GSI me gw- 40 {D Q) S w QGQO G GS|—94-10 PREPRINT FEBRUAR 1994 COVARIANT KINETIC EQUATIONS AND RELAXATION PROCESSES IN RELATIVISTIC HEAVY-ION COLLISIONS M. scHONHOFEr~1, M. CUBERO, B.1.. FRIMAN w. NORENBERG, Gy. w0t.F scam Lteaaatas, sawava w||l\II\\\\||\\|||\l\|I\\|\\||\\\|\I\\|\1\||||\\|\\|| emmaatava Gesellschaft fur Sehvverienenterschung mbI—I Postfach 11 O5 52 · D—6422O Darmstadt · Germany OCR Output OCR OutputCovariant Kinetic Equations and Relaxation Processes 1n Relativistic Heavy-Ion Collisions M. Sch6nhofen“, M. Cubero“, B.L. Frimau“*°, W. N6renberg“·° and Gy. Wolf“ Gesellschaft fiir Schwerionenforschung (GSI) Postfach 110552, D—6422O Darmstadt, Germany Institut fiir Kernphysik, TH Darmstadt Schl0f5gartenstraf5e 9, D-64289 Darmstadt, Germany February 4, 1994 Abstract We present a derivation of covariant kinetic equations for nucleons and deltas within a model of quantum hadrodynamics, which includes scalar, vec tor and pseudo—scalar mesons. By generalizing the Kadanoff-Baym equations to relativistic fields and using the gradient expansion as well as the quasi particle approximation for the baryons, we find kinetic equations with co variant collision integrals of the Uehling-Uhlenbeck type. These relativistic Landau-Vlasov equations for nucleons and deltas are solved numerically for two counter-streaming nuclear-matter currents. Using the microscopic ex pressions we find ~ due to modifications of the baryon properties in nuclear matter ~ strong medium effects on relaxation processes in intermediate—energy heavy-ion collisions. *Dedicated to Professor Klaus Dietrich on the occasion of his 60th birthday OCR Output 1 Introduction and summary Over thc last decade relativistic hcavy—i0n collisions have been extensively in vestigated in order to gain information on nuclear matter at high densities and temperatures. A main objective is to determine the nuclear equation of state, i.e., the thermodynamical properties of nuclear matter, e.g. the pressure as function of temperature and baryon density. However, it is questionable whether ther modynamical equilibrium is attained in transient states such as those formed in relativistic heavy—ion collisions. Therefore, a consistent and reliable theoretical treatment must describe the complete dynamical evolution of nucleus—nucleus col lisions. Present theoretical descriptions of relativistic heavy—ion collisions are rather crude. The main activities have been concentrated on phase—space simulations using phenomenological collision integrals. The development of these approaches, i.e. cascade models [1] and Boltzmann-Uehling-Uhlenbeck transport theories [2 10], has been stimulated by their success in reproducing available experimental data. However, problems related to relativistic dynamics and to medium effects in relativistic heavy—ion collisions have not been studied systematically. The purpose of our paper is therefore threefold: to present a comprehensive derivation of covariant kinetic equations (rela tivistic Landau-Vlasov equations) including the description of all inevitable approximations (cf. section 2), to illustrate the medium effects on relaxation phenomena in a simple model (two counter—streaming nuclear—matter currents) which can be treated nu merically, to correlate them with the stiffness of the nuclear equation of state (cf. section 3), and thereby to establish a basis for further approximations leading e.g. to more macro scopic descriptions, like the relativistic multi-fluid models [11], which allow numerical solutions for realistic heavy—ion collisions (cf. the discussion in section 4). A rigorous covariant description of strongly interacting many—particle systems is possible within quantum field theories, where the interacting meson and baryon fields are treated explicitly. The use of hadronic degrees of freedom rather than quarks and gluons is justified for laboratory energies Elab § 2 GeV / u, cf. ref. [12]. Such theories are referred to as quantum hadrodynamics [13]. In the simplest version of QHD one considers two isoscalar meson fields, i.e. a Lorentz scalar field <I> and a Lorentz-vector field w", which interact with the nucleon field \1/N. In the present work we use a generalization of this model, which includes the isovector—pseudoscalar pion field if and the delta field W2. The lagrangian and the corresponding field equations are described in subsect. 2.1. Several authors have derived covariant kinetic equations for nuclear matter based on similar models. Elze et al. [14] have obtained a covariant mean—field OCR Output equation without collision term for nucleons interacting via 0- and w-mesons. This equation has been solved numerically by several groups [7-9] with the goal of eventually describing heavy-ion collisions. Ko et al. [8] have derived a covariant kinetic equation with a medium-dependent collision term for a system of nucle ons interacting via 0- and w-mesons using the non—equilibrium Green-function formalism of Kadanoff and Baym. However, in the numerical application they have simplified the kinetic equation by introducing a phenomenological medium independent parametrization of the cross section in the collision term [8, 9]. Re cently, Mrowczyriski and Heinz [15] have presented a detailed study of the trans port equations for the Walecka model, which is similar to ours. They have gone beyond our considerations by taking spin degrees of freedom explicitly into ac count. However, their study does not include the delta resonance, and they have not attempted to find solutions to their kinetic equations. In section 2 we present the derivation of covariant kinetic equations for baryons within quantum hadrodynamics. The starting point is the covariant formulation of the equations of motion for non-equilibrium Green functions on a real time contour following the non-relativistic formalism developed by Kadanoff and Baym [16] as well as the work of Martin and Schwinger [17] and Keldysh [18]. For a comprehensive and systematic study of non-relativistic Green functions on a time contour we refer the reader to refs. [19, 20]. In subsect. 2.2 we discuss the Kadanoff-Baym formalism, generalized to the relativistic baryonic fields of nucleons and deltas. The resulting equations of mo tion for the Green functions are far too complicated for a straightforward numer ical treatment of nucleus-nucleus collisions. Therefore, in subsect. 2.3, we carry out a gradient expansion in order to reduce the complexity of the coupled set of equations. Performing a Fourier transformation with respect to the relative coor dinates one obtains equations of motion for the Green functions in the phase-space representation (Wigner transforms). These equations are further simplified by in troducing the quasi—particle approximation for the baryons, which implies that the quasi—particles are assumed to propagate on mass shell between collisions. Hence, the original eight—dimensional phase space is reduced to a seven—dimensional hy persurface. The quasi—particle approximation is reliable, when the mean free path of the scattering particles is large compared to the de Broglie wave length of the particles. The final form of the covariant kinetic equations (relativistic Landau-Vlasov equations) for the phase-space densities of the nucleons contain two different con tributions from the nucleon—meson interaction: the mean fields and the collision term. The matrix elements for the scattering amplitudes in the collision integrals are calculated in Born approximation with mean-field spinors (cf. subsect. 2.4) Al ready at this level of the perturbation expansion, the calculation of the scattering amplitudes is quite involved. Thus, the inclusion of higher-order terms does not seem to be practicable. To some extent we compensate for the short—comings of the Born approximation by fitting the coupling constants and form factors to the nucleon—nucleon scattering cross sections. A drawback of the Born approximaOCR Output tion is that the corresponding scattering amplitude is not unitary. We shall not address this problem here, but note that to cure this problem a T- or K-matrix approach would be needed. In subsect. 2.5 the technical details and extensions are discussed, which are needed in the derivation of the kinetic equation for the deltas (spin = gi and isospin = Thus we finally obtain kinetic equations which allow us to study medium effects on the equilibration processes in intermediate-energy heavy-ion collisions. The numerical treatment of nucleus-nucleus collisions in a kinetic model, such as the one developed here, is quite complicated, because it requires the solution of an initial-value problem in 6 + 1 dimensions (coordinate space, momentum space and time). Therefore, we explore the relaxation process in the infinite system of two counter-streaming nuclear-matter currents (cf. section 3). For such a model the kinetic equations can be treated numerically and the relevance of medium effects in the relaxation process can be studied. In subsect. 3.1 we determine the model parameters by fitting the elastic diff`er ential nucleon-nucleon cross section at Elab = 1 GeV. The coupling constant and the form factor of the pion-nucleon-delta vertex are taken from refs. [21, 22]. The dependence on angle and energy of the elastic and inelastic cross sections agrees quite well with experimental data. In subsect. 3.2 we solve the coupled set of covariant kinetic equations for the baryons in the system of two counter-streaming nuclear-matter currents, where the initial distance of the two Fermi