau gauge: infrared propagators in harmony with the lattice results, Phys. Rev. D 78 (2008) 065047 [arXiv:0806.4348] [INSPIRE].

    ADS  Google Scholar 

  16. P. Boucaud, J.-P. Leroy, A.L. Yaouanc, J. Micheli, O. Pene, et al., IR finiteness of the ghost dressing function from numerical resolution of the ghost SD equation, JHEP 06 (2008) 012 [arXiv:0801.2721] [INSPIRE].

    Article  ADS  Google Scholar 

  17. A. Aguilar, D. Binosi and J. Papavassiliou, Gluon and ghost propagators in the Landau gauge: deriving lattice results from Schwinger-Dyson equations, Phys. Rev. D 78 (2008) 025010 [arXiv:0802.1870] [INSPIRE].

    ADS  Google Scholar 

  18. A. Cucchieri, A. Maas and T. Mendes, Three-point vertices in Landau-gauge Yang-Mills theory, Phys. Rev. D 77 (2008) 094510 [arXiv:0803.1798] [INSPIRE].

    ADS  Google Scholar 

  19. R. Alkofer, M.Q. Huber and K. Schwenzer, Infrared behavior of three-point functions in Landau gauge Yang-Mills theory, Eur. Phys. J. C 62 (2009) 761 [arXiv:0812.4045] [INSPIRE].

    Article  ADS  Google Scholar 

  20. R. Alkofer, M.Q. Huber and K. Schwenzer, Infrared singularities in Landau gauge Yang-Mills theory, Phys. Rev. D 81 (2010) 105010 [arXiv:0801.2762] [INSPIRE].

    ADS  Google Scholar 

  21. C.S. Fischer, A. Maas and J.M. Pawlowski, On the infrared behavior of Landau gauge Yang-Mills theory, Annals Phys. 324 (2009) 2408 [arXiv:0810.1987] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. L. von Smekal, Landau gauge QCD: functional methods versus lattice simulations, plenary talk at the 13th International Conference on Selected Problems of Modern Theoretical Physics (SPMTP 08), Dubna Russia, 23-27 June 2008 [arXiv:0812.0654] [INSPIRE].

  23. C.S. Fischer and J.M. Pawlowski, Uniqueness of infrared asymptotics in Landau gauge Yang-Mills theory II, Phys. Rev. D 80 (2009) 025023 [arXiv:0903.2193] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  24. M.Q. Huber, R. Alkofer and S.P. Sorella, Infrared analysis of Dyson-Schwinger equations taking into account the Gribov horizon in Landau gauge, Phys. Rev. D 81 (2010) 065003 [arXiv:0910.5604] [INSPIRE].

    ADS  Google Scholar 

  25. D. Zwanziger, Goldstone bosons and fermions in QCD, Phys. Rev. D 81 (2010) 125027 [arXiv:1003.1080] [INSPIRE].

    ADS  Google Scholar 

  26. D. Binosi and J. Papavassiliou, Pinch technique: theory and applications, Phys. Rept. 479 (2009)1 [arXiv:0909.2536] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. P. Boucaud, J. Leroy, A.L. Yaouanc, J. Micheli, O. Pene, et al., The infrared behaviour of the pure Yang-Mills Green functions, Few Body Syst. 53 (2012) 387 [arXiv:1109.1936] [INSPIRE].

    Article  ADS  Google Scholar 

  28. A. Maas, Describing gauge bosons at zero and finite temperature, arXiv:1106.3942 [accepted for publication by Phys. Rep.] [INSPIRE].

  29. N. Vandersickel and D. Zwanziger, The Gribov problem and QCD dynamics, arXiv:1202.1491 [INSPIRE].

  30. A. Maas, J. Wambach, B. Gruter and R. Alkofer, High-temperature limit of Landau-gauge Yang-Mills theory, Eur. Phys. J. C 37 (2004) 335 [hep-ph/0408074] [INSPIRE].

    Article  ADS  Google Scholar 

  31. M.Q. Huber, R. Alkofer, C.S. Fischer and K. Schwenzer, The infrared behavior of Landau gauge Yang-Mills theory in d = 2, D = 3 and D = 4 dimensions, Phys. Lett. B 659 (2008) 434 [arXiv:0705.3809] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  32. D. Dudal, J. Gracey, S. Sorella, N. Vandersickel and H. Verschelde, The Landau gauge gluon and ghost propagator in the refined Gribov-Zwanziger framework in 3 dimensions, Phys. Rev. D 78 (2008) 125012 [arXiv:0808.0893] [INSPIRE].

    ADS  Google Scholar 

  33. A. Aguilar, D. Binosi and J. Papavassiliou, Nonperturbative gluon and ghost propagators for D=3 Yang-Mills,Phys. Rev. D 81(2010)125025 [arXiv:1004.2011][INSPIRE].

    ADS  Google Scholar 

  34. A. Maas, Two and three-point Green’s functions in two-dimensional Landau-gauge Yang-Mills theory, Phys. Rev. D 75 (2007) 116004 [arXiv:0704.0722] [INSPIRE].

    ADS  Google Scholar 

  35. D. Dudal, S. Sorella, N. Vandersickel and H. Verschelde, The effects of Gribov copies in 2D gauge theories, Phys. Lett. B 680 (2009) 377 [arXiv:0808.3379] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  36. A. Cucchieri and T. Mendes, The saga of Landau-gauge propagators: gathering new ammo, AIP Conf. Proc. 1343 (2011) 185 [arXiv:1101.4779] [INSPIRE].

    Article  ADS  Google Scholar 

  37. A. Cucchieri, D. Dudal and N. Vandersickel, The no-pole condition in Landau gauge: properties of the Gribov ghost form-factor and a constraint on the 2D gluon propagator, Phys. Rev. D 85 (2012) 085025 [arXiv:1202.1912] [INSPIRE].

    ADS  Google Scholar 

  38. J.M. Pawlowski, The QCD phase diagram: results and challenges, AIP Conf. Proc. 1343 (2011)75 [arXiv:1012.5075] [INSPIRE].

    Article  ADS  Google Scholar 

  39. J. Berges, N. Tetradis and C. Wetterich, Nonperturbative renormalization flow in quantum field theory and statistical physics, Phys. Rept. 363 (2002) 223 [hep-ph/0005122] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. J.M. Pawlowski, Aspects of the functional renormalisation group, Annals Phys. 322 (2007) 2831 [hep-th/0512261] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. H. Gies, Introduction to the functional RG and applications to gauge theories, presented at the ECT* School on Renormalization Group and Effective Field Theory Approaches to Many-Body Systems, Trento Italy, 27 February - 10 March 2006 [hep-ph/0611146] [INSPIRE].

  42. O.J. Rosten, Fundamentals of the exact renormalization group, Phys. Rept. 511 (2012) 177 [arXiv:1003.1366] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  43. C.D. Roberts and A.G. Williams, Dyson-Schwinger equations and their application to hadronic physics, Prog. Part. Nucl. Phys. 33 (1994) 477 [hep-ph/9403224] [INSPIRE].

    Article  ADS  Google Scholar 

  44. C.D. Roberts and S.M. Schmidt, Dyson-Schwinger equations: density, temperature and continuum strong QCD, Prog. Part. Nucl. Phys. 45 (2000) S1 [nucl-th/0005064] [INSPIRE].

    Article  ADS  Google Scholar 

  45. R. Alkofer and L. von Smekal, The infrared behavior of QCD Green’s functions: confinement dynamical symmetry breaking and hadrons as relativistic bound states, Phys. Rept. 353 (2001)281 [hep-ph/0007355] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  46. C.S. Fischer, Infrared properties of QCD from Dyson-Schwinger equations, J. Phys. G 32 (2006) R253 [hep-ph/0605173] [INSPIRE].

    ADS  Google Scholar 

  47. R. Alkofer, M.Q. Huber and K. Schwenzer, Algorithmic derivation of Dyson-Schwinger equations, Comput. Phys. Commun. 180 (2009) 965 [arXiv:0808.2939] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. A. Maas, J.M. Pawlowski, D. Spielmann, A. Sternbeck and L. von Smekal, Strong-coupling study of the Gribov ambiguity in lattice Landau gauge, Eur. Phys. J. C 68 (2010) 183 [arXiv:0912.4203] [INSPIRE].

    Article  ADS  Google Scholar 

  49. A. Cucchieri, D. Dudal, T. Mendes and N. Vandersickel, Modeling the gluon propagator in Landau gauge: lattice estimates of pole masses and dimension-two condensates, Phys. Rev. D 85 (2012)094513 [arXiv:1111.2327] [INSPIRE].

    ADS  Google Scholar 

  50. D. Zwanziger, Some exact properties of the gluon propagator, arXiv:1209.1974 [INSPIRE].

  51. M.Q. Huber and J. Braun, Algorithmic derivation of functional renormalization group equations and Dyson-Schwinger equations, Comput. Phys. Commun. 183 (2012) 1290 [arXiv:1102.5307] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  52. M.Q. Huber, K. Schwenzer and R. Alkofer, On the infrared scaling solution of SU(N ) Yang-Mills theories in the maximally abelian gauge, Eur. Phys. J. C 68 (2010) 581 [arXiv:0904.1873] [INSPIRE].

    Article  ADS  Google Scholar 

  53. D. Dudal, O. Oliveira and J. Rodriguez-Quintero, Nontrivial ghost-gluon vertex and the match of RGZ, DSE and lattice Yang-Mills propagators, arXiv:1207.5118 [INSPIRE].

  54. S. Wolfram, The Mathematica book, Wolfram Media and Cambridge University Press, Cambridge U.K. (2004).

    Google Scholar 

  55. J. Taylor, Ward identities and charge renormalization of the Yang-Mills field, Nucl. Phys. B 33 (1971)436 [INSPIRE].

    Article  ADS  Google Scholar 

  56. C. Fischer, R. Alkofer and H. Reinhardt, The elusiveness of infrared critical exponents in Landau gauge Yang-Mills theories, Phys. Rev. D 65 (2002) 094008 [hep-ph/0202195] [INSPIRE].

    ADS  Google Scholar 

  57. L. von Smekal, K. Maltman and A. Sternbeck, The strong coupling and its running to four loops in a minimal MOM scheme, Phys. Lett. B 681 (2009) 336 [arXiv:0903.1696] [INSPIRE].

    ADS  Google Scholar 

  58. C.S. Fischer and L. von Smekal, Scaling, decoupling and transversality of the gluon propagator, AIP Conf. Proc. 1343 (2011) 247 [arXiv:1011.6482] [INSPIRE].

    Article  ADS  Google Scholar 

  59. C.S. Fischer, Nonperturbative propagators, running coupling and dynamical mass generation in ghost - Anti-ghost symmetric gauges in QCD, Ph.D. Thesis, Eberhard-Karls-Universität zu Tübingen, Germany (2003) [hep-ph/0304233] [INSPIRE].

  60. B. Alles, D. Henty, H. Panagopoulos, C. Parrinello, C. Pittori, et al., αs from the nonperturbatively renormalised lattice three gluon vertex, Nucl. Phys. B 502 (1997) 325 [hep-lat/9605033] [INSPIRE].

    Article  ADS  Google Scholar 

  61. C.S. Fischer and R. Alkofer, Nonperturbative propagators, running coupling and dynamical quark mass of Landau gauge QCD, Phys. Rev. D 67 (2003) 094020 [hep-ph/0301094] [INSPIRE].

    ADS  Google Scholar 

  62. A. Cucchieri, T. Mendes and A. Mihara, Numerical study of the ghost-gluon vertex in Landau gauge, JHEP 12 (2004) 012 [hep-lat/0408034] [INSPIRE].

    Article  ADS  Google Scholar 

  63. A. Cucchieri, A. Maas and T. Mendes, Exploratory study of three-point Green’s functions in Landau-gauge Yang-Mills theory, Phys. Rev. D 74 (2006) 014503 [hep-lat/0605011] [INSPIRE].

    ADS  Google Scholar 

  64. E.-M. Ilgenfritz, M. Muller-Preussker, A. Sternbeck, A. Schiller and I. Bogolubsky, Landau gauge gluon and ghost propagators from lattice QCD, Braz. J. Phys. 37 (2007) 193 [hep-lat/0609043] [INSPIRE].

    Article  Google Scholar 

  65. W. Schleifenbaum, A. Maas, J. Wambach and R. Alkofer, Infrared behaviour of the ghost-gluon vertex in Landau gauge Yang-Mills theory, Phys. Rev. D 72 (2005) 014017 [hep-ph/0411052] [INSPIRE].

    ADS  Google Scholar 

  66. P. Boucaud, D. Dudal, J. Leroy, O. Pene and J. Rodriguez-Quintero, On the leading OPE corrections to the ghost-gluon vertex and the Taylor theorem, JHEP 12 (2011) 018 [arXiv:1109.3803] [INSPIRE].

    Article  ADS  Google Scholar 

  67. L. Fister and J.M. Pawlowski, Yang-Mills correlation functions at finite temperature, arXiv:1112.5440 [INSPIRE].

  68. M.Q. Huber and M. Mitter, CrasyDSE: a framework for solving Dyson-Schwinger equations, Comput. Phys. Commun. 183 (2012) 2441 [arXiv:1112.5622] [INSPIRE].

    Article  ADS  Google Scholar 

  69. J.C. Bloch, Two loop improved truncation of the ghost gluon Dyson-Schwinger equations: multiplicatively renormalizable propagators and nonperturbative running coupling, Few Body Syst. 33 (2003) 111 [hep-ph/0303125] [INSPIRE].

    Article  ADS  Google Scholar 

  70. R. Alkofer, M.Q. Huber, V. Mader and A. Windisch, On the infrared behaviour of QCD Green functions in the maximally abelian gauge, PoS QCD-TNT-II (2011) 003 [arXiv:1112.6173] [INSPIRE].

  71. D. Binosi and L. Theussl, JaxoDraw: a graphical user interface for drawing Feynman diagrams, Comput. Phys. Commun. 161 (2004) 76 [hep-ph/0309015] [INSPIRE].

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Kernphysik, Technische Universität Darmstadt, Schlossgartenstr. 2, 64289, Darmstadt, Germany

    Markus Q. Huber & Lorenz von Smekal

  2. Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D-07743, Jena, Germany

    Axel Maas

Authors
  1. Markus Q. Huber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Axel Maas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lorenz von Smekal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus Q. Huber.

Additional information

ArXiv ePrint: 1207.0222

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huber, M.Q., Maas, A. & von Smekal, L. Two- and three-point functions in two-dimensional Landau-gauge Yang-Mills theory: continuum results. J. High Energ. Phys. 2012, 35 (2012). https://doi.org/10.1007/JHEP11(2012)035

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP11(2012)035

Keywords

Search

Navigation