Collinear Divergences at One-loop Order for External Particles in a Heat-bath

In hard interactions between external particles incident on a heatbath, we show that large logarithms are generated when a radiated or absorbed gauge boson is collinear with the initial fermion momentum. These logarithms can be absorbed into process independent splitting/absorption probabilities. Unlike the zero-temperature case, however, they depend explicitly on the temperature and the scale of the interaction. Theory Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India, Email— [email protected] Theory Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India, Email— [email protected] Theory Group, Tata Institute of Fundamental Research, Navrangpura, Ahmedabad 380009, India, Email— [email protected] In recent years the problem of calculating scattering cross sections or rates for external particles travelling in a thermal medium has become phenomenologically interesting [1]. In a previous paper we have shown that to one-loop order there are no infrared divergences when two external particles collide in a heatbath kept at a temperature T [2]. We were able to resum the one-loop result and calculate the distribution of the mismatch in initial and final momenta due to soft radiation. The result was finite and contained some large logarithms of the type which usually arise from collinear emissions. In this paper we study this collinear behaviour in detail. We find that divergences exist, and are signalled by the familiar “large terms” of the order log(Q/m) (Q is the scale of the process and m is a regulating mass). We show that the collinear part gives rise to a certain universal splitting/absorption probability which explicitly depends on Q and T . There has been recent interest in collinear singularities associated with soft particles [3], because they might spoil the hard thermal loop resummation [4]. Since we study a hard particle in a heat-bath, we have nothing further to say about this. Consider the simplest scattering process in which the collinear singularities make their appearance. This is the inclusive cross section for the scattering of a charged electron with a space-like photon, γ, in a QED heat-bath, which we consider in a perturbation theory of the gauge coupling. The initial electron momentum is denoted by p, the γ momentum by q and the final electron momentum by p. The 4-velocity of the heat-bath in any frame, u (u = 1), is a new vector in the problem [5]. Compared to the T = 0 case, there are extra scalars p · u and q · u which have to be taken into account. For the rest, we use the standard notation q = −Q and p · q = Q 2x , (1) and work in the limit Q ≫ m, where p = p = m. We also take Q ≫ T 2 where T is the temperature of the heat-bath. The inclusive cross section can be written as a perturbation expansion in the form σ = α ( σ0 + ασ1 + α 2σ2 + · · · ) , (2) where α is the gauge coupling. At the lowest order in perturbation theory, one has only the interaction between e and γ, and hence σ0 is independent