Conformal symmetry and the chiral anomaly

Joshua Erlich; Daniel Z. Freedman

doi:10.1103/PhysRevD.55.6522

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Conformal symmetry and the chiral anomaly

Joshua Erlich

Daniel Z. Freedman

Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Department of Mathematics and Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

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Phys. Rev. D 55, 6522 – Published 15 May, 1997

DOI: https://doi.org/10.1103/PhysRevD.55.6522

Abstract

Two-loop contributions to the anomalous correlation function $〈 J_{μ} {(x) J}_{ν} {(y) J}_{ρ} (z) 〉$ of three chiral currents are calculated by a method based on the conformal properties of massless field theories. The method was previously applied to virtual photon diagrams in quantum electrodynamics, and it is extended here to diagrams with scalars and chiral spinors in the Abelian Higgs model and in the $SU (3) \times SU (2) \times U (1)$ standard model. In each case there are nonvanishing contributions to the gauge current correlator from self-energy insertions, vertex insertions, and nonplanar diagrams, but their sum exactly vanishes. The two-loop contribution to the anomaly, therefore, also vanishes, in agreement with the Adler-Bardeen theorem. An application of the method to the correlator $〈 R_{μ} {(x) R}_{ν} {(y) K}_{ρ} (z) 〉$ of the $R$ and Konishi axial vector currents in supersymmetric gauge theories which was reported by Anselmi et al. is discussed here. The net two-loop contribution to this correlator also vanishes.

References (26)

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H. Cheng and S. P. Li, “The radiative corrections of the triangular anomaly,” Report No. (unpublished); “How to treat $γ_{5}$ ,” Report No. (unpublished).
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D. Z. Freedman, K. Johnson, and J. I. Latorre, Nucl. Phys. B371, 353 (1992).
D. Z. Freedman, G. Grignani, K. Johnson, and N. Rius, Ann. Phys. (N.Y.) 218, 75 (1992).
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[c1] S. L. Adler, Phys. Rev. 177, 2426 (1969).

[c2] J. S. Bell and R. Jackiw, Nuovo Cimento A 60, 47 (1969).

[c3] M. F. Atiyah in Supersymmetry and Supergravity Nonperturbative QCD, proceedings of the Winter School held in Mahabaleshwur, India, 1984, edited by P. Roy and V. Singh, Lecture Notes in Physics, Vol. 208 (Springer-Verlag, Berlin, 1984).

[c4] S. L. Adler and W. A. Bardeen, Phys. Rev. 182, 1517 (1969).

[c5] A. Zee, Phys. Rev. Lett. 29, 1198 (1972).

[c6] C. Lucchesi, O. Piguet, and S. P. Sorella, Int. J. Mod. Phys. A 2, 385 (1987); O. Piguet and S. P. Sorella, Nucl. Phys. B381, 373 (1992); B395, 661 (1993).

[c7] L. L. DeRaad, Jr., K. A. Milton, and W.-Y. Tsai, Phys. Rev. D 6, 1766 (1972); 6, 3491 (1972).

[c8] J. Schwinger, Particles, Sources and Fields (Addison-Wesley, Reading, MA, 1989), Vol. 3, Chaps. 5–9.

[c9] A. A. Ansel’m and A. A. Johansen, Sov. Phys. JETP 69, 670 (1989).

[c10] B. A. Faizullaev, M. M. Musakhanov, and N. K. Pak, Phys. Lett. B 361, 343 (1996).

[c11] H. Cheng and S. P. Li, “The radiative corrections of the triangular anomaly,” Report No. (unpublished); “How to treat $γ_{5}$ ,” Report No. (unpublished).

[c12] M. Baker and K. Johnson, Physica 96A, 120 (1979).

[c13] D. Z. Freedman, K. Johnson, and J. I. Latorre, Nucl. Phys. B371, 353 (1992).

[c14] D. Z. Freedman, G. Grignani, K. Johnson, and N. Rius, Ann. Phys. (N.Y.) 218, 75 (1992).

[c15] D. Anselmi, D. Z. Freedman, M. T. Grisaru, and A. A. Johansen, “Universality of the operator product expansions of SCFT $_{4}$ ,” Report No. (to appear in Phys. Lett. B).

[c16] H. Osborn and A. Petkou, Ann. Phys. (N.Y.) 231, 311 (1994).

[c17] E. J. Schreier, Phys. Rev. D 3, 980 (1971).

[c18] J. Erdmenger and H. Osborn, Nucl Phys. B483, 431 (1997).

[c19] K. Johnson, as reported by D. Z. Freedman in Strings and Symmetries 1991, edited by N. Berkovitz et al., Proceedings, Stony Brook, 1991 (World Scientific, Singapore, 1992).

[c20] H. Sonoda, Phys. Rev. D 55, 5245 (1997).

[c21] R. Jackiw, in S. B. Treiman, R. Jackiw, B. Zumino, and E. Witten, Current Algebra and Anomalies (Princeton University Press, Princeton, 1985).

[c22] S. L. Adler, R. W. Brown, T. F. Wong, and B.-L. Young, Phys. Rev. D 4, 1787 (1971).

[c23] H. Cheng (private communication).

[c24] J. I. Rosner, Ann. Phys. (N.Y.) 44, 11 (1967).

[c25] H. Nicolai, Nucl. Phys. B140, 294 (1978).

[c26] I. Jack, Nucl. Phys. B253, 323 (1985).

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Conformal symmetry and the chiral anomaly

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