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 s vanishes. Writing Re+e−(s)=3∑fQf2(1+R(s)), with Qf denoting the electric charge of quark flavour f, we find that for Nf<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=(33−2Nf)/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.