Uniqueness of quark and lepton representations in the standard model from the anomalies viewpoint

C. Q. Geng; R. E. Marshak

doi:10.1103/PhysRevD.39.693

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Uniqueness of quark and lepton representations in the standard model from the anomalies viewpoint

C. Q. Geng

R. E. Marshak

TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, Canada V6T 2A3

Physics Department, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

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Phys. Rev. D 39, 693 – Published 15 January, 1989

DOI: https://doi.org/10.1103/PhysRevD.39.693

Abstract

The uniqueness of the Weyl representations of the standard gauge group is reexamined. We find that, prior to spontaneous breaking of the electroweak subgroup, the minimal Weyl representations and their charges are uniquely determined by insisting on all three known chiral gauge anomaly-free conditions in four dimensions: (1) cancellation of triangular anomalies; (2) absence of the global SU(2) anomaly; and (3) cancellation of the mixed-gauge-gravitational anomaly. The uniqueness question for the left-right-symmetric group and the simple (grand-unified-theory) group are discussed from the anomalies viewpoint.

Comments & Replies

Reply to "Comment on anomaly cancellation in the standard model"

C. Q. Geng and R. E. Marshak

Phys. Rev. D 41, 717 (1990)

Comment on anomaly cancellation in the standard model

J. A. Minahan, P. Ramond, and R. C. Warner

Phys. Rev. D 41, 715 (1990)

References (17)

S. L. Glashow, Nucl. Phys. 22, 579 (1961); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1964); A. Salam, in Elementary Particle Theory: Relativistic Groups and Analyticity (Nobel Symposium No. 8), edited by N. Svartholm (Almqvist and Wiksell, Stockholm, 1968).
H. Fritzsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. 47B, 365 (1973); S. Weinberg, Phys. Rev. Lett. 31, 494 (1973); D. Gross and F. Wilczek, Phys. Rev. D 8, 3633 (1973).
S. Adler, Phys. Rev. 177, 2426 (1969); J. Bell and R. Jackiw, Nuovo Cimento 51A, 47 (1969).
E. Witten, Phys. Lett. 117B, 324 (1982).
R. Delbourgo and A. Salam, Phys. Lett. 40B, 381 (1972); see T. Eguchi and P. Freund, Phys. Rev. Lett. 37, 1251 (1976), who also discuss ``gravitational instantons.''
L. Alvarez-Gaumé and E. Witten, Nucl. Phys. B234, 269 (1983).
C. Bouchiat, J. Illiopoulos and Ph. Meyer, Phys. Lett. 38B, 519 (1972).
D. J. Gross and R. Jackiw, Phys. Rev. D 6, 477 (1972).
S. Adler, Phys. Rev. 177, 2426 (1969); J. Bell and R. Jackiw, Nuovo Cimento 51A, 47 (1969); W. Bardeen, Phys. Rev. 184, 1848 (1969).
One doublet fermion cannot form a mass term without breaking SU(2) since 2 vec times 2 vec -> 1 ${vec}_{A}$ + 3 ${vec}_{S}$ where A and S stand for antisymmetric and symmetric, respectively.
For reviews, see S. L. Glashow, in Quarks and Leptons, proceedings of the Cargése Summer Institute, Cargése, 1979, edited by Jean-Luis Basdevant et al. (NATO Advanced Study Institute Series, Series B: Physics, Vol. 61) (Plenum, New York, 1980); in Fundamental Interactions: Cargése 1981, edited by J. S. Basdevant et al. (NATO Advanced Study Institute, Series B, Physics: Vol. 85) (Plenum, New York, 1982). In the second reference, Glashow remarks (p. 674): ``At this pre-unification level, the curious values of quark and lepton charges remain mysterious.''
Adding arbitrary nonfundamental and singlet representations in (1) does not change our conclusions.
It has been pointed out [H. Georgi (private communication)] that the U(1) $_{Y}$ charges can be fixed by considering the fermion mass generated by the breaking of SU(2) ${}_{L}{times}$ U (1 $)_{Y}$ to U(1) $_{EM}$ . That is true but we are interested in fixing the U(1) $_{Y}$ charges prior to breaking and we show how the mixed anomaly-free condition achieves this.
J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974); R. N. Mohapatra and J. C. Pati, ibid. 11, 566 (1975); ibid. 11, 2559 (1975); G. Senjanović and R. N. Mohapatra, ibid. 12, 1502 (1975); R. E. Marshak and R. N. Mohapatra, Phys. Lett. 91B, 222 (1980); M. A. B. Bég and A. Sirlin, Phys. Rep. 88, 1 (1982).
Cf. R. N. Mohapatra, Unification and Supersymmetry (Springer, New York, 1986).
C. Q. Geng, R. E. Marshak, Z. Y. Zhao and S. Okubo, Phys. Rev. D 36, 1953 (1987); S. Okubo, C. Q. Geng, R. E. Marshak and Z. Y. Zhao, ibid. 36, 3268 (1987). These papers give a group-theoretic proof; a ``topological'' proof will be found in S. Elitzur and V. P. Nair, Nucl. Phys. B243, 205 (1984).
Cf. R. E. Marshak, Pramana (in press).

[c1] S. L. Glashow, Nucl. Phys. 22, 579 (1961); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1964); A. Salam, in Elementary Particle Theory: Relativistic Groups and Analyticity (Nobel Symposium No. 8), edited by N. Svartholm (Almqvist and Wiksell, Stockholm, 1968).

[c2] H. Fritzsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. 47B, 365 (1973); S. Weinberg, Phys. Rev. Lett. 31, 494 (1973); D. Gross and F. Wilczek, Phys. Rev. D 8, 3633 (1973).

[c3] S. Adler, Phys. Rev. 177, 2426 (1969); J. Bell and R. Jackiw, Nuovo Cimento 51A, 47 (1969).

[c4] E. Witten, Phys. Lett. 117B, 324 (1982).

[c5] R. Delbourgo and A. Salam, Phys. Lett. 40B, 381 (1972); see T. Eguchi and P. Freund, Phys. Rev. Lett. 37, 1251 (1976), who also discuss ``gravitational instantons.''

[c6] L. Alvarez-Gaumé and E. Witten, Nucl. Phys. B234, 269 (1983).

[c7] C. Bouchiat, J. Illiopoulos and Ph. Meyer, Phys. Lett. 38B, 519 (1972).

[c8] D. J. Gross and R. Jackiw, Phys. Rev. D 6, 477 (1972).

[c9] S. Adler, Phys. Rev. 177, 2426 (1969); J. Bell and R. Jackiw, Nuovo Cimento 51A, 47 (1969); W. Bardeen, Phys. Rev. 184, 1848 (1969).

[c10] One doublet fermion cannot form a mass term without breaking SU(2) since 2 vec times 2 vec -> 1 ${vec}_{A}$ + 3 ${vec}_{S}$ where A and S stand for antisymmetric and symmetric, respectively.

[c11] For reviews, see S. L. Glashow, in Quarks and Leptons, proceedings of the Cargése Summer Institute, Cargése, 1979, edited by Jean-Luis Basdevant et al. (NATO Advanced Study Institute Series, Series B: Physics, Vol. 61) (Plenum, New York, 1980); in Fundamental Interactions: Cargése 1981, edited by J. S. Basdevant et al. (NATO Advanced Study Institute, Series B, Physics: Vol. 85) (Plenum, New York, 1982). In the second reference, Glashow remarks (p. 674): ``At this pre-unification level, the curious values of quark and lepton charges remain mysterious.''

[c12] Adding arbitrary nonfundamental and singlet representations in (1) does not change our conclusions.

[c13] It has been pointed out [H. Georgi (private communication)] that the U(1) $_{Y}$ charges can be fixed by considering the fermion mass generated by the breaking of SU(2) ${}_{L}{times}$ U (1 $)_{Y}$ to U(1) $_{EM}$ . That is true but we are interested in fixing the U(1) $_{Y}$ charges prior to breaking and we show how the mixed anomaly-free condition achieves this.

[c14] J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974); R. N. Mohapatra and J. C. Pati, ibid. 11, 566 (1975); ibid. 11, 2559 (1975); G. Senjanović and R. N. Mohapatra, ibid. 12, 1502 (1975); R. E. Marshak and R. N. Mohapatra, Phys. Lett. 91B, 222 (1980); M. A. B. Bég and A. Sirlin, Phys. Rep. 88, 1 (1982).

[c15] Cf. R. N. Mohapatra, Unification and Supersymmetry (Springer, New York, 1986).

[c16] C. Q. Geng, R. E. Marshak, Z. Y. Zhao and S. Okubo, Phys. Rev. D 36, 1953 (1987); S. Okubo, C. Q. Geng, R. E. Marshak and Z. Y. Zhao, ibid. 36, 3268 (1987). These papers give a group-theoretic proof; a ``topological'' proof will be found in S. Elitzur and V. P. Nair, Nucl. Phys. B243, 205 (1984).

[c17] Cf. R. E. Marshak, Pramana (in press).

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Uniqueness of quark and lepton representations in the standard model from the anomalies viewpoint

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