January 4, 2019
Multiplicity Fluctuations in Au+Au Collisions at RHIC
V.P. Konchakovski,1, 2 M.I. Gorenstein,1, 3 and E.L. Bratkovskaya3
1Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
2Helmholtz Research School, University of Frankfurt, Frankfurt, Germany
3Frankfurt Institute for Advanced Studies, Frankfurt, Germany
Abstract
The preliminary data of the PHENIX collaboration for the scaled variances of charged hadron
multiplicity fluctuations in Au+Au at
s = 200 GeV are analyzed within the model of independent
sources. We use the HSD transport model to calculate the participant number fluctuations and
the number of charged hadrons per nucleon participant in different centrality bins. This combined
picture leads to a good agreement with the PHENIX data and suggests that the measured multi-
plicity fluctuations result dominantly from participant number fluctuations. The role of centrality
selection and acceptance is discussed separately.
PACS numbers: 24.10.Lx, 24.60.Ky, 25.75.-q
Keywords: nucleus-nucleus collisions, fluctuations, transport models
http://arxiv.org/abs/0704.1831v1
The event-by-event fluctuations in high energy nucleus-nucleus (A+A) collisions (see e.g.,
the reviews [1, 2]) are expected to provide signals of the transition between different phases
(see e.g., Refs. [3, 4]) and the QCD critical point [5]. In the present letter we study the
charged multiplicity fluctuations in Au+Au collisions at RHIC energies. The preliminary
data of the PHENIX collaboration [6] at
s = 200 GeV are analyzed within the model
of independent sources while employing the microscopic Hadron-String-Dynamics (HSD)
transport model [7, 8] to define the centrality selection and to calculate the properties of
hadron production sources.
The centrality selection is an important aspect of fluctuation studies in A+A collisions. At
the SPS fixed target experiments the samples of collisions with a fixed number of projectile
participants N
P can be selected to minimize the participant number fluctuations in the
sample of collision events. This selection is possible due to a measurement of the number of
nucleon spectators from the projectile, N
S , in each individual collision by a calorimeter
which covers the projectile fragmentation domain. However, even in the sample with N
const the number of target participants fluctuates considerably. In the following the variance,
V ar(n) ≡ 〈n2〉 − 〈n〉2, and scaled variance, ω ≡ V ar(n)/〈n〉, where n stands for a given
random variable and 〈· · · 〉 for event-by-event averaging, will be used to quantify fluctuations.
In each sample with N
P = const the number of target participants fluctuates around its
mean value, 〈N targP 〉 = N
P , with the scaled variance ω
P . Within the HSD and UrQMD
transport models it was found in Ref. [9] that the fluctuations of N
P strongly influence
the charged hadron fluctuations. The constant values of N
P and fluctuations of N
P lead
also to an asymmetry between the fluctuations in the projectile and target hemispheres. The
consequences of this asymmetry depend on the A+A dynamics as discussed in Ref. [10].
In Au+Au collisions at RHIC a different centrality selection is used. There are two kinds
of detectors which define the centrality of Au+Au collision, Beam-Beam Counters (BBC)
and Zero Degree Calorimeters (ZDC). At the c.m. pair energy
s = 200 GeV, the BBC
measure the charged particle multiplicity in the pseudorapidity range 3.0 < |η| < 3.9, and
the ZDC – the number of neutrons with |η| > 6.0 [6]. These neutrons are part of the nucleon
spectators. Due to technical reasons the neutron spectators can be only detected by the ZDC
(not protons and nuclear fragments), but in both hemispheres. The BBC distribution will be
used in the HSD calculations to divide Au+Au collision events into 5% centrality samples.
HSD does not specify different spectator groups – neutrons, protons, and nuclear fragments
such that we can not use the ZDC information. In Fig. 1 (left) the HSD results for the BBC
distribution and centrality classes in Au+Au collisions at
s =200 GeV are shown. We
find a good agreement between the HSD shape of the BBC distribution and the PHENIX
data [6]. The experimental estimates of 〈NP 〉 for different centrality classes are based on
the Glauber model. These estimates vary by less than 0.5% depending on the shape of the
cut in the ZDC/BBC space or whether the BBC alone is used as a centrality measure [6].
Note, however, that the HSD 〈NP 〉 numbers are not exactly equal to the PHENIX values.
It is also not obvious that different definitions for the 5% centrality classes give the same
values of the scaled variance ωP for the participant number fluctuations.
0 200 400 600 800
Charge in BBC
0 100 200 300 400
FIG. 1: HSD model results for Au+Au collisions at
s = 200 GeV. Left: Centrality classes defined
via the BBC distribution. Right: The average number of participants, 〈NP 〉, and the scaled variance
of the participant number fluctuations, ωP , calculated for the 5% BBC centrality classes.
Defining the centrality selection via the HSD transport model (which is similar to the
BBC in the PHENIX experiment) we calculate the mean number of nucleon participants,
〈NP 〉, and the scaled variance of its fluctuations, ωP , in each 5% centrality sample. The
results are shown in Fig. 1, right. The Fig. 2 (left) shows the HSD results for the mean
number of charged hadrons per nucleon participant, ni = 〈Ni〉/〈NP 〉, where the index i
stands for “−”, “+”, and “ch”, i.e negatively, positively, and all charged final hadrons.
Note that the centrality dependence of ni is opposite to that of ωP : ni increases with 〈NP 〉,
whereas ωP decreases.
The PHENIX detector accepts charged particles in a small region of the phase space with
pseudorapidity |η| < 0.26 and azimuthal angle φ < 245o and the pT range from 0.2 to 2.0
GeV/c [6]. The fraction of the accepted particles qi = 〈Nacci 〉/〈Ni〉 calculated within the
HSD model is shown in the r.h.s. of Fig. 2. According to the HSD results only 3÷ 3.5% of
charged particles are accepted by the mid-rapidity PHENIX detector.
0 100 200 300 400
0 100 200 300 400
0.030
0.035
FIG. 2: HSD results for different BBC centrality classes in Au+Au collisions at
s = 200 GeV.
Left: The mean number of charged hadrons per participant, ni = 〈Ni〉/〈NP 〉. Right: The fraction
of accepted particles, qi = 〈Nacci 〉/〈Ni〉.
To estimate the role of the participant number event-by-event fluctuations we use the
model of independent sources (see e.g., Refs [1, 9, 10]),
ωi = ω
i + ni ωP . (1)
The first term in the r.h.s. of Eq. (1) corresponds to the fluctuations of the hadron mul-
tiplicity from one source, and the second term, ni ωP , gives additional fluctuations due to
the fluctuations of the number of sources. As usually, we have assumed that the number of
sources is proportional to the number of nucleon participants. The value of ni in Eq. (1) is
then the average number of i’th particles per participant, ni = 〈Ni〉/〈NP 〉. We also assume
that nucleon-nucleon collisions define the fluctuations ω∗i from a single source. To calculate
the fluctuations ωacci in the PHENIX acceptance we use the acceptance scaling formula (see
e.g., Refs. [1, 9, 10]):
ωacci = 1 − qi + qi ωi , (2)
where qi is the fraction of the accepted i’th hadrons by the PHENIX detector. Using Eq. (1)
for ωi one finds,
ωacci = 1 − qi + qi ω∗i + qi ni ωP . (3)
The HSD results for ωP (Fig. 1, right), ni (Fig. 2, left), qi (Fig. 2, right), together with the
HSD nucleon-nucleon values, ω∗
= 3.0, ω∗+ = 2.7, and ω
ch = 5.7 at
s = 200 GeV, define
completely the results for ωacci according to Eq. (3). We find a surprisingly good agreement
of the results given by Eq. (3) with the PHENIX data shown in Fig. 3. Note that the
centrality dependence of ωacci stems from the product, ni · ωP , in the last term of the r.h.s.
of Eq. (3).
100 200 300 400
 PHENIX
 Eq. (3)
FIG. 3: The scaled variance of charged particle fluctuations in Au+Au collisions at
s = 200 GeV
with the PHENIX acceptance. The circles are the PHENIX data [6] while the open points (con-
nected by the solid line) correspond to Eq. (3) with the HSD results for ωP , ni, and qi.
In summary, the preliminary PHENIX data [6] for the scaled variances of charged hadron
multiplicity fluctuations in Au+Au collisions at
s = 200 GeV have been analyzed within
the model of independent sources. Assuming that the number of hadron sources are propor-
tional to the number of nucleon participants, the HSD transport model was used to calculate
the scaled variance of participant number fluctuations, ωP , and the number of i’th hadrons
per nucleon accepted by the mid-rapidity PHENIX detector, qini, in different Beam-Beam
Counter centrality classes. The HSD model for nucleon-nucleon collisions was also used to
estimate the fluctuations from a single source, ω∗i . We find that this model description is in
a good agreement with the PHENIX data [6]. In different (5%) centrality classes ωP goes
down and qini goes up with increasing 〈NP 〉. This results in non-monotonic dependence of
ωacci on 〈NP 〉 as seen in the PHENIX data.
We conclude that both qualitative and quantitative features of the centrality dependence
of the fluctuations seen in the present PHENIX data are the consequences of participant
number fluctuations. To avoid a dominance of the participant number fluctuations one needs
to analyze most central collisions with a much more rigid (≤ 1%) centrality selection. The
statistical model then predicts ω± < 1 [11], whereas the HSD transport model predicts the
values of ω± much larger than unity at
s = 200 GeV [12]. To allow for a clear distinction
between these predictions it is mandatory to enlarge the acceptance of the mid-rapidity
detector up to about 10% (see the discussion in Ref. [12]).
Acknowledgments: We like to thank V.V. Begun, W. Cassing, M. Gaździcki,
W. Greiner, M. Hauer, B. Lungwitz, I.N. Mishustin and J.T. Mitchell for useful discus-
sions. One of the author (M.I.G.) is thankful to the Humboldt Foundation for financial
support.
[1] H. Heiselberg, Phys. Rep. 351, 161 (2001).
[2] S. Jeon and V. Koch, Review for Quark-Gluon Plasma 3, eds. R.C. Hwa and X.-N. Wang,
World Scientific, Singapore, 430-490 (2004) [arXiv:hep-ph/0304012].
[3] M. Gaździcki, M. I. Gorenstein, and S. Mrowczynski, Phys. Lett. B 585, 115 (2004);
M. I. Gorenstein, M. Gaździcki, and O. S. Zozulya, Phys. Lett. B 585, 237 (2004).
[4] I.N. Mishustin, Phys. Rev. Lett. 82, 4779 (1999); Nucl. Phys. A 681, 56 (2001); H. Heiselberg
and A.D. Jackson, Phys. Rev. C 63, 064904 (2001).
[5] M.A. Stephanov, K. Rajagopal, and E.V. Shuryak, Phys. Rev. Lett. 81, 4816 (1998); Phys.
Rev. D 60, 114028 (1999); M.A. Stephanov, Acta Phys. Polon. B 35, 2939 (2004);
[6] S.S. Adler et al., [PHENIX Collaboration], arXiv: nucl-ex/0409015; J.T. Mitchell [PHENIX
http://arxiv.org/abs/hep-ph/0304012
http://arxiv.org/abs/nucl-ex/0409015
Collaboration], J. Phys. Conf. Ser. 27, 88 (2005); J.T. Mitchell, private communications.
[7] W. Ehehalt and W. Cassing, Nucl. Phys. A 602, 449 (1996); E.L. Bratkovskaya and
W. Cassing, Nucl. Phys. A 619 413 (1997); W. Cassing and E.L. Bratkovskaya, Phys. Rep.
308, 65 (1999); W. Cassing, E.L. Bratkovskaya, and A. Sibirtsev, Nucl. Phys. A 691, 753
(2001); W. Cassing, E. L. Bratkovskaya, and S. Juchem, Nucl. Phys. A 674, 249 (2000).
[8] H. Weber, et al., Phys. Rev. C67, 014904 (2003); E.L. Bratkovskaya, et al., Phys. Rev. C 67,
054905 (2003); ibid, 69, 054907 (2004); Prog. Part. Nucl. Phys. 53, 225 (2004).
[9] V.P. Konchakovski, et al., Phys. Rev. C 73, 034902 (2006); ibid. C 74, 064911 (2006).
[10] M. Gaździcki and M.I. Gorenstein, Phys. Lett. B 640, 155 (2006).
[11] V.V. Begun, et al., Phys. Rev. C 74, 044903 (2006); nucl-th/0611075.
[12] V.P. Konchakovski, E.L. Bratkovskaya, and M.I. Gorenstein, nucl-th/0703052.
http://arxiv.org/abs/nucl-th/0611075
http://arxiv.org/abs/nucl-th/0703052
	References

  The preliminary data of the PHENIX collaboration for the scaled variances of
charged hadron multiplicity fluctuations in Au+Au at $\sqrt{s}=200$ GeV are
analyzed within the model of independent sources. We use the HSD transport
model to calculate the participant number fluctuations and the number of
charged hadrons per nucleon participant in different centrality bins. This
combined picture leads to a good agreement with the PHENIX data and suggests
that the measured multiplicity fluctuations result dominantly from participant
number fluctuations. The role of centrality selection and acceptance is
discussed separately.

January 4, 2019
Multiplicity Fluctuations in Au+Au Collisions at RHIC
V.P. Konchakovski,1, 2 M.I. Gorenstein,1, 3 and E.L. Bratkovskaya3
1Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
2Helmholtz Research School, University of Frankfurt, Frankfurt, Germany
3Frankfurt Institute for Advanced Studies, Frankfurt, Germany
Abstract
The preliminary data of the PHENIX collaboration for the scaled variances of charged hadron
multiplicity fluctuations in Au+Au at
s = 200 GeV are analyzed within the model of independent
sources. We use the HSD transport model to calculate the participant number fluctuations and
the number of charged hadrons per nucleon participant in different centrality bins. This combined
picture leads to a good agreement with the PHENIX data and suggests that the measured multi-
plicity fluctuations result dominantly from participant number fluctuations. The role of centrality
selection and acceptance is discussed separately.
PACS numbers: 24.10.Lx, 24.60.Ky, 25.75.-q
Keywords: nucleus-nucleus collisions, fluctuations, transport models
http://arxiv.org/abs/0704.1831v1
The event-by-event fluctuations in high energy nucleus-nucleus (A+A) collisions (see e.g.,
the reviews [1, 2]) are expected to provide signals of the transition between different phases
(see e.g., Refs. [3, 4]) and the QCD critical point [5]. In the present letter we study the
charged multiplicity fluctuations in Au+Au collisions at RHIC energies. The preliminary
data of the PHENIX collaboration [6] at
s = 200 GeV are analyzed within the model
of independent sources while employing the microscopic Hadron-String-Dynamics (HSD)
transport model [7, 8] to define the centrality selection and to calculate the properties of
hadron production sources.
The centrality selection is an important aspect of fluctuation studies in A+A collisions. At
the SPS fixed target experiments the samples of collisions with a fixed number of projectile
participants N
P can be selected to minimize the participant number fluctuations in the
sample of collision events. This selection is possible due to a measurement of the number of
nucleon spectators from the projectile, N
S , in each individual collision by a calorimeter
which covers the projectile fragmentation domain. However, even in the sample with N
const the number of target participants fluctuates considerably. In the following the variance,
V ar(n) ≡ 〈n2〉 − 〈n〉2, and scaled variance, ω ≡ V ar(n)/〈n〉, where n stands for a given
random variable and 〈· · · 〉 for event-by-event averaging, will be used to quantify fluctuations.
In each sample with N
P = const the number of target participants fluctuates around its
mean value, 〈N targP 〉 = N
P , with the scaled variance ω
P . Within the HSD and UrQMD
transport models it was found in Ref. [9] that the fluctuations of N
P strongly influence
the charged hadron fluctuations. The constant values of N
P and fluctuations of N
P lead
also to an asymmetry between the fluctuations in the projectile and target hemispheres. The
consequences of this asymmetry depend on the A+A dynamics as discussed in Ref. [10].
In Au+Au collisions at RHIC a different centrality selection is used. There are two kinds
of detectors which define the centrality of Au+Au collision, Beam-Beam Counters (BBC)
and Zero Degree Calorimeters (ZDC). At the c.m. pair energy
s = 200 GeV, the BBC
measure the charged particle multiplicity in the pseudorapidity range 3.0 < |η| < 3.9, and
the ZDC – the number of neutrons with |η| > 6.0 [6]. These neutrons are part of the nucleon
spectators. Due to technical reasons the neutron spectators can be only detected by the ZDC
(not protons and nuclear fragments), but in both hemispheres. The BBC distribution will be
used in the HSD calculations to divide Au+Au collision events into 5% centrality samples.
HSD does not specify different spectator groups – neutrons, protons, and nuclear fragments
such that we can not use the ZDC information. In Fig. 1 (left) the HSD results for the BBC
distribution and centrality classes in Au+Au collisions at
s =200 GeV are shown. We
find a good agreement between the HSD shape of the BBC distribution and the PHENIX
data [6]. The experimental estimates of 〈NP 〉 for different centrality classes are based on
the Glauber model. These estimates vary by less than 0.5% depending on the shape of the
cut in the ZDC/BBC space or whether the BBC alone is used as a centrality measure [6].
Note, however, that the HSD 〈NP 〉 numbers are not exactly equal to the PHENIX values.
It is also not obvious that different definitions for the 5% centrality classes give the same
values of the scaled variance ωP for the participant number fluctuations.
0 200 400 600 800
Charge in BBC
0 100 200 300 400
FIG. 1: HSD model results for Au+Au collisions at
s = 200 GeV. Left: Centrality classes defined
via the BBC distribution. Right: The average number of participants, 〈NP 〉, and the scaled variance
of the participant number fluctuations, ωP , calculated for the 5% BBC centrality classes.
Defining the centrality selection via the HSD transport model (which is similar to the
BBC in the PHENIX experiment) we calculate the mean number of nucleon participants,
〈NP 〉, and the scaled variance of its fluctuations, ωP , in each 5% centrality sample. The
results are shown in Fig. 1, right. The Fig. 2 (left) shows the HSD results for the mean
number of charged hadrons per nucleon participant, ni = 〈Ni〉/〈NP 〉, where the index i
stands for “−”, “+”, and “ch”, i.e negatively, positively, and all charged final hadrons.
Note that the centrality dependence of ni is opposite to that of ωP : ni increases with 〈NP 〉,
whereas ωP decreases.
The PHENIX detector accepts charged particles in a small region of the phase space with
pseudorapidity |η| < 0.26 and azimuthal angle φ < 245o and the pT range from 0.2 to 2.0
GeV/c [6]. The fraction of the accepted particles qi = 〈Nacci 〉/〈Ni〉 calculated within the
HSD model is shown in the r.h.s. of Fig. 2. According to the HSD results only 3÷ 3.5% of
charged particles are accepted by the mid-rapidity PHENIX detector.
0 100 200 300 400
0 100 200 300 400
0.030
0.035
FIG. 2: HSD results for different BBC centrality classes in Au+Au collisions at
s = 200 GeV.
Left: The mean number of charged hadrons per participant, ni = 〈Ni〉/〈NP 〉. Right: The fraction
of accepted particles, qi = 〈Nacci 〉/〈Ni〉.
To estimate the role of the participant number event-by-event fluctuations we use the
model of independent sources (see e.g., Refs [1, 9, 10]),
ωi = ω
i + ni ωP . (1)
The first term in the r.h.s. of Eq. (1) corresponds to the fluctuations of the hadron mul-
tiplicity from one source, and the second term, ni ωP , gives additional fluctuations due to
the fluctuations of the number of sources. As usually, we have assumed that the number of
sources is proportional to the number of nucleon participants. The value of ni in Eq. (1) is
then the average number of i’th particles per participant, ni = 〈Ni〉/〈NP 〉. We also assume
that nucleon-nucleon collisions define the fluctuations ω∗i from a single source. To calculate
the fluctuations ωacci in the PHENIX acceptance we use the acceptance scaling formula (see
e.g., Refs. [1, 9, 10]):
ωacci = 1 − qi + qi ωi , (2)
where qi is the fraction of the accepted i’th hadrons by the PHENIX detector. Using Eq. (1)
for ωi one finds,
ωacci = 1 − qi + qi ω∗i + qi ni ωP . (3)
The HSD results for ωP (Fig. 1, right), ni (Fig. 2, left), qi (Fig. 2, right), together with the
HSD nucleon-nucleon values, ω∗
= 3.0, ω∗+ = 2.7, and ω
ch = 5.7 at
s = 200 GeV, define
completely the results for ωacci according to Eq. (3). We find a surprisingly good agreement
of the results given by Eq. (3) with the PHENIX data shown in Fig. 3. Note that the
centrality dependence of ωacci stems from the product, ni · ωP , in the last term of the r.h.s.
of Eq. (3).
100 200 300 400
 PHENIX
 Eq. (3)
FIG. 3: The scaled variance of charged particle fluctuations in Au+Au collisions at
s = 200 GeV
with the PHENIX acceptance. The circles are the PHENIX data [6] while the open points (con-
nected by the solid line) correspond to Eq. (3) with the HSD results for ωP , ni, and qi.
In summary, the preliminary PHENIX data [6] for the scaled variances of charged hadron
multiplicity fluctuations in Au+Au collisions at
s = 200 GeV have been analyzed within
the model of independent sources. Assuming that the number of hadron sources are propor-
tional to the number of nucleon participants, the HSD transport model was used to calculate
the scaled variance of participant number fluctuations, ωP , and the number of i’th hadrons
per nucleon accepted by the mid-rapidity PHENIX detector, qini, in different Beam-Beam
Counter centrality classes. The HSD model for nucleon-nucleon collisions was also used to
estimate the fluctuations from a single source, ω∗i . We find that this model description is in
a good agreement with the PHENIX data [6]. In different (5%) centrality classes ωP goes
down and qini goes up with increasing 〈NP 〉. This results in non-monotonic dependence of
ωacci on 〈NP 〉 as seen in the PHENIX data.
We conclude that both qualitative and quantitative features of the centrality dependence
of the fluctuations seen in the present PHENIX data are the consequences of participant
number fluctuations. To avoid a dominance of the participant number fluctuations one needs
to analyze most central collisions with a much more rigid (≤ 1%) centrality selection. The
statistical model then predicts ω± < 1 [11], whereas the HSD transport model predicts the
values of ω± much larger than unity at
s = 200 GeV [12]. To allow for a clear distinction
between these predictions it is mandatory to enlarge the acceptance of the mid-rapidity
detector up to about 10% (see the discussion in Ref. [12]).
Acknowledgments: We like to thank V.V. Begun, W. Cassing, M. Gaździcki,
W. Greiner, M. Hauer, B. Lungwitz, I.N. Mishustin and J.T. Mitchell for useful discus-
sions. One of the author (M.I.G.) is thankful to the Humboldt Foundation for financial
support.
[1] H. Heiselberg, Phys. Rep. 351, 161 (2001).
[2] S. Jeon and V. Koch, Review for Quark-Gluon Plasma 3, eds. R.C. Hwa and X.-N. Wang,
World Scientific, Singapore, 430-490 (2004) [arXiv:hep-ph/0304012].
[3] M. Gaździcki, M. I. Gorenstein, and S. Mrowczynski, Phys. Lett. B 585, 115 (2004);
M. I. Gorenstein, M. Gaździcki, and O. S. Zozulya, Phys. Lett. B 585, 237 (2004).
[4] I.N. Mishustin, Phys. Rev. Lett. 82, 4779 (1999); Nucl. Phys. A 681, 56 (2001); H. Heiselberg
and A.D. Jackson, Phys. Rev. C 63, 064904 (2001).
[5] M.A. Stephanov, K. Rajagopal, and E.V. Shuryak, Phys. Rev. Lett. 81, 4816 (1998); Phys.
Rev. D 60, 114028 (1999); M.A. Stephanov, Acta Phys. Polon. B 35, 2939 (2004);
[6] S.S. Adler et al., [PHENIX Collaboration], arXiv: nucl-ex/0409015; J.T. Mitchell [PHENIX
http://arxiv.org/abs/hep-ph/0304012
http://arxiv.org/abs/nucl-ex/0409015
Collaboration], J. Phys. Conf. Ser. 27, 88 (2005); J.T. Mitchell, private communications.
[7] W. Ehehalt and W. Cassing, Nucl. Phys. A 602, 449 (1996); E.L. Bratkovskaya and
W. Cassing, Nucl. Phys. A 619 413 (1997); W. Cassing and E.L. Bratkovskaya, Phys. Rep.
308, 65 (1999); W. Cassing, E.L. Bratkovskaya, and A. Sibirtsev, Nucl. Phys. A 691, 753
(2001); W. Cassing, E. L. Bratkovskaya, and S. Juchem, Nucl. Phys. A 674, 249 (2000).
[8] H. Weber, et al., Phys. Rev. C67, 014904 (2003); E.L. Bratkovskaya, et al., Phys. Rev. C 67,
054905 (2003); ibid, 69, 054907 (2004); Prog. Part. Nucl. Phys. 53, 225 (2004).
[9] V.P. Konchakovski, et al., Phys. Rev. C 73, 034902 (2006); ibid. C 74, 064911 (2006).
[10] M. Gaździcki and M.I. Gorenstein, Phys. Lett. B 640, 155 (2006).
[11] V.V. Begun, et al., Phys. Rev. C 74, 044903 (2006); nucl-th/0611075.
[12] V.P. Konchakovski, E.L. Bratkovskaya, and M.I. Gorenstein, nucl-th/0703052.
http://arxiv.org/abs/nucl-th/0611075
http://arxiv.org/abs/nucl-th/0703052
	References
