K^-/K^+ ratio in heavy-ion collisions 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 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 K¯ spectral density, including both s- and p-wave components of the 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 ``broadband equilibration'' advocated by Brown, Rho, and Song due to the additional strength of the spectral function of the K- at low energies.
KÀÕK¿ratio in heavy-ion collisions with an antikaon self-energy in hot and dense matter
Laura Tolo ´s, Artur Polls, and Angels Ramos
Departament d’Estructura i Constituents de la Mate `ria, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain
Ju ¨rgen Schaffner-Bielich
Institut fu ¨r Theoretische Physik, J. W. Goethe-Universita ¨t, D-60054 Frankfurt am Main, Germany
?Received 21 February 2003; published 15 August 2003?
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 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 K¯spectral density,
including both s- and p-wave components of the K¯N interaction, lowers substantially the freeze-out tempera-
ture 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 ‘‘broadband equilibration’’ advocated by
Brown, Rho, and Song due to the additional strength of the spectral function of the K?at low energies.
DOI: 10.1103/PhysRevC.68.024903PACS number?s?: 13.75.Jz, 14.40.Ev, 25.75.Dw, 24.10.Pa
The study of the properties of hadrons in hot and dense
matter is receiving a lot of attention in recent years to under-
stand 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 supernovae and the composition of neutron
A special effort has been invested to understand the prop-
erties of antikaons in the medium, especially after the specu-
lation of the possible existence of an antikaon condensed
phase was put forward in Ref. ?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 ef-
fects on the antikaon-nucleon scattering amplitude, the be-
havior of which is governed by the isospin zero ?(1405)
resonance, naturally explain the evolution from repulsion in
free space to attraction in the nuclear medium ?9,10?. How-
ever, the presence of the resonance makes the size of the
antikaon-nucleus potential very sensitive to the many-body
treatment of the medium effects. The phenomenological at-
tempts of extracting information about the antikaon-nucleus
potential from kaonic-atom data favored very strongly attrac-
tive 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 attrac-
tive kaon-nucleus interaction. In fact, recent analyses of ka-
onic atoms ?18–20? are able to find a reasonable reproduc-
tion 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 poten-
tial at nuclear matter saturation density.
On the other hand, heavy-ion collisions at energies around
1–2AGeV 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 such as antikaons is available. Since antika-
ons 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 ad-
dressed in Refs. ?14,29?.
Event generators trying to analyze heavy-ion collision
data ?30–36? need to implement the modified properties of
the hadrons in the medium where they are produced. Trans-
port models have shown, for instance, that the multiplicity
distributions of kaons and antikaons are much better repro-
duced 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
experiments have been performed with Au?Au, Ni?Ni,
C?C at energies between 0.6 and 2.0AGeV ?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 energy needed to produce the par-
ticle ?2.5 GeV for K?via pp→?K?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 two to three orders of
magnitude different. This has been interpreted to be a mani-
festation of the enhancement of the K?mass and the reduc-
tion 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-
PHYSICAL REVIEW C 68, 024903 ?2003?
0556-2813/2003/68?2?/024903?11?/$20.00 ©2003 The American Physical Society
medium enhanced ??→K?p production has also been
given ?14?. Another interesting observation is that at incident
energies of 1.8 and 1.93AGeV, the K?and K?multiplicities
have the same impact parameter dependence ?42–44?. Equal
centrality dependence for K?and K?and, hence, indepen-
dence of centrality for the K?/K?ratio have also been ob-
served in Au?Au and Pb?Pb reactions between 1.5AGeV
and RHIC energies (?s?200AGeV) ?44,49–52?. This inde-
pendence of centrality is astonishing, since at low energies
one expects that as centrality increases—and with it the par-
ticipating 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 signaling the lack of in-medium
effects. A recent interesting interpretation of this phenom-
enon is given in Ref. ?53?, where it is shown that the K?are
predominantly produced via ?Y collisions (Y??,?) and,
hence, the K?multiplicity is strongly correlated with the K?
one, since kaons and hyperons are mainly produced together
via the reaction NN→KYN.
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 chemi-
cal and thermal equilibrium and common freeze-out param-
eters for all particles, are quite successful in describing par-
ticle 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 parameters at freeze-
out, the baryonic chemical potential ?B?720 MeV and the
temperature T?70 MeV, which also reproduce a variety of
particle multiplicity ratios ?60,61?. On the other hand, the
centrality independence of the K?/K?ratio is automatically
obtained in statistical models within the canonical or grand-
canonical schemes because the terms depending on the sys-
tem 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 ex-
perimental 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 ‘‘broadband equilibra-
tion’’ 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 broadband relies on the fact that the
baryonic chemical potential ?Bincreases with density by an
amount which coincides roughly with the reduction of the
K?mass. However, as mentioned above, the antikaon prop-
erties are very sensitive to the type of model for the K¯N
interaction used and of the in-medium effects included. The
purpose of the present work is precisely to investigate to
which extent the broadband equilibration concept holds
when more sophisticated models for the in-medium antikaon
properties are used. We explore within the context of a sta-
tistical model what is the behavior 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 noninteracting 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 calculation in symmetric
nuclear matter at finite temperature ?29?, taking as bare
meson-baryon interaction the meson-exchange potential of
the Ju ¨lich 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 on the approxi-
mation adopted for the antikaon self-energy.
The paper is organized as follows. In Sec. II, the formal-
ism of the thermal model is described and the different
ingredients used in the determination of the off-shell proper-
ties of the K?are given. The results are presented and dis-
cussed in Sec. III. Finally, our concluding remarks are given
in Sec. IV.
In this section, we present a brief description of the ther-
mal 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, such as baryon number and electric charge conser-
vation. 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 associated with these other
quantum numbers are satisfied on average, allowing for fluc-
tuations around the corresponding mean values.
To restrict the ensemble according to the exact strange-
ness conservation law, as done in Refs. ?56–61?, one
has to project the grand-canonical partition function
Z¯(T,V,?B,?S,?Q) onto a fixed value of strangeness S,
d? e?iS?Z ¯?T,V,?B,?S,?Q?,
where ?B,?S,?Qare the baryon, strangeness, and charge
fugacities, respectively, and where ?Sstands explicitly for
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 probability
than particles with S??2,?3. The grand-canonical parti-
tion function is calculated assuming an independent particle
behavior and the Boltzmann approximation for the one-
TOLO´S, POLLS, RAMOS, AND SCHAFFNER-BIELICHPHYSICAL REVIEW C 68, 024903 ?2003?
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 dress-
ing of the hadrons present in the gas, especially the antika-
ons, affects the observables such as the ratio of kaon and
antikaon particle multiplicities, in particular for the condi-
tions of the heavy-ion collisions at SIS/GSI energies.
Within the approximations mentioned above, the grand-
canonical partition function reads as follows:
where NS?0,?1 is the sum over one-particle partition
functions of all particles and resonances with strangeness
The expressions ZBi
tion function for baryons and mesons, respectively, while ZRk
is the one-particle partition function associated with a bary-
onic or mesonic resonance. In the latter case, however, the
factor e?B(Rk)/Twould not be present in Eq. ?6?. Notice that
the resonance is described by means of a Breit-Wigner pa-
rametrization. The quantity V is the interacting volume of the
system, gB, gM, and gRare spin-isospin degeneracy factors,
and ?Band ?Qare the baryonic and charge chemical poten-
tials of the system. For Ni?Ni system at SIS energies, ?Q
can be omitted because it is associated with the isospin
asymmetry of the system and, in this case, the deviation from
the isospin-symmetric case is only 4% ?see Ref. ?58??. Under
these conditions, ?Bcoincides with the nucleonic one, ?N.
Since the abundance of nucleons is much larger than the one
for the other baryons produced, ?Ncan be safely obtained by
a purely nucleonic equation of state, as done in the present
work. The energies EB,EMrefer to the in-medium single-
particle energies of the hadrons present in the system at a
Following Ref. ?58?, the canonical partition function for
total strangeness S?0 is
indicate the one-particle parti-
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 appropri-
ate 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-
According to statistical mechanics, to compute the num-
ber of kaons and antikaons ?63? one has to differentiate the
partition function with respect to the particle fugacity,
Expanding ZS?0in the small volume limit, NK? and NK? are
where antibaryons have not been considered because the ra-
tio B¯/B?e?2?B/Tis negligibly small at GSI/SIS colliding
energies, where B and B¯represent the number of baryons
and antibaryons, respectively. The expression for NK? (NK?)
indicates that the number of K?(K?) has to be balanced
with all particles and resonances with S??1 (S?1). It can
be observed from Eq. ?9? that the ratio K?/K??NK?/NK?
in the canonical ensemble does not depend on the volume
because it cancels out exactly.
K?/K?RATIO IN HEAVY-ION COLLISIONS WITH AN . . .PHYSICAL REVIEW C 68, 024903 ?2003?
At the other extreme, i.e., in the thermodynamic limit
?large volumes?, since it is known that the canonical treat-
ment is equivalent to the grand-canonical one, one can com-
pute the ratio explicitly from the grand-canonical partition
tion functions for baryons ?mesons? with S??1. Then, by
imposing strangeness conservation on average,
) is the sum of one-particle parti-
one can easily obtain ?S,
one obtains the ratio
The condition ?S??0 and dealing with strange particles of
S?0,?1 make the ratio independent of the volume.
Although the expression obtained is the same as that from
the canonical ensemble in the small volume limit, the proof
that the ratio K?/K?is independent of the volume has to be
obtained from a general intermediate size situation. This was
shown to be the case in Ref. ?60?, where expanding
ZS?0(T,V,?B) of Eq. ?7? in power series, it was expressed as
ZS?0?Z0I0(x1), where Z0is the partition function that in-
cludes all particles and resonances with S?0, I0(x1) is the
modified Bessel function, and x1?2?NS?1NS??1. The
computed K?/K?ratio gave precisely the same expression
as those given here for small ?Eq. ?9?? and large ?Eq. ?14??
volumes. Therefore, as noted in Ref. ?59?, the K?/K?ratio
is independent of the volume and, consequently, independent
of whether it is calculated in the canonical or grand-
In-medium effects in KÀÕK¿ratio at finite T
In this section, we study how the in-medium modifica-
tions of the properties of the hadrons at finite temperature
affect the value of the K?/K?ratio, focusing our attention
on the properties of the antikaons in hot and dense matter.
For consistency with previous papers, we prefer to compute
the inverse ratio K?/K?. As it was mentioned before, the
number of K?(K?) has to be balanced by particles and
resonances with S??1 (S??1) in order to conserve
strangeness exactly. For balancing the number of K?, the
main contribution in the S??1 sector comes from the ?
and ? hyperons and, in a smaller proportion, from the K?
mesons. In addition, the effect of the ?*?1385? resonance is
also considered because it is comparable to that of the K?
mesons. On the other hand, the number of K?is balanced
only by the presence of K?. Then, we can write the K?/K?
where the Z’s indicate the different one-particle partition
functions for K?, K?, ?, ?, and ?*, and for baryons, they
now contain the corresponding fugacity. It is clear from Eq.
?15? that the relative abundance of ?, ?, and ?* baryons
with respect to that of K?mesons determines the value of
In order to introduce the in-medium and temperature ef-
fects, the particles involved in the calculation of the ratio are
dressed according to their properties in the hot and dense
medium in which they are embedded. For the ? and ? hy-
perons, the partition function
is constructed using a mean-field dispersion relation for the
For U?(?), we take the parametrization of Ref. ?64?,
U?(?)??340??1087.5?2. For U?(?), we take a repulsive
potential, U?(?)?30?/?0, extracted from the analysis of ?
?0.17 fm?3is the saturation density of symmetric nuclear
matter. A repulsive ? potential is compatible with the ab-
sence of any bound state or narrow peaks in the continuum in
a recent ?-hypernuclear search done at BNL ?67?. The
?*?1385? resonance is described by a Breit-Wigner shape,
TOLO´S, POLLS, RAMOS, AND SCHAFFNER-BIELICHPHYSICAL REVIEW C 68, 024903 ?2003?
with m?*?1385 MeV and ??37 MeV, where ?s is inte-
grated from m?*?2? to m?*?2?.
In the case of K?, we take
where UK?(?)?32?/?0is obtained from a t? approxima-
tion, as discussed in Refs. ?10,68?. One could also take for
the kaon potential the recent experimental value of 20 MeV
at ?0?69?. However, we note that in the present approach the
properties of the kaons do not play a direct role in the deter-
mination of the K?/K?ratio. The reason is that the negative
strangeness of the antikaons is balanced only with S??1
kaons and, as seen in Eq. ?15?, the corresponding partition
function cancels out in performing the ratio.
A particular effort has been invested in studying the anti-
kaon properties in the medium, since the K¯N has a particu-
larly rich structure due to the presence of the ??1405? reso-
nance ?12–15?. The antikaon optical potential in hot and
dense nuclear matter has recently been obtained ?29? within
the framework of a coupled-channel self-consistent calcula-
tion taking, as bare meson-baryon interaction, the meson-
exchange potential of the Ju ¨lich group ?17?. In order to un-
derstand the influence of the in-medium antikaon properties
on the K?/K?ratio, two different prescriptions for the
single-particle energy of the antikaons have been considered.
First, the so-called on-shell approximation to the antikaon
single-particle energy has been adopted. The antikaon parti-
tion function in this approach reads
where UK¯(T,?,EK?,p) is the K¯single-particle potential in
the Brueckner-Hartree-Fock approach given by
which is built from a self-consistent effective K¯N interaction
in nuclear symmetric matter, averaging over the occupied
nucleonic states according to the Fermi distribution at a
given temperature, n(k,T).
The second approach incorporates the complete energy-
and-momentum dependent K¯self-energy
via the corresponding K¯spectral density
stands for the K¯ propagator. In this case, the K¯ partition
We note, however, that only the s-wave contribution of
the Ju ¨lich K¯N interaction has been kept. The reason is that
this potential presents some shortcomings in the L?1 partial
wave, which manifest themselves especially in the low-
energy region of the K¯self-energy. Specifically, the ? and ?
poles of the K¯N T matrix come out lower about 100 MeV
than the physical values and, consequently, the correspond-
ing strength in the antikaon spectral function due to hyperon-
hole excitations appears at too low energies, a region very
important for the calculation we are conducting here. In ad-
dition, the role of the ?*?1385? pole, which lies below the
K¯N threshold, is not included in the Ju ¨lich K¯N interaction. In
order to overcome these problems, we have added to our K¯
self-energy the p-wave contribution as calculated in Ref.
?68?. In this model, the p-wave self-energy comes from the
coupling of the K¯meson to hyperon-hole (YN?1) excita-
tions, where Y stands for ?, ?, and ?*. In symmetric
nuclear matter at T?0, this self-energy reads
The quantities gN?K¯ , gN?K¯ , and gN?*K¯ are the N?K¯,
N?K¯, and N?*K¯coupling constants, while f?, f?, f?*are
the ?, ?, ?* relativistic recoil vertex corrections and U?,
U?, U?*the Lindhard functions at T?0. For more details,
see Ref. ?68?. The relevance of this low-energy region makes
it advisable to extend this p-wave contribution to finite
The hyperon-hole Lindhard functions at finite temperature
are easily obtained from the ?-hole one given in Eq. ?A16?
of Ref. ?29?, by ignoring, due to strangeness conservation,
the crossed-term contribution. The spin-isospin degeneracy
factors and coupling constants need to be accommodated to
K?/K?RATIO IN HEAVY-ION COLLISIONS WITH AN . . .PHYSICAL REVIEW C 68, 024903 ?2003?
the notation used in Eq. ?27? and this amounts to replacing
(2/3)(f?*/fN) by 3/2. Finally, the width ? in Eq. ?A16? of
Ref. ?29? is taken to zero, not only for the stable ? and ?
hyperons but also, for simplicity, for the ?* hyperon, which
allows one to obtain the imaginary part of the Lindhard func-
In this section, we discuss the effects of dressing the K?
mesons in hot and dense nuclear matter on the K?/K?ratio
around the value found in Ni?Ni collisions at an energy of
1.93AGeV. A preliminary study was already reported in Ref.
?70?. As previously mentioned, we prefer to discuss the re-
sults for the inverted ratio K?/K?.
The K?/K?ratio is shown in Fig. 1 as a function of
density at two given temperatures, T?50 and 80 MeV, cal-
culated for the three ways of dressing of the K?: free ?dot-
dashed lines?, the on-shell or mean-field approximation of
Eq. ?21? ?dotted lines?, and using the K¯spectral density in-
cluding s-wave ?long-dashed lines? or both s-wave and
p-wave contributions ?solid lines?. The two chosen tempera-
tures roughly delimit the range of temperatures which have
been claimed to reproduce, in the framework of the thermal
model, not only the K?/K?ratio but also all the other
particle ratios involved in the Ni?Ni collisions at SIS
Since the baryonic chemical potential ?Bgrows with den-
sity, the factor e?B/Tin the partition functions of Eqs. ?16?
and ?18? allows one to understand why the ratio increases so
strongly with density in the free gas approximation ?dot-
dashed lines?. The same is true when the particles are
dressed. In this case, however, the K?feels an increasing
attraction with density which tends to compensate the varia-
tion of ?Band the curves bend down after the initial in-
crease. This effect is particularly notorious when the full K¯
spectral density is used. The results are in qualitative agree-
ment with the broadband equilibration notion introduced by
Brown et al. ?62?. However, in this present model, the gain
in binding energy in the on-shell approximation for K?
?thick dotted line in Fig. 1? when the density grows does not
completely compensate the increase of ?B, as was the case
in Ref. ?62?. To illustrate this fact, we note that the variation
of the K?single-particle energy at zero momentum at T
?70 MeV changes in our model ?29? from 434 MeV to 375
MeV when the density grows from 1.2?0to 2.1?0, while ?B
changes from 873 MeV to 962 MeV. Therefore, the relevant
quantity to understand the behavior of the K?/K?ratio with
density in the on-shell approximation, i.e., the sum of ?Band
EK? „see Eq. ?6? of Ref. ?62?…, suffers in our model a varia-
tion of about 30 MeV. On the other hand, the model of Ref.
?62? assumed a slower variation of ?B, from 860 MeV to
905 MeV, which was almost canceled by the change of EK?
from 380 MeV to 332 MeV, giving therefore a practically
density independent ratio. We note that our chemical poten-
tial is derived from a nucleonic energy spectrum obtained
in a Walecka ?-? model, using density dependent scalar
and vector coupling constants fitted to reproduce Dirac-
Brueckner-Hartree-Fock calculation ?see Table 10.9 in
The results obtained in the present microscopic calcula-
tion show that the broadband equilibration only shows up
clearly when the full spectral function is used ?solid line in
Fig. 1?. After an increase at low densities, the K?/K?ratio
remains constant at intermediate and high densities. The use
of the spectral density implicitly amounts to an additional
gain in binding energy for the antikaons and, as density in-
creases, it compensates rather well the variation of ?B.
To understand the origin of this additional effective attrac-
tion when the full spectral density is used, we show in Fig. 2
the two functions that contribute to the integral over the en-
ergy in the definition of the K?partition function ?Eq. ?26??,
namely, the Boltzmann factor e??s/Tand the K?spectral
functions including L?0 ?long-dashed line? and L?0?1
?solid line? components of the K¯N interaction, for a momen-
tum q?500 MeV at ??0.17 fm?3and T?80 MeV. As it is
clearly seen in the figure, the overlap of the Boltzmann factor
with the quasiparticle peak of the K?spectral function is
small for this momentum. It is precisely the overlap with the
strength in the low-energy region that acts as a source of
attraction in the contribution to the K?partition function.
This effect is particularly pronounced when the p waves are
included, due to the additional low-energy components in the
spectral function coming from the coupling of the K?meson
to hyperon-hole (YN?1) excitations, where Y stands for ?,
?, and ?*. Assigning these low-energy components to real
antikaons in the medium is not clear, since one should inter-
pret them as representing the production of hyperons through
onshell (µ free)
FIG. 1. K?/K?ratio as a function of the density for T
?50 MeV ?left panel? and T?80 MeV ?right panel? calculated in
different approaches: the free Fermi gas ?dot-dashed line?, on-shell
self-energies ?dotted line?, on-shell self-energies with ? from a free
?noninteracting? Fermi gas ?thin dotted line?, dressing the K?with
its single-particle spectral function, with the L?0 contribution
?long-dashed line? and taking into account the additional L?1 par-
tial wave ?solid line?.
TOLO´S, POLLS, RAMOS, AND SCHAFFNER-BIELICH PHYSICAL REVIEW C 68, 024903 ?2003?
K¯N→Y conversion. While this is certainly true, it may also
happen that, once these additional hyperons are present in
the system, they can subsequently interact with fast non-
strange particles ?pions, nucleons? to create new antikaons. A
clear interpretation on what fraction of the low-energy
strength will emerge as antikaons at freeze-out is certainly an
interesting question and its investigation will be left here for
Once the integral over the energy is performed, the deter-
mination of the K?partition function still requires an inte-
gral over the momentum. The integrand as a function of
momentum is plotted in Fig. 3 for the same density and
temperature than in the previous figure. As expected, the
integrand is larger when the full spectral density is consid-
ered. In this case, the K?partition function is enhanced and
therefore the K?/K?ratio is smaller than that in the on-shell
approximation and also in the case where only the L?0
contributions to the spectral density are used. Notice the be-
havior at large q, which decays very quickly in the on-shell
approximation but has a long tail for the L?0?1 spectral
density originated from the coupling of the K?meson to
Another aspect that we want to consider is how the dress-
ing of the ? hyperon affects the value of the ratio. In Fig. 4,
the value of the K?/K?ratio at T?50 MeV is shown as a
function of density for different situations. In all calculations
displayed in the figure, the partition function associated with
K?has been obtained using the full K?spectral density. The
dotted line corresponds to the case where only the ? hyper-
ons, dressed with the attractive mean-field potential given in
the preceding section, are included to balance strangeness.
When the ? hyperon is incorporated with a moderately at-
tractive potential of the type U???30?/?0MeV, the
K?/K?ratio is enhanced substantially ?dashed line?. This
enhancement is more moderate when one uses the repulsive
potential U???30?/?0instead ?dot-dashed line?. Finally,
the additional contribution of the ?* resonance produces
only a small increase of the ratio ?solid line? due to its higher
mass. We have checked that heavier strange baryonic or me-
sonic resonances do not produce visible changes in our re-
sults. Notice also that, although the ratios obtained with both
prescriptions for the mean-field potential of the ? meson
differ appreciably, the present uncertainties in the ratio
0100200300 400500600700800 900 1000
FIG. 2. The Boltzmann factor ?dotted line? and the K?spectral
function, including s-wave ?dashed line? or s- and p-wave ?solid
line? components of the K¯N interaction, as functions of the energy,
for a momentum q?500 MeV at saturation density and temperature
0 200 400600800 100012001400
2 ∫ exp(s
FIG. 3. The integrand which defines the K?partition function
?Eq. ?26?? as a function of momentum, at saturation density and T
?80 MeV, for different approaches: on-shell prescription ?dotted
line?, using the K?spectral function with the L?0 components of
the K¯N interaction ?long-dashed line? and including also the L?1
partial waves ?solid line?.
0 0.050.10.15 0.20.25 0.3
FIG. 4. The K?/K?ratio as a function of density at T
?50 MeV. The dotted line shows the results when only the ?
hyperons are considered in the determination of the ratio. The
dashed ?dot-dashed? line includes also the contribution of the ?
hyperon dressed with an attractive ?repulsive? mean-field potential.
The solid line includes the effect of the ?* resonance.
K?/K?RATIO IN HEAVY-ION COLLISIONS WITH AN . . .PHYSICAL REVIEW C 68, 024903 ?2003?
would not permit to discriminate between them.
In the framework of the statistical model, one obtains a
relation between the temperature and the chemical potential
of the hadronic matter produced in the heavy-ion collisions
by fixing the value of the K?/K?ratio which was measured
for Ni?Ni collisions at GSI to be on the average K?/K?
?0.031?0.005 ?44?. We compare our results with a corre-
sponding inverse ratio of K?/K??30 in the following. The
temperatures and chemical potentials compatible with that
ratio are shown in Fig. 5 for different approaches. The dot-
dashed line stands for a free gas of hadrons, similar to the
calculations reported in Refs. ?58,59?. The dotted line shows
the T(?) curve obtained with the on-shell or mean-field ap-
proximation ?see Eqs. ?21? and ?22??, while the dashed and
solid lines correspond to the inclusion of the off-shell prop-
erties of the K?self-energy by using its spectral density
?Eqs. ?23?, ?24? and ?26??, including L?0 or L?0?1 com-
In the free gas limit, the temperatures compatible with a
ratio K?/K??30 imply a narrow range of values for the
baryonic chemical potentials, namely, ?B??665,740? MeV
for temperatures in the range of 20–100 MeV. These values
translate into density ranges of ???6?10?7?0,0.9?0?.
As it can be seen from the dotted line in Fig. 5, the at-
tractive mean-field potential of the antikaons compen-
sates the effect of increasing baryochemical potential ?B. As
a consequence, the density at which the freeze-out tempera-
ture compatible with the measured ratio takes place also
grows. But this attraction is not enough to get the same
K?/K?ratio for a substantially broader range of density
compared to the free case. So we do not see a clear indica-
tion of broadband equilibration in our self-consistent mean-
field calculation in contrast to the results of Brown, Rho, and
The influence of the antikaon dressing on the ratio is
much more evident when the spectral density is employed
?long-dashed and solid lines?. From the preceding discus-
sions, it is easy to understand that the low-energy behavior of
the spectral density enhances the K?contribution to the ra-
tio, having a similar role as an attractive potential and, hence,
the value of ?Bcompatible with a ratio at a given tempera-
ture increases. Moreover, due to the bending of the K?/K?
ratio with density and its evolution with temperature ob-
served in Fig. 1, it is clear that there will be a maximum
value of T compatible with a given value of the ratio. Below
this maximum temperature, there will be two densities or,
equivalently, two chemical potentials compatible with the ra-
tio. For example, the ratio K?/K??30 will in fact not be
realized with the temperatures of T?50 MeV and T
?80 MeV displayed in Fig. 1, if the antikaon is dressed with
the spectral density containing L?0 and L?1 components.
As shown in Fig. 5, only temperatures lower than 34 MeV
are compatible with ratio values of 30.
We note that the flat regions depicted by the solid lines in
Fig. 5 could be considered to be in correspondence with the
notion of broadband equilibration of Brown et al. ?62?, in the
sense that a narrow range of temperatures and a wide range
of densities are compatible with a particular value of the
K?/K?ratio. Nevertheless, the temperature range is too low
to be compatible with the measured slope parameter of the
pion spectra. Explicitly, for K?/K??30, we observe a
nearly constant ratio observed in the range of 30–34 MeV
covering a range of chemical potentials in between 680 and
815 MeV which translates into a density range ???1.5
?10?4?0,0.02?0?. Note that in this case, we can hardly
speak of a broadband equilibration in the sense of that intro-
duced by Brown, Rho, and Song in Ref. ?62?, where a ratio
K?/K??30 holds over a large range of densities in between
4?0and 2?0for T?70 MeV. However, as we indicated at
the beginning of this section, this result was obtained in the
framework of a mean-field model and our equivalent on-
shell results ?dashed lines in Figs. 1 and 5?, based on a stron-
ger variation of the ?Bwith density and on a less attractive
UK¯ , seem to be very far from producing the broadband
As pointed out before, our nucleon chemical potential is
obtained in the framework of a relativistic model and varies
with density more strongly than that used in Ref. ?62?, which
shows values close to those for a free Fermi gas. If we now
calculate the K?/K?ratio using a ?B(?,T) for a free ?non-
interacting? system in the on-shell approximation, we obtain
the thin dotted line in Fig. 1. At T?80 MeV, we now ob-
serve a tendency of a broadband equilibration but for ratios
higher than 30, of around 50. This is connected to the par-
ticular on-shell potential of the antikaon which, in our self-
consistent procedure, turns to be moderately attractive. Only
if the attraction was larger would the broadband be realized
in this on-shell picture for smaller values of the ratio as
found by Brown et al. ?62?.
Figure 6 shows the K?/K?ratio for the full model calcu-
lation as a contour plot for different temperatures and bary-
ochemical potentials. We note that the ratio is substantially
lower at the temperature and density range of interest, being
more likely around 15 or so for a moderately large region of
baryochemical potential. Note that this reduced ratio trans-
650 700750 800 850900950
on shell approach
L=0 spectral density
L=0+1 spectral density
FIG. 5. Relation between the temperature and the baryochemical
potential of hadronic matter produced in heavy-ion collisions for
fixed K?/K?ratio of 30, calculated within different approaches as
discussed in the text.
TOLO´S, POLLS, RAMOS, AND SCHAFFNER-BIELICHPHYSICAL REVIEW C 68, 024903 ?2003?
lates into an overall enhanced production of K?by a factor 2
compared to the experimentally measured value. At pion
freeze-out, the medium can hold twice as many K?as
needed to explain the measured enhanced production of K?
if one considers the full spectral features of the K?in the
medium. We stress again that this enhancement is not due to
an increased attraction in the sense of a mean-field calcula-
tion. It is a consequence of the additional strength of the
spectral function at low energies, which emerges when tak-
ing into account p-wave hyperon-hole excitations. The Bolt-
zmann factor amplifies the contribution of the low-energy
region of the spectral function so that these excitations are
becoming the main reason for the overall enhanced produc-
tion of the K?in the medium.
We have studied, within the framework of a statistical
model, the influence of considering the modified properties
in hot and dense matter of the hadrons involved in the deter-
mination of the K?/K?ratio produced in heavy-ion colli-
sions at SIS/GSI. We have focused our attention on incorpo-
rating the effects of the antikaon self-energy, which was
derived within the framework of a self-consistent coupled-
channel calculation taking, as bare interaction, the meson-
exchange potential of the Ju ¨lich group for the s-wave and
adding the p-wave components of the Yh excitations, with
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 effect of dressing the K?with a spectral function includ-
ing both s- and p-wave components of the K¯N interaction
lowers considerably the effective temperature while increas-
ing the chemical potential ?Bup to 850 MeV, compared to
calculations for a noninteracting hadron gas.
When the free or on-shell properties of the antikaon are
considered, the ratio at a given temperature shows a strong
dependence on the density. This is in contrast with the broad-
band equilibration advocated by Brown et al. ?62?, which
was established, in the context of a mean-field picture,
through a compensation between the increased attraction of
the mean-field K¯potential as density grows with the increase
in the baryon chemical potential. Our mean-field properties
do not achieve such a compensation due to a stronger in-
crease of the nucleon chemical potential ?Bwith density.
When taking into account the full features of the spectral
function of the K?via hyperon-hole excitations, we find that
the K?/K?ratio exhibits broadband equilibration. Neverthe-
less, the ratio is even in excess of the measured ratio at
temperatures which are deduced from the slope parameter for
pions. The incompatibility with the experimentally measured
ratio can be resolved in several ways. First, the spectral func-
tion has too much strength below threshold and there is no
broadband equilibration. Second, the simple statistical ansatz
cannot be used for a medium modified K?.
Admitting the plausibility of our approach, one would still
need, in order to see the particle ratios of the medium, to
assume an instantaneous substantial change of the spectral
function from the in-medium one to the free one at freeze-
out, i.e., it must be fast enough so that inelastic collisions are
not possible anymore. If that process is too slow, particle
ratios will be adjusted accordingly yielding a final ratio
which is compatible with that of a free gas of noninteracting
particles. The K?interacts quite strongly with the medium,
transforming nucleons to hyperons and pions. If that inelastic
process ceases later than the ones for pions, the correspond-
ing particle ratios will be fixed at a different value of tem-
perature and baryochemical potential than that deduced from
the slope of the transverse mass spectrum of the pions. In-
deed, different slope parameters for K?and K?have been
measured in heavy-ion experiments at GSI by the KaoS Col-
laboration ?72? where the slope parameter for the antikaons
is significantly lower than that for kaons. In addition, K?are
emitted isotropically for central collisions, while there is a
strong forward/backward enhancement in the angular distri-
bution for produced K??23?. From microscopical transport
models, it is also well known that the antikaons freeze-out
much later than kaons due to their substantially shorter
mean-free path ?see, e.g., Ref. ?73??. These observations in-
dicate that the statistical description of subthreshold antikaon
production has to be taken with care. We note that the dif-
ferent slope parameters for kaons and antikaons can indeed
be attributed to medium effects, especially to a sizable
attraction in the medium felt by the antikaons ?see, e.g., Refs.
The question how the antikaon gets on shell and what
fraction of the antikaon strength emerges as real antikaons
are interesting ones. This investigation is outside the scope of
the present study and is left to be addressed in forthcoming
work. We note, however, that the first question is addressed
recently in an off-shell transport model simulation and refer
the interested reader to Ref. ?76? which uses essentially the
same spectral function for antikaons as in the present work.
There it is shown that the antikaon spectral function reaches
550600650 700750 800 850900950
FIG. 6. The K?/K?ratio plotted for the full spectral density of
the K?as a contour plot for different temperatures and baryochemi-
K?/K?RATIO IN HEAVY-ION COLLISIONS WITH AN . . . PHYSICAL REVIEW C 68, 024903 ?2003?
the vacuum asymptotically. The necessary energy to lift the
in-medium antikaons to the free one is taken from the trans-
verse expansion of the surrounding matter. Most importantly
for our discussion outlined here, the transport model simula-
tion demonstrates that the expansion is sufficiently fast to
preserve the in-medium information of antikaons and
shows a significant different slope parameter for kaons and
We are very grateful to V. Koch, L. Alvarez-Ruso, and E.
Oset for useful discussions. This work was partially sup-
ported by DGICYT Project No. BFM2002-01868 and by the
Generalitat de Catalunya Project No. 2001SGR00064. L.T.
also wishes to acknowledge support from the Ministerio de
Educacio ´n y Cultura ?Spain?.
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