Ultrasoft Amplitudes in Hot QCD

Abstract

By using the Boltzmann equation describing the relaxation of colour excitations in the QCD plasma, we obtain effective amplitudes for the ultrasoft colour fields carrying momenta of order $g^2 T$. These amplitudes are of the same order in $g$ as the hard thermal loops (HTL), which they generalize by including the effects of the collisions among the hard particles. The ultrasoft amplitudes share many of the remarkable properties of the HTL's: they are gauge invariant, obey simple Ward identities, and, in the static limit, reduce to the usual Debye mass for the electric fields. However, unlike the HTL's, which correspond effectively to one-loop diagrams, the ultrasoft amplitudes resum an infinite number of diagrams of the bare perturbation theory. By solving the linearized Boltzmann equation, we obtain a formula for the colour conductivity which accounts for the contributions of the hard and soft modes beyond the leading logarithmic approximation.