Gauge-invariant on-shellZ2 in QED, QCD and the effective field theory of a static quark

  • D. J. Broadhurst
  • N. Gray
  • K. Schilcher
Article

Abstract

We calculate theon-shell fermion wave-function renormalization constantZ2 of a general gauge theory, to two loops, inD dimensions and in an arbitrary covariant gauge, and find it to be gauge-invariant. In QED this is consistent with the dimensionally regularized version of the Johnson-Zumino relation: d logZ2/da0=i(2π)De02∫dDk/k4=0. In QCD it is, we believe, a new result, strongly suggestive of the cancellation of the gauge-dependent parts of non-abelian UV and IR anomalous dimensions to all orders. At the two-loop level, we find that the anomalous dimension γF of the fermion field in minimally subtracted QCD, withNL light-quark flavours, differs from the corresponding anomalous dimension\(\tilde \gamma _F \) of the effective field theory of a static quark by the gauge-invariant amount
$$\begin{gathered} \gamma _F - \tilde \gamma _F \equiv \mu \frac{d}{{d\mu }}\log \left( {\frac{{Z_2^{MS} (\mu )}}{{\tilde Z_2^{MS} (\mu )}}} \right) \hfill \\ = 2\frac{{\bar \alpha _s (\mu )}}{\pi } + \left( {\frac{{41}}{4} - \frac{{11}}{{18}}N_L } \right)\frac{{\bar \alpha _s^2 (\mu )}}{{\pi ^2 }} + O(\bar \alpha _s^3 ) \hfill \\ \end{gathered} $$
.

A complete description of two-loop on-shell renormalization of one-lepton QED, inD dimensions, is also given. More generally, we show that there is no need of integration in the two-loop calculation of on-shell two-and three-point functions.

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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • D. J. Broadhurst
    • 1
  • N. Gray
    • 1
  • K. Schilcher
    • 2
  1. 1.Physics DepartmentOpen UniversityMilton KeynesUK
  2. 2.Institut für Physik der Universität MainzMainzFederal Republic of Germany

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