- Email: [email protected]
Comparison of hadronic rescattering calculations of elliptic flow and HBT with measurements from RHIC
ELSEVIER
Nuclear
Physics A7 15 (2003)
64 1~~644~ www.elsevier.comllocateinpe
Comparison of hadronic rescattering with measurements from RHIC T. J. Humani? aDepartment
calculations
of elliptic
flow and HBT
* of Physics.
The Ohio State IIniversity,
Columbus,
OH 43210
Results fro111 the data obtainctl in the first physics run of the Relativistic Heavy Ion Collider (RHIC) have shown suprisingly large elliptic flow and suprisingly small HBT radii. Attempts to explain both results in a consistant picture have so far been unsuccessful. The present work shows tha.t a simple thermal-like initial state model coupled to a hadronic rescattering calculation can explain reasonably well both elliptic flow and HBT result,s from RHIC. The calculation suggests A very early hadronization time of about 1 fm/c after t,he initial collision of the nuclei. The rescattering calculation method used is similar to that used in previous calculations for CERN SPS energies [l]. Rescattering is simulated with a semi-classical Monte Carlo calculation which assumes strong binary collisions between isospin-averaged hadrons. RelAll calculations are made to simulate RHICativistic kinematics is used throughout. energy Au+Au collisions in order to compare with the results of the Year-l RHIC data. -4 detailed description of the present calculations can be found elsewhere [a]. The hadronization model used employs simple parameterizations to describe the initial momenta and space-time of the hadrons. The initial momenta are assumed to follow a thermal-like transverse momentum distribution for all particles, (l/mT)dN/dmT
= CmT/[exp(mT/T)
where T is a “temperature
dN/dy = Dew
[-(Y
-
f
parameter”,
11
(1)
and a gaussian
rapidity
distribution
for mesons,
YO)~/(~~,~)]
(‘4
where gV is the rapidity width. Two rapidity distributions for baryons have been tried: 1) flat and then falling off near beam rapidity and 2) peaked at central rapidity and falling off until beam rapidity. Both baryon distributions give about the same results. The initial longitudinal particle hadronization position @had) and time (thad) are determined by the relativistic equations, Z,& = Thhedsinh y;
thnd = ?-hadcod1 y
(3)
where ?-had is the hadronization proper time. From Equations 2 and 3, it is seen that longitudinal invariance is not assumed in the initial conditions for the present calculations. ‘This
work
was supported
by the U. S. National
0375.9474/03/$ - see front matter 81 2003 Elsevier doi:l0.l016/S0375-9474(02)01551-X
Science Science R.V
Foundation
under
All rights reserved.
grant
PHY-0099476
Z.1 Humanic
642c
-6
-4
-2
0
/Nuclrar
2
Physics
4
6
11
Pseudorapidity Figure 1. summed over pions. kaons, from rescattering calculations and 8 fm.
distributions and nucleons for h = 0,5,
A715
(2003)
64/c-644~
Iilj_ll
200 mT - m. (MeV)
d
,_ I
400
Figure 2. Transverse mass distributions from the rescattering model. The lines are exponential fits to the distributions and the slope parameters are shown.
Calculations were carried out using isospinsummed events containing at freezeout about 3000 pions, 500 kaons, and 650 nucleons (R’s were decayed). The ha,dronization rnodel parameters used were T = 300 MeV, cr,=2.4, and q&=1 fm/c. Rescattering calculation results for spectra are presented in Figures l-3. Figure 1 shows normalized pseudorapidity distributions summed over pions. kaons, and nucleons for three impact parameters, b = 0,5, and 8 fm. The widths of the distributions and the flattening near n = 0 are similar to data from PHOBOS [3]. Figure 2 shows mT distributions for pions, kaons, and nucleons for b = 0 fm near midrapidity (-1 < y < 1) fitted to exponentials of the form exp (-mT/B), where B is the slope parameter. The extracted slope parameters shown in Figure 2 are close in value to preliminary measurements from the STAR experiment for the K, K-, and anti-proton of 190 f 10, 300 If 30, and 565 & 50 MeV, respectively [4]. F’g1 ure 3 shows transverse momentum distributions for pions, kaons, and nucleons for b = 8 fm which extend to high-pT, i.e. 6 GeV/c. As observed in PHENIX data [5], the pion and nucleon distributions merge for PT > 2 GeV/c and the kaon distribution crosses that for nucleons at around 1 GeV/c. Results for elliptic flow from rescattering calculations are given in Figures 4-6. Figure 4 shows the rapidity dependence of v2 for pions for b = 8 fm. As seen, the rescattering calculation predicts that v2 is peaked at mid-rapidity and falls off to zero near y = f6. This same behavior is seen experimentally by PHOBOS in their ‘~2 vs. pseudorapidity measurements for charged hadrons with a minimum bias trigger [6]. Figure 5 shows the pi dependence of 02 for pions and nucleons for b = 8 fm compared with the trends of the STAR measurements for X+ + ST- and p + p - bar at 11 - 45% centrality [7], which roughly corresponds to this impact parameter. Figure 6 compares the pT dependence of r’s for kaons for b = 8 fm with the STAR measurements for K”, at 11 - 45% centrality [8]. As seeu, the rescattering calculation values are in reasonable agreement with the STAR
TJ Humanic/Nuciear
7-mT--,
Physics
A715
(2003)
I
Rescatt.
641c-644~
643c
-m
/
pT dist. (b=8 fm)
I pions
0
0
c
0.1 g -+
0.01
e 5
0.001
0
1
2
3
4
5
6
P, (@V/c)
Figure 3. Transverse momentum tions for pions, kaons, and nucleons b = 8 fm rescattering calculation.
-6
-4
-2
0 Y
distribufrom the
Figure 4. Rapidity dependence of w2 for pions from the 6 = S fm rescattering calculation.
measurements. The flattening out of the pion and nucleon u2 distributions for pi > 2 GeV/c is consistant with that seen in STAR and PHENIX results for minimum-bias hadrons [9,5]. The pion source parameters extracted from HBT analyses of rescattering calculations for three different, impact parameters, b = 0, 5, and 8 fm, are compared with STAR Y measurements at three centrality bins [lo] in Figure 7. In the left panel, the centrality dependence of the HBT parameters is plotted for a pi bin of 0.125 - 0.225 GeV/c. In the right panel, the mT dependence of the HBT parameters is plotted for centrality bin 3, for the STAR measurments, or b = 0 fm, for the rescattering calculations. Although there are differences in some of the details, the trends of the STAR HBT measurements are seen to be described reasonably well by the rescattering calculation.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
T. T. B. C. W. B. C. C. R. C.
J. Humanic, Phys. Rev. C 5’7, 866 (1998). J. Humanic, arXiv:nucl-th/0205053. B. Back et al. [PHOBOS C o 11a b oration], Nucl. Phys. A 698, 88 (2002). Adler ef al. [STAR Collaboration], Nucl. Phys. A 698, 64c (2002). A. Zajc e2 al. [PHENIX Collaboration], Nud. Phys. A 698, 39 (2002). B. Back et al. [PHOBOS Collaboration], arXiv:nucl-ex/0205021. Adler et al. [STAR C o 11a b oration], Phys. Rev. Lett. 87, 182301 (2001). Adler et al. [STAR C o 11a b oration], preprint (submitted to Phys. Rev. Lett). J. Snellings [STAR Collaboration], Nucl. Phys. A 698, 193 (2002). Adler el al. [STAR C o 11a b oration], Phys. Rev. Lett. 87, 082301 (2001).
r
644c
T.1 Humanrc/Nucleur
Physics
Figure $5. Calculations of v2 from the reseattering model for b = 8 fnr for pions and nucleans compared with STt-\K measurements at 1 l-45% centrality.
A715
(2003)
641~~644c,
Calculation of v2 from the Figure 6. rescattering model for kaons at b = 8 fm compared with STAR measurements for Z<“‘o, at ll-4ri% centrality.
8 s 25 .-2
6
b E a’-
4 2 0
iLvL 1
i--l.2 STAR Centrality
Ji.
.Luu>
3 bin
0.2
_0.3 mT (GW
0.4
Figure 7. C:omparison of HRT source parameters from rescattering with STAR measurements as a function of centrality bin (see text) and m,T. The STAR measurements arc the solid symbols and the rescattering calculations are the open symbols. The errors on the STAR measurements are statisticalfsystematic.
Nuclear
Physics A7 15 (2003)
64 1~~644~ www.elsevier.comllocateinpe
Comparison of hadronic rescattering with measurements from RHIC T. J. Humani? aDepartment
calculations
of elliptic
flow and HBT
* of Physics.
The Ohio State IIniversity,
Columbus,
OH 43210
Results fro111 the data obtainctl in the first physics run of the Relativistic Heavy Ion Collider (RHIC) have shown suprisingly large elliptic flow and suprisingly small HBT radii. Attempts to explain both results in a consistant picture have so far been unsuccessful. The present work shows tha.t a simple thermal-like initial state model coupled to a hadronic rescattering calculation can explain reasonably well both elliptic flow and HBT result,s from RHIC. The calculation suggests A very early hadronization time of about 1 fm/c after t,he initial collision of the nuclei. The rescattering calculation method used is similar to that used in previous calculations for CERN SPS energies [l]. Rescattering is simulated with a semi-classical Monte Carlo calculation which assumes strong binary collisions between isospin-averaged hadrons. RelAll calculations are made to simulate RHICativistic kinematics is used throughout. energy Au+Au collisions in order to compare with the results of the Year-l RHIC data. -4 detailed description of the present calculations can be found elsewhere [a]. The hadronization model used employs simple parameterizations to describe the initial momenta and space-time of the hadrons. The initial momenta are assumed to follow a thermal-like transverse momentum distribution for all particles, (l/mT)dN/dmT
= CmT/[exp(mT/T)
where T is a “temperature
dN/dy = Dew
[-(Y
-
f
parameter”,
11
(1)
and a gaussian
rapidity
distribution
for mesons,
YO)~/(~~,~)]
(‘4
where gV is the rapidity width. Two rapidity distributions for baryons have been tried: 1) flat and then falling off near beam rapidity and 2) peaked at central rapidity and falling off until beam rapidity. Both baryon distributions give about the same results. The initial longitudinal particle hadronization position @had) and time (thad) are determined by the relativistic equations, Z,& = Thhedsinh y;
thnd = ?-hadcod1 y
(3)
where ?-had is the hadronization proper time. From Equations 2 and 3, it is seen that longitudinal invariance is not assumed in the initial conditions for the present calculations. ‘This
work
was supported
by the U. S. National
0375.9474/03/$ - see front matter 81 2003 Elsevier doi:l0.l016/S0375-9474(02)01551-X
Science Science R.V
Foundation
under
All rights reserved.
grant
PHY-0099476
Z.1 Humanic
642c
-6
-4
-2
0
/Nuclrar
2
Physics
4
6
11
Pseudorapidity Figure 1. summed over pions. kaons, from rescattering calculations and 8 fm.
distributions and nucleons for h = 0,5,
A715
(2003)
64/c-644~
Iilj_ll
200 mT - m. (MeV)
d
,_ I
400
Figure 2. Transverse mass distributions from the rescattering model. The lines are exponential fits to the distributions and the slope parameters are shown.
Calculations were carried out using isospinsummed events containing at freezeout about 3000 pions, 500 kaons, and 650 nucleons (R’s were decayed). The ha,dronization rnodel parameters used were T = 300 MeV, cr,=2.4, and q&=1 fm/c. Rescattering calculation results for spectra are presented in Figures l-3. Figure 1 shows normalized pseudorapidity distributions summed over pions. kaons, and nucleons for three impact parameters, b = 0,5, and 8 fm. The widths of the distributions and the flattening near n = 0 are similar to data from PHOBOS [3]. Figure 2 shows mT distributions for pions, kaons, and nucleons for b = 0 fm near midrapidity (-1 < y < 1) fitted to exponentials of the form exp (-mT/B), where B is the slope parameter. The extracted slope parameters shown in Figure 2 are close in value to preliminary measurements from the STAR experiment for the K, K-, and anti-proton of 190 f 10, 300 If 30, and 565 & 50 MeV, respectively [4]. F’g1 ure 3 shows transverse momentum distributions for pions, kaons, and nucleons for b = 8 fm which extend to high-pT, i.e. 6 GeV/c. As observed in PHENIX data [5], the pion and nucleon distributions merge for PT > 2 GeV/c and the kaon distribution crosses that for nucleons at around 1 GeV/c. Results for elliptic flow from rescattering calculations are given in Figures 4-6. Figure 4 shows the rapidity dependence of v2 for pions for b = 8 fm. As seen, the rescattering calculation predicts that v2 is peaked at mid-rapidity and falls off to zero near y = f6. This same behavior is seen experimentally by PHOBOS in their ‘~2 vs. pseudorapidity measurements for charged hadrons with a minimum bias trigger [6]. Figure 5 shows the pi dependence of 02 for pions and nucleons for b = 8 fm compared with the trends of the STAR measurements for X+ + ST- and p + p - bar at 11 - 45% centrality [7], which roughly corresponds to this impact parameter. Figure 6 compares the pT dependence of r’s for kaons for b = 8 fm with the STAR measurements for K”, at 11 - 45% centrality [8]. As seeu, the rescattering calculation values are in reasonable agreement with the STAR
TJ Humanic/Nuciear
7-mT--,
Physics
A715
(2003)
I
Rescatt.
641c-644~
643c
-m
/
pT dist. (b=8 fm)
I pions
0
0
c
0.1 g -+
0.01
e 5
0.001
0
1
2
3
4
5
6
P, (@V/c)
Figure 3. Transverse momentum tions for pions, kaons, and nucleons b = 8 fm rescattering calculation.
-6
-4
-2
0 Y
distribufrom the
Figure 4. Rapidity dependence of w2 for pions from the 6 = S fm rescattering calculation.
measurements. The flattening out of the pion and nucleon u2 distributions for pi > 2 GeV/c is consistant with that seen in STAR and PHENIX results for minimum-bias hadrons [9,5]. The pion source parameters extracted from HBT analyses of rescattering calculations for three different, impact parameters, b = 0, 5, and 8 fm, are compared with STAR Y measurements at three centrality bins [lo] in Figure 7. In the left panel, the centrality dependence of the HBT parameters is plotted for a pi bin of 0.125 - 0.225 GeV/c. In the right panel, the mT dependence of the HBT parameters is plotted for centrality bin 3, for the STAR measurments, or b = 0 fm, for the rescattering calculations. Although there are differences in some of the details, the trends of the STAR HBT measurements are seen to be described reasonably well by the rescattering calculation.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
T. T. B. C. W. B. C. C. R. C.
J. Humanic, Phys. Rev. C 5’7, 866 (1998). J. Humanic, arXiv:nucl-th/0205053. B. Back et al. [PHOBOS C o 11a b oration], Nucl. Phys. A 698, 88 (2002). Adler ef al. [STAR Collaboration], Nucl. Phys. A 698, 64c (2002). A. Zajc e2 al. [PHENIX Collaboration], Nud. Phys. A 698, 39 (2002). B. Back et al. [PHOBOS Collaboration], arXiv:nucl-ex/0205021. Adler et al. [STAR C o 11a b oration], Phys. Rev. Lett. 87, 182301 (2001). Adler et al. [STAR C o 11a b oration], preprint (submitted to Phys. Rev. Lett). J. Snellings [STAR Collaboration], Nucl. Phys. A 698, 193 (2002). Adler el al. [STAR C o 11a b oration], Phys. Rev. Lett. 87, 082301 (2001).
r
644c
T.1 Humanrc/Nucleur
Physics
Figure $5. Calculations of v2 from the reseattering model for b = 8 fnr for pions and nucleans compared with STt-\K measurements at 1 l-45% centrality.
A715
(2003)
641~~644c,
Calculation of v2 from the Figure 6. rescattering model for kaons at b = 8 fm compared with STAR measurements for Z<“‘o, at ll-4ri% centrality.
8 s 25 .-2
6
b E a’-
4 2 0
iLvL 1
i--l.2 STAR Centrality
Ji.
.Luu>
3 bin
0.2
_0.3 mT (GW
0.4
Figure 7. C:omparison of HRT source parameters from rescattering with STAR measurements as a function of centrality bin (see text) and m,T. The STAR measurements arc the solid symbols and the rescattering calculations are the open symbols. The errors on the STAR measurements are statisticalfsystematic.