Fluctuations of the $K/π$ Ratio in Nucleus-Nucleus Collisions: Statistical and Transport Models

0.36 MB·English

by  M. I. Gorenstein

Fluctuations of the K/π Ratio in Nucleus-Nucleus Collisions:
Statistical and Transport Models
M.I. Gorenstein,1,2 M. Hauer,3 V.P. Konchakovski,2,3 and E.L. Bratkovskaya1
1Frankfurt Institute for Advanced Studies, Frankfurt, Germany
9 2Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
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0 3Helmholtz Research School, University of Frankfurt, Frankfurt, Germany
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n Abstract
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Event-by-eventfluctuationsofthekaontopionnumberratioinnucleus-nucleuscollisionsarestudied
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within the statistical hadron-resonance gas model (SM) for different statistical ensembles and in the

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h Hadron-String-Dynamics (HSD) transport approach. We find that the HSD model can qualitatively
t
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l reproduce the measured excitation function for the K/π ratio fluctuations in central Au+Au (or
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Pb+Pb)collisions from low SPS up to top RHIC energies. Substantial differences in the HSD and SM
n
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resultsarefoundforthefluctuationsandcorrelations ofthekaonandpionnumbers. Thesepredictions

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v impose a challenge for future experiments.
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PACS numbers: 24.10.Lx,24.60.Ky, 25.75.-q
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1 Keywords:
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I. INTRODUCTION
The study of event-by-event fluctuations in high energy nucleus-nucleus (A+A) collisions
opensnewpossibilitiestoinvestigatethephasetransitionbetweenhadronicandpartonicmatter
as well as the QCD critical point (cf. the reviews [1]). By measuring the fluctuations one might
observe anomalies from the onset of deconfinement [2] and dynamical instabilities when the
expanding system goes through the 1-st order transition line between the quark-gluon plasma
andthehadrongas[3]. Furthermore, theQCD criticalpoint may besignaled bya characteristic
pattern in the fluctuations as pointed out in Ref. [4]. However only recently, due to a rapid
development of experimental techniques, first measurements of the event-by-event fluctuations
of particle multiplicities [5, 6, 7, 8] and transverse momenta [9] in nucleus-nucleus collisions
have been performed.
From the theoretical side such event-by-event fluctuations for charged hadron multiplicities
(in nucleus-nucleus collisions) have been studied in statistical models [10, 11, 12, 13, 14, 15,
16, 17, 18, 19] and in dynamical transport approaches [20, 21, 22, 23], which have been used
as important tools to investigate high-energy nuclear collisions. We recall that the statistical
models reproduce the mean multiplicities of the produced hadrons (see e.g. Refs. [24, 25, 26]),
whereas the transport models (see, e.g., Refs. [27, 28, 29]) provide, in addition, a dynamical
description of the various bulk properties of the system. By studying the various fluctuations
within statistical and transport models we have found out that fluctuations provide an ex-
tremely sensitive observable - depending on the details of the models - which are partly washed
out by looking at general quantities such as ensemble averages.
In particular, there is a qualitative difference in the properties of the mean multiplicity
and the scaled variance of the multiplicity distribution in statistical models. In the case of
mean multiplicities the results obtained within the grand canonical ensemble (GCE), canonical
ensemble (CE), and micro-canonical ensemble (MCE) approach each other in the large vol-
ume limit. One refers here to the thermodynamical equivalence of the statistical ensembles.
However, it was recently found [10, 14] that corresponding results for the scaled variances are
different in the GCE, CE and MCE ensembles, and thus the scaled variance is sensitive to
global conservation laws obeyed by a statistical system. These differences are preserved in the
thermodynamic limit.
Also there is a qualitative difference in the behavior of the scaled variances of multiplicity
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distributions in statistical and transport models. The transport models predict [21, 22] that the
scaled variances in central nucleus-nucleus collisions remain close to the corresponding values in
proton-protoncollisions andincrease withcollision energy in thesame way asthe corresponding
multiplicities, whereas in the statistical models the scaled variances approach finite values at
high collision energy, i.e. become independent of energy. Accordingly, the differences in the
scaled variance of charged hadrons can be about factor of 10 at the top RHIC energy [21]. Only
upcoming experimental data can clarify the situation.
The QGP stage may form a specific set of primordial fluctuation signals. A well known
example is the equilibrium electric charge fluctuation in QGP which is about a factor 2-3
smaller than in an equilibrium hadron gas [30, 31]. To observe primordial QGP fluctuations
they should be frozen out during expansion, hadronization, and further hadron-hadron re-
scatterings. Evolution and survival of the conserved charge fluctuations in systems formed in
nucleus-nucleus collisions at the SPS and RHIC energies were discussed in Refs. [32, 33]. Note
that both the statistical models and the HSD approach used in our study do not include the
quark-gluon degrees of freedom. Thus, the fluctuations in the QGP are outside of the scope of
the present paper.
The measurement of the fluctuations in the kaon to pion ratio by the NA49 Collaboration [5]
was the first event-by-event measurement in nucleus-nucleus collisions. It was suggested that
this ratio might allow to distinguish the enhanced strangeness production attributed to the
QGP phase. Nowadays, the excitation function for this observable is available in a wide range
of energies: from the NA49 collaboration in Pb+Pb collisions at the CERN SPS [7] and from
the STAR collaboration in Au+Au collisions at RHIC [8]. First statistical model estimates of
the K/π fluctuations have been reported in Refs. [34, 35], and results from the transport model
UrQMD in Ref. [36].
In this paper we present a systematic study of statistical model results (in different ensem-
bles) in comparison to HSD transport model results for the fluctuations in the kaon to pion
number ratio. The paper is organized as follows: In Section II the characteristic definitions for
fluctuations in particle number ratios are introduced. In Section III the relevant formulas of
the statistical models (in different ensembles) are presented. Statistical and HSD model results
for the fluctuations in the kaon to pion ratio for central nucleus-nucleus collisions are compared
in Section IV. In Section V the HSD transport model results are additionally confronted with
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the available data on K/π fluctuations. A summary closes the paper in Section VI.
II. MEASURES OF PARTICLE RATIO FLUCTUATIONS
A. Notations and Approximations
Let us introduce some notations. We define the deviation ∆N from the average number
A
N of the particle species A by N = N +∆N . Then we define covariance for species A
A A A A
h i h i
and B
∆(N ,N ) ∆N ∆N = N N N N , (1)
A B A B A B A B
≡ h i h i − h ih i
scaled variance
∆(N ,N ) (∆N )2 N2 N 2
ω A A = h A i = h Ai−h Ai , (2)
A
≡ N N N
A A A
h i h i h i
and correlation coefficient
∆N ∆N
A B
ρ h i . (3)
AB ≡ 2 2 1/2
(∆N ) (∆N )
A B
h i h i
(cid:2) (cid:3)
The fluctuations of the ratio R N /N will be characterised by [34, 35]
AB A B
≡
2
(∆R )
σ2 h AB i . (4)
≡ R 2
AB
h i
Using the expansion,
2
N N +∆N N +∆N ∆N ∆N
A A A A A B B
= h i = h i 1 + , (5)
N N +∆N N × − N N − ···
B h Bi B h Bi " h Bi (cid:18)h Bi(cid:19) #
one finds to second order in ∆N / N and ∆N / N the average value and the fluctuations
A A B B
h i h i
of the A to B ratio:
N ω ∆(N ,N )
A B A B
R = h i 1 + , (6)
AB ∼
h i N N − N N
h Bi (cid:20) h Bi h Aih Bi (cid:21)
∆(N ,N ) ∆(N ,N ) ∆(N ,N )
σ2 = A A + B B 2 A B
∼ N 2 N 2 − N N
A B A B
h i h i h ih i
1/2
ω ω ω ω
A B A B
= + 2ρ . (7)
AB
N N − N N
h Ai h Bi (cid:20)h Aih Bi(cid:21)
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If species A and B fluctuate independently according to Poisson distributions (this takes place,
for example, in the GCE for an ideal Boltzmann gas) one finds ω = ω = 1 and ρ = 0.
A B AB
Equation (7) then reads
1 1
σ2 = + . (8)
N N
A B
h i h i
In a thermal gas, the average multiplicities are proportional to the system volume V. Equation
(8) demonstrates then a simple dependence of σ2 1/V on the system volume.
∝
A few examples concerning to Eq. (7) are appropriate here. When N N , e.g.,
B A
h i ≫ h i
A = K+ +K− and B = π+ +π−, the quantity σ2 (7) is dominated by the fluctuations of less
abundant particles. When N = N , e.g., A = π+ and B = π−, the correlation term in
A ∼ B
h i h i
Eq. (7) may become especially important. A resonance decaying always into a π+π−-pair does
not contribute to σ2 (7), but contributes to the π+ and π− average multiplicities. This leads
[35] to a suppression of σ2 (7) in comparison to its value given by Eq. (8). For example, if
all π+ and π− particles come in pairs from the decay of resonances, one finds the correlation
coefficient ρπ+π− = 1 in Eq. (7), and thus σ2 = 0. In this case, the numbers of π+ and π−
fluctuate as the number of resonances, but the ratio π+/π− does not fluctuate.
B. Mixed Events Procedure
The experimental data forN /N fluctuations areusually presented in terms ofthe so called
A B
dynamical fluctuations [37]1
σ sign σ2 σ2 σ2 σ2 1/2 , (9)
dyn ≡ − mix − mix
(cid:0) (cid:1)(cid:12) (cid:12)
where σ2 is defined by Eq. (7) and σ2 corresponds(cid:12)to the follow(cid:12)ing mixed events procedure2.
mix
One takes a large number of nucleus-nucleus collision events and measures the numbers of N
A
and N in each event. Then all A and B particles from all events are combined into one set.
B
The construction of mixed events is done as follows: One fixes a random number N = N +N
A B
according to the experimental probability distribution P(N), takes randomly N particles (A
1 Other dynamical measures, such as Φ [38, 39] and F [35], can be also used.
2 We describe the idealized mixed events procedure appropriate for the model analysis. The real experimen-
tal mixed events procedure is more complicated and includes experimental uncertainties, such as particle
identification etc.
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and/or B) from the whole set, fixes the values of N and N , and returns these N particles into
A B
the set. This is the mixed event number one. Then one constructs event number 2, number 3,
etc.
Note that the number of events is much larger than the number of hadrons, N, in any single
event. Therefore, the probabilities p and p = 1 p , to take the A and B species from
A B A
−
the whole set, can be considered as constant values during the event construction. Another
consequence of a large number of events is the fact that A and B particles in any constructed
mixed event belong to different physical events of nucleus-nucleus collisions. Therefore, the
correlations between the N and N numbers in a physical event are expected to be destroyed
B A
in a mixed event. This is the main purpose of the mixed events construction. For any function
f(N ,N ) the mixed events averaging is then defined as
A B
(N +N )!
f(N ,N ) = P(N) f(N ,N ) δ(N N N ) A B pNApNB . (10)
h A B imix A B − A − B N !N ! A B
A B
XN NXA,NB
The straightforward calculations of mixed averages (10) can be simplified by introducing the
generating function Z(x,y),
(N +N )!
Z(x,y) P(N) δ(N N N ) A B (xp )NA (yp )NB
A B A B
≡ − − N ! N !
A B
XN NXA,NB
= P(N)(xp +yp )N , (11)
A B
N
X
which depends on auxiliary variables x and y. The averages (10) are then expressed as x- and
y-derivatives of Z(x,y) at x = y = 1. One finds:
∂Z ∂Z
N = = p N , N = = p N , (12)
A mix A B mix B
h i ∂x h i h i ∂y h i
(cid:18) (cid:19)x=y=1 (cid:18) (cid:19)x=y=1
∂2Z
N (N 1) = = p2 N(N 1) , (13)
h A A − imix ∂2x A h − i
(cid:18) (cid:19)x=y=1
∂2Z
N (N 1) = = p2 N(N 1) , (14)
h B B − imix ∂2y B h − i
(cid:18) (cid:19)x=y=1
∂2Z
N N N N = = p p ω N , (15)
A B mix A mix B mix A B N
h i − h i h i ∂x∂y h i
(cid:18) (cid:19)x=y=1
where
N2 N 2
N N P(N) , N2 N2 P(N) , ω h i − h i . (16)
N
h i ≡ h i ≡ ≡ N
N N h i
X X
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Calculating the N /N fluctuations for mixed events according to Eq. (7) one gets:
A B
∆ (N ,N ) ∆ (N ,N ) ∆ (N ,N )
σ2 mix A A + mix B B 2 mix A B
mix ≡ N 2 N 2 − N N
A B A B
h i h i h ih i
1 ω 1 1 ω 1 ω 1
N N N
= + − + + − 2 −
N N N N − N
(cid:20)h Ai h i (cid:21) (cid:20)h Bi h i (cid:21) h i
1 1
= + . (17)
N N
A B
h i h i
A comparison of the final result in Eq. (17) with Eq. (8) shows that the mixed events procedure
gives the same σ2 for N /N fluctuations as in the GCE formulation for an ideal Boltzmann
A B
gas, i.e. for ω = ω = 1 and ρ = 0. If ω = 1 (e.g. for the Poisson distribution P(N)), one
A B AB N
indeed finds ωmix = ωmix = 1 and ρmix = 0. Otherwise, if ω = 1, the mixed events procedure
A B AB N 6
leads to ωmix = 1, ωmix = 1, and to non-zero N N correlations, as seen from the second line of
A 6 B 6 A B
Eq. (17). Thus, if, e.g., event-by-event fluctuations in the total number of pions and kaons are
stronger than Poissonian ones, i.e. ω > 1, positive pion-kaon correlations appear in the mixed
N
events. They lead to larger (smaller) N in the sample of mixed events with larger (smaller)
K
N . However, the final result for σ2 (17) is still the same as for ω = 1, it does not depend
π mix N
on the specific form of P(N). Non-trivial (ωmix = 1) fluctuations of N and N as well as
A,B 6 A B
non-zero ρmix correlations may exist in the mixed events procedure, but they are cancelled out
AB
in σ2 .
mix
III. FLUCTUATIONS OF RATIOS IN STATISTICAL MODELS
A. Quantum Statistics and Resonance Decays
The occupation numbers, n , of single quantum states (with fixed projection of particle
p,j
spin) labelled by the momentum vector p are equal to n = 0,1,..., for bosons and
p,j
∞
n = 0,1 for fermions. Their average values are
p,j
1
n = , (18)
p,j
h i exp[(ǫ µ )/T] α
pj j j
− −
and their fluctuations read
(∆n )2 (n n )2 = n (1 + α n ) v2 , (19)
h p,j igce ≡ h p,j − h p,ji igce h p,ji j h p,ji ≡ p,j
where T is the system temperature, m is the mass of a particle j, ǫ = p2 +m2 is the single
j pj j
particle energy. The value of α depends on quantum statistics, i.e. +1qfor bosons and 1 for
j
−
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fermions, while α = 0 gives the Boltzmann approximation. The chemical potential µ of a
j j
species j equals to: µ = q µ + b µ + s µ , where q , b , s are the particle electric
j j Q j B j S j j j
charge, baryon number, and strangeness, respectively, while µ , µ , µ are the corresponding
Q B S
chemical potentials which regulate the average values of these global conserved charges in the
GCE.
In the equilibrium hadron-resonance gas model the mean number of primary particles (or
resonances) is calculated as:
∞
g V
N∗ n = j p2dp n , (20)
h ji ≡ h p,ji 2π2 h p,ji
p Z0
X
where V is the system volume and g is the degeneracy factor of a particle of species j (the
j
number of spin states). In the thermodynamic limit, V , the sum over the momentum
→ ∞
states can be substituted by a momentum integral.
It is convenient to introduce a microscopic correlator, ∆n ∆n , which in the GCE has
p,j k,i
h i
the simple form:
∆n ∆n = υ2 δ δ . (21)
h p,j k,iigce p,j ij pk
Hence there are no correlations between different particle species, i = j, and/or between dif-
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ferent momentum states, p = k. Only the Bose enhancement, v2 > n for α = 1, and the
6 p,j h p,ji j
Fermi suppression, v2 < n for α = 1, exist for fluctuations of primary particles in the
p,j h p,ji j −
GCE. The correlator (1) can be presented in terms of microscopic correlators (21):
∆N∗ ∆N∗ = ∆n ∆n = δ v2 . (22)
h j i igce h p,j k,iigce ij p,j
p,k p
X X
In the case i = j equation (22) gives the variance of primordial particles (before resonance
decays) in the GCE.
For the hadron resonance gas formed in relativistic A+A collisions the corrections due to
quantum statistics (Bose enhancement and Fermi suppression) are small3. For the pion gas
at T = 160 MeV, one finds ω = 1.1, instead of ω = 1 for Boltzmann particles. The quan-
π ∼
tum statistics effects are even smaller for heavier particles like kaons and almost negligible for
resonances.
3 Possible strong Bose effects are discussed in Ref. [12]
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The average final (after resonance decays) multiplicities N are equal to:
i
h i
∗
N = N + N n . (23)
h ii h i i h Rih iiR
R
X
∗
In Eq. (23), N denotes the number of stable primary hadrons of species i, the summation
i R
runs over all types of resonances R, and hniiR ≡ rbRr nRi,r is the average over resonance dePcay
channels. The parameters bR are the branching rPatios of the r-th branches, nR is the number
r i,r
of particles of species i produced in resonance R decays via a decay mode r. The index r runs
over all decay channels of a resonance R with the requirement bR = 1. In the GCE the
r r
correlator (1) after resonance decays can be calculated as [35]: P
∆N ∆N = ∆N∗∆N∗ + ∆N2 n n + N ∆n ∆n , (24)
h A Bigce h A Bigce h Rih AiRh BiR h Rih A BiR
R
X(cid:2) (cid:3)
where ∆n ∆n bRnR nR n n .
h A BiR ≡ r r A,r B,r − h AiRh BiR
P
B. Global Conservation Laws
In the MCE, the energy and conserved charges are fixed exactly for each microscopic state of
thesystem. ThisleadstotwomodificationsincomparisonwiththeGCE.First,additionalterms
appearfortheprimordialmicroscopiccorrelatorsintheMCE.Theyreflectthe(anti)correlations
between different particles, i = j, and different momentum levels, p = k, due to charge and
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energy conservation in the MCE [14],
υ2 v2
∆n ∆n = υ2 δ δ p,j k,i [ q q M +b b M +s s M
h p,j k,iimce p,j ij pk − A i j qq i j bb i j ss
| |
+ (q s +q s )M (q b +q b )M (b s +b s )M
i j j i qs i j j i qb i j j i bs
− −
+ ǫ ǫ M (q ǫ +q ǫ )M + (b ǫ +b ǫ )M (s ǫ +s ǫ )M ] , (25)
pj ki ǫǫ i pj j ki qǫ i pj j ki bǫ i pj j ki sǫ
− −
where A is the determinant and M are the minors of the following matrix,
ij
| |
∆(q2) ∆(bq) ∆(sq) ∆(ǫq)
∆(qb) ∆(b2) ∆(sb) ∆(ǫb)
A = , (26)
∆(qs) ∆(bs) ∆(s2) ∆(ǫs)
 
 
∆(qǫ) ∆(bǫ) ∆(sǫ) ∆(ǫ2)
 
 
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with the elements, ∆(q2) q2υ2 , ∆(qb) q b υ2 , ∆(qǫ) q ǫ υ2 , etc.
≡ p,j j p,j ≡ p,j j j p,j ≡ p,j j pj p,j
The sum, , means intePgration over momentuPm p, and the summatioPn over all hadron-
p,j
resonance sPpecies j contained in the model. The first term in the r.h.s. of Eq. (25) corresponds
to the microscopic correlator (21) in the GCE. Note, that the presence of the terms containing
the single particle energy ǫ = p2 +m2 in Eq. (25) is a consequence of energy conservation.
pj j
In the CE, only charges are conqserved, thus the terms containing ǫ in Eq. (25) are absent.
pj
The matrix A in Eq. (26) then becomes a 3 3 matrix (see Ref. [13]). An important property
×
of the microscopic correlator method is that the particle number fluctuations and the corre-
lations in the MCE or CE, although being different from those in the GCE, are expressed by
quantities calculated within the GCE. The microscopic correlator (25) can be used to calculate
the primordial particle (or resonances) correlator in the MCE (or in the CE):
∆N ∆N = ∆n ∆n . (27)
i j mce p,i k,j mce
h i h i
p,k
X
A second feature of the MCE (or CE) is the modificationof the resonance decay contribution
to the fluctuations in comparison to the GCE (24). In the MCE (or CE) it reads[13, 14]:
∗ ∗ ∗
∆N ∆N = ∆N ∆N + N ∆n ∆n + ∆N ∆N n
h A Bimce h A Bimce h Ri h A BiR h A Rimce h BiR
R R
X X
∗
+ h∆NB ∆NRimce hnAiR + h∆NR ∆NR′imce hnAiR hnBiR′ . (28)
R R,R′
X X
Additional terms in Eq. (28) compared to Eq. (24) are due to the correlations (for primordial
particles) induced by energy and charge conservations in the MCE. Eq. (28) has the same form
in the CE [13] and MCE [14], the difference between these two ensembles appears because
of different microscopic correlators (25). The microscopic correlators of the MCE together
∗ ∗ ∗
with Eq. (27) should be used to calculate the correlators ∆N ∆N , ∆N ∆N ,
h A Bimce h A Rimce
∗ ∗
h∆NA ∆NRimce , h∆NB ∆NRimce , and h∆NR ∆NR′imce entering in Eq. (28). The correlators
(28) define finally the scaled variances ω and ω (2) and correlations ρ (3) between the
A B AB
N and N numbers. Together with the average multiplicities N and N they define
A B A B
h i h i
completely the fluctuations σ2 (7) of the A to B number ratio.
IV. STATISTICAL AND HSD MODEL RESULTS FOR THE K/π RATIO
In this section we present the results of the hadron-resonance gas statistical model (SM)
and the HSD transport model for the fluctuations of the K/π ratio in central nucleus-nucleus
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