Note on the derivation of chiral anomalies from fermion determinants

L. Bonora; M. Bregola; P. Pasti

doi:10.1103/PhysRevD.31.2665

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Note on the derivation of chiral anomalies from fermion determinants

L. Bonora

M. Bregola

P. Pasti

Dipartimento di Fisica ‘‘G. Galilei’’ dell’Università di Padova e Sezione, Istituto Nazionale di Fisica Nucleare, Padova, Italy

Istituto di Fisica Università di Ferrara e Sezione, Istituto Nazionale de Fisica Nucleare di Padova, Padova, Italy

Dipartimento di Fisica ‘‘G. Galilei’’ dell’Università di Padova e Sezione, Istituto Nazionale di Fisica Nucleare, Padova, Italy

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Phys. Rev. D 31, 2665 – Published 15 May, 1985

DOI: https://doi.org/10.1103/PhysRevD.31.2665

Abstract

We present an improved method for deriving chiral anomalies, in particular Bardeen’s anomaly, from regularized fermion determinants by means of nonperturbative calculations.

References (26)

S. Adler, Phys. Rev. 177, 2426 (1969); J. Bell and R. Jackiw, Nuovo Cimento 60 A, 47 (1969).
S. Adler, in Lectures on Elementary Particles and Quantum Field Theory, edited by S. Deser et al. (MIT Press, Cambridge, MA, 1970); R. Jackiw, in Lectures on Current Algebra and Its Applications, edited by S. Treiman et al. (Princeton Univ. Press, Princeton, NJ, 1972).
W. Bardeen, Phys. Rev. 184, 1848 (1969).
B. Zumino, Wu Yong Shi and A. Zee, Nucl. Phys. B239, 477 (1984); B. Zumino, Les Houches Lectures, 1983 (unpublished).
R. Stora, Cargese Lectures, 1983 (unpublished).
H. F. Atiyah and I. M. Singer, Proc. Nat. Acad. Sci. 81, 2597 (1984).
L. Bonora and P. Cotta-Ramusino, Phys. Lett. 107 B, 87 (1981); Commun. Math. Phys. 87, 589 (1983).
L. D. Faddeev, Phys. Lett. 145B, 81 (1984).
See references quoted in Ref. 5; L. Bonora and P. Pasti, Phys. Lett. 132 B, 75 (1983); L. Baulieu, Nucl. Phys. B 241, 557 (1984).
A. Andrianov, L. Bonora and P. Pasti, Ann. Phys. (N.Y.) 158, 379 (1984).
L. Alvarez-Gaume and E. Witten, Nucl. Phys. B 234, 269 (1985).
The conventions for gamma matrices are $γ_{μ}$ = $γ_{μ}^{dag}$ , $γ_{5}$ = $γ_{0}$ $γ_{1}$ $γ_{2}$ $γ_{3}$ . Therefore $γ_{5}^{dag}$ = $γ_{5}$ , $γ_{5}$ ${}^{2}$ =1.
A. Andrianov and L. Bonora, Nucl. Phys. B 233, 232 (1984).
K. Fujikawa, Phys. Rev. Lett. 42, 2195 (1979); Phys. Rev. D 21, 2848 (1980).
D. McKay and B.-L. Young, Phys. Rev. D 28, 1039 (1983); M. Einhorn and D. Jones, ibid. 29, 331 (1984).
A. P. Balachandran, G. Marmo, V. P. Nair and C. G. Trahern, Phys. Rev. D 25, 2713 (1982).
J. Wess and B. Zumino, Phys. Lett. 37 B, 95 (1972).
From the point of view of the effective Lagrangian approach see N. K. Pak and P. Rossi, CERN Report No. TH3831, 1984 (unpublished).
T. Kato, Perturbation Theory for Linear Operators, 2nd ed. (Springer, New York, 1976); N. Dunford and J. T. Schwartz, Linear Operators (Wiley-Interscience, New York, 1971), Part III.
See T. Kato, Ref. 19, pp. 212–214: the norm of the perturbation must become smaller and smaller as the eigenvalue considered approaches infinity.
F. Treves, Topological Vector Spaces, Distributions and Kernels (Academic, New York, 1967).
A term proportional to $Λ^{2}$ appears in this approach too. The integrals that define the relative numerical coefficients contain logarithmic singularities. This depends on the fact that when introducing continuous integrals over $k_{mu}$ , we have surreptitiously taken the infinite-volume limit (see Ref. 13). The value of the above integral depends on the way we take this limit, or, in other words, on the way we regularize it. Here, for simplicity we have chosen a dimensional regularization, which gives a vanishing result.
A. Andrianov and L. Bonora, Nucl. Phys. B 233, 247 (1984).
J. Ambjoørn, J. Greensite and C. Petersen, Nucl. Phys. B 221, 381 (1983).
D. J. Gross and R. Jackiw, Phys. Rev. D 6, 477 (1972).
K. Fujikawa, Phys. Rev. D 29, 275 (1984).

[c1] S. Adler, Phys. Rev. 177, 2426 (1969); J. Bell and R. Jackiw, Nuovo Cimento 60 A, 47 (1969).

[c2] S. Adler, in Lectures on Elementary Particles and Quantum Field Theory, edited by S. Deser et al. (MIT Press, Cambridge, MA, 1970); R. Jackiw, in Lectures on Current Algebra and Its Applications, edited by S. Treiman et al. (Princeton Univ. Press, Princeton, NJ, 1972).

[c3] W. Bardeen, Phys. Rev. 184, 1848 (1969).

[c4] B. Zumino, Wu Yong Shi and A. Zee, Nucl. Phys. B239, 477 (1984); B. Zumino, Les Houches Lectures, 1983 (unpublished).

[c5] R. Stora, Cargese Lectures, 1983 (unpublished).

[c6] H. F. Atiyah and I. M. Singer, Proc. Nat. Acad. Sci. 81, 2597 (1984).

[c7] L. Bonora and P. Cotta-Ramusino, Phys. Lett. 107 B, 87 (1981); Commun. Math. Phys. 87, 589 (1983).

[c8] L. D. Faddeev, Phys. Lett. 145B, 81 (1984).

[c9] See references quoted in Ref. 5; L. Bonora and P. Pasti, Phys. Lett. 132 B, 75 (1983); L. Baulieu, Nucl. Phys. B 241, 557 (1984).

[c10] A. Andrianov, L. Bonora and P. Pasti, Ann. Phys. (N.Y.) 158, 379 (1984).

[c11] L. Alvarez-Gaume and E. Witten, Nucl. Phys. B 234, 269 (1985).

[c12] The conventions for gamma matrices are $γ_{μ}$ = $γ_{μ}^{dag}$ , $γ_{5}$ = $γ_{0}$ $γ_{1}$ $γ_{2}$ $γ_{3}$ . Therefore $γ_{5}^{dag}$ = $γ_{5}$ , $γ_{5}$ ${}^{2}$ =1.

[c13] A. Andrianov and L. Bonora, Nucl. Phys. B 233, 232 (1984).

[c14] K. Fujikawa, Phys. Rev. Lett. 42, 2195 (1979); Phys. Rev. D 21, 2848 (1980).

[c15] D. McKay and B.-L. Young, Phys. Rev. D 28, 1039 (1983); M. Einhorn and D. Jones, ibid. 29, 331 (1984).

[c16] A. P. Balachandran, G. Marmo, V. P. Nair and C. G. Trahern, Phys. Rev. D 25, 2713 (1982).

[c17] J. Wess and B. Zumino, Phys. Lett. 37 B, 95 (1972).

[c18] From the point of view of the effective Lagrangian approach see N. K. Pak and P. Rossi, CERN Report No. TH3831, 1984 (unpublished).

[c19] T. Kato, Perturbation Theory for Linear Operators, 2nd ed. (Springer, New York, 1976); N. Dunford and J. T. Schwartz, Linear Operators (Wiley-Interscience, New York, 1971), Part III.

[c20] See T. Kato, Ref. 19, pp. 212–214: the norm of the perturbation must become smaller and smaller as the eigenvalue considered approaches infinity.

[c21] F. Treves, Topological Vector Spaces, Distributions and Kernels (Academic, New York, 1967).

[c22] A term proportional to $Λ^{2}$ appears in this approach too. The integrals that define the relative numerical coefficients contain logarithmic singularities. This depends on the fact that when introducing continuous integrals over $k_{mu}$ , we have surreptitiously taken the infinite-volume limit (see Ref. 13). The value of the above integral depends on the way we take this limit, or, in other words, on the way we regularize it. Here, for simplicity we have chosen a dimensional regularization, which gives a vanishing result.

[c23] A. Andrianov and L. Bonora, Nucl. Phys. B 233, 247 (1984).

[c24] J. Ambjoørn, J. Greensite and C. Petersen, Nucl. Phys. B 221, 381 (1983).

[c25] D. J. Gross and R. Jackiw, Phys. Rev. D 6, 477 (1972).

[c26] K. Fujikawa, Phys. Rev. D 29, 275 (1984).

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Note on the derivation of chiral anomalies from fermion determinants

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