Chiral symmetry breaking in a uniform external magnetic field

D.-S. Lee; C. N. Leung; Y. J. Ng

doi:10.1103/PhysRevD.55.6504

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Chiral symmetry breaking in a uniform external magnetic field

D.-S. Lee

C. N. Leung

Y. J. Ng

Institute of Physics, Academia Sinica, Taipei, Taiwan;
Department of Physics, Soochow University, Taipei, Taiwan;
Institute of Field Physics, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, North Carolina 27599

Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716

Institute of Field Physics, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, North Carolina 27599

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Phys. Rev. D 55, 6504 – Published 15 May, 1997

DOI: https://doi.org/10.1103/PhysRevD.55.6504

Abstract

Using the nonperturbative Schwinger-Dyson equation, we show that chiral symmetry is dynamically broken in QED at weak gauge couplings when an external magnetic field is present and that chiral symmetry is restored above a critical temperature which is much smaller than $\sqrt{| eH |}$ , where $H$ is the magnetic-field strength.

References (19)

C. N. Leung, Y. J. Ng, and A. W. Ackley, Phys. Rev. D 54, 4181 (1996).
V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. D 52, 4747 (1995); Nucl. Phys. B462, 249 (1996).
Contribution of V. I. Ritus, in Issues in Intense-Field Quantum Electrodyanamics, edited by V. L. Ginzburg (Nova Science, Commack, New York, 1987).
See, e.g., Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1964).
See, e.g., E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge University Press, Cambridge, England, 1927), pp. 350–351.
D. K. Hong, Y. Kim, and S.-J. Sin, Phys. Rev. D 54, 7879 (1996).
See, e.g., contribution of Y. J. Ng and Y. Kikuchi, in Vacuum Structure in Intense Fields, edited by H. M. Fried and B. Muller (Plenum, New York, 1991); contribution of Y. J. Ng, in Tests of Fundamental Laws in Physics, edited by O. Fackler and J. Tran Thanh Van (Editions Frontières, Gif-sur-Yvette, France, 1989), and references therein.
See, e.g., L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974).
The orthonormality and the completeness relations are replaced, respectively, by $\int_{x} {\bar{E}}_{p^{'}} {(x) E}_{p} (x) = (2 π)^{4} δ ̂_{T}^{(4)} (p$ $- p^{'})$ and $Σ \int_{p} E_{p} (x) {\bar{E}}_{p} (y) = (2 π)^{4} δ^{(4)} (x - y)$ .
J. Schwinger, L. L. DeRaad, Jr., and K. A. Milton, Ann. Phys. (N.Y.) 115, 1 (1978).
For a more accurate estimate of the critical temperature, see the Note added in proof.
See, e.g., M. S. Turner and L. M. Widrow, Phys. Rev. D 37, 2743 (1988); G. Baym, D. Bodeker, and L. McLerran, 53, 662 (1996); A. Brandenburg, K. Enqvist, and P. Olesen, 54, 1291 (1996).
Note that a constant magnetic field, in general, is equivalent to a field configuration with constant and mutually perpendicular electric and magnetic fields in a different inertial frame.
See E. Dagotto and H. W. Wyld, Phys. Lett. B 205, 73 (1988); see also Ref. [7].
W. A. Bardeen, C. N. Leung, and S. T. Love, Phys. Rev. Lett. 56, 1230 (1986); Nucl. Phys. B273, 649 (1986); B323, 493 (1989).
V. P. Gusynin and I. A. Shovkovy (private communication).
D. Ebert and V. Ch. Zhukovsky, .
I. A. Shushpanov and A. V. Smilga, .
V. A. Miransky, Dynamical Symmetry Breaking in Quantum Field Theories (World Scientific, Singapore, 1993).

[c1] C. N. Leung, Y. J. Ng, and A. W. Ackley, Phys. Rev. D 54, 4181 (1996).

[c2] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. D 52, 4747 (1995); Nucl. Phys. B462, 249 (1996).

[c3] Contribution of V. I. Ritus, in Issues in Intense-Field Quantum Electrodyanamics, edited by V. L. Ginzburg (Nova Science, Commack, New York, 1987).

[c4] See, e.g., Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1964).

[c5] See, e.g., E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge University Press, Cambridge, England, 1927), pp. 350–351.

[c6] D. K. Hong, Y. Kim, and S.-J. Sin, Phys. Rev. D 54, 7879 (1996).

[c7] See, e.g., contribution of Y. J. Ng and Y. Kikuchi, in Vacuum Structure in Intense Fields, edited by H. M. Fried and B. Muller (Plenum, New York, 1991); contribution of Y. J. Ng, in Tests of Fundamental Laws in Physics, edited by O. Fackler and J. Tran Thanh Van (Editions Frontières, Gif-sur-Yvette, France, 1989), and references therein.

[c8] See, e.g., L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974).

[c9] The orthonormality and the completeness relations are replaced, respectively, by $\int_{x} {\bar{E}}_{p^{'}} {(x) E}_{p} (x) = (2 π)^{4} δ ̂_{T}^{(4)} (p$ $- p^{'})$ and $Σ \int_{p} E_{p} (x) {\bar{E}}_{p} (y) = (2 π)^{4} δ^{(4)} (x - y)$ .

[c10] J. Schwinger, L. L. DeRaad, Jr., and K. A. Milton, Ann. Phys. (N.Y.) 115, 1 (1978).

[c11] For a more accurate estimate of the critical temperature, see the Note added in proof.

[c12] See, e.g., M. S. Turner and L. M. Widrow, Phys. Rev. D 37, 2743 (1988); G. Baym, D. Bodeker, and L. McLerran, 53, 662 (1996); A. Brandenburg, K. Enqvist, and P. Olesen, 54, 1291 (1996).

[c13] Note that a constant magnetic field, in general, is equivalent to a field configuration with constant and mutually perpendicular electric and magnetic fields in a different inertial frame.

[c14] See E. Dagotto and H. W. Wyld, Phys. Lett. B 205, 73 (1988); see also Ref. [7].

[c15] W. A. Bardeen, C. N. Leung, and S. T. Love, Phys. Rev. Lett. 56, 1230 (1986); Nucl. Phys. B273, 649 (1986); B323, 493 (1989).

[c16] V. P. Gusynin and I. A. Shovkovy (private communication).

[c17] D. Ebert and V. Ch. Zhukovsky, .

[c18] I. A. Shushpanov and A. V. Smilga, .

[c19] V. A. Miransky, Dynamical Symmetry Breaking in Quantum Field Theories (World Scientific, Singapore, 1993).

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