Dynamical Screening and Radiative Parton Energy Loss in a Quark-gluon Plasma

Dynamical screening in the magnetic part of the one-gluon exchange interaction is included in the study of radiative energy loss of a fast parton propagating inside a quark-gluon plasma. As a result the final radiative energy loss is about twice as large as when only the electric part of one-gluon exchange interaction is considered. A non-perturbative magnetic screening mass is also used in the estimate of the mean-free-path of parton scattering in a hot QCD matter. 24.85.+p, 12.38.Mh, 25.75.+r, 12.38.Bx Typeset using REVTEX 1 Radiative energy loss of a fast parton inside a hot QCD matter has been proposed as a good probe of the medium and should lead to observable consequences such as jet quenching in high-energy heavy-ion collisions [1–4]. Theoretical estimate of the radiative energy loss suffered by a fast parton in a hot QCD medium has attracted a lot of interests because it helps us to understand the dependence of the energy loss on the properties of the medium and in particular the difference between parton energy loss in a cold nuclear medium and a hot quark-gluon plasma. Radiative energy loss has been estimated in various approaches, from uncertainty principle analysis [5] to calculation of induced radiation in a multiple scattering model [6]. A very interesting feature of the radiative energy loss found by a recent study in Ref. [7], referred to as BDMPS in this paper, is that the energy loss depends quadratically on the distance that the parton travels through. BDMPS demonstrated that such a nonlinear dependence arises from the non-abelian gluon rescattering in the medium. Most of these studies used the screened static-potential model for multiple scattering in a hot medium as proposed by Gyulassy and Wang (GW) [6]. In the GW model for multiple scattering, the interaction suffered by the propagating parton is assumed to be by a static potential with Debye screening. Such a screened static potential model gives finite cross section and average transverse momentum broadening. Even though BDMPS and Zakharov [8,9] later on generalized the study to other models of parton scattering, the problem of the magnetic part of one-gluon exchange interaction in a medium and its effect on the radiative energy loss remains unexplored. In this paper we will study the radiative energy loss of a fast parton inside a quark-gluon plasma including both the electric and magnetic part of the strong interaction. The magnetic part of the one-gluon exchange interaction is not screened perturbatively in the static limit in a hot QCD plasma. One therefore has to introduce a non-perturbative magnetic screening mass μmag in order to calculate the parton scattering cross section or the mean-free-path of a propagating parton similar to the calculation of the gluon damping rate [10]. For the calculation of some transport quantities, like the average momentum transfer per interaction, the dynamical screening provided by the imaginary part of the self-energy in the magnetic 2 interaction is enough to regulate the infrared behavior of the magnetic interaction and gives finite results. In both cases, correlation scales provided by the static and dynamics magnetic screening are somewhat different from the static electric screening. They should have significant effect on radiative parton energy loss in a quark-gluon plasma. According to BDMPS [7], the radiative energy loss of a fast parton inside a medium with finite size L is dE dz = αsNc 4 〈p ⊥W 〉, (1) for any model of multiple parton scattering, where 〈p ⊥W 〉 is the total accumulated momentum broadening during the parton’s propagation inside the medium which grows linearly with the media length L, i.e., 〈p ⊥W 〉 = Ld〈p 2 ⊥ 〉/dL. The momentum broadening per unit distance is d〈p ⊥ 〉 dL = ρ ∫ μ/B 0 dqq dσ dq2 , (2) where ρ is the media parton density, B = λ/L with λ being the mean-free-path of the propagating parton and μ is the typical momentum transfer in a parton scattering which is the Debye screening mass μD in the GW model of multiple scattering. In a hot quark-gluon plasma, one should include both the electric and magnetic interaction of one-gluon exchange. One should also replace Eq. (2) with its thermal averaged value, d〈p ⊥ 〉a dL = ∑ b νb ∫ dpb (2π)3 f(pb)(1± f(pc)) dpc (2π)3 dpd (2π)3 q|Mab| (2π)δ(p+ pb − pc − pd) (3) where we use the index a to denote the flavor of the fast parton and f(p) is the BoseEinstein fBS (Fermi-Dirac fFD) distribution for the thermal gluons (quarks) in the medium. We will only consider the elastic channels that are dominant at small angles. The statistical factor ν2 is 2(N 2 c − 1) for gluons and 4Ncnf for nf flavors of quarks. In this paper we will assume nf = 2. We neglect the quantum statistical effect for the fast partons. We denote the energy and momentum transfer of the parton scattering by ω and q, respectively. The above integral is dominated by contributions from small angle scattering. In this small-angle approximation, i.e., ω, q ≪ E,Eb, energy-momentum conservation leads to 3 ~ pc = ~ p+ ~q , ~pd = ~ pb − ~q Ec = E + ω , Ed = Eb − ω ω ≈ ~v · ~q ≈ ~vb · ~q, (4) where ~v = ~ p/E and ~vb = ~pb/Eb. In the small-angle scattering limit, the effective matrix element for parton scattering is [10],