Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction II: Plasmon - hard particle scattering

TL;DR

This paper extends a Hamiltonian formalism to describe the scattering of plasmons off hard particles in a quark-gluon plasma, providing detailed mathematical structures and transformations for analyzing such interactions.

Contribution

It generalizes the Hamiltonian formalism to include hard particle interactions and derives the most comprehensive form of interaction Hamiltonians and canonical transformations.

Findings

Derived general interaction Hamiltonians up to fourth order.

Presented a new ansatz for separating color and momentum degrees of freedom.

Analyzed approximation of scattering amplitudes for large hard momenta.

Abstract

It is shown that the Hamiltonian formalism proposed previously in [1] to describe the nonlinear dynamics of only {\it soft} fermionic and bosonic excitations contains much more information than initially assumed. In this paper, we have demonstrated in detail that it also proved to be very appropriate and powerful in describing a wide range of other physical phenomena, including the scattering of colorless plasmons off {\it hard} thermal (or external) color-charged particles moving in hot quark-gluon plasma. A generalization of the Poisson superbracket including both anticommuting variables for hard modes and normal variables of the soft Bose field, is presented for the case of a continuous medium. The corresponding Hamilton equations are defined, and the most general form of the third- and fourth-order interaction Hamiltonians is written out in terms of the normal boson field variables and hard momentum modes of the quark-gluon plasma. The canonical transformations involving both bosonic and hard mode degrees of freedom of the system under consideration, are discussed. The canonicity conditions for these transformations based on the Poisson superbracket, are derived. The most general structure of canonical transformations in the form of integro-power series up to sixth order in a new normal field variable and a new hard mode variable, is presented. For the hard momentum mode of quark-gluon plasma excitations, an ansatz separating the color and momentum degrees of freedom, is proposed. The question of approximation of the total effective scattering amplitude when the momenta of hard excitations are much larger than those of soft excitations of the plasma, is considered.