PHYSICAL REVIEW C 70, 021903(R) (2004) Model or the spacetime evolution o 200A GeV Au-Au collisions Thorsten Renk Department o Physics, Duke University, P.O. Box 90305, Durham, North Carolina 27708, USA (Received 16 April 2004; published 31 August 2004) We investigate the space-time evolution o ultrarelativistic Au-Au collisions at ull relativistic heavy-ion collider energy using a schematic model o the expansion. Assuming a thermally equilibrated system, we can adjust the essential scale parameters o this model such that the measured transverse momentum spectra and Hanbury-Brown Twiss correlation parameters are well described. We ind that the experimental data strongly constrain the dynamics o the evolution o the emission source although hadronic observables or the most part relect the inal breakup o the system. DOI: 10.1103/PhysRevC.70.021903 PACS number(s): 25.75. q Numerical simulations o inite temperature quantum chromodynamics (QCD) on discrete space-time lattices suggest that the theory exhibits a transition to a new state o matter, the quark-gluon plasma (QGP), at temperatures o the order o 170 MeV [1]. Experimentally, hadronic matter at this temperature can only be created or a short time in ultrarealtivistic heavy-ion collisions, and currently there are ongoing eorts rom both experiment and theory to ind conclusive evidence or the creation o the QGP and ultimately to study its properties. A wealth o data on hadronic single particle distributions and two particle correlations has been assembled so ar. In the interpretation o experimental data o ultrarelativistic heavy-ion collisions, however, one is oten aced with the challenge to disentangle signals o new physics indicating the production o a QGP rom known hadronic eects. This question can only be reliably addressed i the space-time evolution o the ireball created in the collision is suiciently known. In this note, we present an attempt to determine the evolution o the bulk hadronic matter by itting the essential scales o a schematic model or the ireball expansion dynamics to the measured hadronic data. The resulting scenario can then be used in the calculation o other observables (not used in the it) which are sensitive to the bulk matter expansion, such as the emission o dileptons and photons or jet quenching, thus reducing or eliminating the inherent ambiguity in the interpretation mentioned above. The main assumption or the model is that an equilibrated system is ormed a short time 0 ater the onset o the collision. Furthermore, we assume that this thermal ireball subsequently expands isentropically until the mean ree path o particles exceeds (at a timescale ) the dimensions o the system and particles move without signiicant interaction to the detector. In addition to this inal breakup (reeze-out) o the ireball, particles are emitted throughout the expansion period whenever they cross the boundary o the thermalized ireball matter. For simplicity we restrict the discussion to a system exhibiting radial symmetry around the beam z -axis corresponding to a central b=0 collision. For the entropy density at a given proper time we make the ansatz s, s,r = NR r, H s, 1 with the proper time as measured in a rame co-moving with a given volume element and s = 1 2ln t+z / t z the spacetime rapidity and R r,,h s, two unctions describing the shape o the distribution and N a normalization actor. We use Woods-Saxon distributions R r, =1 1 + exp r R c, d ws H s, =1 1 + exp s H c, 2 ws to describe the shapes or a given. Thus, the ingredients o the model are the skin thickness parameters d ws and ws and the parametrizations o the expansion o the spatial extensions R c,h c as a unction o proper time. For simplicity, we assume or the moment a radially nonrelativistic expansion and constant acceleration, thereore we ind R c =R 0 + a /2 2. H c is obtained by integrating orward in a trajectory originating rom the collision center which is characterized by a rapidity c = 0 +a with c =a tanh v c z where v c z is the longitudinal expansion velocity or that trajectory. Since the relation between proper time as measured in the co-moving rame and lab time is determined by the rapidity at a given time, the resulting integral is in general nontrivial and solved numerically (see [2] or details). R 0 is determined in overlap calculations using Glauber theory, the initial size o the rapidity interval occupied by the ireball matter. 0 is a ree parameter and we choose to use the transverse velocity v =a and rapidity at decoupling proper time = 0 +a as parameters. Thus, speciying 0,, v, and sets the essential scales o the spacetime evolution and d ws and ws speciy the detailed distribution o entropy density. For simplicity, we do not discuss a (possible) time dependence o the shapes [e.g., parametrized by d ws and ws ] at this point. We require that the parameter set, ws reproduces the experimentally observed rapidity distribution o particles [3] but we do not make any assumptions about the initial rapid- 0556-2813/2004/70(2)/021903(5)/$22.50 70 021903-1 2004 The American Physical Society
THORSTEN RENK PHYSICAL REVIEW C 70, 021903(R) (2004) FIG. 1. The longitudinal low proile s as a unction o spacetime rapidity or both the RHIC scenario itted in the present paper (solid) and or a boost-invariant scenario at reeze-out time. The boost-invariant case has been chosen such that it leads to the same inal distribution o particles in. In the accelerated scenario the extension in spacetime rapidity is always smaller than in the boost-invariant one or a given proper time, hence =3.5 is reached or a value o s =2.7 already. Note that the low proile in the accelerated case is nevertheless to a good approximation linear in s. ity interval characterized by 0. This allows or the possibility o accelerated longitudinal expansion and implies in general s. Here, = 1 2 ln p 0+ p z / p 0 p z denotes the longitudinal rapidity o a volume element moving with momentum p. We require that the longitudinal low proile is such that an initially homogeneous distribution remains homogeneous. For a general accelerated motion, there is no simple analytical expression or the spacetime position o sheets o given or the low proile such as = t 2 z 2 and = s which are valid or the non-accelerated case. In [2] however, we have investigated such an expansion pattern with a constant acceleration and argued that or rapidities 4 the acceleration leads to an approximately linear relation s at given (see Fig. 1) and that sheets o constant proper time are approximately hyperbolae as in the nonaccelerated case. This mismatch between and s leads to additional Lorentz contraction actors in volume integrals at given, hence the longitudinal extension o matter on a sheet o given in the interval ront ront s s s must then be calculated as L 2 sinh 1 ront s. 3 1 In the ollowing, we use these results in our computations whenever 0. For = 0, the model reduces to the wellknown expressions o the Bjorken expansion scenario, e.g., L =2 0 with = t 2 z 2. For transverse low we assume a linear relation between radius r and transverse rapidity =a tanh v =r/r c c with c =a tanh a. For the net baryon density inside the ireball matter we assume a transverse distribution (apart rom a normalization actor) given by R r,, but its longitudinal distribution we parametrize such as to describe the measured data [4]. We proceed by speciying the equation o state EoS o the thermalized matter. In the QGP phase, we use an equation o state based on a quasiparticle interpretation o lattice QCD data (see [5]). In the hadronic phase, we adopt the picture o subsequent chemical reeze-out (at the transition temperature T c ) and thermal reeze-out (at breakup temperature T F ) and consequently use a resonance gas EoS which depends on the local net baryon density. The reason or this is that a inite baryochemical potential B leads to an increased number o heavy resonances at the phase transition point, and decay pions rom these resonance decays lead in turn to a inite pion chemical potential in the late evolution phases which implies an overpopulation o pion phase space and aster cooling as compared to a scenario in chemical equilibrium. We calculate the pion chemical potential as a unction o the local baryon and entropy density using the statistical hadronization ramework outlined in [6]. We ind to be small O 30 MeV in the midrapidity region at relativistic heavey-ion collider (RHIC), however the corrections to the chemically equilibrated case are important. With the help o the EoS, we can ind the local temperature T s,r, o a volume element rom its entropy density s s,r, and net baryon density B s,r,. We calculate particle emission throughout the whole lietime o the ireball by selecting a reeze-out temperature T, inding the hypersurace characterized by T s,r, =T and evaluating the Cooper-Frye ormula E d3 N d 3 p = g 2 3 d p exp p u i T = d 4 xs x,p with p the momentum o the emitted particle and g its degeneracy actor. Note that the actor d p contains the spacetime rapidity s and the actor p u the rapidity. Since these are in general not the same in our model, the analytic expressions valid or a boost-invariant scenario [7] do not apply. In our case, the parameter denotes the reeze-out time or the last volume element to reach the temperature T. Volume elements reeze-out throughout the whole evolution time (and are boosted with a time- and position-dependent low velocity) whenever they cross the Cooper-Frye hypersurace. Thereore the transverse expansion parameter v also does not necessarily relect the typical transverse velocity o a volume element since reeze-out may occur earlier than and not at the radius r=r. Using this emission unction, we calculate the Hanbury- Brown Twiss (HBT) parameters as [8,9] 2 R side 4 = ỹ 2 2 R out = x t 2 R long = z 2 5 021903-2
MODEL FOR THE SPACETIME EVOLUTION OF HEAVY- PHYSICAL REVIEW C 70, 021903(R) (2004) FIG. 2. Let panel: Time evolution o the average ireball temperature or the itted set o parameters (solid line) and a model assuming boost-invariant longitudinal expansion (dotted). Right panel: Measured transverse momentum spectra or (circles), K (squares), and p (diamonds) as compared to the model results. with x =x x and x K = d 4 x x S x,k. 6 d 4 xs x,k In order to parametrize the spacetime evolution o a central 200A GeV Au-Au collision at RHIC we it the remaining set o parameters d ws, 0, T F, and v to the experimentally obtained single particle transverse momentum spectra [10] and HBT parameters [11]. For the description o the HBT correlation parameters which are measured or 30% central collisions, we scale down the entropy content and initial overlap radius guided by overlap calculations and neglect the angular asymmetry. We ind that the choice o parameters T =110 MeV, d ws 1.0 m, 0 =1.8 and v =0.67 is able to give a good description o the data. =3.5 and ws =0.6 are determined by the experimentally observed rapidity distribution o particles. In particular, ws inluences results in the orward rapidity region and the present investigation is not very sensitive to changes o this parameter. =19 m/c ends the ireball expansion by the time all volume elements have cooled down to the temperature T. The resulting average cooling curve (computed by averaging the entropy density over the ireball volume at a given proper time and determining the corresponding temperature, where we deine the ireball volume at given as the 3-volume bounded by the Cooper-Frye surace) and the transverse momentum spectra or, K and nucleons are shown in Fig. 2, the HBT correlation parameters in Fig. 3. The it apparently misses the low p t part o the transverse momentum spectra or the heavier particles but describes the + + correlation radii well, with the exception o the low p t part o R out where the calculation lies somewhat above the data. The rather steep allo o R side, R out as a unction o k t avors a large amount o low. However, demanding simultaneous agreement with the slope o the momentum spectra implies that large transverse low has to be accompanied by low reeze-out temperatures. Thereore, the volume at reeze-out has to be large in order to reach small entropy densities. The inclusion o a inite pion chemical potential (originating rom resonance decays in chemical nonequilibrium) is crucial to reduce the volume corresponding to a given average temperature. Since the radial expansion is rather constrained by the normalization o the HBT parameters, a large volume implies sizeable longitudinal extension and hence a long-lived system. Such a long lietime in combination with a standard boost-invariant longitudinal expansion leads to large values o R long which are clearly incompatible with the data. A sizeable longitudinal compression and re-expansion however leads to a reasonable description o R long as well. It is not possible to identiy a single eature o the model as being responsible or the good quality o the it but one can illustrate certain trends. Neglecting the additional cooling caused by a inite value o, it is still possible to describe the spectra and the slope o the HBT data well or a low reeze-out temperature, however a larger volume is needed to get to this temperature and thereore the normalization o the correlation radii is systematically above the data (see Fig. 4, let panel). On the other hand, retaining the idea o a boost-invariant scenario without longitudinal acceleration implies a short lietime o the system as apparent rom the estimate R long = T /m t 1/2 based on such an expansion pattern. Such a short lietime can be achieved by a large value o T. The transverse expansion velocity can then be adjusted to it the m t spectra, but the resulting solution is characterized by small low velocities which leave both normalization and allo o R side in disagreement with the data (see Fig. 4, let panel). Retaining a low reeze-out temperature in a boostinvariant scenario allows to describe the transverse spectra and correlations well but signiicantly overpredicts R long (see Fig. 4, right panel). 021903-3
THORSTEN RENK PHYSICAL REVIEW C 70, 021903(R) (2004) FIG. 3. The HBT correlation parameters R side, R out, R long, and the ratio R out /R side in the model calculation as compared to PHENIX data [10]. Concluding, we ind that the m t spectra and HBT parameters measured at RHIC can be simultaneously described assuming a scenario with small reeze-out temperature T F 110 MeV and a sizeable initial longitudinal compression and re-expansion. The data strongly constrain alternative scenarios. An upcoming publication will investigate the dependence o the results on the dierent model parameters in more detail. The calculation o FIG. 4. Let panel: R side in a scenario with vanishing pion chemical potential (dotted line) and a dierence scenario assuming T =140 MeV (dashed line) as compared to the standard calculation (solid) Right panel: R long in a scenario with T =110 MeV but assuming no longitudinal acceleration (dotted) compared with the standard scenario (solid). Note that the transverse momentum spectra are well described by all o the scenarios shown in this igure. 021903-4
MODEL FOR THE SPACETIME EVOLUTION OF HEAVY- PHYSICAL REVIEW C 70, 021903(R) (2004) urther observables relecting dierent properties o the ireball expansion such as the emission o electromagnetic probes or the suppression o high p t jets within the same ramework will help to conirm or disprove the outlined scenario. I would like to thank S. A. Bass and B. Müller or helpul discussions, comments, and their support during the preparation o this paper. This work was inancially supported by DOE Grant No. DE-FG02-96ER40945 and the Alexander von Humboldt Foundation. [1] F. Karsch, E. Laermann and A. Peikert, Phys. Lett. B 478, 447 (2000). [2] T. Renk, hep-ph/0403239. [3] I. G. Bearden et al., BRAHMS Collaboration, Phys. Rev. Lett. 88, 202301 (2002). [4] I. G. Bearden et al., BRAHMS Collaboration, nucl-ex/ 0312023. [5] R. A. Schneider and W. Weise, Phys. Rev. C 64, 055201 (2001); M. A. Thaler, R. A. Schneider, and W. Weise, Phys. Rev. C 69, 035210 (2004). [6] T. Renk, Phys. Rev. C 68, 064901 (2003). [7] E. Schnedermann, J. Sollrank, and U. W. Heinz, Phys. Rev. C 48, 2462 (1993). [8] U. A. Wiedemann and U. W. Heinz, Phys. Rep. 319, 145 (1999). [9] B. Tomasik and U. A. Wiedemann, in Quark Gluon Plasma, edited by R. C. Hwa and X.-N. Wang (World Scientiic, River Edge, 2004), p. 715. [10] S. S. Adler et al., PHENIX Collaboration, nucl-ex/0401003. [11] S. S. Adler et al., PHENIX Collaboration, Phys. Rev. C 69, 034909 (2004). 021903-5