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

Abstract

We consider the behaviour of the perturbative QCD corrections to the R e + e − ratio, in the limit that the c.m. energy s vanishes. Writing R e +e − (s)=3∑ fQ 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=(33−2 N 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.