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Journal of Mathematical Physics / Volume 53 / Issue 2 / ARTICLES / Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

J. Math. Phys. 53, 022305 (2012); http://dx.doi.org/10.1063/1.3686000 (24 pages)

Regularized path integrals and anomalies: U(1) chiral gauge theory

Christoph Kopper and Benjamin Lévêque

Centre de Physique Théorique, CNRS, UMR 7644 Ecole Polytechnique, F-91128 Palaiseau, France

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(Received 24 October 2011; accepted 29 January 2012; published online 23 February 2012)

We analyze the origin of the Adler–Bell–Jackiw anomaly of chiral U(1) gauge theory within the framework of regularized path integrals. Momentum or position space regulators allow for mathematically well-defined path integrals but violate local gauge symmetry. It is known how (nonanomalous) gauge symmetry can be recovered in the renormalized theory in this case [Kopper, C. and Müller, V. F., “Renormalization of spontaneously broken SU(2) Yang-Mills theory with flow equations,” Rev. Math. Phys. 21, 781 (2009)]10.1142/S0129055X0900375X. Here we analyze U(1) chiral gauge theory to show how the appearance of anomalies manifests itself in such a context. We show that the three-photon amplitude leads to a violation of the Slavnov-Taylor identities which cannot be restored on taking the UV limit in the renormalized theory. We point out that this fact is related to the nonanalyticity of this amplitude in the infrared region.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. THE CLASSICAL ACTION OF CHIRAL U (1) GAUGE THEORY
  3. REGULARIZED PATH INTEGRALS AND RENORMALIZATION
    1. The regularized effective action
    2. Inserted Schwinger functions
    3. Proper vertex functions
    4. Weak renormalizability
  4. THE VIOLATED SLAVNOV-TAYLOR IDENTITIES
    1. Deduction of the VSTI from the path integral
    2. The relevant contributions to the VSTI
      1. The relevant contributions to the functional Γ
      2. The relevant contributions to the functional Γ θ

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KEYWORDS and PACS

Keywords

chiral symmetries, gauge field theory, renormalisation

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3686000

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    Adler, S. L., “Axial-vector vertex in spinor electrodynamics,” Phys. Rev. 177, 2426 (1968).

    Adler, S. L. and Bardeen, W. A., “Absence of higher-order corrections in the anomalous axial-vector divergence equation,” Phys. Rev. 182, 1517 (1969).

    Falco, P., “Vector and axial anomaly in the Thirring–Wess model,” J. Math. Phys. 51, 082306 (2010)JMAPAQ000051000008082306000001.

    Fujikawa, K., “Path integral measure for gauge-invariant theories,” Phys. Rev. Lett. 42, 1195 (1979).

    Gross, D. J. and Jackiw, R., “Effect of anomalies in quasi-renormalizable theories,” Phys. Rev. D 6, 477 (1972).

    Mastropietro, V., “Non perturbative Adler–Bardeen theorem,” J. Math. Phys. 48, 022302 (2007)JMAPAQ000048000002022302000001.


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