Classical scattering matrix for hard and soft Bose-excitations in a non-Abelian plasma within the Hamiltonian formalism

TL;DR

This paper develops a Hamiltonian-wave framework to study elastic scattering of a fast color-charged particle off soft excitations in a non-Abelian quark-gluon plasma, defining a classical scattering matrix $\mathcal{S}$ in the Zakharov-Schulman formalism and working within the HTL regime. It derives the explicit fourth-order interaction Hamiltonian, constructs the classical $\mathcal{S}$-matrix as the exponential of a generator $\mathcal{T}$, and obtains the leading contribution to the energy loss through an effective current associated with the scattering process. The authors formulate the energy-loss expression in terms of HTL propagators and plasmon densities, showing that the colorless plasmon channel contributes while color factors cancel in the leading order. Order-of-magnitude estimates illuminate the regime where this elastic scattering mechanism is relevant, and the work outlines a path to extend the framework to fermionic excitations and higher-plasmon processes, offering a complementary perspective to perturbative QCD transport results.

Abstract

Within the framework of the Zakharov-Schulman approach, in close analogy with the methods of quantum field theory, the classical scattering matrix for the simplest process of interaction between hard and soft excitations in a quark-gluon plasma (QGP), is determined. The classical $\mathcal{S}$-matrix is defined in the form of the most general integro-power series expansion in the asymptotic values as $t\rightarrow-\infty$ of normal bosonic variables $c^{-\,a}_{\hspace{0.02cm}{\bf k}}(t)$ and $(c^{-\,a}_{\hspace{0.02cm}{\bf k}}(t))^{\ast}$, describing the soft gluon excitations of the system, and a color charge $\mathcal{Q}^{-\hspace{0.03cm}a}(t)$ of a hard particle. The first nontrivial contribution to this matrix is calculated. The quantum commutator of quantum field operators is replaced by the so-called Lie-Poisson bracket depending on the classical asymptotic variables. The developed approach is used to derive a general formula for energy loss of a fast color-charged particle during its scattering off soft bosonic excitations of QGP in the framework of the classical Hamiltonian formalism. For this purpose, the notion of an effective current of the scattering process under consideration is introduced and its relation to the classical $\mathcal{S}$-matrix is determined. With the help of the known form of the classical scattering matrix, the desired effective current is recovered, which in turn allowed us to determine the formula for energy loss of the hard color particle. The rough estimates of energy loss at the order-of-magnitude level is provided and their comparison with the well-known results on the radiation and collision losses is performed.

Figures (3)

• Figure 1: Effective amplitude $\widetilde{T}^{\,(2)\,a_{1}\space a_{2}\, a}_{\,{\bf k}_{1},\, {\bf k}_{2}}$ for the elastic plasmon scattering process off a hard colored particle. The double line denotes the hard particle carrying a color charge $\mathcal{Q}^{\space a}$ and the black dots denote the HTL summation
• Figure 2: Graphical representation of two coupled integral equations (\ref{['eq:4q']}) and (\ref{['eq:4w']})
• Figure 3: The elastic tree level scattering processes involving four and more plasmons