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Gauge-invariant on-shellZ 2 in QED,QCD and the effective field theory of a static quark

Authors:

Institution:

1. Physics Department, Open University, MK 7 6AA, Milton Keynes, UK
2. Institut für Physik der Universit?t Mainz, Postfach 3980, W-6500, Mainz, Federal Republic of Germany

Abstract:

We calculate theon-shell fermion wave-function renormalization constantZ ₂ 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 logZ ₂/da ₀=i(2 pgr

)^–D e ₀ ² int

d^D k/k ⁴=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 gamma

_F of the fermion field in minimally subtracted QCD, withN _L 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

$\begin{gathered} \gamma _F - \tilde \gamma _F \equiv \mu \frac{d}{{d\mu }}\log \left( {\frac{{Z_2^{MS} (\mu )}}{{\tilde Z_2^{MS} (\mu )}}} \right) \hfill \\ = 2\frac{{\bar \alpha _s (\mu )}}{\pi } + \left( {\frac{{41}}{4} - \frac{{11}}{{18}}N_L } \right)\frac{{\bar \alpha _s^2 (\mu )}}{{\pi ^2 }} + O(\bar \alpha _s^3 ) \hfill \\ \end{gathered}$

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