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Propagation of Vector-Meson Spectral-Functions in a BUU Type Transport Model: Application to Di-Electron Production

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Propagation of Vector-Meson Spectral-Functions in a BUU Type Transport Model: Application to Di-Electron Production The Open Nuclear & Particle Physics Journal, 2010, 3, 1-15 1 Open Access Propagation of Vector-Meson Spectral-Functions in a BUU Type Transport Model: Application to Di-Electron Production 1 1, 2 2 H.W. Barz , B. Kämpfer *, Gy. Wolf and M. Zétényi 1Forschungszentrum Dresden-Rossendorf, Institut für Strahlenphysik, PF 510119, 01314 Dresden, Germany 2KFKI RMKI, H-1525 Budapest, POB 49, Hungary Abstract: The time evolution of vector-meson spectral-functions is studied within a kinetic theory approach. We implement this formalism in a BUU type transport model. Applications focus on ! and ! mesons being important pieces for the interpretation of the di-electron invariant mass spectrum measured by the HADES collaboration for the reaction C + C at 2 AGeV bombarding energy. Since the evolution of the spectral functions is driven by the local density, the in-medium modifications are tiny for small collision systems within this approach, as the life time of the compressed stage is too short. Keywords: Heavy-ion collisions, compressed nuclear matter, di-electron emission, broad resonance. 1. INTRODUCTION electron, j µ the is electromagnetic current operator, and Di-electrons serve as direct probes of dense and hot !!…"" denotes the ensemble average encoding the nuclear matter stages during the course of heavy-ion dependence on temperature and density (chemical potential), collisions [1-3]. The superposition of various sources, respectively. This rate may be combined with a global however, requires a deconvolution of the spectra by means dynamic model which provides us the space averaged of models. Of essential interest are the contributions of the quantities as a function of time [12] or even adopting a time light vector mesons and . The spectral functions of ! ! average [13-16]. both mesons are expected to be modified in a strongly interacting environment in accordance with chiral dynamics, (ii) Some sophistication can be achieved by employing a QCD sum rules etc. [1, 2, 4-6]. After the first pioneering detailed model for the space-time evolution of baryon experiments with the Dilepton Spectrometer DLS [7, 8] now density and temperature. e.g., as delivered by hydrodynamics. improved measurements with the High-Acceptance Di- One may also extract from transport models such parameters, Electron Spectrometer HADES [9-11] start to explore where, however, local off-equilibrium and/or anisotropic systematically the baryon-dense region accessible in fixed- momentum distributions hamper a reliable definition of target heavy-ion experiments at beam energies in the 1 - 2 density and temperature. Nevertheless, once the rate is given AGeV region. The invariant mass spectra of di-electrons for one has a very concise approach, as realized, e.g., in [17]. A the reaction C + C at 1 and 2 AGeV are now available [7, 10, similar approach has been presented in [18]. 11] allowing us to hunt for interesting many-body effects. (iii) Microscopic or kinetic transport models do not There are several approaches for describing the emission require isotropic momentum distributions or local equilibrium. of real and virtual photons off excited nuclear matter: Once general principles are implemented, transport models also provide a detailed treatment of the emission of (i) A piece of matter at rest in thermal equilibrium at electromagnetic radiation in heavy-ion collisions. temperature T emits e+ e! pairs with total momentum q at Our approach belongs to item (iii). The time evolution of 4 4 2 2 3 a rate dN / d xd q = !" / (3M # ) fB Im$em [1, 2], where single particle distribution functions of various hadrons are f (q ,T ) is the bosonic thermal distribution function and evaluated within the framework of a kinetic theory. We B 0 focus on the vector mesons ! and ! . The ! meson is denotes the retarded electromagnetic current-current !em already a broad resonance in vacuum, while the ! meson correlator, ! = "i d 4 xeiqx$(x )%%[ j µ (x), j (0)]&& , the may acquire a noticeable width in nuclear matter [19]. em # 0 µ Therefore, we are forced to treat dynamically these imaginary part of which determines directly the thermal resonances and their decays into di-electrons. Resorting to emission rate. Here, ! stands for the electromagnetic fine consider only pole-mass dynamics is clearly insufficient in a structure constant, M means the invariant mass of the di- microscopic approach [20]. Instead, one has to propagate properly the spectral functions of the ! and ! mesons. This is the main goal of our paper. We consider our work as being *Address correspondence to this author at the Forschungszentrum Dresden- on an explorative level, not yet as a firm and deep Rossendorf, Institut für Strahlenphysik, PF 510119, 01314 Dresden, theoretically founded prescription of dealing with di-electron Germany; Tel: +49-351-260-3258; Fax: +49-351-260-3600; emission from excitations with quantum numbers of ! and E-mail: [email protected] ! mesons off excited nuclear matter. 1874-415X/10 2010 Bentham Open 2 The Open Nuclear & Particle Physics Journal, 2010, Volume 3 Barz et al. Our paper is organized as follows. Essential features of other hand, when calculating such quantities as densities, our transport model are outlined in section 2. In Subsection Pauli blocking factors etc. we average over the ensembles in 2.1 we describe how the mean field potentials enter in the each time step. This method transforms the partial relativistic transport equation. Subsection 2.2 introduces the differential-integro equations (1) into a set of ordinary dynamics of broad resonances and its implementation in the differential equations (looking like equations of motion) for test-particle method. The crucial quantities for the spectral a number of test particles. A default version of the code has functions are the self-energies dealt with in subsection 2.3. been applied to strangeness dynamics [33-35]. Subsections 2.4 and 2.5 are devoted to particle production and di-electron emission, respectively. Numerical results of 2.2. Off-Shell Transport of Broad Resonances our simulations and a tentative comparison with published Recently theoretical progress has been made in HADES data are presented in sections 3 (2 AGeV) and 4 describing the in-medium properties of particles starting (1 AGeV). Discussion and summary can be found in section 5. from the Kadanoff-Baym equations [36] for the Green The present analysis supersedes [21], where a first functions of particles. Applying first-order gradient preliminary comparison of transport model results with the expansion after a Wigner transformation one arrives at a C(2 AGeV) + C data [10] has been presented. Further transport equation for the retarded Green function [37, 38]. analyses of HADES data have been performed in [22-27]. In the medium, particles acquire a self-energy !(x, p) which 2. TREATMENT OF HEAVY-ION COLLISIONS depends on position and momentum as well as the local properties of the surrounding medium. They have a finite life 2.1. The Standard BUU Treatment time which is described by the width ! related to the The employed BRoBUU computer code for heavy-ion imaginary part of the self-energy. Their properties are collisions developed by a Budapest-Rossendorf cooperation described by the spectral function being the imaginary part solves a set of coupled Boltzmann-Ühling-Uhlenbeck (BUU) of the retarded propagator A(p) = !2ImGret (x, p) . For equations in the quasi-particle limit [28, 29] ret bosons the spectral function is related to the self-energy ! !F !H !F !H !F via i + i " i = C , t p x x p # ij ! ! ! ! ! j (1) !ˆ (x, p) A(p) = , (3) 2 2 2 ! 2 2 ret 2 1 2 H = (mi +U(p, x)) + p ˆ (E " p " m0 " Re# (x, p)) + !(x, p) 4 for the one-body distribution functions Fi (x, p,t) of the where the resonance widths ! and !ˆ obey various hadron species i , each with rest mass m , in a i !ˆ (x, p) = "2Im#ret $ 2m ! , and m is the vacuum pole momentum and density dependent mean field U . The scalar 0 0 mean field U is chosen in such a manner that the mass of the respective particle. The spectral function (3) is 2 2 2 nr normalized as dp A = 2" . (We omit here the label Hamiltonian H equals H = mi + p +Ui with a potential ! nr denoting the particle type for simplicity). Ui calculated in the local rest frame as To solve numerically the Kadanoff-Baym equations one ' n ! n $ 2 d 3 p) F (x, p)) may exploit the above test-particle ansatz for a modified U nr = A + B + C N , (2) i # & ( 3 2 retarded Green function (see Refs. [37, 38]). This function n0 " n0 % n0 (2* ) ! p + p'$ 1+ can be interpreted as a product of particle number density "# %& , multiplied with the spectral function A . The spectral where the parameters A , B , C , ! , ! define special types function can significantly change in the course of the heavy- of potentials, while n , n and F stand for the baryon ion collision process. Therefore, the standard test-particle 0 N method, where the test-particle mass is a constant of motion, number density, saturation density and nucleon distribution 0 function. We use the momentum dependent soft potential must be extended by treating the energy E = p of the four- defined by A =-0.120 GeV, B =0.151 GeV, ! =1.23, C = momentum p as an independent variable. -0.0645 GeV, ! =2.167 GeV. The BRoBUU code Equations of motion for test particles follow from the propagates in the baryon sector the nucleons and 24 ! and transport equation. We use the relativistic version of the * N resonances and additionally !,",#,$ and ! mesons. equations which have been derived in Ref. [37]: Different particle species are coupled by the collision integral ! % ! 2 2 ret ! ( dx 1 1 ! ret m ! m0 ! Re# ˆ Cij which also contains the Ühling-Uhlenbeck terms respon- = '2 p + "p Re# + "p$*, (4) dt 1! C 2E & $ˆ ) sible for Pauli blocking of spin-1/2 baryons in the collision as well as particle creation and annihilation processes.
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