Few-Body Systems

, 46:229 | Cite as

Gauge Invariance of a Critical Number of Flavours in QED3

  • A. Bashir
  • A. Raya
  • S. Sánchez-Madrigal
  • C. D. Roberts


The fermion propagator in an arbitrary covariant gauge can be obtained from the Landau gauge result via a Landau–Khalatnikov–Fradkin transformation. This transformation can be written in a practically useful form in both configuration and momentum space. It is therefore possible to anticipate effects of a gauge transformation on the propagator’s analytic properties. These facts enable one to establish that if a critical number of flavours for chiral symmetry restoration and deconfinement exists in noncompact QED3, then its value is independent of the gauge parameter. This is explicated using simple forms for the fermion–photon vertex and the photon vacuum polarisation. The illustration highlights pitfalls that must be avoided in order to arrive at valid conclusions. Landau gauge is seen to be the covariant gauge in which the propagator avoids modification by a non-dynamical gauge-dependent exponential factor, whose presence can obscure truly observable features of the theory.


  1. 1.
    Göpfert M., Mack G.: Proof of confinement of static quarks in three-dimensional U(1) lattice gauge theory for all values of the coupling constant. Commun. Math. Phys. 82, 545 (1981)CrossRefGoogle Scholar
  2. 2.
    Burden C.J., Praschifka J., Roberts C.D.: Photon polarization tensor in three-dimensional quantum electrodynamics. Phys. Rev. D46, 2695–2702 (1992)ADSGoogle Scholar
  3. 3.
    Roberts C.D.: Hadron properties and Dyson-Schwinger equations. Prog. Part. Nucl. Phys. 61, 50–65 (2008)CrossRefADSGoogle Scholar
  4. 4.
    Franz M., Tesanovic Z., Vafek O.: QED3 theory of pairing pseudogap in cuprates I: from d-wave superconductor to antiferromagnet via algebraic Fermi liquid. Phys. Rev. B66, 054535 (2002)ADSGoogle Scholar
  5. 5.
    Herbut I.F.: QED3 theory of underdoped high temperature superconductors. Phys. Rev. B66, 094504 (2002)ADSGoogle Scholar
  6. 6.
    Thomas I.O., Hands S.: Chiral symmetry restoration in anisotropic QED(3). Phys. Rev. B75, 134516 (2007)ADSGoogle Scholar
  7. 7.
    Novoselov K.S. et al.: Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197 (2005)CrossRefADSGoogle Scholar
  8. 8.
    Gusynin V.P., Sharapov S.G., Carbotte J.P.: AC conductivity of graphene: from tight-binding model to 2+1-dimensional quantum electrodynamics. Int. J. Mod. Phys. B21, 4611–4658 (2007)ADSGoogle Scholar
  9. 9.
    Krein G., Roberts C.D., Williams A.G.: On the implications of confinement. Int. J. Mod. Phys. A7, 5607–5624 (1992)ADSGoogle Scholar
  10. 10.
    Roberts C.D., Williams A.G.: Dyson-Schwinger equations and their application to hadronic physics. Prog. Part. Nucl. Phys. 33, 477–575 (1994)CrossRefADSGoogle Scholar
  11. 11.
    Roberts C.D., Schmidt S.M.: Dyson-Schwinger equations: density, temperature and continuum strong QCD. Prog. Part. Nucl. Phys. 45, S1–S103 (2000)CrossRefADSGoogle Scholar
  12. 12.
    Maris P.: Confinement and complex singularities in QED in three-dimensions. Phys. Rev. D52, 6087–6097 (1995)MathSciNetADSGoogle Scholar
  13. 13.
    Bashir A., Raya A., Cloët I.C., Roberts C.D.: Regarding confinement and dynamical chiral symmetry breaking in QED3. Phys. Rev. C78, 055201 (2008)ADSGoogle Scholar
  14. 14.
    Chernodub M.N., Ilgenfritz E.-M., Schiller A.: Confinement and the photon propagator in 3D compact QED: a lattice study in Landau gauge at zero and finite temperature. Phys. Rev. D67, 034502 (2003)ADSGoogle Scholar
  15. 15.
    Appelquist T., Nash D., Wijewardhana L.C.R.: Critical behavior in (2+1)-dimensional QED. Phys. Rev. Lett. 60, 2575 (1988)CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Sannino, F.: (Near) Conformal technicolor: What is really new? arXiv:0806.3575 [hep-ph]Google Scholar
  17. 17.
    Gies H., Jaeckel J.: Chiral phase structure of QCD with many flavors. Eur. Phys. J. C46, 433–438 (2006)CrossRefADSGoogle Scholar
  18. 18.
    Kurachi M., Shrock R.: Behavior of the S parameter in the crossover region between walking and QCD-like regimes of an SU(N) gauge theory. Phys. Rev. D74, 056003 (2006)ADSGoogle Scholar
  19. 19.
    Hands S.J., Kogut J.B., Scorzato L., Strouthos C.G.: Non-compact QED(3) with N(f) = 1 and N(f) = 4. Phys. Rev. B70, 104501 (2004)ADSGoogle Scholar
  20. 20.
    Strouthos, C., Kogut, J.B.: The phases of non-compact QED(3). PoS LAT2007:278 (2007)Google Scholar
  21. 21.
    Appelquist T., Cohen A.G., Schmaltz M.: A new constraint on strongly coupled field theories. Phys. Rev. D 60, 045003 (1999)CrossRefADSGoogle Scholar
  22. 22.
    Mavromatos, N.E., Papavassiliou, J.: Novel phases and old puzzles in QED3 and related models. cond-mat/0311421Google Scholar
  23. 23.
    Bashir A., Raya A.: On gauge independent dynamical chiral symmetry breaking. Few-Body Syst. 41, 185–199 (2007)CrossRefADSGoogle Scholar
  24. 24.
    Goecke T., Fischer C.S., Williams R.: Finite volume effects and dynamical chiral symmetry breaking in QED3. Phys. Rev. B79, 064513 (2009)ADSGoogle Scholar
  25. 25.
    Landau, L.D., Khalatnikov, I.M.: The gauge transformation of the Green function for charged particles. Sov. Phys. JETP 2, 69 (1956), Zh. Eksp. Teor. Fiz. 29, 89 (1955)Google Scholar
  26. 26.
    Fradkin E.S.: Concerning some general relations of quantum electrodynamics. Zh. Eksp. Teor. Fiz. 29, 258–261 (1955)MathSciNetGoogle Scholar
  27. 27.
    Johnson K., Zumino B.: Gauge dependence of the wave-function renormalization constant in quantum electrodynamics. Phys. Rev. Lett. 3, 351–352 (1959)MATHCrossRefADSGoogle Scholar
  28. 28.
    Zumino B.: Gauge properties of propagators in quantum electrodynamics. J. Math. Phys. 1, 1–7 (1960)MATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Bashir A.: Nonperturbative fermion propagator for the massless quenched QED3. Phys. Lett. B491, 280–284 (2000)ADSGoogle Scholar
  30. 30.
    Bashir A., Delbourgo R.: The nonperturbative propagator and vertex in massless quenched QED(d). J. Phys. A37, 6587–6598 (2004)MathSciNetADSGoogle Scholar
  31. 31.
    Pascual P., Tarrach R.: QCD: renormalization for the practitioner. Lect. Notes Phys. 194, 1–277 (1984)CrossRefADSGoogle Scholar
  32. 32.
    Davydychev A.I., Osland P., Saks L.: Quark gluon vertex in arbitrary gauge and dimension. Phys. Rev. D63, 014022 (2001)ADSGoogle Scholar
  33. 33.
    Bashir A., Raya A.: Landau-Khalatnikov-Fradkin transformations and the fermion propagator in quantum electrodynamics. Phys. Rev. D66, 105005 (2002)ADSGoogle Scholar
  34. 34.
    Bashir A., Raya A.: Dynamical fermion masses and constraints of gauge invariance in quenched QED3. Nucl. Phys. B709, 307–328 (2005)CrossRefADSGoogle Scholar
  35. 35.
    Aitchison I.J.R., Fraser C.M.: Gauge invariance and the effective potential. Ann. Phys. 156, 1 (1984)CrossRefMathSciNetADSGoogle Scholar
  36. 36.
    Elias V., Scadron M., Tarrach R.: Gauge independence of subleading contributions to the operator product pole mass. Phys. Lett. B173, 184 (1986)ADSGoogle Scholar
  37. 37.
    Johnston, D.: Gauge independence of the quark mass pole in perturbative and nonperturbative QCD. Unpublished, LPTHE Orsay 86/49 (1986)Google Scholar
  38. 38.
    Ball J.S., Chiu T.-W.: Analytic properties of the vertex function in gauge theories. 1. Phys. Rev. D22, 2542 (1980)ADSGoogle Scholar
  39. 39.
    Burden C.J., Roberts C.D.: Light cone regular vertex in quenched QED in three-dimensions. Phys. Rev. D44, 540–550 (1991)ADSGoogle Scholar
  40. 40.
    Burden C.J., Roberts C.D.: Gauge covariance and the fermion–photon vertex in three-dimensional and four-dimensional, massless quantum electrodynamics. Phys. Rev. D47, 5581–5588 (1993)ADSGoogle Scholar
  41. 41.
    Dong Z.-H., Munczek H.J., Roberts C.D.: Gauge covariant fermion propagator in quenched, chirally symmetric quantum electrodynamics. Phys. Lett. B333, 536–544 (1994)ADSGoogle Scholar
  42. 42.
    Bashir A., Pennington M.R.: Gauge independent chiral symmetry breaking in quenched QED. Phys. Rev. D50, 7679–7689 (1994)ADSGoogle Scholar
  43. 43.
    Hawes F.T., Williams A.G., Roberts C.D.: Renormalization and chiral symmetry breaking in quenched QED in arbitrary covariant gauge. Phys. Rev. D54, 5361–5372 (1996)ADSGoogle Scholar
  44. 44.
    Maris P.: The influence of the full vertex and vacuum polarization on the fermion propagator in QED3. Phys. Rev. D54, 4049–4058 (1996)ADSGoogle Scholar
  45. 45.
    Bashir A., Raya A.: Constructing the fermion boson vertex in QED3. Phys. Rev. D64, 105001 (2001)ADSGoogle Scholar
  46. 46.
    Kızılersü A., Pennington M.R.: Building the full fermion-photon vertex of QED by imposing multiplicative renormalizability of the Schwinger-Dyson equations for the fermion and photon propagators. Phys. Rev. D79, 125020 (2009)ADSGoogle Scholar
  47. 47.
    Fischer C.S., Alkofer R., Dahm T., Maris P.: Dynamical chiral symmetry breaking in unquenched QED(3). Phys. Rev. D70, 073007 (2004)ADSGoogle Scholar
  48. 48.
    Hawes F.T., Roberts C.D., Williams A.G.: Dynamical chiral symmetry breaking and confinement with an infrared vanishing gluon propagator. Phys. Rev. D49, 4683–4693 (1994)ADSGoogle Scholar
  49. 49.
    Hollenberg L.C.L., Roberts C.D., McKellar B.H.J.: Two loop calculation of the ω-ρ mass splitting. Phys. Rev. C46, 2057–2065 (1992)ADSGoogle Scholar
  50. 50.
    Bender A., Blaschke D., Kalinovsky Y., Roberts C.D.: Continuum study of deconfinement at finite temperature. Phys. Rev. Lett. 77, 3724–3727 (1996)CrossRefADSGoogle Scholar
  51. 51.
    Bender A., Poulis G.I., Roberts C.D., Schmidt S.M., Thomas A.W.: Deconfinement at finite chemical potential. Phys. Lett. B 431, 263–269 (1998)CrossRefADSGoogle Scholar
  52. 52.
    Chen H. et al.: Chemical potential and the gap equation. Phys. Rev. D78, 116015 (2008)ADSGoogle Scholar
  53. 53.
    Blaschke D., Roberts C.D., Schmidt S.M.: Thermodynamic properties of a simple, confining model. Phys. Lett. B425, 232–238 (1998)ADSGoogle Scholar
  54. 54.
    Sinclair D.K.: Separating the scales of confinement and chiral-symmetry breaking in lattice QCD with fundamental quarks. Phys. Rev. D78, 054512 (2008)ADSGoogle Scholar
  55. 55.
    Binosi D., Papavassiliou J.: Gauge-invariant truncation scheme for the Schwinger-Dyson equations of QCD. Phys. Rev. D77, 061702 (2008)ADSGoogle Scholar
  56. 56.
    Chang, L., Roberts, C.D.: Sketching the Bethe-Salpeter kernel. Phys. Rev. Lett. 103, 081601 (4 pages) (2009)Google Scholar

Copyright information

© US Government 2009

Authors and Affiliations

  • A. Bashir
    • 1
  • A. Raya
    • 1
  • S. Sánchez-Madrigal
    • 1
  • C. D. Roberts
    • 2
    • 3
  1. 1.Instituto de Física y MatemáticasUniversidad Michoacana de San Nicolás de HidalgoMoreliaMexico
  2. 2.Physics DivisionArgonne National LaboratoryArgonneUSA
  3. 3.Department of PhysicsPeking UniversityBeijingChina

Personalised recommendations