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Journal of Nuclear and Particle Physics 2015, 5(2): 38-43 DOI: 10.5923/j.jnpp.20150502.03 Transverse Mass Spectra for Charged Pions in Nucleus–Nucleus Collisions in Terms of Relativistic Quantum Molecular Dynamics Model A. M. Abdalla Department of Mathematical and Physical Engineering, Faculty of Engineering in Shoubra Benha University, Cairo, Egypt Abstract Transverse mass spectrum is one of the experimental parameter used to describe relativistic kinematic and mechanisms for hadron-hadron collisions and extended to nucleus-nucleus collisions. Charged pions emitted from Au+Au and Pb+Pb collisions at central rapidity (|𝑦𝑦| < 0.5) and impact parameter 𝑏𝑏 ≤ 3.4 𝑓𝑓𝑓𝑓 at different values of lab energies are investigated. The transverse mass distributions of secondaries depend on the energy of the interacting system. The experimental results compared with predictions by Quantum Molecular Dynamic model QMD and it ultra-relativistic implementation Ur-QMD using art version Ur-QMD-2.3. This model can give suitable distribution for transverse mass and describe experimental data for collision energy below 40 GeV. Above this energy, the hadron productions are not systematic which may due to the creation of different nuclear states of non-thermal equilibrium and high energy densities. The inverse slop parameter represents the effective temperature of the mechanism responsible for hadron productions. The temperature increases in regular tool with collision energy up to 40 GeV. Keywords Relativistic nucleus-nucleus collisions, Transverse mass, Effective temperature, Charged pions production 1. Introductions One of the major goals of research on high-energy heavy-ion collisions is to explore properties of strongly interacting matter, particularly its phase structure [1]. Nucleus-nucleus collisions such as, Au+Au collisions at √𝑠𝑠𝑁𝑁𝑁𝑁 = 200 GeV at Relativistic Heavy Ion Collider (RHIC), allow to study strongly interacting nuclear matter at extreme temperatures and densities. At this condition, nuclear matter will reach to new phase quark gluon plasma (QGP) [2]. Several signatures of the formation of transient QGP state during the early stage of a nucleus-nucleus collision at high energies were proposed in the past [3]. Quantum chromodynamics QCD, the main theory of strong interactions, believed that a gas of hadrons when its temperature exceeds a critical value [4] undergo a transition to state of quasi-free quarks and gluons [1]. In this state of matter, the normal confinement of quarks and gluons in hadrons removed and the partons can exist as quasi-free particles in an extended region of space-time. In the above given example for high-energy nucleus-nucleus collisions, could give many signals point to the formation of quark-gluon plasma (QGP) [5-9]. One of the first signatures proposed was an enhancement of strange particle production * Corresponding author: [email protected] (A. M. Abdalla) Published online at http://journal.sapub.org/jnpp Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved in A+A with respect to p+p collisions [10]. Theoretical analysis of these data is still in progress. RHIC continues to study the detailed properties of the strongly interacting matter using p+p, d+Au and Au+Au systems at various colliding energies from 7.7 GeV up to 200 GeV. Measurements of transverse momentum spectra for particles emerging from p+p collisions used as a baseline to which similar measurements from heavy ion collisions are compared to study collective effects. Explanation of transverse-mass spectra of hadrons in collisions of heavy nuclei turned out to be one of the most difficult tasks [11-13]. Experimental data on transverse-mass spectra of pions produced in central Au+Au [14] and Pb+Pb [15, 16] collisions reveal peculiar dependence on the incident energy. The inverse-slope parameter (called effective temperature T) of these spectra at mid-rapidity increases with incident energy in the energy domain of BNL Alternating Gradient Synchrotron (AGS) and then saturates at energies of CERN Super Proton Synchrotron (SPS). At these values of collision energies, nuclear materials reach to a state of nuclear saturation. It was assumed that this saturation is associated with the de-confinement phase transition [17, 18]. Many theoretical models are partly able to give new concepts on nuclear materials under these conditions and special formulations for treatments of the assumed processes that are responsible for the creations and production of experimentally observed secondary particles. One of the important of these models is the Quantum Molecular Journal of Nuclear and Particle Physics 2015, 5(2): 38-43 Dynamic model QMD and it ultra-relativistic implementation Ur-QMD and recent version is the art version Ur-QMD-2.3 [19, 20]. This model is a microscopic many body approach to pp, pA, and AA interactions at relativistic energies. It can be applied for studying hadron-hadron, hadron-nucleus and heavy ion collisions from Elab = 100 A MeV to √𝑠𝑠𝑁𝑁𝑁𝑁 = 200 GeV. Its microscopic transport approach is based on the covariant propagation of color strings, constituent quarks and di-quarks (as string ends) accompanied by mesonic and baryonic degrees of freedom. It simulates multiple interactions of ingoing and newly produced particles, the excitation and fragmentation of color strings and the formation and decay of hadronic resonances. At higher energies, the treatment of sub-hadronic degrees of freedom is of major importance. In the present model, these degrees of freedom enter via the introduction of a formation time for hadrons produced in the fragmentation of strings [21-23]. The major aspects and formulation of Ur-QMD are discussed in Refs. [24, 25]. Details on transverse mass distributions will discussed in section 2. A comparison and discussions for transverse mass distributions of charged pions produced from central and mid rapidity from Pb-Pb and Au-Au collisions at relativistic energies and results suggested by Ur-QMD-2.3 model will be in section 3. Transverse mass 𝑚𝑚 𝑇𝑇 (momentum 𝑝𝑝𝑇𝑇 ) is one of the variable parameter used to describe the relativistic kinematic for nucleon-nucleon collisions and extended to hadron-nucleus and nucleus-nucleus collisions. At hadron colliders, a significant and unknown proportion of the energy of the incoming hadrons in each event escapes down the beam-pipe. Consequently, if invisible particles are created in the final state, their net momentum can only be constrained in the plane transverse to the beam direction. Defining the z-axis as the beam direction, this net momentum is equal to the missing transverse energy vector 𝐸𝐸𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = − ∑𝑖𝑖 𝑝𝑝𝑇𝑇(𝑖𝑖) quantity 𝑚𝑚 𝑇𝑇 defined by 𝑚𝑚2𝑇𝑇 ≡ [𝐸𝐸𝑇𝑇 (1) + 𝐸𝐸𝑇𝑇 (2)]2 − [𝑝𝑝𝑇𝑇 (1) + 𝑝𝑝𝑇𝑇 (2)]2 where = 𝑚𝑚12 + 𝑚𝑚22 + 2[𝐸𝐸𝑇𝑇 (1)𝐸𝐸𝑇𝑇 (2) − 𝑝𝑝𝑇𝑇 (1) ∙ 𝑝𝑝𝑇𝑇 (2)] (2) 𝑝𝑝𝑇𝑇 (1) = 𝐸𝐸𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 (3) 𝑚𝑚2𝑇𝑇 = 2|𝑝𝑝𝑇𝑇 (1)||𝑝𝑝𝑇𝑇 (2)|(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝜑𝜑12 ) (4) This quantity 𝑚𝑚 𝑇𝑇 is called the ‘transverse mass’ by hadron collider experimentalists. The distribution of event = 𝑚𝑚 . If 𝑚𝑚 𝑇𝑇 values possesses an end-point at 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇 𝑚𝑚1 = 𝑚𝑚2 = 0 then where, 𝜑𝜑𝑖𝑖𝑖𝑖 is defined as the angle between particles i, and j, in the transverse plane. The transverse momentum 𝑝𝑝𝑇𝑇 is the projection of the particle's 3-momentum onto the plane that is transverse to the collision axis z: 𝑝𝑝𝑇𝑇 = 𝑝𝑝 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠, where 𝜃𝜃 is the initial polar angle of the particle with respect to the event vertex position along the collision axis z. The transverse energy of the particle with rest mass 𝑚𝑚0 is 𝑚𝑚 𝑇𝑇 = �𝑝𝑝𝑇𝑇2 + 𝑚𝑚02 (1) where the sum runs over the transverse momenta of all visible final state particles. (5) Along the beam axis z, a quantity called rapidity y defines the longitudinal motion scale for a particle with a known mass 𝑚𝑚0 : 𝑦𝑦 = 2. Theoretical Treatments 39 1 2 𝑙𝑙𝑙𝑙𝑙𝑙 � 𝐸𝐸+𝑝𝑝 𝑧𝑧 𝐸𝐸−𝑝𝑝 𝑧𝑧 � (6) where 𝑝𝑝𝑧𝑧 = 𝑚𝑚 𝑇𝑇 sinh(𝑦𝑦). The four-momentum of a particle can be described in terms of its transverse momentum 𝑝𝑝𝑇𝑇 , its rapidity 𝑦𝑦, and transverse energy 𝑚𝑚 𝑇𝑇 and 𝑝𝑝𝜇𝜇 = (𝑚𝑚 𝑇𝑇 cosh 𝑦𝑦, 𝑝𝑝𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝜑𝜑0 , 𝑝𝑝𝑇𝑇 sin 𝜑𝜑0 , 𝑚𝑚 𝑇𝑇 𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝑦𝑦) (7) The transport models Ur-QMD that employ hadronic and string degrees of freedom, [21-23] are used for our study. They take into account the formation and multiple re-scattering of hadrons and thus dynamically describe the generation of pressure in the hadronic expansion phase. The Ur-QMD transport approach [21-23] includes all baryonic resonances up to masses of 2 GeV as well as mesonic resonances up to 1.9 GeV. For hadronic continuum excitations, a string model used with hadron formation times in the order of 1-2 fm/c depending on the momentum and energy of the created hadron. The transport approach is matching to reproduce the nucleon-nucleon, meson nucleon and meson-meson cross section data in a wide kinematic range [26]. 3. Comparison with Experimental Results and Discussions Figure 1. Two body decay Consider a single heavy particle of mass m produced in association with visible particles which decays as in Fig.1 to two particles, of which one (labeled particle 1) is invisible. The mass of the parent particle can be constrained with the The pion is one of the particles that mediate the interaction between a pair of nucleons. This interaction is attractive pulls the nucleons together. Pions are mesons with zero spin, and they are composed of first-generation quarks. In the quark model, an up quark and an anti-down quark make up a π+, whereas a down quark and an anti-up quark make up the π−, and these are the antiparticles of one another. The neutral pion π0 is a combination of an up quark with an anti-up quark 40 A. M. Abdalla: Transverse Mass Spectra for Charged Pions in Nucleus–Nucleus Collisions in Terms of Relativistic Quantum Molecular Dynamics Model or a down quark with an anti-down quark. The two combinations have identical quantum numbers, and hence they are only found in superposition. The lowest-energy superposition of these is the π0, which is its own antiparticle. Together, the pions form a triplet of isospin. Each pion has isospin (I = 1) and third-component isospin equal to its charge (Iz = +1,0 or −1) [27]. Figure 2 shows the transverse mass spectra for positive pions at central rapidity (|𝑦𝑦| < 0.5) and impact parameter 𝑏𝑏 ≤ 3.4 𝑓𝑓𝑓𝑓 for Pb+Pb collisions at lab energies 4, 6, 8 and 11 GeV. The solid lines represent the corresponding calculations using QMD model with new art version Ur-QMD-2.3. The corresponding data from different experiments [28-33] are depicted with symbols. The same comparison applied for negative pions from Pb+Pb/Au+Au at energies 30, 40, 80, 160 and 200 GeV shown in Fig. 3. The UrQMD calculations give valuable representation for 𝑚𝑚 𝑇𝑇 spectra especially for low values of transverse mass. The Ur-QMD transport model successfully used to predict and interpret experimental data at various energies and for a multitude of observables and reaction systems, e.g. hadron yields, transverse spectra [34-36]. It treats the initial nucleon-nucleon interactions within a string-hadronic framework. In addition, this model includes effects such as string-string interactions and hadronic re-scattering, which are expected to be relevant in A+A collisions. Figure 2. Transverse mass distribution for secondary positive pions emitted from Au-Au collisions at 4, 6, 8 and 11 lab energies. The solid line the corresponding predictions by UrQMD version 2.3 Journal of Nuclear and Particle Physics 2015, 5(2): 38-43 41 Figure 3. Transverse mass distribution for secondary negative pions emitted from Pb-Pb collisions at 30, 40, 80, 160 and 200 lab energies. The solid line the corresponding predictions by UrQMD version 2.3 42 A. M. Abdalla: Transverse Mass Spectra for Charged Pions in Nucleus–Nucleus Collisions in Terms of Relativistic Quantum Molecular Dynamics Model Table 1. The temperature of positive pion production system at lab collision energies. The magnitudes in parentheses are the corresponding predicted by Ur-QMD 𝑇𝑇(𝜋𝜋 +) MeV 4 GeV 6 GeV 8 GeV 11 GeV 326.57±4.19 (372.81±5,29) 341.20±6.19 (393.11±3.40) 379.84±9.27 (407.07±3.11) 369.79±8.82 (415.11±4.16) Table 2. The temperature of negative pion production system at lab collision energies. The magnitudes in parentheses are the corresponding predicted by Ur-QMD 𝑇𝑇(𝜋𝜋 − ) MeV 30 GeV 40 GeV 80 GeV 160 GeV 200 GeV 375.90±12.98 (426.07±3.59) 409.83±14.41 (428.38±4.20) 321.82±31.88 (455.16±27.29) 397.90±16.87 (431.53±3.63) 514.32±9.10 (474.67±3.50) the two-pion correlation function [38]. This may be due to decrease of collision formation time and suitable conditions of high energy density. The processes become more difficult and confirm of creation states of non-thermal equilibrium of nuclear matter and non-regular mechanisms of hadron productions. 4. Conclusions Figure 4. Temperature of the charged pion production system as a function of the energies for Au-Au and Pb-Pb collisions The 𝑚𝑚 𝑇𝑇 spectra for hadrons like 𝜋𝜋 + and 𝜋𝜋 − production described by a power law relation in the full phase space accessible to the experiment with 1 𝑑𝑑𝑑𝑑 𝑚𝑚 𝑇𝑇 𝑑𝑑𝑚𝑚 𝑇𝑇 𝑑𝑑𝑑𝑑 = 𝐶𝐶 𝑒𝑒𝑒𝑒𝑒𝑒 �− 𝑚𝑚 𝑇𝑇 𝑇𝑇 � (8) In Eq. 8 the inverse slope parameter T and the normalization parameter C are free parameters and their values are extracted from the least square fits to the experimental spectra. The magnitude of T is the effective temperature of the system responsible for the emissions of secondary hadrons. The values of T, for system responsible for production of positive pions at above energies are given in table 1. The same values obtained for negative pion given in table 2. The values in parentheses are the corresponding theoretical values. The behavior of this dependence, as a function of collision energies in the lab system are shown in Fig. 4. The solid line represents the linear fitting of the experimental data. For energies below 40 GeV the temperature of the system responsible for charged pion production is uniformly changes with collision energies from Pb+Pb (Au+Au) collisions. Above this energies there is a suddenly changes in T and UrQMD assumption is however difficult to justify with the dynamical models of the collision process [20, 37] and the results on the energy dependence of Transverse mass is one of the experimental parameter used to study kinematic of nucleus-nucleus collision at relativistic energies. The ultra-relativistic Quantum Molecular Dynamic Model using version Ur-QMD-2.3 give suitable transverse mass distribution for charged pions production compared with experimental data for collision energies below 40 GeV and above the hadrons production is not systematic which may due to the creation of nuclear states of non-thermal equilibrium and states of high energy density. The inverse slop parameter represents the effective temperature of the mechanism responsible for hadrons production. The effective temperatures are regular increase with collision energy up to 40 GeV. ACKNOWLEDGEMENTS I would like to thank RQMD group for answering the questions and their suitable advice to make the best use of the model. I am grateful to the group for their supply by the program of calculations. REFERENCES [1] R. Stock, J. Phys. G 30, 633 (2004). [2] J. C. Collins and M. J. Perry, Phys. Rev. Lett. 34, 1353 (1975), E. V. Shuryak, Phys. Rep. 61, 71 (1980) and 115, 151(1984). [3] J. 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