Dispersive derivation of the triangle anomaly

Jiří Hořejší

doi:10.1103/PhysRevD.32.1029

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Dispersive derivation of the triangle anomaly

Jiří Hořejší^*

Nuclear Center, Faculty of Mathematics and Physics, Charles University Prague, V Holešovičkách 2, Praha 8, Czechoslovakia

^*Present address: Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141 980 Dubna, U.S.S.R.

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Phys. Rev. D 32, 1029 – Published 15 August, 1985

DOI: https://doi.org/10.1103/PhysRevD.32.1029

Abstract

We present a straightforward generalization of the results of some previous treatments, in which the Adler-Bell-Jackiw anomaly has been recovered with the help of dispersion relations. We consider the absorptive part of the $VVA$ triangle diagram with the external momenta $k$ , $p$ at vector vertices such that $k^{2} = p^{2} <~ 0$ and the fermion mass $m >~ 0$ . An integral of the imaginary part of the relevant invariant amplitude is calculated explicitly and shown to produce the desired anomalous contribution to the axial Ward identity. This also enables one to demonstrate the $δ$ -like behavior of such an imaginary part in the limit $k^{2} \to 0$ , $m \to 0$ .

References (13)

H. Fukuda and Y. Miyamoto, Prog. Theor. Phys. 4, 347 (1949) H. Fukuda et al., ibid. 4, 477 (1949) S. Ozaki, S. Oneda, and S. Sasaki, ibid. 4, 524 (1949) J. Steinberger, Phys. Rev. 76, 1180 (1949)
J. Schwinger, Phys. Rev. 82, 664 (1951)
S. Adler, Phys. Rev. 177, 2426 (1969)
J. Bell and R. Jackiw, Nuovo Cimento 60A, 47 (1969)
A. D. Dolgov and V. I. Zakharov, Nucl. Phys. B27, 525 (1971)
Y. Frishman et al., Nucl. Phys. B177, 157 (1981)
K. Huang, Quarks, Leptons, and Gauge Fields (World Scientific, Singapore, 1982)
L. Brown, R. Carlitz, and C. Lee, Phys. Rev. D 16, 417 (1977) R. Jackiw and C. Rebbi, ibid. 16, 1052 (1977)
K. Fujikawa, Phys. Rev. D 21, 2848 (1980) G. Christos, Z. Phys. C 18, 156 (1983) G. Siopsis, Caltech. Report No. CALT-68-1187, 1984 (unpublished)
V. N. Gribov, Budapest Report No. KFKI-1981-66, 1981 (unpublished) J. Ambjørn, J. Greensite, and C. Peterson, Nucl. Phys. B221, 381 (1983)
J. Bjorken and S. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965)
J. Hořejší, Dubna Report No. E2-84-783, 1984 (unpublished)
C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980)

[c1] H. Fukuda and Y. Miyamoto, Prog. Theor. Phys. 4, 347 (1949) H. Fukuda et al., ibid. 4, 477 (1949) S. Ozaki, S. Oneda, and S. Sasaki, ibid. 4, 524 (1949) J. Steinberger, Phys. Rev. 76, 1180 (1949)

[c2] J. Schwinger, Phys. Rev. 82, 664 (1951)

[c3] S. Adler, Phys. Rev. 177, 2426 (1969)

[c4] J. Bell and R. Jackiw, Nuovo Cimento 60A, 47 (1969)

[c5] A. D. Dolgov and V. I. Zakharov, Nucl. Phys. B27, 525 (1971)

[c6] Y. Frishman et al., Nucl. Phys. B177, 157 (1981)

[c7] K. Huang, Quarks, Leptons, and Gauge Fields (World Scientific, Singapore, 1982)

[c8] L. Brown, R. Carlitz, and C. Lee, Phys. Rev. D 16, 417 (1977) R. Jackiw and C. Rebbi, ibid. 16, 1052 (1977)

[c9] K. Fujikawa, Phys. Rev. D 21, 2848 (1980) G. Christos, Z. Phys. C 18, 156 (1983) G. Siopsis, Caltech. Report No. CALT-68-1187, 1984 (unpublished)

[c10] V. N. Gribov, Budapest Report No. KFKI-1981-66, 1981 (unpublished) J. Ambjørn, J. Greensite, and C. Peterson, Nucl. Phys. B221, 381 (1983)

[c11] J. Bjorken and S. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965)

[c12] J. Hořejší, Dubna Report No. E2-84-783, 1984 (unpublished)

[c13] C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980)

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