6533b831fe1ef96bd12982c5

RESEARCH PRODUCT

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

subject

description

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(2π)−D e 0 2 ∫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 γ 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 $$\begin{gathered} \gamma _F - \tilde \gamma _F \equiv \mu \frac{d}{{d\mu }}\log \left( {\frac{{Z_2^{MS} (\mu )}}{{\tilde Z_2^{MS} (\mu )}}} \right) \hfill \\ = 2\frac{{\bar \alpha _s (\mu )}}{\pi } + \left( {\frac{{41}}{4} - \frac{{11}}{{18}}N_L } \right)\frac{{\bar \alpha _s^2 (\mu )}}{{\pi ^2 }} + O(\bar \alpha _s^3 ) \hfill \\ \end{gathered} $$ . 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.