Formulation of transverse mass distributions in Au-Au collisions at 200 GeV/nucleon

The transverse mass spectra of light mesons produced in Au-Au collisions at 200 GeV/nucleon are analyzed in Tsallis statistics. In high energy collisions, it has been found that the spectra follow a generalized scaling law. We applied Tsallis statistics to the description of different particles using the scaling properties. The calculated results are in agreement with experimental data of PHENIX Collaboration. And, the temperature of emission sources is extracted consistently.


I. INTRODUCTION
Multiparticle production is an important experimental phenomenon at Relativistic Heavy Ion Collider (RHIC) in Brookhaven National Laboratory (BNL). In Au-Au collisions, identified particle yields per unity of rapidity integrated over transverse momentum p T ranges have provided information about temperature T and chemical potential µ at the chemical freeze-out by using a statistical investigation [1]. It brings valuable insight into properties of quark-gluon plasma (QGP) created in the collisions. A much broader and deeper study of QGP will be done at Large Hadron Collider (LHC) at the European Organization for Nuclear Research (CERN) and the Facility for Antiproton and Ion Research (FAIR) at the Gesellschaft für Schwerionenforschung mbH (GSI).
In order to estimate hadronic decay backgrounds in photon, single lepton and dilepton spectra which are penetrating probes of QGP, m T spectra of identified mesons have been studied in detail [2][3][4][5][6], where m T = m 2 0 + p 2 T is transverse mass of a particle with rest mass m 0 at a given p T . In Ref. [7], m T spectral shapes of pions and η mesons in S-S and S-Au collisions are identical. Such behaviors are caused by m T scaling properties, which help us to predict new m T spectra and understand the mechanism of meson production. Statistical analysis of m T spectra is extremely useful to extract information of particle production process and interaction in hadronic and QGP phases. In the CGS (Color Glass Condensate) description, the total hadron multiplicity follows a scaling behavior motivated by the gluon saturation.
Different phenomenological models of initial coherent multiple interactions and particle transport have been introduced to describe the production of final-state particles [8,9] in Au-Au collisions. With Tsallis statistics' development and success in dealing with non-equilibrated complex systems in condensed matter research [10], it has been utilized to understand the particle production in high-energy physics [11][12][13]. In our previous work [14], the temperature information of emission sources was understood indirectly by a excitation degree, which varies with location in a cylinder. We have obtained emission source location dependence of the exciting degree specifically.
From central axis to side-surface of the cylinder, the excitation degree of the emission source decreases linearly with the direction of radius. In this work, we parametrize experimentally measured m T spectra of pions in Tsallis statistics. Using the m T scaling properties in the spectrum calculation, we reproduce m T spectra of other light * libc2010@163.com, s6109@sxu.edu.cn 2 mesons and obtain the temperature of emission sources directly.

II. THE FORMULATION AND COMPARISON WITH PHENIX RESULTS
At the initial stage of nucleus-nucleus collisions, plenty of primary nucleon-nucleon collisions happen. The primary nucleon-nucleon collision can be regarded as an emission source (a compound hadron fireball) at intermediate energy or a few sources (wounded partons and woundless partons) at high energy. The participant nucleons in primary collisions have probabilities to take part in cascade collisions with latter nucleons. Meanwhile, the particles produced in primary or cascade nucleon-nucleon collisions have probabilities to take part in secondary collisions with latter nucleons and other particles. Each cascade (or secondary) collision is also regarded as an emission source or a few sources. Many emission sources of final-state particles are expected to be formed in the collisions.
According to Tsallis statistics [10], the total number of the mesons is given by where p, E, T , µ, V and g are the momentum, the energy, the temperature, the chemical potential, the volume and the degeneracy factor, respectively, a parameter q is used to characterize the degree of nonequilibrium. The corresponding momentum distribution is We have the transverse mass m T distribution, At midrapidity y = 0, for zero chemical potential, the transverse mass spectrum in terms of y and m T is which is only a m T distribution of particles emitted in the emission source at midrapidity y = 0.
Considering a width of the corresponding rapidity distribution of final-state particles, the m T spectrum is rewritten as where C = gV (2π) 2 is a normalization constant and Y(−Y) is the maximum (minimum) value of the observed rapidity. Generally speaking, the temperature T and q can be fixed for different event centralities (or impact parameters) by fitting the experimental data of pions. The temperature T of emission sources is calculated naturally and consistently in the current formulation. The symbols represent experimental data of PHENIX Collaboration [15,16] and the curves are fitting results by using Eq.(5). By fitting the experimental data, values of T and q are given in Table I [15,16] in different centrality cuts indicated in the figure. The curves are the results obtained by fitting the data. We fit the spectra using Tsallis distributions and obtain the values of T and q which are given in Table I with χ 2 /dof. It is found that the temperature T increase with the increase of the centrality. Fig. 6 and Fig. 7 show invariant yields of K ± , J/ψ, φ, ω and η as a function of m T − m in corresponding centrality cuts. The symbols represent experimental data of PHENIX Collaboration [16-18, 20, 21]. The curves are the results calculated by using m T scaling properties. The values of χ 2 /dof are shown in Table II. For different centralities, the m T scaled results are in agreement with the experimental data of different mesons.   Fig.2−Fig.4, Fig.6 and Fig.7.    Table III. The ratios are helpful to understand the contribution of hadronic decay in photonic and leptonic channels.
Final-state particles produced in high-energy nuclear collisions have attracted much attention, since attempt have been made to understand the properties of strongly coupled QGP by studying the possible production mechanisms [22,23]. Thermal-statistical models have been successful in describing particle yields in various systems at different energies [14,[24][25][26]. The temperature T of emission sources is very important for understanding the matter evolution in Au-Au collisions at RHIC. In the rapidity space, different sources of final-state particles stay at different positions due to stronger longitudinal flow [27][28][29] . In our previous work, we have studied the trans-