# 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.

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**ABSTRACT:**The in-medium modifications on the K−/K+K−/K+ ratio produced at GSI are studied. Particular attention is paid to the properties of antikaons, which determine the chemical potential and temperature at freeze-out conditions. Different approaches have been considered: non-interacting K−K−, on-shell self-energy and single-particle spectral density. We observe that the full off-shell approach to the spectral density reproduces the Brown et al. “broad-band equilibration” which is crucial to explain an enhanced K−/K+K−/K+ ratio.Nuclear Physics A 12/2003; 754:356–360. · 2.50 Impact Factor - SourceAvailable from: ArXiv[Show abstract] [Hide abstract]

**ABSTRACT:**We study kaon production in hot and dense hypernuclear matter with a conserved zero total strangeness and a conserved small negative isospin charge fraction in order to make our results relevant to relativistic heavy-ion collisions. The baryons and kaons are treated as MIT bags in the context of the modified quark–meson coupling model. They interact with each other via the scalar mesons σ, σ∗ and the vector mesons ω, ϕ as well as the isovector meson ρ. We adopt realistic sets of hyperon–hyperon interactions based on several versions of the Nijmegen core potential models. Our results indicate that the hyperons as well as the kaons are produced abundantly when the temperature increases and approaches the critical temperature for the phase transition to a quark–gluon plasma. Moreover we find that the kaons are only produced thermally and we find no kaon condensation in the regime explored by relativistic heavy-ion collisions.Nuclear Physics A 01/2005; 759(1):201-226. · 2.50 Impact Factor - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**The production of strange particles in a hot medium as produced in collisions of heavy ions is considered one of the most important signals for the phase transition to a quark-gluon plasma. In the first part of this lecture, the theoretical description of strangeness production in hot matter is outlined for a gas of quarks and gluons and for a hadronic gas and its impact on the deconfinement phase transition. Then in the second part, constraints from the underlying chiral symmetry of quantum chromodynamics (QCD) are utilized to extract signals with strangeness for the chiral phase transition in hot matter. Based on a student lecture given at the International Conference on Strangeness in Quark Matter, Atlantic Beach, NC, USA, 12-17 March 2003.Journal of Physics G Nuclear and Particle Physics 01/2004; 30(10). · 2.84 Impact Factor

Page 1

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

I. INTRODUCTION

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

stars ?4?.

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?

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68 024903-1

Page 2

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.

II. FORMALISM

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,

2??

0

ZS?T,V,?B,?Q??

1

2?

d? e?iS?Z ¯?T,V,?B,?S,?Q?,

?1?

where ?B,?S,?Qare the baryon, strangeness, and charge

fugacities, respectively, and where ?Sstands explicitly for

?S?ei?.

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?

024903-2

Page 3

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:

Z¯?T,V,?B,?S,?Q??exp?NS?0?NS??1e?i??NS?1ei??,

?2?

where NS?0,?1 is the sum over one-particle partition

functions of all particles and resonances with strangeness

S?0,?1,

NS?0,?1??

Bi

ZBi

1??

Mj

ZMj

1??

Rk

ZRk

1,

?3?

ZBi

1?gBiV?

d3p

?2??3e?EBi/Te?Bi/Te?Q(Bi)/T,

?4?

ZMj

1?gMjV?

d3p

?2??3e?EMj/Te?Q(Mj)/T,

?5?

ZRk

1?gRkV?

d3p

?2??3?

m?2?

m?2?

dse??p2?s/T

?1

?

m?

?s?m2?2?m2?2?e?B(Rk)/T?e?Q(Rk)/T.

?6?

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

given temperature.

Following Ref. ?58?, the canonical partition function for

total strangeness S?0 is

1and ZMj

1

indicate the one-particle parti-

1

ZS?0?T,V,?B??

1

2??

0

2?

d?

?exp?NS?0?NS??1e?i??NS?1ei??.

?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 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-

canonical treatments.

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,

NK?(K?)

???K?(K?)

?

??K?(K?)

lnZS?0??K?(K?)??

?K?(K?)?1

. ?8?

Expanding ZS?0in the small volume limit, NK? and NK? are

given by

NK??gK?V?

???

d3p

?2??3e?EK? /T

gBiV?

i

d3p

?2??3e(?EBi(S??1)??Bi)/T

??

j

gMjV?

d3p

?2??3e?EMj(S??1)/T??

k

ZRk(S??1)

1

?,

NK??gK?V?

???

d3p

?2??3e?EK? /T

gMjV?

j

d3p

?2??3e?EMj(S??1)/T??

k

ZRk(S??1)

1

?,

?9?

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?

024903-3

Page 4

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

function Z¯(T,V,?B,?S),

lnZ¯?T,V,?B,?S???SZK?

1?

1

?SZK?

1??B

1

?SZB,S??1

1

??SZM,S??1

1

?

1

?SZM,S??1

1

,

?10?

where ZB,S??1

tion functions for baryons ?mesons? with S??1. Then, by

imposing strangeness conservation on average,

1

(ZM,S??1

1

) is the sum of one-particle parti-

?S???S

?

??SlnZ¯?0,

?11?

one can easily obtain ?S,

?S

2?

ZK?

1??BZB,S??1

ZK?

1

?ZM,S??1

1

1?ZM,S??1

1

.

?12?

Therefore, from

?NK????SZK?

1,

?NK???

1

?SZK?

1,

?13?

one obtains the ratio

K?

K??

ZK?

ZK?

1

1

ZK?

1?ZM,S??1

1??BZB,S??1

1

ZK?

1

?ZM,S??1

1

.

?14?

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-

canonical schemes.

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?

ratio as

K?

K??NK?

NK??

ZK?

1

?ZK?

1?Z?

ZK?

1?Z?

1ZK?

1?Z?*

1?

1

?1?

Z?

1?Z?

1?Z?*

ZK?

1

1

,

?15?

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

the ratio.

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

Z?,?

1

?g?,?V?

d3p

?2??3e(?E?,???B)/T

?16?

is constructed using a mean-field dispersion relation for the

single-particle energies:

E?,???m?,?

2

?p2?U?,????.

?17?

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 ?

atoms and

?-nucleus scattering

?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,

?65,66?, where

?0

Z?*

1?g?*V?

d3p

?2??3? dse??p2?s/T

?1

?

m?*?

2?2?m?*

?s?m?*

2?2e?B/T,

?18?

TOLO´S, POLLS, RAMOS, AND SCHAFFNER-BIELICHPHYSICAL REVIEW C 68, 024903 ?2003?

024903-4

Page 5

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

ZK?

1?gK?V?

d3p

?2??3e?EK? /T,

?19?

EK???mK?

2?p2?UK????,

?20?

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

ZK?

1?gK?V?

d3p

?2??3e?EK? /T,

EK???mK?

2?p2?UK¯?T,?,EK?,p?,

?21?

where UK¯(T,?,EK?,p) is the K¯single-particle potential in

the Brueckner-Hartree-Fock approach given by

UK¯?T,?,EK?,p?

?Re?d3kn?k,T??K¯N?GK¯N→K¯N???EN?EK¯ ,T??K¯N?,

?22?

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

?K¯?T,?,?,p??2?p2?mK¯

2UK¯?T,?,?,p?,

?23?

via the corresponding K¯spectral density

SK¯?T,?,?,p???1

?ImDK¯?T,?,?,p?,

?24?

where

DK¯?T,?,?,p??

1

?2?p2?mK¯

2??K¯?T,?,?,p?

?25?

stands for the K¯ propagator. In this case, the K¯ partition

function reads

ZK?

1?gK?V?

d3p

?2??3? dsSK¯?T,?,???s,p?e??s/T,

?26?

where s??2.

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

2?

2?

2?

?K¯

p??,?,p???1

gN?K¯

2M?

2M?

2M?

2

p?2f?

2U???,?,p??

?3

gN?K¯

2

p?2f?

2U???,?,p??

?1

gN?*K¯

2

p?2f?*

2U?*??,?,p??. ?27?

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

temperature.

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?

024903-5

Page 6

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-

tion analytically.

III. RESULTS

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

energies ?58,59?.

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

Ref. ?71??.

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

00.050.10.15 0.2

ρ (fm

3)

free gas

onshell

onshell (µ free)

spectral L=0

spectral L=0+1

00.05 0.10.15

ρ (fm

3)

0

25

50

75

100

125

150

175

200

225

250

K

+/K

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?

024903-6

Page 7

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

forthcoming work.

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

YN?1configurations.

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

s

1/2(MeV)

0

0.2

0.4

0.6

0.8

1

SK(q=500 MeV,s

1/2)(GeV

2)/10

exp(s

1/2/T)

L=0

L=0+1

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

T?80 MeV.

0 200 400600800 100012001400

qK (MeV)

0

1000

2000

3000

4000

5000

4π qK

2 ∫ exp(s

1/2/T) SK(qK,s

1/2) ds

L=0+1

L=0

onshell

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

ρ (fm

3)

0

5

10

15

20

25

30

K

+/K

onlyΛ

Λ+Σ attractive

Λ+Σrepulsive

Λ+Σ repulsive+Σ

*

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?

024903-7

Page 8

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-

ponents, respectively.

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

Song ?62?.

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

1

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

equilibration behavior.

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

µB (MeV)

20

30

40

50

60

70

80

90

100

T (MeV)

free gas

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?

024903-8

Page 9

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.

IV. CONCLUSIONS

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

Y??,?,?*.

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.

?74,75??.

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

µB (MeV)

20

30

40

50

60

70

80

90

100

T (MeV)

K

K

K

K

K

+/K =5

+/K =10

+/K =15

+/K =20

+/K =30

FIG. 6. The K?/K?ratio plotted for the full spectral density of

the K?as a contour plot for different temperatures and baryochemi-

cal potentials.

K?/K?RATIO IN HEAVY-ION COLLISIONS WITH AN . . . PHYSICAL REVIEW C 68, 024903 ?2003?

024903-9

Page 10

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

antikaons.

ACKNOWLEDGMENTS

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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