Notes

(1) Renormalization group improvement would be helpful only if the theory were asymptotically free.

(2) For quark loops, higher-order QCD corrections are governed by an effective coupling evaluated on the scale of (see eq. (3)), and may therefore safely be ignored.

(3) For the purposes of computing higher-order corrections to the effective , we have defined .

(4) Similar conclusions have been reached by demanding that the high-energy interactions of Higgs particles in the Born approximation should not violate unitarity [3].

(5) If dimensional regularization is used, then the counter-terms generated at each order in the perturbation series must be proportional to the bare , since the renormalization mass (which allows the coupling constant to attain dimensions away from ) can enter only in logarithms. Hence the vanishing of the renormalized in which was suggested in ref. [2] may be preserved naturally to all orders, despite the fact that no symmetry requires it. It would naively be guaranteed by scale invariance, but this is violated by renormalization. Nevertheless, the violations in perturbation theory are logarithmic and do not provide a scale for the mass.

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