$K^-/K^+$ ratio in heavy-ion collisions at GSI with an antikaon self-energy in hot and dense matter

Abstract

The $K^-/K^+$ ratio produced in heavy-ion collisions at GSI energies is studied. The in-medium properties at finite temperature of the hadrons involved are included, paying a special attention to the in-medium properties of antikaons. Using a statistical approach, it is found that the determination of the temperature and chemical potential at freeze-out conditions compatible with the ratio $K^-/K^+$ is very delicate, and depends very strongly on the approximation adopted for the antikaon self-energy. The use of an energy dependent $\bar{K}$ spectral density, including both s and p-wave components of the $\bar{K}N$ interaction, lowers substantially the freeze-out temperature compared to the standard simplified mean-field treatment and gives rise to an overabundance of $K^-$ production in the dense and hot medium. Even a moderately attractive antikaon-nucleus potential obtained from our self-consistent many-body calculation does reproduce the ``broad-band equilibration'' advocated by Brown, Rho and Song due to the additional strength of the spectral function of the $K^-$ at low energies. Comment: 20 pages, 6 figures, sent to Phys. Rev. C

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arXiv:nucl-th/0302082v1 28 Feb 2003
K−/K+ratio in heavy-ion collisions at GSI with an antikaon
self-energy in hot and dense matter
Laura Tol´ os, Artur Polls, Angels Ramos
Departament d’Estructura i Constituents de la Mat` eria, Universitat de Barcelona,
Diagonal 647, 08028 Barcelona, Spain
J¨ urgen Schaffner-Bielich
Institut f¨ ur Theoretische Physik, J. W. Goethe-Universit¨ at
D-60054 Frankfurt am Main, Germany
(February 8, 2008)
Abstract
The K−/K+ratio produced in heavy-ion collisions at GSI energies is stud-
ied. The in-medium properties at finite temperature of the hadrons involved
are included, paying a special attention to the in-medium properties of an-
tikaons. Using a statistical approach, it is found that the determination of
the temperature and chemical potential at freeze-out conditions compatible
with the ratio K−/K+is very delicate, and depends very strongly on the
approximation adopted for the antikaon self-energy. The use of an energy de-
pendent¯K spectral density, including both s and p-wave components of the
¯KN interaction, lowers substantially the freeze-out temperature compared to
the standard simplified mean-field treatment and gives rise to an overabun-
dance of K−production in the dense and hot medium. Even a moderately
attractive antikaon-nucleus potential obtained from our self-consistent many-
body calculation does reproduce the “broad-band equilibration” advocated by
Brown, Rho and Song due to the additional strength of the spectral function
of the K−at low energies.
PACS: 13.75.Jz, 14.40.Ev, 25.75-q, 25.75.Dw, 24.10.Pa, 24.60.-k
Keywords:¯KN interaction, Kaon-nucleus potential, Λ(1405), Finite temper-
ature, Heavy-ion collisions
I. INTRODUCTION
The study of the properties of hadrons in hot and dense matter is receiving a lot of
attention in recent years to understand fundamental aspects of the strong interaction, such
as the partial restoration of chiral symmetry [1–3], as well as a variety of astrophysical
phenomena, such as the dynamical evolution of supernovas and the composition of neutron
stars [4].
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A special effort has been invested to understand the properties of antikaons in the
medium, especially after the speculation of the possible existence of an antikaon condensed
phase was put forward in [5] which would soften the equation-of-state producing, among
other phenomena, lower neutron star masses [6–8].
While it is well established that the antikaons should feel an attractive interaction when
they are embedded in a nuclear environment, the size of this attraction has been the subject
of intense debate. Theoretical models including medium effects on the antikaon-nucleon scat-
tering amplitude, the behaviour of which is governed by the isospin zero Λ(1405) resonance,
naturally explain the evolution from repulsion in free-space to attraction in the nuclear
medium [9,10]. However, the presence of the resonance makes the size of the antikaon-
nucleus potential be very sensitive to the many-body treatment of the medium effects. The
phenomenological attempts of extracting information about the antikaon-nucleus potential
from kaonic-atom data favoured very strongly attractive well depths [11], but recent self-
consistent in-medium calculations [12–15] based on chiral Lagrangian’s [10,16] or meson-
exchange potentials [17] predict a moderately attractive kaon-nucleus interaction. In fact,
recent analysis of kaonic atoms [18–20] are able to find a reasonable reproduction of the data
with relative shallow antikaon-nucleus potentials, therefore indicating that kaonic atom data
cannot really pin down the strength of the antikaon-nucleus potential at nuclear matter sat-
uration density.
On the other hand, heavy-ion collisions at energies around 1−2 AGeV offer the possibility
of studying experimentally the properties of a dense and hot nuclear system [21–23]. In
particular, a considerable amount of information about strange particles like antikaons is
available. Since antikaons are produced at finite density and finite momentum, the chiral
models have recently incorporated the higher partial waves of the antikaon-nucleon scattering
amplitude both in free space [24–26] and in the nuclear medium [27,28]. The complete
scenario taking into account finite density, finite momentum and finite temperature has
recently been addressed in Refs. [14,29].
Event generators trying to analyse heavy-ion collision data [30–36] need to implement
the modified properties of the hadrons in the medium where they are produced. Transport
models have shown, for instance, that the multiplicity distributions of kaons and antikaons
are much better reproduced if in-medium masses rather than bare masses are used [37,38].
Production and propagation of kaons and antikaons have been investigated with the Kaon
Spectrometer (KaoS) of the SIS heavy-ion synchrotron at GSI (Darmstadt). The exper-
iments have been performed with Au+Au, Ni+Ni, C+C at energies between 0.6 and 2.0
AGeV [39–48]. One surprising observation in C+C and Ni+Ni collisions [42–44] is that, as
a function of the energy difference√s −√sth, where√sthis the needed energy to produce
the particle (2.5 GeV for K+via pp → ΛK+p and 2.9 GeV for K−via pp → ppK−K+),
the number of K−balanced the number of K+in spite of the fact that in pp collisions the
production cross-sections close to threshold are 2-3 orders of magnitude different. This has
been interpreted to be a manifestation of the enhancement of the K+mass and the reduction
of the K−one in the nuclear medium, which in turn influence the corresponding production
thresholds [7,35–38], although a complementary explanation in terms of in-medium enhanced
πΣ → K−p production has also been given [14]. Another interesting observation is that at
incident energies of 1.8 and 1.93 AGeV, the K−and K+multiplicities have the same impact
parameter dependence [42–44]. Equal centrality dependence for K+and K−and, hence,
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independence of centrality for the K−/K+ratio has also been observed in Au+Au and
Pb+Pb reactions between 1.5 AGeV and RHIC energies (√s = 200 AGeV) [44,49–52]. This
independence of centrality is astonishing, since at low energies one expects that as centrality
increases — and with it the participating system size and the density probed — the K−/K+
ratio should also increase due to the increased reduction of the K−mass together with the
enhancement of the K+mass. In fact, the independence of the K−/K+ratio on centrality
has often been advocated as signalling the lack of in-medium effects. A recent interesting
interpretation of this phenomenon is given in Ref. [53], where it is shown that the K−are
predominantly produced via πY collisions (Y = Λ,Σ) and, hence, the K−multiplicity is
strongly correlated with the K+one, since kaons and hyperons are mainly produced together
via the reaction NN → KY N.
Although the transport model calculations show that strangeness equilibration requires
times of the order of 40 − 80 fm/c [54,55], statistical models, which assume chemical and
thermal equilibrium and common freeze-out parameters for all particles, are quite success-
ful in describing particle yields including strange particles [56–61]. The kaon and antikaon
yields in the statistical models are based on free masses and no medium effects are needed
to describe the enhanced in-medium K−/K+ratio or its independence with centrality. The
increased value of the K−/K+ratio is simply obtained by choosing a particular set of pa-
rameters at freeze-out, the baryonic chemical potential µB≃ 720 MeV and the temperature
T ≃ 70 MeV, which also reproduce a variety of particle multiplicity ratios [60,61]. On the
other hand, centrality independence of the K−/K+ratio is automatically obtained in statis-
tical models within the canonical or grand-canonical schemes because the terms depending
on the system size drop out [61]. However, as shown by Brown et al. in Ref. [62], using
the reduced in-medium K−mass in the statistical model would force, in order to reproduce
the experimental value of the K−/K+ratio, a larger value of the chemical potential and
hence a larger and more plausible baryonic density for strangeness production. In addition,
Brown et al. introduce the concept of “broad-band equilibration” according to which the
K−mesons and the hyperons are produced in an essentially constant ratio independent of
density, hence explaining also the centrality independence of the K−/K+ratio but including
medium effects. In essence, the establishment of a broad-band relies on the fact that the
baryonic chemical potential µBincreases with density an amount which coincides roughly
with the reduction of the K−mass. However, as mentioned above, the antikaon properties
are very sensitive to the type of model for the¯KN interaction used and of the in-medium
effects included. The purpose of the present work is precisely to investigate to which extent
the “broad-band equilibration” concept holds when more sophisticated models for the in-
medium antikaon properties are used. We explore within the context of a statistical model
what is the behaviour of the K−/K+ratio as a function of the nuclear density when the
hadrons are dressed, paying special attention to different ways of dressing the antikaons:
either treating them as non-interacting, or dressing them with an on-shell self-energy, or,
finally, considering the complete antikaon spectral density. We use the antikaon self-energy
which has been derived within the framework of a coupled-channel self-consistent calcula-
tion in symmetric nuclear matter at finite temperature [29], taking as bare meson-baryon
interaction the meson exchange potential of the J¨ ulich group [17]. We will show that the
determination of temperature and chemical potential at freeze-out conditions compatible
with the experimental value of the K−/K+ratio is very delicate, and depends very strongly
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on the approximation adopted for the antikaon self-energy.
The paper is organised as follows. In Sec. II the formalism of the thermal model is
described and the different ingredients used in the determination of the off-shell properties
of the K−are given.The results are presented and discussed in Sec. III. Finally, our
concluding remarks are given in Sec. IV.
II. FORMALISM
In this section we present a brief description of the thermal models to account for
strangeness production in heavy-ion collisions. The basic hypothesis is to assume that the
relative abundance of kaons and antikaons in the final state of relativistic nucleus-nucleus
collisions is determined by imposing thermal and chemical equilibrium [56–61,63]. The fact
that the number of strange particles in the final state is small requires a strict treatment
of the conservation of strangeness and, for this quantum number, one has to work in the
canonical scheme. Other conservation laws must also be imposed, like baryon number and
electric charge conservation. Since the number of baryons and charged particles is large,
they can be treated in the grand-canonical ensemble. In this way, the conservation laws as-
sociated to these other quantum numbers are satisfied on average, allowing for fluctuations
around the corresponding mean values.
To restrict the ensemble according to the exact strangeness conservation law, as done in
Refs. [56–61], one has to project the grand-canonical partition function,¯Z(T,V,λB,λS,λQ),
onto a fixed value of strangeness S,
ZS(T,V,λB,λQ) =
1
2π
?2π
0
dφ e−iSφ ¯Z(T,V,λB,λS,λQ) , (1)
where λB,λS,λQare the baryon, strangeness and charge fugacities, respectively, and where
λSstands explicitly for λS= eiφ.
Only particles with S = 0,±1 are included in the grand-canonical partition function
because, in the range of energies achieved at GSI, they are produced with a higher proba-
bility than particles with S = ±2,±3. The grand-canonical partition function is calculated
assuming an independent particle behaviour and the Boltzmann approximation for the one-
particle partition function of the different particle species. In principle, one deals with a
dilute system, so the independent particle model seems justified. However, medium effects
on the particle properties can be relevant. As mentioned in the Introduction, the aim of
this paper is to study how the dressing of the hadrons present in the gas, especially the an-
tikaons, affects the observables such as the ratio of kaon and antikaon particle multiplicities,
in particular for the conditions of the heavy-ion collisions at SIS/GSI energies.
Within the approximations mentioned above, the grand-canonical partition function
reads as follows,
¯Z(T,V,λB,λS,λQ) = exp(NS=0+ NS=−1e−iφ+ NS=1eiφ) , (2)
where NS=0,±1is the sum over one-particle partition functions of all particles and resonances
with strangeness S = 0,±1,
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NS=0,±1=
?
Bi
Z1
Bi+
?
Mj
Z1
Mj+
?
Rk
Z1
Rk, (3)
Z1
Bi= gBiV
?
?
?
d3p
(2π)3e
d3p
(2π)3e
d3p
(2π)3
−EBi
T
e
µBi
T
e
µQ(Bi)
T
, (4)
Z1
Mj= gMjV
−EMj
T
e
µQ(Mj)
T
, (5)
Z1
Rk= gRkV
?m+2Γ
m−2Γ
ds e
−√
p2+s
T
1
π
mΓ
(s − m2)2+ m2Γ2
?
e
µB(Rk)
T
?
e
µQ(Rk)
T
. (6)
The expressions Z1
mesons respectively, while Z1
Biand Z1
Mjindicate the one-particle partition function for baryons and
Rkis the one-particle partition function associated to a baryonic
or mesonic resonance. In the latter case, however, the factor e
Eq. (6). Notice that the resonance is described by means of a Breit-Wigner parameterisation.
The quantity V is the interacting volume of the system, gB, gM and gR are spin-isospin
degeneracy factors and µBand µQare the baryonic and charge chemical potentials of the
system. For Ni+Ni system at SIS energies, µQ can be omitted because it is associated
to the isospin-asymmetry of the system and, in this case, the deviation from the isospin-
symmetric case is only 4% (see Ref. [58]). On the other hand, µB will be fixed to the
nucleonic chemical potential, µN, because the abundance of nucleons is larger than the one
for the other baryons produced. The energies EB, EMrefer to the in-medium single-particle
energies of the hadrons present in the system at a given temperature.
Following Ref. [58], the canonical partition function for total strangeness S = 0 is
µB(Rk)
T
would not be present in
ZS=0(T,V,λB) =
1
2π
?2π
0
dφ exp(NS=0+ NS=−1e−iφ+ NS=1eiφ) .(7)
In this work, as well as in Ref. [59], the small and large volume limits of the particle
abundances were studied. These limits were performed to show that the canonical treatment
of strangeness in obtaining the particle abundances gives completely different results in
comparison to the grand-canonical scheme, demonstrating at the same time that, for the
volume considered, the canonical scheme is the appropriate one. The aim of going through
these limits again in the following is to remind the reader that, for the specific case of the
ratio K−/K+, the result is independent of the size of the system and is the same for both
the canonical and grand-canonical treatments.
According to statistical mechanics, to compute the number of kaons and antikaons [63]
one has to differentiate the partition function with respect to the particle fugacity
NK−(K+)≡
?
λK−(K+)
∂
∂λK−(K+)lnZS=0(λK−(K+))
?
λK−(K+)=1
.(8)
Expanding ZS=0in the small volume limit, NK− and NK+ are given by
NK+ = gK+V
?
d3p
(2π)3e
−EK+
T
×



?
i
giV
?
d3p
(2π)3e
−EBi(S=−1)+µBi
T
+
?
j
gjV
?
d3p
(2π)3e
−EMj(S=−1)
T
+
?
k
ZRk(S=−1)


,
5