Triviality, anomalies, and canonical electroweak theory at small distances

M. A. B. Bég

doi:10.1103/PhysRevD.39.2373

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Triviality, anomalies, and canonical electroweak theory at small distances

M. A. B. Bég

The Rockefeller University, New York, New York 10021

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Phys. Rev. D 39, 2373 – Published 15 April, 1989

DOI: https://doi.org/10.1103/PhysRevD.39.2373

Abstract

We consider two scenarios for rescuing canonical electroweak theory from the trap of triviality. Finite renormalization of asymptotically nonfree couplings is a feature common to both. The first is formally consistent in that it is cutoff-free; we show, however, that it does not accommodate Adler-Bell-Jackiw anomalies and, in consequence, does not allow the process $π^{0}$ →2γ to proceed at the observed rate. The second entails explicit introduction of a cutoff parameter $Λ_{s}$ to stabilize the theory; here we find, contrary to naive expectation, that the anomalies may be retrieved if $Λ_{s}$ ≫250 GeV and there need be no problem with $π^{0}$ decay. Some implications of these results are discussed.

References (33)

M. A. B. Bég, Comments Nucl. Part. Phys. 17, 119 (1987), and papers cited therein.
For a comprehensive review and all relevant references, see D. J. E. Callaway, Phys. Rep. 167, 241 (1989).
L. D. Landau, Niels Bohr and the Development of Physics (McGraw-Hill, New York, 1955).
M. A. B. Bég, C. Panagiotakopoulos and A. Sirlin, Phys. Rev. Lett. 52, 883 (1984); D. J. E. Callaway, Nucl. Phys. B233, 189 (1984).
D. J. E. Callaway and R. Petronzio, Nucl. Phys. B277, 50 (1986); see also ibid. B292, 497 (1987), and references cited therein.
See, for example, A. Sirlin, in Lepton and Photon Interactions, proceedings of the International Symposium on Lepton and Photon Interactions at High Energies, Hamburg, West Germany, 1987, edited by W. Bartel and R. Rückl [Nucl. Phys. B, Proc. Suppl. 3 (1987)].
M. A. B. Bég, in Particles and the Universe, proceedings of the International Symposium on Particles and Cosmology, Thessaloniki, Greece, 1985, edited by G. Lazarides and Q. Shafi (North-Holland, New York, 1986), p. 61.
S. L. Adler, Phys. Rev. 177, 2426 (1969); J. S. Bell and R. Jackiw, Nuovo Cimento A 60, 47 (1969).
D. G. Sutherland, Nucl. Phys. B2, 433 (1967).
T. D. Lee, S. Weinberg and B. Zumino, Phys. Rev. Lett. 18, 1029 (1967); J. Dooher, ibid. 19, 600 (1967); S. Okubo, Phys. Rev. 179, 1629 (1969).
P. H. Ginsparg and K. G. Wilson, Phys. Rev. D 25, 2649 (1982).
H. B. Nielsen and M. Ninomiya, Phys. Lett. 105B, 219 (1981); Nucl. Phys. B193, 173 (1981) ibid. B185, 20 (1981).
R. J. Crewther, Phys. Rev. Lett. 28, 1421 (1972).
G. Källén, Helv. Phys. Acta 25, 417 (1952); H. Lehmann, Nuovo Cimento 11, 342 (1954).
J. Frohlich, Nucl. Phys. B200, [FS4], 281 (1982).
See, for example, M. A. B. Bég and A. Sirlin, Annu. Rev. Nucl. Sci. 24, 379 (1974).
This possibility was first considered in the classic paper of M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300 (1954).
B. Schroer, Lett. Nuovo Cimento 2, 867 (1971).
S. L. Adler, C. G. Callan, Jr., D. J. Gross and R. Jackiw, Phys. Rev. D 6, 2982 (1972).
K. G. Wilson, Phys. Rev. 179, 1499 (1969).
E. J. Schreier, Phys. Rev. D 3, 980 (1971).
M. A. B. Bég and S.-S. Shei, Phys. Rev. D 12, 3092 (1975).
For a perturbative proof of the theorem, in spinor electrodynamics, see S. L. Adler and W. A. Bardeen, Phys. Rev. 182, 1517 (1969).
Okubo (Ref. 10). See also the penultimate paragraph of Sec. VIID in Ref. 20.
M. A. B. Bég, Phys. Rev. D 11, 1165 (1975).
S.-S. Shei and H.-S. Tsao, Phys. Rev. D 15, 3049 (1977).
J. M. Cornwall, Phys. Rev. Lett. 16, 1174 (1966).
To obtain $S_{v}$ one measures the rate for ``gamma''->isovector hadrons, ``gamma'' being a hard virtual timelike photon; to ascertain K one measures the rate for ``gamma''->``gamma''+ $π^{0}$ at fixed $π^{0}$ momentum. Such photons can be created in $e^{+}$ $e^{-}$ annihilation and detected, in the second process, via muon pair creation. [See D. J. Gross and S. B. Treiman, Phys. Rev. D 4, 2105 (1971). The right-hand side of Eq. (25) of this paper should be multiplied by $K^{2}$ .] This experimental handle would not be available, however, if the first scenario were to be valid with α ${bar}_{\inf}$ app1; it would be impossible then to separate a one-photon process from a multiphoton process.
I am indebted to Professor L. S. Brown for emphasizing this point.
M. A. B. Bég, Phys. Rev. 134, B1386 (1964). See also the Appendix of the present paper.
Crewther, Nielsen, and others [see R. J. Crewther and N. K. Nielsen, Nucl. Phys. B87, 52 (1975), and references cited therein] have managed to retrieve the ABJ anomaly in finite spinor QED by allowing the axial-vector current to have an anomalous dimension. Justification for this somewhat radical possibility is to be found in perturbation theory, which is not compatible with $Z_{γ}$ != 0, unless the cutoff is kept finite, as in our second scenario, or in summation of a judiciously selected infinite subset of all contributing graphs. However, these authors concede, in the above-cited paper, that one should remain suspicious of the validity of the summation procedures in, and thus the internal consistency of, finite QED. Our insistence on canonical dimensionality for the observable axial-vector current cuts the Gordian knot and leads to the demise of the anomaly. See also Ref. 32 below.
Cf. M. Baker and K. Johnson, Physica (Utrecht) 96A, 120 (1979). The incompatibility between finite QED and the Adler-Bardeen theorem (Ref. 23) led these authors to state that ``the finite theory does not seem to be a real possibility.''
See, for example, A. Chodos, Comments Nucl. Part. Phys. 17, 211 (1987).

[c1] M. A. B. Bég, Comments Nucl. Part. Phys. 17, 119 (1987), and papers cited therein.

[c2] For a comprehensive review and all relevant references, see D. J. E. Callaway, Phys. Rep. 167, 241 (1989).

[c3] L. D. Landau, Niels Bohr and the Development of Physics (McGraw-Hill, New York, 1955).

[c4] M. A. B. Bég, C. Panagiotakopoulos and A. Sirlin, Phys. Rev. Lett. 52, 883 (1984); D. J. E. Callaway, Nucl. Phys. B233, 189 (1984).

[c5] D. J. E. Callaway and R. Petronzio, Nucl. Phys. B277, 50 (1986); see also ibid. B292, 497 (1987), and references cited therein.

[c6] See, for example, A. Sirlin, in Lepton and Photon Interactions, proceedings of the International Symposium on Lepton and Photon Interactions at High Energies, Hamburg, West Germany, 1987, edited by W. Bartel and R. Rückl [Nucl. Phys. B, Proc. Suppl. 3 (1987)].

[c7] M. A. B. Bég, in Particles and the Universe, proceedings of the International Symposium on Particles and Cosmology, Thessaloniki, Greece, 1985, edited by G. Lazarides and Q. Shafi (North-Holland, New York, 1986), p. 61.

[c8] S. L. Adler, Phys. Rev. 177, 2426 (1969); J. S. Bell and R. Jackiw, Nuovo Cimento A 60, 47 (1969).

[c9] D. G. Sutherland, Nucl. Phys. B2, 433 (1967).

[c10] T. D. Lee, S. Weinberg and B. Zumino, Phys. Rev. Lett. 18, 1029 (1967); J. Dooher, ibid. 19, 600 (1967); S. Okubo, Phys. Rev. 179, 1629 (1969).

[c11] P. H. Ginsparg and K. G. Wilson, Phys. Rev. D 25, 2649 (1982).

[c12] H. B. Nielsen and M. Ninomiya, Phys. Lett. 105B, 219 (1981); Nucl. Phys. B193, 173 (1981) ibid. B185, 20 (1981).

[c13] R. J. Crewther, Phys. Rev. Lett. 28, 1421 (1972).

[c14] G. Källén, Helv. Phys. Acta 25, 417 (1952); H. Lehmann, Nuovo Cimento 11, 342 (1954).

[c15] J. Frohlich, Nucl. Phys. B200, [FS4], 281 (1982).

[c16] See, for example, M. A. B. Bég and A. Sirlin, Annu. Rev. Nucl. Sci. 24, 379 (1974).

[c17] This possibility was first considered in the classic paper of M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300 (1954).

[c18] B. Schroer, Lett. Nuovo Cimento 2, 867 (1971).

[c19] S. L. Adler, C. G. Callan, Jr., D. J. Gross and R. Jackiw, Phys. Rev. D 6, 2982 (1972).

[c20] K. G. Wilson, Phys. Rev. 179, 1499 (1969).

[c21] E. J. Schreier, Phys. Rev. D 3, 980 (1971).

[c22] M. A. B. Bég and S.-S. Shei, Phys. Rev. D 12, 3092 (1975).

[c23] For a perturbative proof of the theorem, in spinor electrodynamics, see S. L. Adler and W. A. Bardeen, Phys. Rev. 182, 1517 (1969).

[c24] Okubo (Ref. 10). See also the penultimate paragraph of Sec. VIID in Ref. 20.

[c25] M. A. B. Bég, Phys. Rev. D 11, 1165 (1975).

[c26] S.-S. Shei and H.-S. Tsao, Phys. Rev. D 15, 3049 (1977).

[c27] J. M. Cornwall, Phys. Rev. Lett. 16, 1174 (1966).

[c28] To obtain $S_{v}$ one measures the rate for ``gamma''->isovector hadrons, ``gamma'' being a hard virtual timelike photon; to ascertain K one measures the rate for ``gamma''->``gamma''+ $π^{0}$ at fixed $π^{0}$ momentum. Such photons can be created in $e^{+}$ $e^{-}$ annihilation and detected, in the second process, via muon pair creation. [See D. J. Gross and S. B. Treiman, Phys. Rev. D 4, 2105 (1971). The right-hand side of Eq. (25) of this paper should be multiplied by $K^{2}$ .] This experimental handle would not be available, however, if the first scenario were to be valid with α ${bar}_{\inf}$ app1; it would be impossible then to separate a one-photon process from a multiphoton process.

[c29] I am indebted to Professor L. S. Brown for emphasizing this point.

[c30] M. A. B. Bég, Phys. Rev. 134, B1386 (1964). See also the Appendix of the present paper.

[c31] Crewther, Nielsen, and others [see R. J. Crewther and N. K. Nielsen, Nucl. Phys. B87, 52 (1975), and references cited therein] have managed to retrieve the ABJ anomaly in finite spinor QED by allowing the axial-vector current to have an anomalous dimension. Justification for this somewhat radical possibility is to be found in perturbation theory, which is not compatible with $Z_{γ}$ != 0, unless the cutoff is kept finite, as in our second scenario, or in summation of a judiciously selected infinite subset of all contributing graphs. However, these authors concede, in the above-cited paper, that one should remain suspicious of the validity of the summation procedures in, and thus the internal consistency of, finite QED. Our insistence on canonical dimensionality for the observable axial-vector current cuts the Gordian knot and leads to the demise of the anomaly. See also Ref. 32 below.

[c32] Cf. M. Baker and K. Johnson, Physica (Utrecht) 96A, 120 (1979). The incompatibility between finite QED and the Adler-Bardeen theorem (Ref. 23) led these authors to state that ``the finite theory does not seem to be a real possibility.''

[c33] See, for example, A. Chodos, Comments Nucl. Part. Phys. 17, 211 (1987).

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Triviality, anomalies, and canonical electroweak theory at small distances

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