\documentclass[epj,final,nopacs]{svjour} \usepackage{latexsym} \usepackage{graphics} %\input{tcilatex} \begin{document} \title{Coulomb interaction, isospin symmetry and $\frac{\pi^-}{\pi^+}$ ratio in Au-Au collisions at $\sqrt{s_{NN}}=200GeV$} \author{Bucharest Group} \institute{} \date{} \abstract{Coulomb effects on $\pi^{\pm}$ spectra in $Au-Au$ collisions at $\sqrt{s_{NN}}=200GeV$ are investigate. The Coulomb effect provides information on source size (freeze-out radius), flow velocity and freeze-out time. We observe a reduced Coulomb interaction than obtained in AGS and SPS at lower energies.} %\PACS{} \maketitle \section{Introduction} The nuclear matter formed in relativistic heavy ion collisions evolves to freeze-out state through complex processes defined by the collision energy and the collision geometry. In ultrarelativistic heavy ion collisions a large number of particles are produced, mostly pions. The pion ratios reflect the equilibrium degree of isospin in the reaction and also the effects of the Coulomb repulsion, because the charged particles, especially the pions, are highly influenced by the Coulomb field produced by the net charge of the reaction protons. Information on the charge distribution in the participant region could be obtained from analysing the dependence of the negative to positive ratios of the produced pions on the transverse energy. Since the Coulomb forces influence the matter essentially after freeze-out, the asymmetry in the number of charged particles can be directly related to the freeze-out parameters. In this work we analyse the Coulomb interaction using the $\frac{\pi^+}{\pi^-}$ ratio obtained in Au-Au collisions at $\sqrt{s_{NN}}=200GeV$. \section{Experimental Setup} The BRAHMS Experimental set-up consists from two independent small aperture magnetic spectrometers (Forward Spectrometer FS and Mid-Rapidity Spectrometer MRS) and a set of detectors for global characterization of the events []. The Mid-Rapidity Spectrometer (MRS) has a solid angle of $6.5 msr$ and it covers the angular range of $30^o<\theta<95^o$. A single dipole magnet (D5) of the MRS is placed between two TPC's, which are used for tracking. Particle identification is obtained with a time of flight wall. $\pi-K$ separation is achieved up to a momentum of $2.3 GeV/c$ and K-p separation up to $3.9 GeV/c$. The Forward Spectrometer (FS) has a solid angle of $0.8 msr$ and it covers the angular range of $2.3^o<\theta<30^o$. Four magnets (D1-D4) are used for sweeping and analyzing particles emerging from the reaction. The FS contains two Time Projection Chambers (T1 and T2), which provide good three-dimensional track recognition and rejection of background in a high multiplicity enviroment, three Drift Chambers (T3-T5), two time-of-flight hodoscopes (H1-H2) and two Cerenkov detectors (a threshold Ceren\-kov detector (C1) placed behind D2 magnet and a ring imaging Cerenkov detector (RICH) placed behind H2). H1 (at $8.6 m$) and H2 (at $18 m$) allow for K-p separation up to $p=5.5$ and $8 Gev/c$, respectively. C1 identifies pions in the range from $p=3$ to $9 GeV/c$ and RICH allows $\pi-K$ separation up to $p=25 GeV/c$ and K-p separation from $p=10$ to $35 GeV/c$. \section{Experimental results} The Coulomb effect was studied in heavy ions collisions at much lower energies, at $1 A GeV$ (SIS - Darmstadt), $11.4 AGeV$ (AGS - Brookhaven National Laboratory) and $158 A GeV$ (SPS - CERN) [5-7]. From the analysis that was made, it have been observed that the Coulomb effect is much stronger at low transverse momentum, because the Coulomb interaction potential energy is smaller than the kinetic energy of particles and because of the low velocity of expansion for the net charge. At lower energies, the nuclei are fully stopped. Under these conditions, the total charge stays together sufficient time to significantly accelerate or decelerated the produced charged pions. Because of that, the pion spectra are modified, leading to nonuniform pion yield ratios. At higher energies, the colliding nuclei are no longer completely stopped and in these collisions can be achieved a degree of transparency of the colliding nuclei. In these collisions one could expect a smaller Coulomb effect. It has been proposed the introduction of the Coulomb momentum (\textit{"Coulomb kick"}) in order to study the $\pi^+$ - $\pi^-$ asymmetry at AGS and SPS energies [1-4]. The authors consider that the interaction between charged pions and the net charge of the participant protons changes transverse momentum with the quantity: \begin{equation} p_c{\equiv}|p_{\perp}-p_{{\perp},0}|{\cong}2e^2{\frac{dN^{ch}}{dy}{\frac{1}{R_f}}} \end{equation} where $p_{{\perp},0}$ is the transverse momentum at freeze-out and $p_{\perp}$ is the final momentum. The authors predicted the pion ratio can be described by the following relationship: \begin{equation} \frac{\pi^-}{\pi^+}=\left\langle\frac{\pi^-}{\pi^+}\right\rangle\frac{p_{\perp}+p_c}{p_{\perp}-p_c}\exp{\left(\frac{m_{\perp}^--m_{\perp}^+}{T}\right)} \end{equation} where $m_\perp^\pm=\sqrt{m^2+\left(p_{\perp}{\pm}p_c\right)^2}$. The authors assume that the radius at freeze-out is: \begin{equation} R_f=R_{geom}+0.5{\cdot}\beta_\perp\cdot\tau_f \end{equation} Here, $R_{geom}$ is the initial (geometrical) radius of the overlap zone at the collision time and $\beta$ is the transverse flow velocity. The experimental data from AGS ($11.4 A GeV$) and SPS ($158 A GeV$) are well reproduced by $p_c^{AGS}{\cong}20MeV/c$, $p_c^{SPS}{\cong}10MeV/c$, respectively. Using the measured proton rapidity distributions, $dN_p^{AGS}/{dy}{\cong}70$, $dN_p^{SPS}/{dy}{\cong}70$, respectively, it was calculated the size of the systems at freeze-out, namely: $R_f^{AGS}{\cong}10fm$ and $R_f^{SPS}{\cong}9fm$. We analysed the $\frac{\pi^-}{\pi^+}$ ratio as a function of transverse mass ($m_\perp-mass$) for $Au+Au$ collisions at $\sqrt{s_{NN}}=200GeV$. The pions were detected with MRS at $90^o$, at rapidity $y=0$ [8]. The ratios were obtained for different centrality cuts ($0-10$\%, $10-20$\%, $20-40$\%, $40-60$\%). From central to peripheral data the $\frac{\pi^-}{\pi^+}$ ratios are fitted using the the expression (2), and assuming that the temperature is obtained as below. The transverse mass distributions of the pions were fit to the expression: \begin{equation} \frac{1}{N_{ev}}\frac{1}{2{\pi}m_T}\frac{dN}{dm_Tdy}=\frac{1}{2\pi}\frac{dN}{dy}\frac{1}{T(T+m_0)}e^\frac{m_T-m_0}{T} \end{equation} with the slope parameter $T$ and yield $dN/dy$ set as free parameters, for different centrality cuts. The obtained results are listed in Table I. \begin{center} $\begin{array}{|r|c|c|} \hline centrality & T(\pi^+)[MeV] & T(\pi^-)[MeV] \\ \hline \hline 0-10\% & 243.4\pm0.9 & 242.6\pm0.9 \\ \hline 10-20\% & 245.5\pm1.1 & 241.6\pm1.1 \\ \hline 20-40\% & 241.2\pm1.2 & 240.7\pm1.2 \\ \hline 40-60\% & 230.3\pm1.9 & 231.5\pm2.0 \\ \hline \end{array}$\\ \scriptsize{\textbf{Table I}: Effective temperatures obtained from \\ fits to pion spectra for four centrality cuts} \end{center} These temperatures are greater than those obtained at SPS. At RHIC energies, the mass dependence of the slope parameters seems to be stronger than that at the SPS energies, indicating a larger colective flow in higher energy nuclear collisions. The obtained temperatures show a weak centrality dependence for the first three centrality class. The results of the fit of the $\frac{\pi^-}{\pi^+}$ yields within each centrality cut are shown in Table II. No strong centrality dependence is observed in the centrality range measured. The $\chi^2$ contour levels for the two parameters of the fit (Coulomb kick and the total ratio) are shown for each centrality cut (Fig.2, Fig.4, Fig.6, Fig.8). The $\chi^2/DOF$ is in the range $(1.1\div1.5)$ for all centrality cuts. \begin{center} $\begin{array}{|r|c|} \hline centrality & p_c[MeV/c] \\ \hline \hline 0-10\% & 6.50\pm1.59 \\ \hline 10-20\% & 6.81\pm1.09 \\ \hline 20-40\% & 5.04\pm1.52 \\ \hline 40-60\% & 5.06\pm2.52 \\ \hline \end{array}$\\ \scriptsize{\textbf{Table II}: Coulomb kick obtained from fits to \\ $\pi^-/\pi^+$ yields for four centrality cuts} \end{center} Fig. 1. shows the fit to the $\frac{\pi^-}{\pi^+}$ ratios for the most central collisions $0-10$\%. The obtained Coulomb kick value is $6.5 MeV/c$. This value of Coulomb momentum is smaller than values obtained at AGS and SPS. The result shows that at RHIC energies, the system expands collectively under strong internal pressure, the collective flow is faster and the pions experience a smaller Coulomb kick from the net charge of the protons. Therefore, the Coulomb repulsion is reduced than at AGS and SPS energies. %\resizebox*{5cm}{3cm}{\includegraphics{./figuri/1.eps}} \resizebox*{5cm}{3cm}{\includegraphics{./figuri/2.eps}} In these collisions, the two nuclei pass through each other forming a barion poor region in the middle, at these energies is obtained a smaller stopping than at SPS and AGS energies, and therefore the charge density is smaller at midrapidity. Fig. 9. shows the transverse mass spectrum for the protons at $y=0$. The proton rapidity density was obtained by fitting the proton transverse mass spectrum with the (4) and the value obtained from the $0-10$\% most central events is $dN/dy=29.79\pm0.38$. Using the relation (1) the freeze-out radius obtained is $R_f=11.15fm$. This value which is greater than the geometrical radius indicate that significant expansion takes place before freeze-out (for $0-10$\% centrality). In order to estimate the geometrical radius we use a phenomenological geometrical model [9,10], which consider that a very hot region is created from the overlapping region of the two colliding nuclei. The spectator regions slow down the flow and can absorb some particles created in the hot region. Different physical quantities can be calculated in the following working assumptions: $\bullet$ the nucleons are spheres of radii $r_0$ and nuclei are spheres of radii $R=r_0A^{1/3}$; $\bullet$ initially, in the target nucleus a geometrical spherical zone occurs, the volume of the geometrical spherical zone depends on the impact parameter $b$ and on the beam energy; $\bullet$ the ratio $(Z_P+Z_T)/(A_P+A_T)$ remains constant for the very hot region $\bullet$ the geometrical spherical zone evolves in a very hot sphere and the volume of the sphere is equal to the volume of the geometrical spherical zone. In these assumptions the radius of the very hot sphere is \begin{equation} R_{geom}=2^{2/3}c(\gamma)d^{1/3}(3r_1^2+3r_2^2+h^2)^{1/3} \end{equation} where $r_{1,2}^2=|R_T^2-(b{\pm}R_P)^2|$, $h=2R_P$, $R_P=R_T=R$, $R=r_0A^{1/3}$. The factor $c(\gamma)$ is a quantity depending on the time evolution of the fireball. This evolution is related to the contraction Lorentz factor, $\gamma$. For the estimation of the impact parameter we use the simulated data with the HIJING code. For the most central collisions, $0-5$\%, average impact parameter is $1.42fm$. For $c(\gamma)=2^{-2/3}$, we find the geometrical radius $R_{geom}=6.87fm$. In $Au+Au$ collisions at RHIC energies, the density of produced particles is high; following this, the number of secondary collisions among the produced particles or the number of rescatterings is high and thus collective transverse motion is very strong. At midrapidity $(y=0)$, the observed particle density at is about $2.1$ times greater than in $Pb+Pb$ collisions at $17.2GeV$[11]. Assuming a transverse flow velocity of $0.6c$, using the relation (3), and the value for $R_{geom}=6.87fm$, we obtained the freeze-out time: $\tau_f=14fm/c$. This freeze-out time is higher than the value obtained at SPS, indicating that in these collisions the source is more expanded longitudinal and that the freeze-out occurs much later (for $0-10$\% centrality). At $10-20$\% centrality, the obtained value is very close to the value obtained for the most central events (Fig.3.). The pions multiplicities decrease slowly for the first two centrality cuts ($0-10$\%, $10-20$\%); it follows that Coulomb kick is almost constant for these centrality cuts. For the $20-40$\% centrality, the Coulomb kick slowly decrease (Fig. 5). For the peripheral collisions ($40-60$\% centrality), there are less produced particles and thus the collective flow is much reduced; in these collisins the rapidity density of the protons is much smaller. Because of that the obtained values could be affected (Coulomb kick could be obtained with much greater errors) (Fig.7). \section{Final remarks} The Coulomb effects in pion spectra are sensitive to the degree of stopping and the distribution of positive charge, as well as at the flow velocity of the participant region. Using the experimental results obtained by BRAHMS experimental set-up, a modification in the transverse momentum due to the Coulomb interaction smaller than obtained in other experiments at lower energies is observed. This value can reflect a reduced Coulomb effect at higher flow velocities of the nuclear matter from participant region. \begin{figure}[tbp] \resizebox{1.\hsize}{!}{\includegraphics{figuri_final/fig1.eps}} \caption{Transverse mass dependence of the $\pi^-/\pi^+$ yields. The data are at y=0 and for 0-10\% centrality. The full curve is for the fit, as described in text.} \label{fig:1} \end{figure} \begin{figure}[tbp] \resizebox{1.\hsize}{!}{\includegraphics{figuri_final/fig2.eps}} \caption{The $\chi^2$ contour levels of the parameters of the fit: Coulomb kick (vertical) and the total pion ratio (the total production)(horizontal) for 0-10\% centrality.} \label{fig:2} \end{figure} \begin{figure}[tbp] \resizebox{1.\hsize}{!}{\includegraphics{figuri_final/fig3.eps}} \caption{Transverse mass dependence of the $\pi^-/\pi^+$ yields. The data are at y=0 and for 10-20\% centrality. The full curve is for the fit to the data.} \label{fig:3} \end{figure} \begin{figure}[tbp] \resizebox{1.\hsize}{!}{\includegraphics{figuri_final/fig4.eps}} \caption{The $\chi^2$ contour levels of the parameters of the fit: Coulomb kick (vertical) and the total pion ratio (horizontal) for 10-20\% centrality.} \label{fig:4} \end{figure} \begin{figure}[tbp] \resizebox{1.\hsize}{!}{\includegraphics{figuri_final/fig5.eps}} \caption{Transverse mass dependence of the $\pi^-/\pi^+$ yields. The data are at y=0 and for 20-40\% centrality. The full curve is for the fit to the data.} \label{fig:5} \end{figure} \begin{figure}[tbp] \resizebox{1.\hsize}{!}{\includegraphics{figuri_final/fig6.eps}} \caption{The $\chi^2$ contour levels of the parameters of the fit: Coulomb kick (vertical) and the total pion ratio (horizontal) for 20-40\% centrality.} \label{fig:6} \end{figure} \begin{figure}[tbp] \resizebox{1.\hsize}{!}{\includegraphics{figuri_final/fig7.eps}} \caption{Transverse mass dependence of the $\pi^-/\pi^+$ yields. The data are at y=0 and for 40-60\% centrality. The full curve is for the fit to the data.} \label{fig:7} \end{figure} \begin{figure}[tbp] \resizebox{1.\hsize}{!}{\includegraphics{figuri_final/fig8.eps}} \caption{The $\chi^2$ contour levels of the parameters of the fit: Coulomb kick (vertical) and the total pion ratio (horizontal) for 40-60\% centrality.} \label{fig:8} \end{figure} \begin{figure}[tbp] \resizebox{1.\hsize}{!}{\includegraphics{figuri_final/fig9.eps}} \caption{Transverse mass spectra of protons from Au-Au collisions at 200 AGeV for central events (0-10\%) at y=0.} \label{fig:9} \end{figure} \begin{thebibliography}{99} \bibitem{1} H. W. Barz, J. P. Bondorf, J. J. Gaardoje and H. Heiselberg - Coulomb effects on particle spectra in relativistic nuclear collisions - nucl-th/9711064 \bibitem{2} H. Heiselberg - Freeze-out from HBT and Coulomb effects - nucl-th/9802002 \bibitem{3} H. W. Barz, J. P. Bondorf, J. J. Gaardoje and H. Heiselberg - Phys. Rev. C 56, 3 (1997) \bibitem{4} H. W. Barz, J. P. Bondorf, J. J. Gaardoje and H. Heiselberg - nucl-th/9704045 \bibitem{5} H. Boggild et al. (NA 44) - Phys. Lett. B 372, 343 (1996) \bibitem{6} D. Pelte et al. - Z. Phys. A357 (1997) 215 \bibitem{7} L. Ahle et al - Nucl. Phys. A610 (1996) 139c \bibitem{9} C. Besliu, Al. Jipa - Rom. J. Phys.37(1992)1011 \bibitem{10} Al. Jipa - J. Phys. G22(1996)231 \bibitem{11} F.Rami for the BRAHMS Collaboration "Charged Particle Production at RHIC energies" - Proceedings of 'QCD and high energy hadronic interactions', Moriond, Les Arcs, March 2002. \end{thebibliography} % %Figure caption: % %1 Transverse mass dependence of the ?- / ?+ yields. The data are at y=0 and for 0-10% centrality. %The full curve is for the fit, as described in text. % %2 The ?2 contour levels of the parameters of the fit: Coulomb kick (vertical) and the total pion %ratio (the total production)(horizontal) for 0-10% centrality % %3 Transverse mass dependence of the ?- / ?+ yields. The data are at y=0 and for 10-20% centrality. The full %curve is for the fit to the data. % %4The ?2 contour levels of the parameters of the fit: Coulomb kick (vertical) and the total pion ratio (horizontal) %for 10-20% centrality % %5Transverse mass dependence of the ?- / ?+ yields. The data are at y=0 and for 20-40% centrality. %The full curve is for the fit to the data. % %6The ?2 contour levels of the parameters of the fit: Coulomb kick (vertical) and the total pion %ratio (horizontal) for 20-40% centrality % %7Transverse mass dependence of the ?- / ?+ yields. The data are at y=0 and for 40-60% centrality. %The full curve is for the fit to the data. % %8The ?2 contour levels of the parameters of the fit: Coulomb kick (vertical) and the total pion %ratio (horizontal) for 40-60% centrality % %9 Transverse mass spectra of protons from Au-Au collisions at 200 AGeV for central %events (0-10%) at y=0 % \end{document}