Consistent quantization and symmetry structure of a non-Abelian chiral gauge theory

Ken-ichi Shizuya

doi:10.1103/PhysRevD.40.1260

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Consistent quantization and symmetry structure of a non-Abelian chiral gauge theory

Ken-ichi Shizuya

Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11794-3840

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Phys. Rev. D 40, 1260 – Published 15 August, 1989

DOI: https://doi.org/10.1103/PhysRevD.40.1260

Abstract

The SU(N) chiral Schwinger model with a Wess-Zumino term is studied by use of non-Abelian bosonization, the Becchi-Rouet-Stora formalism, and a dual transformation, and it is confirmed that this model is a sensible quantum theory in a certain range of the anomaly parameter a. The SU(N) gauge symmetry restored by the inclusion of the Wess-Zumino term gets spontaneously broken and the gauge field becomes massive. Left-handed fermions are found to be confined while right-handed fermions remain free and massless. For the specific value a=2, the symmetry of the model enlarges [to a U(N)×U(N) Kac-Moody symmetry]. It is shown by fermionization of the Wess-Zumino field that for a=2 this model is equivalent to massless two-dimensional QCD ( ${QCD}_{2}$ ) in the sense that they share the same gauge field and the same left-handed fermions. A dual transformation is used to cast the model into an equivalent nonlinear system of scalar fields only, which reveals the particle spectrum of the model.

References (24)

L. Faddeev, Phys. Lett. 145B, 81 (1984); L. Faddeev and S. Shatashvili, ibid. 167B, 225 (1986).
J. Wess and B. Zumino, Phys. Lett. 37B, 95 (1971).
R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54, 1219 (1985); ibid. 54, 2060(E) (1985); R. Rajaraman, Phys. Lett. 154B, 305 (1985).
R. Rajaraman, Phys. Lett. 162B, 148 (1985).
I. Halliday, E. Rabinovici, A. Schwimmer and M. Chanowitz, Nucl. Phys. B268, 413 (1986); M. Chanowitz, Phys. Lett. B 171, 280 (1986); H. Girotti, M. Rothe and K. Rothe, Phys. Rev. D 33, 514 (1986); ibid. 34, 592 (1986); D. Boyanovsky, Nucl. Phys. B294, 223 (1987).
O. Babelon, F. Schaposnik and C. Viallet, Phys. Lett. B 177, 385 (1986); K. Harada and I. Tsutsui, ibid. 183, 311 (1986); R. Banerjee, Phys. Rev. Lett. 56, 1889 (1986); J. Webb, Z Phys. C 31, 301 (1986).
S. Miyake and K. Shizuya, Phys. Rev. D 36, 3781 (1987); ibid. 37, 2282 (1988).
K. Shizuya, Phys. Lett. B 213, 298 (1988).
S. Miyake and K. Shizuya, Mod. Phys. Lett. A 4, 775 (1989).
K. Funakubo and T. Kashiwa, Phys. Rev. Lett. 60, 2113 (1988); S. Aoki, BNL report, 1987 (unpublished); T. Berger, N. Falck, and G. Kramer, Report No. DESY 88-009, 1988 (unpublished).
E. Witten, Commun. Math. Phys. 92, 455 (1984).
A. Polyakov and P. Wiegmann, Phys. Lett. 131B, 121 (1983); V. Knizhnik and A. Zamolodchikov, Nucl. Phys. B247, 83 (1984).
C. Becchi, A. Rouet and R. Stora, Ann. Phys. (N.Y.) 98, 287 (1976); T. Kugo and I. Ojima, Suppl. Prog. Theor. Phys. 66, 1 (1978), and references therein.
A. Sugamoto, Phys. Rev. D 19, 1820 (1979), and earlier references therein.
K. Fujikawa, Phys. Rev. Lett. 42, 1195 (1979); Phys. Rev. D 21, 2848 (1980).
W. Bardeen, Phys. Rev. 184, 1848 (1969); W. Bardeen and B. Zumino, Nucl. Phys. B244, 421 (1984).
P. Goddard and D. Olive, Int. J. Mod. Phys. A 1, 303 (1986).
S. Coleman, Commun. Math. Phys. 31, 259 (1973).
B. Klaiber, 1976 Boulder Lectures in Theoretical Physics (Gordon and Breach, New York, 1968), p. 141; N. Nakanishi, Prog. Theor. Phys. 57, 269 (1977).
J. Schwinger, Phys. Rev. Lett. 3, 296 (1959); T. Goto and I. Imamura, Prog. Theor. Phys. 14, 196 (1955).
G.'t Hooft, Nucl. Phys. B75, 461 (1974); E. Witten, ibid. B160, 57 (1979).
In deriving this equivalent theory within the path-integral framework, one may include the source term $J^{μ}$ $A_{mu}$ and integrate over $A_{μ}^{sprime}$ . Then the resulting action is at most quadratic in $J_{mu}$ . The linear term gives the identification of $A_{mu}$ as in Eq. (5.5). The quadratic term is of the form - 1/2 $κ^{- 2}$ $δ^{2}$ (0) int $d^{2}$ x JG [ φ ] J. On the other hand, the $A_{μ}^{sprime}$ inte- gration yields a term prop $δ^{2}$ (0) tr ( ln G [ φ ]). As is well known, such $δ^{2}$ (0) terms arise from the use of the $T^{star}$ products and they eventually cancel in Feynman-diagram calculations.
Omitted end note.
To establish renormalizability a careful analysis based on the Ward-Takahashi identities is needed; we do not address this problem here.

[c1] L. Faddeev, Phys. Lett. 145B, 81 (1984); L. Faddeev and S. Shatashvili, ibid. 167B, 225 (1986).

[c2] J. Wess and B. Zumino, Phys. Lett. 37B, 95 (1971).

[c3] R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54, 1219 (1985); ibid. 54, 2060(E) (1985); R. Rajaraman, Phys. Lett. 154B, 305 (1985).

[c4] R. Rajaraman, Phys. Lett. 162B, 148 (1985).

[c5] I. Halliday, E. Rabinovici, A. Schwimmer and M. Chanowitz, Nucl. Phys. B268, 413 (1986); M. Chanowitz, Phys. Lett. B 171, 280 (1986); H. Girotti, M. Rothe and K. Rothe, Phys. Rev. D 33, 514 (1986); ibid. 34, 592 (1986); D. Boyanovsky, Nucl. Phys. B294, 223 (1987).

[c6] O. Babelon, F. Schaposnik and C. Viallet, Phys. Lett. B 177, 385 (1986); K. Harada and I. Tsutsui, ibid. 183, 311 (1986); R. Banerjee, Phys. Rev. Lett. 56, 1889 (1986); J. Webb, Z Phys. C 31, 301 (1986).

[c7] S. Miyake and K. Shizuya, Phys. Rev. D 36, 3781 (1987); ibid. 37, 2282 (1988).

[c8] K. Shizuya, Phys. Lett. B 213, 298 (1988).

[c9] S. Miyake and K. Shizuya, Mod. Phys. Lett. A 4, 775 (1989).

[c10] K. Funakubo and T. Kashiwa, Phys. Rev. Lett. 60, 2113 (1988); S. Aoki, BNL report, 1987 (unpublished); T. Berger, N. Falck, and G. Kramer, Report No. DESY 88-009, 1988 (unpublished).

[c11] E. Witten, Commun. Math. Phys. 92, 455 (1984).

[c12] A. Polyakov and P. Wiegmann, Phys. Lett. 131B, 121 (1983); V. Knizhnik and A. Zamolodchikov, Nucl. Phys. B247, 83 (1984).

[c13] C. Becchi, A. Rouet and R. Stora, Ann. Phys. (N.Y.) 98, 287 (1976); T. Kugo and I. Ojima, Suppl. Prog. Theor. Phys. 66, 1 (1978), and references therein.

[c14] A. Sugamoto, Phys. Rev. D 19, 1820 (1979), and earlier references therein.

[c15] K. Fujikawa, Phys. Rev. Lett. 42, 1195 (1979); Phys. Rev. D 21, 2848 (1980).

[c16] W. Bardeen, Phys. Rev. 184, 1848 (1969); W. Bardeen and B. Zumino, Nucl. Phys. B244, 421 (1984).

[c17] P. Goddard and D. Olive, Int. J. Mod. Phys. A 1, 303 (1986).

[c18] S. Coleman, Commun. Math. Phys. 31, 259 (1973).

[c19] B. Klaiber, 1976 Boulder Lectures in Theoretical Physics (Gordon and Breach, New York, 1968), p. 141; N. Nakanishi, Prog. Theor. Phys. 57, 269 (1977).

[c20] J. Schwinger, Phys. Rev. Lett. 3, 296 (1959); T. Goto and I. Imamura, Prog. Theor. Phys. 14, 196 (1955).

[c21] G.'t Hooft, Nucl. Phys. B75, 461 (1974); E. Witten, ibid. B160, 57 (1979).

[c22] In deriving this equivalent theory within the path-integral framework, one may include the source term $J^{μ}$ $A_{mu}$ and integrate over $A_{μ}^{sprime}$ . Then the resulting action is at most quadratic in $J_{mu}$ . The linear term gives the identification of $A_{mu}$ as in Eq. (5.5). The quadratic term is of the form - 1/2 $κ^{- 2}$ $δ^{2}$ (0) int $d^{2}$ x JG [ φ ] J. On the other hand, the $A_{μ}^{sprime}$ inte- gration yields a term prop $δ^{2}$ (0) tr ( ln G [ φ ]). As is well known, such $δ^{2}$ (0) terms arise from the use of the $T^{star}$ products and they eventually cancel in Feynman-diagram calculations.

[c23] Omitted end note.

[c24] To establish renormalizability a careful analysis based on the Ward-Takahashi identities is needed; we do not address this problem here.

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Consistent quantization and symmetry structure of a non-Abelian chiral gauge theory

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