All-orders infra-red freezing of R(e+e-) in perturbative QCD

We consider the behaviour of the perturbative QCD corrections to the R(e+e-) ratio in the limit that the c.m. energy sqrt(s) vanishes. We find that for N(f)<9 flavours of massless quarks, the perturbative correction to the parton model result freezes to the infra-red limit 2/b as s-->0. Here b=(33-2N(f))/6 is the first QCD beta-function coefficient. The freezing holds to all-orders in perturbation theory. The s-dependence can be written analytically in closed form in terms of the Lambert W function.