All-orders infrared freezing of Re+e- in perturbative QCD

We consider the behaviour of the perturbative QCD corrections to the Re+e- ratio, in the limit that the c.m. energy roots vanishes. Writing Re+e-(s) = 3 Sigma(f) Q(f)(2) (1 + R(s)), with Q(f) denoting the electric charge of quark flavour f, we find that for N-f < 9 flavours of massless quarks, the perturbative correction R(s) to the parton model result smoothly approaches from below the infrared limit R(0) = 2/b, as s --> 0. Here b = (33 - 2N(f))/6 is the first QCD beta-function coefficient. This 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. (C) 2002 Elsevier Science B.V. All rights reserved.