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Equivariant Dimensional Regularization

Stefan Weinzierl

Abstract

The calculation of loop amplitudes with parity violation or spin effects within dimensional regularization needs a consistent definition of gamma5. Also loop calculations in supersymmetric theories need a consistent definition of gamma5. In this paper we develop a new formalism, which allows us to define consistent regularization schemes. We use Grothendieck's K-functor to construct finite-dimensional vectorspaces of non-integer rank. The rank will play the role of the ``4-2eps'' in conventional dimensional regularization. We then define two regularization schemes, one similar to the 't Hooft--Veltman scheme, the other one as a scheme, where all algebra is performed in four dimensions. Lorentz invariance is maintained in both cases. However the structure of the Clifford algebra cannot be preserved. We show that the HV-like scheme and the four-dimensional scheme correspond to two different deformations of the Clifford algebra. It is the purpose of this paper to advocate the four-dimensional scheme for future calculations, since it is easier to use. As a consistency check we performed explicit one-loop calculations of various triangle anomalies in both schemes and we found agreement with Bardeen's results.

Equivariant Dimensional Regularization

Abstract

The calculation of loop amplitudes with parity violation or spin effects within dimensional regularization needs a consistent definition of gamma5. Also loop calculations in supersymmetric theories need a consistent definition of gamma5. In this paper we develop a new formalism, which allows us to define consistent regularization schemes. We use Grothendieck's K-functor to construct finite-dimensional vectorspaces of non-integer rank. The rank will play the role of the ``4-2eps'' in conventional dimensional regularization. We then define two regularization schemes, one similar to the 't Hooft--Veltman scheme, the other one as a scheme, where all algebra is performed in four dimensions. Lorentz invariance is maintained in both cases. However the structure of the Clifford algebra cannot be preserved. We show that the HV-like scheme and the four-dimensional scheme correspond to two different deformations of the Clifford algebra. It is the purpose of this paper to advocate the four-dimensional scheme for future calculations, since it is easier to use. As a consistency check we performed explicit one-loop calculations of various triangle anomalies in both schemes and we found agreement with Bardeen's results.

Paper Structure

This paper contains 19 sections, 192 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Basic K-theory
  3. Definition of the integration
  4. Dirac matrices in even dimensions
  5. The gauge group
  6. Electroweak and supersymmetric theories
  7. The scheme in practice
  8. Examples
  9. The singlet axial-vector current and the triangle anomaly
  10. The AAA-anomaly
  11. The non-singlet axial-vector current
  12. The Ward identity for the vector current
  13. Summary of the examples
  14. Conclusions
  15. Acknowledgements
  16. ...and 4 more sections

Figures (3)

  • Figure 1: A two-loop correction to the gluon propagator
  • Figure 2: The triangle graphs for the anomaly
  • Figure 3: Feynman graphs for the non-singlet axial-vector Ward identity