arXiv:0704.0554v1  [nucl-th]  4 Apr 2007
In-Medium Effects on Particle Production in Heavy Ion
Collisions
V. Prassa a G.Ferini b T. Gaitanos c H.H. Wolter d
G.A. Lalazissis a M. Di Toro b
aDepartment of Theoretical Physics, Aristotle University of Thessaloniki,
Thessaloniki Gr-54124,Greece
bLaboratori Nazionali del Sud INFN, I-95123 Catania, Italy
cInstitut für Theoretische Physik, Justus-Liebig-Universität Giessen, D-35392
Giessen, Germany
dSektion Physik, Universität München, D-85748 Garching, Germany
email: Theodoros.Gaitanos@theo.physik.uni-giessen.de
The effect of possible in-medium modifications of nucleon-nucleon
(NN) cross sections on particle production is investigated in heavy
ion collisions (HIC) at intermediate energies. In particular, using a
fully covariant relativistic transport approach, we see that the den-
sity dependence of the inelastic cross sections appreciably affects
the pion and kaon yields and their rapidity distributions. How-
ever, the (π−/π+)- and (K0/K+)-ratios depend only moderately
on the in-medium behavior of the inelastic cross sections. This is
particularly true for kaon yield ratios, since kaons are more uni-
formly produced in high density regions. Kaon potentials are also
suitably evaluated in two schemes, a chiral perturbative approach
and an effective meson-quark coupling method, with consistent re-
sults showing a similar repulsive contribution for K+ and K0. As
a consequence we expect rather reduced effects on the yield ratios.
We conclude that particle ratios appear to be robust observables
for probing the nuclear equation of state (EoS) at high baryon
density and, particularly, its isovector sector.
Key words: Asymmetric nuclear matter, symmetry energy,
relativistic heavy ion collisions, particle production.
PACS numbers: 25.75.-q, 21.65.+f, 21.30.Fe, 25.75.Dw.
Preprint submitted to Elsevier Preprint 4 November 2018
http://arxiv.org/abs/0704.0554v1
1 Introduction
The knowledge of the properties of highly compressed and heated hadronic
matter is an important issue for the understanding of astrophysical processes,
such as the mechanism of supernovae explosions and the physics of neutron
stars [1,2]. Heavy ion collisions provide the unique opportunity to explore
highly excited hadronic matter, i.e. the high density behavior of the nuclear
EoS, under controlled conditions (high baryon energy densities and tempera-
tures) in the laboratory [3]. Of particular recent interest is also the still poorly
known density dependence of the isovector channel of the EoS.
Suggested observables have been the nucleon collective flows [3,4] and the
distributions of produced particles such as pions and, in particular, particles
with strangeness (kaons) [5,6]. Because of the rather high energy threshold
(Elab = 1.56 GeV for Nucleon-Nucleon collisions), kaon production in HICs
at energies in the range 0.8− 1.8 AGeV is mainly due to secondary processes
involving ∆ resonances and pions (π). On the other hand, secondary processes
require high baryon density. This explains why the kaon production around
threshold is intimately connected to the high density stage of the nucleus-
nucleus collision. Furthermore, the relatively large mean free path of positive
charged (K+) and neutral (K0) kaons inside the hadronic environment causes
hadronic matter to be transparent for kaons [7]. Therefore kaon yields and
generally strangeness ratios have been proposed as important signals for the
investigation of the high density behavior of the nuclear EoS. This idea, as
firstly suggested by Aichelin and Ko [8], has been recently applied in HIC
at intermediate energies in terms of strangeness ratios, e.g. the ratio of the
kaon yields in Au+Au and C+C collisions [5,9]. In these studies it was found
that this ratio is very sensitive to the stiffness of the nuclear EoS. Indeed
comparisons with KaoS data [10] favored a soft behavior of the high density
nuclear EoS, a statement which is particularly consistent with elliptic flow
data of the FOPI collaboration [11].
The idea of studying particle ratios in HICs around the kinematical threshold
has been recently applied in the determination of the isovector channel of the
nuclear EoS, i.e. the high density dependence of the symmetry energy Esym. It
has turned out that particle ratios, such as (π−/π+) [12] or (K0/K+) [13–15],
are sensitive to the stiffness of the symmetry energy and, in particular to the
strength of the vector isovector field. However in medium effects on the kaon
propagation have been neglected so far. Here we will test the robustness of the
yield ratio against the inclusion and the variation of the corresponding kaon
potentials. At the same time in Ref. [16] the role of the in-medium modifica-
tions of NN cross sections has been studied in terms of baryon and strangeness
dynamics. It was found that the pion and kaon yields are sensitively influenced
by the reduced effective NN cross sections for inelastic processes. Here we will
see that the kaon yield ratio appears robust even with respect to the density
dependence of the in-medium inelastic NN cross sections, while at variance
the pion ratio seems to be more sensitive.
The collision dynamics is rather complex and involves the nuclear mean field
(EoS) and binary 2-body collisions. In the presence of a nuclear medium the
treatment of binary collisions represents a non-trivial problem. The NN cross
sections for elastic and inelastic processes, which are the crucial physical pa-
rameters here, are experimentally accessible only in free space and not for
2-body scattering at finite baryon density. Recent microscopic studies, based
on the G-matrix approach, have shown a strong decrease of the elastic NN
cross section [17,18] in a hadronic medium. These in-medium effects of the
elastic NN cross section considerably influence the hadronic reaction dynam-
ics [19]. Obviously the question arises whether similar in-medium effects of the
inelastic NN cross sections may affect the reaction dynamics and, in particular,
the production of particles (pions and kaons).
Furthermore, the strangeness propagation inside the nuclear medium is even
more complex and involves the additional consideration of kaon mean field
potentials in the dynamical description. This is an important issue when com-
paring with experimental kaon data [10]. In a Chiral Perturbation approach
at the lowest order (ChPT Potentials), the kaon (antikaon) potential has an
attractive scalar and a repulsive (attractive) vector part [20]. This leads to
weakly repulsive (strongly attractive) potentials for kaons (antikaons) with
corresponding scalar and vector kaon-nucleon coupling constants depending
on the parametrization [20,21] accounted for. Similar results can be obtained
in an effective meson-coupling model (OBE Potentials, in the RMF spirit),
where the K-meson couplings are simply related to the nucleon-meson ones,
in the spirit of ref. [22]. The latter approach has the advantage of being fully
consistent with the covariant transport equations used to simulate the reaction
dynamics [14,15]. We remind that the high density dependence of the kaon
self energies is still an object of current debate, e.g. see Refs. [23,7] in which
the role of the kaon potential has been investigated in terms of kaon in-plane
and out-of-plane flows. Moreover for studies aimed to the determination of
the symmetry energy from strangeness production one has to consider with
particular care the isospin dependence of the kaon mean field potential.
The main focus of the present work is on a detailed study of the robustness of
the pionic (π−/π+) and, in particular, the strangeness ratio (K0/K+) with re-
spect to the in-medium modifications of the imaginary part of the nucleon self
energy, i.e. the NN cross sections, and to the in-medium variations of the kaon
self energy, i.e. the density dependence of the kaon potential. This analysis,
which goes beyond our previous investigations of [14,15], is also motivated by
new measurements of the FOPI collaboration [24] by means of the strangeness
ratios.
The paper is organized as follows: The next Section describes the theoret-
ical treatment of the reaction dynamics within the Relativistic Boltzmann-
Uheling-Uhlenbeck (RBUU) transport equation. A detailed discussion on the
in-medium modifications of the inelastic NN cross sections is presented. In Sec-
tion 3 we discuss the kaon mean field potentials (in both ChPT and OBE/RMF
schemes) and their expected isospin dependence. Section 4 is devoted to a short
introduction to the dynamical calculations. Results are then shown in Section
5, mostly for central 197Au+197 Au collisions at 1AGeV , in terms of pion and
kaon yields. The initial presentation of the absolute yields is relevant for a de-
tailed discussion as well as for a comparison with theoretical results of other
groups and with experimental data of the KaoS and FOPI collaborations.
All together this intermediate step is important for testing the reliability of
the calculations, since ratios do not do it. Finally we present the pion and
strangeness ratios and discuss their dependence on the in-medium modifica-
tions of the cross NN cross sections and of the kaon potentials, including the
isospin effects. In Section 6 we conclude with a summary and some general
comments and perspectives.
2 Theoretical description of the collision dynamics
In this chapter we briefly discuss the transport equation focusing on the treat-
ment of two features important for kaon dynamics: (a) the collision integral by
means of the cross sections; (b) the kaon mean field potential and its isospin
dependence.
2.1 The RBUU equation
The theoretical description of HICs is based on the semiclassical kinetic the-
ory of statistical mechanics, i.e. the Boltzmann Equation with the Uehling-
Uhlenbeck modification of the collision integral [25]. The relativistic analog of
this equation is the Relativistic Boltzmann-Uehling-Uhlenbeck (RBUU) equa-
tion [26]
k∗µ∂xµ + (k
µν +M∗∂µxM
∗) ∂k
f(x, k∗) =
2(2π)9
W (kk2|k3k4)
f3f4f̃ f̃2 − ff2f̃3f̃4
, (1)
where f(x, k∗) is the single particle distribution function. In the collision term
the short-hand notations fi ≡ f(x, k∗i ) for the particle and f̃i ≡ (1− f(x, k∗i ))
0 50 100 150 200 250 300 350
 [MeV]
=1.1 fm
=1.34 fm
=1.7 fm
pn-data
Fig. 1. Elastic in-medium neutron-proton cross section σel at various Fermi momenta
kF as a function of the laboratory energy Elab. The free cross section (kF = 0) is
compared to the experimental total np cross section [17].
for the hole distributions are used, with E∗
M∗2 + k2. The collision in-
tegral explicitly exhibits the final state Pauli-blocking while the in-medium
scattering amplitude includes the Pauli-blocking of intermediate states.
The dynamics of the drift term, i.e. the lhs of eq.(1), is determined by the
mean field. Here the attractive scalar field Σs enters via the effective mass
M∗ = M − Σs and the repulsive vector field Σµ via the kinetic momenta
k∗µ = kµ − Σµ and via the field tensor F µν = ∂µΣν − ∂νΣµ. The dynamical
description according to Eq.(1) involves the strangeness propagation in the
nuclear medium. This topic will be discussed in more detail at the end of this
section.
2.2 In-medium effects on NN cross sections
The in-medium cross sections for 2-body processes (see below) enter in the
collision integral via the transition amplitude
W = (2π)4δ4 (k + k2 − k3 − k4) (M∗)4|T |2 (2)
with T the in-medium scattering matrix element.
In the kinetic equation (1) both physical input quantities, the mean field (EoS)
and the collision integral (cross sections) should be derived from the same
underlying effective two-body interaction in the medium, i.e. the in-medium
T-matrix; Σ ∼ ℜTρB, σ ∼ ℑT , respectively. However, in most practical
applications phenomenological mean fields and cross sections have been used.
In such approach the strategy is to adjust to the known bulk properties of
nuclear matter around the saturation point, and to try to constrain the models
at supra-normal densities with the help of heavy ion reactions [27,28]. Medium
modifications of the NN cross sections are usually not taken into account.
In spite of that for several observables the comparison to experimental data
appears to work astonishingly well [27–30]. However, in particular kinematical
regimes a sensitivity to the elastic NN cross sections of dynamical observables,
such as collective flows and stopping [19,31] or transverse energy transfer [32],
has been observed.
Microscopic Dirac-Brueckner-Hartree-Fock (DBHF) studies for nuclear matter
above the Fermi energy regime show a strong density dependence of the elastic
[17] and inelastic [18,33] NN cross sections. In such studies one starts from the
bare NN-interaction in the spirit of the One-Boson-Exchange (OBE) model by
fitting the parameters to empirical nucleon-nucleus scattering and solves then
the equations of the nuclear matter many body problem in the T -matrix or
ladder approximation. It is not the aim of the present work to go into further
details on this topic. An important feature of such microscopic calculations is
the inclusion of the Pauli-blocking effect in the intermediate scattering states
of the T -matrix elements and their in-medium modifications, i.e. the density
dependence of the nucleon mass and momenta. Here of particular interest are
the in-medium modifications of the inelastic NN cross sections since they di-
rectly influence the production mechanism of resonances and thus the creation
of pions and kaons according to the channels listed later (see Sect.3). DBHF
studies on inelastic NN cross sections are rare and in limited regions of density
and momentum [18]. For this reason we will first discuss in the following the
in-medium dependence of the elastic NN cross sections, which will be then
used as a starting basis for a detailed analysis of the density dependence of
the inelastic NN cross sections.
The microscopic in-medium dependence of the elastic cross sections can be
seen in Fig. 1, where the energy dependence of the in-medium neutron-proton
(np) cross section at Fermi momenta kF = 0.0, 1.1, 1.34, 1.7fm
−1, correspond-
ing to ρB ∼ 0, 0.5, 1, 2ρ0 (ρ0 = 0.16fm−3 is the nuclear matter saturation den-
sity) is shown. These results are obtained from relativistic Dirac-Brueckner
calculations [17]. The presence of the medium leads to a substantial suppres-
sion of the cross section which is most pronounced at low laboratory energy
Elab and high densities where the Pauli-blocking of intermediate states is most
efficient. At larger Elab asymptotic values of 15-20 mb are reached. Also the
angular distributions are affected by the presence of the medium. E.g. the ini-
tially strongly forward-backward peaked np cross sections become much more
isotropic at finite densities, mainly due to the Pauli suppression of intermedi-
ate soft modes (π-exchange) [17]. As a consequence a larger transverse energy
transfer can be expected.
The case of the inelastic NN cross sections is similar, but more complicated.
The presence of the medium influences not only the matrix elements, but also
the threshold energy Etr, which is an important quantity at beam energies be-
low or near the threshold of particle production. In free space it is calculated
from the invariant quantity s = (p
1 + p
2 )(p1µ + p2µ) with p
i , (i = 1, 2) the
4-momenta of the two particles in the ingoing collision channel, e.g. NN −→
N∆. This quantity is conserved in binary collisions in free space, from which
one determines the modulus of the momenta of the particles in the outgoing
channel. The threshold condition reads Etr ≡
s ≥ M1 +M2. Cross sections
in free space are usually parametrized in terms of
s or the corresponding mo-
mentum in the laboratory system plab within the One-Boson-Exchange (OBE)
model, see e.g. [34] for details.
At finite density, however, particles carry kinetic momenta and effective masses
and obey a dispersion relation p∗µp
∗µ = m∗2 modified with respect to the free
case. These in-medium effects shift the threshold energy in the free space
according to s∗ = (p
2 )(p
2µ) and the threshold condition for inelastic
processes inside the medium reads now E∗
s∗ ≥ m∗
. The requirement
of energy-momentum conservation can be carried out in terms of the quantity
s∗ or s, only as long as the in-medium mean fields or the corresponding self
energies do not change between ingoing and outgoing channels.
The application of free parametrizations of cross sections for inelastic processes
in dynamical situations of HICs at finite density leads thus to an inconsistency,
since the threshold condition is performed in terms of effective quantities,
but the matrix elements are carried out in free space, e.g. by fitting their
parameters to free empirical NN scattering. This effect can be seen in Fig.
2 (left panel) where the free inelastic NN −→ N∆ cross section σinel as a
function of the laboratory energy Elab is displayed, at various baryon densities
ρB. The threshold energy in the free space is Etr =
s = 2.014 GeV (for
M = 0.939 GeV and Mmin = 1.076 for the nucleon and the lower limit mass of
the ∆ resonance). The corresponding threshold value of the laboratory energy
Elab = (E
− 4M2)/2M is 0.32 GeV. However, at finite density the threshold
is shifted towards lower energies, i.e. the free cross section increases, due to
the reduction of the free masses of the outgoing particles in the threshold
condition E∗
. Obviously at higher energies far from threshold the
free cross section does not depend on the density.
A more consistent approach is the determination of the inelastic cross section
under the consideration of in-medium effects, i.e. the Pauli-blocking of inter-
mediate scattering states and in-medium modified spinors in the determina-
tion of the matrix elements within the OBE model. A simultaneous treatment
of the transport equation and the structure equations of DBHF for actual
anisotropic momentum configurations is not possible, due to its high com-
plexity. For this reason we have applied the same method as for the case
0 0.5 1 1.5 2
 (GeV)
0 0.5 1 1.5 2
 (GeV)
=0.5ρ0
free effective
Fig. 2. Inelastic NN −→ N∆ cross section σinel at various baryon densities ρB (in
units of the saturation density ρ0 = 0.16 fm
−3) as a function of the laboratory
energy Elab using the free parametrizations (left) and the in-medium modified ones
from DBHF [18] (right).
of elastic binary processes, i.e. in-medium parametrizations of the inelastic
cross sections of the type NN −→ N∆ within the same underlying DBHF
approach as already used for the elastic processes. Haar and Malfliet [18]
investigated this topic for infinite nuclear matter with the result of a strong
in-medium modification of the inelastic cross sections due to the reasons given
above. However, these studies were performed at various densities but only in
a limited region of momenta. For a practical application in HICs we have
thus extended these DBHF calculations using an extrapolation technique. We
have imposed an exponential decay law of the form ae−bplab on the values of
the in-medium cross sections of the channel NN → N∆ given in ref. [18].
The parameter a normalizes to the last value of the extrapolated cross sec-
tion and b is defined by fitting the slope of the free cross section, since it
does not change with density. For the density dependence we have enforced
a correction of the form f(ρB) = 1 + a0(ρB/ρ0) + a1(ρB/ρ0)
2 + a2(ρB/ρ0)
where a0 = −0.601, a1 = 0223, a2 = −0.0035, with ρ0 saturation density,
are extracted from the results of ref.[18]. The same modification is imposed
on the cross sections of all the inelastic channels, in a form of the type
σeff = σfree(Elab)f(ρB), with σfree taken from the standard free parametriza-
tions of Ref. [34]. Such a procedure is well appropriate at low energies but at
higher momenta can be less accurate. This, however, should not be a problem
at the reaction energies below the kaon production threshold considered in
this work.
Fig. 2 (right panel) shows the energy dependence of the inelastic NN cross
section at various densities as obtained from DBHF calculations [17] for sym-
metric nuclear matter. As in the case of elastic processes (see Fig. 1), the
inelastic one drops with increasing baryon density ρB mainly due to the Pauli
blocking of intermediate scattering states and the in-medium modification of
the effective Dirac mass [17]. There are also phenomenological studies [16,33]
which give similar medium effects on the inelastic cross sections, within the
limitation to isospin symmetric nuclear matter. More suitable results would
come from a DBHF approach to isospin asymmetric nuclear matter. Only re-
cently such studies have been started [35], however, limiting to low momenta
regions, below the threshold energy of inelastic channels.
3 Kaon Potentials
Before starting with the presentation of the results, it is important to analyse
the in-medium kaon potential, since it could be relevant when theoretical re-
sults will be compared with experiments. In fact it has been widely discussed
whether the kaon potential plays a crucial role in describing kaon production
and their dynamics [23,7,9]. Kaplan and Nelson [20] found that the explicit chi-
ral symmetry breaking is not so small forK mesons and this leads to significant
corrections to the free kaon mass at finite baryon density. There are different
models for the description of kaon properties in the nuclear medium. Here we
will briefly discuss two main approaches, one based on Chiral Perturbation
Theory (ChPT ) and a second on effective meson couplings (OBE/RMF ),
more consistent with the general frame of our covariant reaction dynamics.
The results are in good agreement and this is not surprising on the basis of
a simple physics argument. It is well established [7] that kaons (K0,+) feel a
weak repulsive potential in nuclear matter, of the order of 20 − 30 MeV at
normal density. This can be described as the net result of the cancellation of an
attractive scalar and a repulsive vector interaction terms. Such a mechanism
can be reproduced in the ChPT approach through the competition between
an attractive scalar Kaplan-Nelson term [20] and a repulsive vector Weinberg-
Tomozawa [36] term. The same effect can be obtained in an effective meson
field scheme just via a coupling to the attractive σ-scalar and to the repulsive
ω-vector fields.
In this paper antikaons K− and their strong attractive potential will be not
discussed, since for the higher threshold they have been not considered in the
energy range of interest here.
Finally, for studies aimed to the determination of the symmetry energy from
strangeness production one has to treat with particular care the isospin de-
pendence of the kaon mean field potential.
3.1 Chiral Perturbative Results
Starting from an effective chiral Lagrangian for the K mesons one obtains a
density and isospin dependence for the effective kaon (K0,+) masses [7]. In
isospin asymmetric matter we finally get
m∗K =
m2K −
ρs3 + VµV µ (upper sign, K
+), (3)
where ρs, ρs3 are total and isospin scalar densities, with mK = 494MeV the
free kaon mass, fπ = 93MeV the pion decay constant, and ΣKN the kaon-
nucleon sigma term (attractive scalar), here chosen as 450 MeV. The vector
potential is given by:
8f ∗2π
8f ∗2π
jµ3 (upper sign, K
+), (4)
with jµ, jµ3 baryon and isospin currents. The f
π is an in-medium reduced pion
decay constant. It is expected to scale with density in a way similar to the chiral
condensate [37]. This leads to a reduction around normal density f ∗2π ≃ 0.6f 2π .
Such a reduction is compensated in one-loop ChPT by other contributions
in the scalar attractive term so we will use f ∗π only for the vector potential,
with an enhanced repulsive effect [7]. The constant C has been fixed from the
Gell-Mann-Okubo mass formula (i.e. in free space) to a value of 33.5MeV
[22]. In Eqs. (3-4) upper signs hold for K+ and lower signs for K0. As can be
seen, the vector term, which dominates over the scalar one at high density, is
more repulsive for K0 than for K+. This leads to a higher (lower) K0 (K+)
kaon in-medium energy given by the dispersion relation
EK(k) = k0 =
k2 +m∗2K + V0 (5)
The density dependence, evaluated in the chiral approach, of the quantity
EK(k)k=0 = m
K + V0 for K
0,+, that directly influences the in-medium pro-
duction thresholds is shown by the upper curves in Fig. 3 (left panel). In
particular, it can be noted that that K0 and K+ in medium-energy differs by
≈ 5% at ρB = 2ρ0 (with EK0 > EK+), at a fixed isospin asymmetry around
0.2. Therefore, the inclusion of isovector terms favors K+ over K0 production,
with a consequent reduction of the K0/K+ strangeness ratio.
0 1 2 3
0 1 2 3
0 - E
Fig. 3. Density dependence (ρ0 is the saturation density) of in medium kaon energy
(left panel) in unit of the free kaon mass (mK = 0.494GeV ). Upper curves refer
to ChPT model calculations: the central line corresponds to symmetric matter, the
other two give the isospin effect (up K0, down K+). Bottom curves are obtained
in the OBE/RMF approach, the solid one is for symmetric matter. The isospin
splitting is given by the dashed (NLρ) and dotted (NLρδ) lines, again up K0, down
K+. Right panel: relative weight of the isospin splitting, see text. All the curves are
obtained considering an asymmetry parameter α = 0.2.
3.2 Relativistic Mean Field Results
Kaon potentials can be also derived within an effective meson field OBE ap-
proach, fully consistent with the RMF transport scheme used to simulate the
reaction dynamics, see Eq.(1). We will use a simple constituent quark-counting
prescription to relate the kaon-meson couplings to the nucleon-meson cou-
plings, i.e. just a factor 3 reduction. Following the chiral argument discussed
before, only for the scalar vector case we have further increased the kaon
coupling to gωK ≃ 1.4/3gωN . This will ensure the required repulsion around
normal densities for K+s. Consistently the isospin dependence will be directly
derived from the coupling between the kaon fields and the ρ and δ isovector
mesons [22].
The in-medium energy carried by kaons will have the same form as in Eq.(5)
but with effective masses and vector potentials given by
m∗K =
m2K −mK(gσKσ ±
gδKgδN
m2K −mK(gσKσ ±
fδρs3) (6)
gωKgωN
gρKgρN
∗ρB ± fρρB3) (7)
where upper signs are forK+s. The fi ≡ g2iN/m2i , i = σ, ω, ρ, δ are the nucleon-
meson coupling constants used in our RMF Lagrangians and f ∗ω = 1.4fω due
to the enhanced kaon-scalar/vector coupling. σ represents the solution of the
non linear equation for the scalar/isoscalar field which gives the reduction of
the nucleon mass in symmetric matter, therefore we can directly evaluate the
kaon-σ coupling using
gσKσ =
(M −M∗)
where M∗ is the nucleon effective mass at the fixed baryon density.
In this RMF approach we can derive an almost analytical expression for
the isospin effects on the kaon in-medium energy Eq.(5) at k = 0. Using
the approximate form ρs ≃ M∗/E∗FρB for the scalar density, we get a rel-
ative weight of the isospin splitting of the kaon potentials ∆EK(k)k=0 ≡
EK0(k)k=0 −EK+(k)k=0 given by
2α(fρ − M
f ∗ω +
(mK − 16(M −M∗))
with α ≡ ρB3/ρB the asymmetry parameter.
We can now easily estimate the isospin splitting of K0 vs. K+ for the two
isovector mean field Lagrangians used here, NLρ and NLρδ. The effect will
be clearly larger when the δ coupling is included since we have to increase
the ρ-coupling fρ, see [14,15], but still the expected weight is relatively small,
going from about 1.5% (NLρ) to about 3.0% (NLρδ) at ρB = 2ρ0, for a fixed
isospin asymmetry around 0.2. The complete results are also shown in Fig. 3
(right panel). The agreement with the ChPT estimations is rather good, but
in the RMF scheme we see an overall reduced repulsion and a smaller isospin
splitting. Both effects are of interest for our discussion, the first affecting the
K0,+ absolute yields, the second important for the K0/K+ yield ratios.
4 Numerical realization and notations
The Vlasov term of the RBUU equation (1) is treated within the Relativistic
Landau-Vlasov method, in which the phase space distribution function f(x, p∗)
is represented by covariant Gaussians in coordinate and momentum space [38].
For the nuclear mean field or the corresponding EoS in symmetric matter the
fσ (fm
2) fω (fm
2) fρ (fm
2) fδ (fm
2) A (fm−1) B
NLρ 9.3 3.6 1.22 0.0 0.015 -0.004
NLρδ 9.3 3.6 3.4 2.4 0.015 -0.004
Table 1
Coupling parameters in terms of fi ≡ ( gimi )
2 for i = σ, ω, ρ, δ, A ≡ a
and B ≡ b
for the non-linear NL models [14] using the ρ (NLρ) and both, the ρ and δ mesons
(NLρδ) for the description of the isovector mean field.
NL2 parametrization [26] of the non-linear Walecka model [39] is adopted with
a compression modulus of 200 MeV and a Dirac effective mass of m∗ = 0.82 M
(M is the bare nucleon mass) at saturation. The momentum dependence enters
via the relativistic treatment in terms of the vector component of the baryon
self energy. The isovector components in the mean fields are introduced in the
NLρ,NLρδ Lagrangians as in the recent Refs. [14,15]. In Table 1 we report
all the coupling constants and the coefficients of the non-linear σ-terms.
The collision integral is treated within the standard parallel ensemble algo-
rithm imposing energy-momentum conservation. For the elastic NN cross sec-
tions the DBHF calculations of Ref. [17] have been used throughout this work.
At intermediate relativistic energies up to the threshold of kaon (K0,+) pro-
duction, i.e. Elab = 1.56 GeV, the major inelastic channels are (B, Y,K stand
for a baryon (nucleons N or a ∆-resonance), hyperon and kaon, respectively)
– NN ←→ N∆ (∆-production and absorption)
– ∆←→ πN (π-production and absorption)
– BB −→ BYK, Bπ −→ Y K (K-production from BB and Bπ-channels)
The produced resonances propagate in the same mean field as the nucleons,
and their decay is characterized by the energy dependent lifetime Γ which is
taken from Ref. [34]. The produced pions propagate under the influence of the
Coulomb interaction with the charged hadrons. Kaon production is treated
hereby perturbatively due to the low cross sections, taken from Refs. [40].
Kaons undergo elastic scattering and their phase space trajectories are deter-
mined by relativistic equations of motion, if the kaon potential is accounted
In the next section the results of transport calculations in terms of pion and
kaon yields and their rapidity distributions will be presented. The following
cases for the inelastic NN cross sections σinel and the kaon potential ΣK (scalar
and vector) will be particularly discussed:
– free σinel, without ΣK (w/o K-pot σfree)
– free σinel, with ΣK (w K-pot σfree)
– free σinel, with isospin dependent ΣK (w ID K-pot σfree)
– effective σinel, without ΣK (w/o K-pot σeff )
– effective σinel, with ΣK (w K-pot σeff )
– effective σinel, with isospin dependent ΣK (w ID K-pot σeff )
For pions only the different cases of σinel will be labelled, since they do not ex-
perience any potential, apart coulomb. One should note that in all calculations
only inelastic processes including the lowest mass resonance ∆(1232MeV )
have been considered, without accounting for the N∗(1440) resonance. This
will have not appreciable consequences for pions yields, but it slightly reduces
the kaon multiplicities.
5 Results
As mentioned in the introduction, the main topic of the present work is to
study the sensitivity of particle ratios to physical parameters such as in-
medium effects of cross sections and the isospin dependence of the kaon po-
tential. This is an important issue to clarify since there is some evidence sug-
gesting the yield ratios as good observables in determining the high density
behavior of the symmetry energy. In a near future these data will be exper-
imentally accessible with the help of reactions with radioactive ion beams.
However, a comparison of absolute values with experimental data, although it
is not the aim of this work, is essential and it has to be included in order to
show the consistency of our approach. Thus we will start the presentation of
the results first in terms of absolute yields, and comparison with data, before
passing to the main section on the particle ratios. Most calculations refer to
central 197Au+197 Au collisions at 1 AGeV .
5.1 Effects of in-medium inelastic NN cross sections on particle yields
5.1.1 Resonance and Pion Production
Here we study the role of the density dependence of the effective inelastic NN
cross sections on particle yields (pions and kaons). We start with the temporal
evolution of the ∆ resonances and the produced pions, as shown in Fig. 4.
The maximum of the multiplicity of produced ∆-resonances occurs around 15
fm/c which corresponds to the time of maximum compression. Due to their
finite lifetimes these resonances decay into pions (and nucleons) as ∆ −→
πN . Some of these pions are re-absorbed in the inverse process, i.e. πN −→
∆ but chemical equilibrium is never reached, as pointed out in [15]. This
0 10 20 30 40 50 60
time (fm/c)
0 10 20 30 40 50 60
time (fm/c)
Fig. 4. Time evolution of the ∆-resonances (left panel) and total pion yield (right
panel) for a central (b = 0 fm) Au+Au reaction at 1 AGeV incident energy. Cal-
culations with free (solid lines) and effective (DBHF, dashed lines) σinel are shown.
mechanism continues until all resonances have decayed leading to a saturation
of the pion yield for times t ≥ 50 fm/c (the so-called freeze-out time). The
resonance production takes place during the high density phase, where the
in-medium effects of the effective cross sections are expected to dominate.
In fact, the transport results with the in-medium modified σinel reduce the
multiplicity of inelastic processes, and thus the yields of ∆ resonances and
pions. However, the in-medium effect is not so pronounced here with respect
to similar phenomenological studies of Ref. [16,33], which should come from
the moderate density dependence of the effective cross sections, see also again
Fig. 2.
Fig. 5 shows the centrality dependence of the charged pion yields for Au+Au
collisions at 1.0 AGeV incident energy. The degree of centrality is characterized
by the observable Apart, which gives the number of participant nucleons and
can be calculated within a geometrical picture using smooth density profiles for
the nucleus [41]. Obviously Apart increases with decreasing impact parameter
b and its value approaches the total mass number of the two colliding nuclei
in the limiting case of b = 0 fm. As can be seen in Fig. 5, the charged pion
yields are enhanced with increasing Apart, particularly in a non-linear Apart-
dependence. As pointed out in [41], the charged pion multiplicities show a
similar non-linear increase also in the data. However, by directly comparing
the theoretical charged pion yields with the experiments [41] we observe that
our calculations overpredict the data, even when the in-medium reductions in
σinel are accounted for.
This discrepancy is a general feature of the transport models and may lie on
the role of the rescattering processes that take place in the spectator region,
0.0 100.0 200.0 300.0 400.0
 0  0
 5  5
 10  10
 15  15
 20  20
 (FOPI)
 (FOPI)
E=1.0 AGeV
Au+Au
Fig. 5. Centrality dependence (in terms of Apart) of the negative (π
−) and positive
(π+) charged pions for Au+Au collisions at 1 AGeV incident energy. Calculations
with free (solid lines, filled circles) and effective (dashed lines, filled squares) cross
sections are shown as indicated. Experimental data, taken from FOPI collaboration
[41], are also displayed for comparison.
where nuclear surface effects can play a crucial role. In order to check this
point we have performed a selection on pions produced at central rapidity,
where data are also available [41].
In Fig. 6 we present the inclusive (all centralities) pion rapidity distributions
vs. the FOPI data for charged pions. We see that the agreement is rather good
at mid-rapidity while we see a definite overcounting in the spectator sources.
Such a good evaluation of the pion production ad mid-rapidity is confirmed
by the results shown in Fig. 7, where we present the inclusive (all centralities)
pion transverse spectrum at midrapidity (−0.2 < y0 < 0.2). We first note that
this is also not much affected by the inclusion of the in-medium inelastic cross
sections. Moreover we see again that our results are in good agreement with
the experimental values from the FOPI collaboration [41], in the same rapidity
selection. The overestimation of the pion yields shown in Fig. 5 probably
results from other rapidity regions where the role of the spectator sources is
more evident. We have also to say that we are not imposing any experimental
filter to our results. The point is rather delicate since the main discrepancies
appear in high rapidity regions. In any case such a fine agreement at mid-
-2 -1 0 1 2
-2 -1 0 1 2
Fig. 6. Inclusive (all centralities) pion rapidity distributions for a Au+Au reaction
at Ebeam = 1 AGeV incident energy. Comparison with the experimental values given
by FOPI collaboration [41]; as in the data we have used a transverse momentum
cut to pt > 0.1GeV/c.
0 200 400 600 800
 (MeV)
 FOPI
 FOPIπ
0.1*π
Fig. 7. Inclusive transverse spectrum at midrapidity of π−, π+ for a Au+Au reaction
at Ebeam = 1 AGeV incident energy. Comparison with the experimental values given
by FOPI collaboration [41]. The cross sections are normalized to a rapidity interval
dy = 1.
rapidity is very important for the reliability of our results on kaon production,
mostly produced in that rapidity range via secondary πN,∆N channels, see
[15].
The pion reaction dynamics is furthermore not sensitively affected by the
in-medium inelastic cross sections. We restrict here the analysis to central
Au+Au collisions at 1 AGeV. In Fig. 8 we show cross section effects on the
-2 -1 0 1 2
-2 -1 0 1 2
Fig. 8. Rapidity distributions of negative and positive charged pions (left and right
panels, respectively) for a central (b = 0 fm) Au+Au reaction at Ebeam = 1 AGeV
incident energy.
rapidity distributions (normalized to the projectile rapidity in the cm sys-
tem) for π±, an observable which characterizes the degree of stopping or the
transparency of the colliding system. This is due to the fact that the global
dynamics is mainly governed by the total NN cross sections, in which its elas-
tic contribution is the same for all the cases. In previous studies [19,31] the
in-medium effects of the elastic NN cross sections gave important contribu-
tions to the degree of transparency or stopping. It was found that a reduction
of the effective NN cross section particularly at high densities is essential in
describing the experimental data [19], as confirmed by various other analy-
ses [31]. The density effects on the inelastic NN cross section influence only
those nucleons associated with resonance production, and therefore they do
not affect the global baryon dynamics significantly.
5.1.2 Kaon Production
The situation is different for kaon production, see Fig. 9. The influence of the
in-medium dependence of σinel is important, and reduces the kaon abundancies
by a factor of ≈ 30%. This is due to the fact that the leading channels for
kaon production are N∆ −→ BY K and Nπ −→ ΛK. Thus kaon production is
essentially a twostep process and the medium-modified inelastic cross sections
enter twice, leading to an increased sensitivity.
Fig. 10 shows the rapidity distributions of kaons, where the in-medium effect
is more visible with respect to the corresponding pion rapidity distributions
(see Fig. 8). These results seem to show that kaon production could be used
to determine the in-medium dependence of the NN cross section for inelas-
0 10 20 30 40 50 60
time (fm/c)
0 10 20 30 40 50 60
time (fm/c)
w/o K-pot σ
w/o K-pot σ
Fig. 9. Time evolution of the K0 (left panel) and K+ (right panel) multiplicities,
for the same reaction and models as in Fig. 4, with free and in-medium inelastic
cross sections, without the inclusion of the kaon potentials.
-2 -1 0 1 2
w/o K-pot σ
w/o K-pot σ
-2 -1 0 1 2
Fig. 10. Same as in Fig. 9, but for the normalized rapidity distributions.
tic processes. Similar phenomenological studies based on the BUU approach
[16,33] strongly support in-medium modifications of the free cross sections. It
is of great interest to perform an extensive comparison with experimental data
on kaon production, in order to have a more clear image of the effect of the
in-medium cross sections on their production. The point is that kaon absolute
yields are also largely affected by the kaon potentials, see the following, as
expected from the general discussion of the previous section. However since
kaons are mainly produced in more uniform high density regions the effects
of the medium on cross sections tend to disappear in the yield ratios. In the
next section we will show that the same holds true for the K0, K+ potentials.
Our conclusion is that the kaon yield ratios might finally be a rather robust
0 10 20 30 40 50 60
time (fm/c)
0 10 20 30 40 50 60
time (fm/c)
w/o K-pot
w K-pot
w ID K-pot
Fig. 11. Time evolution of the K0 (left panel) and K+ (right panel) multiplicities
for the same reaction as in Fig. 4. Calculations without (w/o K-pot, solid), with (w
K-pot, dashed) and with the isospin dependent (w ID K-pot,dotted-dashed) kaon
potential are shown. In all the cases the free choice for σinel is adopted.
observable to probe the nuclear EoS at high baryon densities.
5.2 The role of the kaon potential
As discussed in the previous sections, the important quantity which influences
the kaon production threshold is the in-medium energy at zero momentum [7].
This quantity rises with increasing baryon density and in the general case of
isospin asymmetric matter shows a splitting between K0 and K+, see Fig. 3.
We are presenting here several K-production results in ab initio collision sim-
ulations using the Chiral determination of the K-potentials, ChPT . Fig. 11
shows the time dependence of the two isospin states of the kaon with respect
to the role of the kaon potential and its isospin dependence. First of all, the
repulsive kaon potential considerably reduces the kaon yields, at least in this
ChPT evaluation.
The inclusion of the isospin dependent part of the kaon potential slightly
modifies the kaon yields, towards a larger K+ production in neutron-excess
matter. However by comparing to the corresponding isospin dependence of
the in-medium kaon energy, see Fig. 3, the effect is less pronounced in the
dynamical situation. This is due to the fact that in heavy ion collisions the local
asymmetry in the interacting region varies with time, see [15]. In particular,
it decreases with respect to the initial asymmetry because of partial isospin
equilibration due to stopping and inelastic processes with associated isospin
exchange. This is reflected also in the kaon rapidity distributions, see Fig. 12,
-2 -1 0 1 2
-2 -1 0 1 2
w/o K-pot
w K-pot
w ID K-pot
Fig. 12. Same as in Fig. 11, but for the rapidity distributions.
-1 0 1 2
normalized rapidity y
-1 0 1 2
normalized rapidity y
-1 0 1 2
normalized rapidity y
w/o K-pot w K-pot w ID K-pot
Fig. 13. K+ rapidity distributions for semi-central(b < 4 fm) Ni+Ni reactions at
1.93 AGeV. Theoretical calculations (as indicated) are compared with the exper-
imental data of FOPI (open triangles) and KaoS (open diamonds) collaborations
[42,43].
where the role of the kaon potential is crucial, but not its isospin dependence.
As we have already seen even in-medium modifications of inelastic cross sec-
tions are affecting the kaon absolute yields, so it appears of interest to look at
the combined effects. For that purpose we have performed calculations for a
semi-central (b < 4fm) Ni+Ni system at 1.93AGeV , where data are existing
from the FOPI [42] and KaoS [43] collaborations. The results for K+ rapidity
distributions, compared to experimental data, are shown in Fig.13.
We observe that although the kaon yields are reduced when using the in-
medium inelastic cross section, we are still rather far away from the data, left
panel of Fig. 13. We note that the reduction due to the density dependence
of the effective inelastic cross sections is rather moderate here with respect
to that of the heavier Au-system (see Fig. 10). for kaons). This is due to the
less compression achieved for the lighter Ni-systems. The inclusion of the kaon
potential, without (central panel) and with (right panel) isospin dependence,
is further suppressing the K+ yield, towards a better agreement with data, as
expected for the repulsive behavior at high density.
In fact the results obtained with kaon potentials and effective cross sections
seem to underestimate the data. This could be an indication that the ChPT
K-potentials are too repulsive at densities around 2ρ0 where kaons are pro-
duced, see [15]. We like to remind that the parameters of ChPT potentials
are essentially derived from free space considerations. When we follow a more
consistent RMF approach, directly linked to the effective Lagrangians used to
describe bulk properties of the nuclear matter as well as the relativistic trans-
port dynamics, we see less repulsion, bottom curves in Fig. 3 (left panel). This
is valid also for the isospin dependent part of the K-potentials, that more di-
rectly will affect the K0/K+ yield ratio. We see from the same Fig. 3 (right
panel) that in the RMF frame this splitting is reduced to a few percent for
all the different isovector interactions. The conclusion is that when kaon po-
tentials are evaluated within a consistent effective field approach we have a
better agreement with data for absolute yields, with a very similar reduction
of the K0 and K+ rates. This is important for the yield ratio, that then should
be not much sensitive to the in-medium effects on kaon propagation.
A similar conclusion on K-potential effects, obtained within the ChPT ap-
proach, can be drawn from the centrality dependence of the K+ yields shown
in Fig. 14 in the case of Au+Au collisions at 1 AGeV beam energy and com-
pared with KaoS data [44]. The trend in centrality can be reproduced by all
theoretical calculations (with different cross sections), however, all of them
seem to underestimate the experimental yields.
In fact we have to mention that another possible source of the discrepancy
with data can be that in all our simulations only the lowest mass resonance
∆(1232MeV ) has been dynamically included. Transport calculations from
other groups, that take care also of the N∗(1440MeV ) resonance, are getting
an enhancement of the K+ yield for Au+Au collisions at 1 AGeV incident
energy [7]. This significant dependence of the kaon yields on the N∗ resonance
comes from the 2-pionic N∗-decay channel, i.e. N∗ −→ ππN . Therefore, since
the most important channels of kaon production are the pionic ones, we can
expect some underestimation of the absolute yields in our calculations. Just
to confirm this point, in Fig. 14 we report also transport results from the
Tübingen group, in which all resonances are accounted for [7]. We finally re-
0,0 0,2 0,4 0,6 0,8
 5,0×10
 1,0×10
 1,5×10
 2,0×10
0,0 0,2 0,4 0,6 0,8 1,0
w K-pot w ID K-pot
Fig. 14. K+ centrality dependence in Au+Au reactions at 1 AGeV incident energy.
Our theoretical calculations (as indicated) are compared with KaoS data from [44]
(open diamonds) and with results of the Tübingen group (open squares).
mark that the inclusion of other nucleon resonances in neutron-rich matter
will further contribute to increase the K0 yield through a larger intermedi-
ate π− production. This can contribute to compensate the opposite effect of
isospin dependent part of the K-potentials on the K0/K+ yield ratios.
5.3 Pionic and Strangeness Ratios
A crucial question is whether particle yield ratios are influenced by in-medium
effects both on inelastic cross section and kaon potentials. This point is of ma-
jor importance particularly for kaons, since ratios of particles with strangeness
have been widely used in determining the nuclear EoS at supra-normal density.
Relative ratios of kaons between different colliding systems have been utilized
in determining the isoscalar sector of the nuclear EoS [5]. More recently, the
(π−/π+)- and (K0/K+)-ratios have been proposed in order to explore the high
density behavior of the symmetry energy, i.e. the isovector part of the nuclear
mean field [14,15,13,12].
Fig. 15 shows the incident energy dependence of the pionic (π−/π+, left panel)
and strangeness (K0/K+, right panel) ratios for the different choices of in-
elastic cross sections and kaon potentials, as widely discussed in the previous
sections.
First of all, a rapid decrease of the pionic ratio with increasing beam energy
0.8 1 1.2 1.4 1.6 1.8 2
 (AGeV)
0.8 1 1.2 1.4 1.6 1.8 2
 (AGeV)
free w/o K-pot
eff w/o K-pot
0.8 1 1.2 1.4 1.6 1.8 2
 (AGeV)
free w K-pot
eff w K-pot
0.8 1 1.2 1.4 1.6 1.8 2
 (AGeV)
tio σ
free w ID K-pot
eff w ID K-pot
Fig. 15. Energy dependence of the π−/π+ (left panel) and K0/K+ (right panel)
ratios for central (b = 0 fm) Au+Au reactions.
is observed, related to the opening of secondary rescattering processes (reab-
sorption/recreation of pions with associated isospin exchange) channels. The
corresponding strangeness ratio depends only moderately on beam energy due
to the absence of secondary interactions with the hadronic environment.
The pionic ratio is partially affected by the in-medium effects of σinel, as it can
be seen in the left panel of Fig.15. Its slope is slightly changing with respect to
beam energy. The situation is similar for the strangeness ratio, which actually
appears even more robust vs. in-medium modifications, even with the kaon
potentials. This can be seen in the right panel of Fig.15, where for all the
considered beam energies the ratio remains almost unchanged. Such a result
is consistent with those of the previous sections, where it was found that the
absolute kaon yields decrease in the same way when the effective σinel are
applied and when the K-potentials are included.
The different sensitivity to variations in the inelastic cross sections of pionic
vs. strangeness ratios can be easily understood. For the large rescattering and
lower masses pions can be produced at different times during the collision,
in different density regions. At variance kaons are mainly produced at early
times in a rather well definite compression stage, i.e. in a source with a more
uniform high density, and so the density dependence of the inelastic cross
sections will affect in the same way neutral and charged kaon yields, leaving
the ratio unchanged. At this level of investigation one could argue that the
strangeness ratio is a very promising observable in determining the nuclear
EoS and particularly its isospin dependent part. This has been also the main
conclusion in Ref. [15].
However, a strong isospin dependence of the kaon potentials could directly
NL NLρ NLρδ
K0/K+ (w/o K-pot) 1.24 (± 0.02) 1.35 (± 0.01) 1.43 (± 0.02)
K0/K+ (w ID K-pot) 1.02 (± 0.03) 1.22 (± 0.04) 1.34 (± 0.05)
Table 2
Sensitivity of the strangeness ratio K0/K+ to the isospin dependent kaon potential
and to the isovector mean field (NL, no isovector fields, NLρ and NLρδ). The
considered reaction is a central (b = 0 fm) Au+Au collision at 1 AGeV incident
energy.
affect the ratio, since the K0 and K+ rates will be modified in opposite ways.
This is shown by the two triangle points at 1 AGeV in the right panel of Fig.
15. As already discussed this large isospin dependence of the kaon potential,
clearly present in the ChPT evaluation, is greatly reduced in a consistent
mean field approach, see Fig. 3 and the arguments presented in Section 3.
In any case this point deserves more detailed studies. We plan to perform
ab initio kaon-production simulations within the OBE/RMF evaluation of
kaon potentials, with an isospin part fully consistent with the isovector fields
of the Hadronic Lagrangians used for the reaction dynamics.
An interesting final comment is that the sensitivity of the strangeness ratio
to the isovector part of the nuclear EoS remains even when strong isospin
dependence of the kaon potentials is inserted, as in the ChPT case.
In order to check this, we have repeated for Au+Au at 1 AGeV incident energy
the calculations by varying the isovector part of the nuclear mean field. As
in Refs. [14,15], three options for the isovector mean field have been applied:
the NL (no isovector fields), NLρ and NLρδ parametrizations, but now in-
cluding the isospin effect in the kaon potential in the ChPT evaluation. The
options of the symmetry energy differ from each other in the high density stiff-
ness. NL gives a relatively soft Esym, NLρδ a relatively stiff one, and NLρ
lies in the middle between the other limiting cases [14]. Table 2 shows the
strangeness ratio as function of these different cases for the isovector mean
field, keeping now constant the other parameters (free σinel, isospin dependent
kaon potential). The ratio, indeed, strongly decreases when the isospin part in
the kaon potential is accounted for. The more interesting result is, however,
that the relative difference between the different choices of the symmetry en-
ergy remains stable. This can be understood from the fact that in the kaon
self energies the isospin sector contains only the isospin densities and currents
without additional parameters such meson-nucleon coupling constants. Since
the local asymmetry does not strongly vary from one case to the other (NL,
NLρ, NLρδ), one would expect a robustness of the EoS dependence. Thus
we conclude that the strangeness ratio appears to be well suited in determin-
ing the isovector EoS, however, a fully consistent mean field approach is still
missing.
6 Conclusions
We have investigated the role of the in-medium modifications of the inelastic
cross section and of the kaon mean field potentials on particle production in
intermediate energy heavy ion collisions within a covariant transport equation
of a Boltzmann type. We have used for both, the elastic and inelastic NN cross
sections the same DBHF approach which provide in a parameter free manner
the in-medium modifications of the imaginary part of the self energy in nu-
clear matter. The kaon potential has been evaluated in two ways, following a
Chiral Perturbative approach and an Effective Field scheme, considering va-
lence quark-meson couplings. We have applied these modifications of the cross
sections and kaon potentials to the collision integral of the transport equation
and analyzed Au+Au and Ni+Ni collisions at intermediate relativistic energies
around the kaon threshold energy.
Our studies have shown a good sensitivity of the particle multiplicities and
rapidity distributions of pions and kaons. In particular, a moderate reduction
for pions has been seen when the in-medium effects in the inelastic cross section
are accounted for. The pion yields are still overestimating the inclusive data
while we have a very nice agreement with the pion spectra and multiplicities at
mid-rapidity. The latter point is important for trusting the kaon production,
mainly due to secondary pion collisions at mid-rapidity.
At variance the kaon (K0,+) yields show a larger sensitivity to the reduction
of the inelastic cross sections, with a decrease of about 30 %. However we
see that the introduction of a repulsive kaon potential is essential in order to
reproduce even inclusive data.
We have then focused our attention on π−/π+ and K0/K+ yield ratios, re-
cently suggested as good probes of the isovector part of the EoS at high den-
sities. The pionic ratios, due to their strong secondary interaction processes
with the hadronic environment, show a dependence on the density behavior
of the inelastic cross sections. A further selection of the production source, i.e.
a transverse momentum discrimination, could be required in order to have a
more reliable probe of the nuclear EoS.
The situation appears more favorable for the kaon ratios. In fact we find that
the multiplicities of K0 and K+ are influenced in such a way that their ratio
is not affected by the density dependence of the inelastic cross sections. This
is due to the long mean free path of the K0,+ that are produced only in the
compression stage of the collision [15]. The effects of the in medium kaon
potentials are also largely compensating in the K0/K+ yield ratio, due to
the similar repulsive field seen by K0 and K+ mesons. Such a result can be
modified by the isospin dependence of the kaon potentials which is expected to
act in opposite directions for neutral and charged kaons rates. Actually this is a
rather stimulating open problem. In our analysis with two completely different
approaches, ChPT vs. RMF , we get a good agreement for the isoscalar kaon
potential but a rather different prediction for the isovector part.
However, a study in terms of the different choices of the isovector kaon field
has shown that the relative dependence of the strangeness ratios on the stiff-
ness of the isovector nuclear EoS remains a well robust observable. This is
an important issue in determining the high density behavior of the symmetry
energy in more systematic analyses in the future, when more experimental
data will be available.
Acknowledgments. This work is supported by BMBF, grant 06LM189 and
the State Scholarships Foundation (I.K.Y.). It is also co-funded by European
Union Social Fund and National funded Pythagoras II - EPEAEK II, under
project 80861. One of the authors (V.P.) would like to thank H.H. Wolter and
M. Di Toro for the warm hospitality during her short stays at their institutes.
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  The effect of possible in-medium modifications of nucleon-nucleon ($NN$)
cross sections on particle production is investigated in heavy ion collisions
($HIC$) at intermediate energies. In particular, using a fully covariant
relativistic transport approach, we see that the density dependence of the {\it
inelastic} cross sections appreciably affects the pion and kaon yields and
their rapidity distributions. However, the $(\pi^{-}/\pi^{+})$- and
$(K^{0}/K^{+})$-ratios depend only moderately on the in-medium behavior of the
inelastic cross sections. This is particularly true for kaon yield ratios,
since kaons are more uniformly produced in high density regions. Kaon
potentials are also suitably evaluated in two schemes, a chiral perturbative
approach and an effective meson-quark coupling method, with consistent results
showing a similar repulsive contribution for $K^{+}$ and $K^{0}$. As a
consequence we expect rather reduced effects on the yield ratios. We conclude
that particle ratios appear to be robust observables for probing the nuclear
equation of state ($EoS$) at high baryon density and, particularly, its
isovector sector.

Introduction
The knowledge of the properties of highly compressed and heated hadronic
matter is an important issue for the understanding of astrophysical processes,
such as the mechanism of supernovae explosions and the physics of neutron
stars [1,2]. Heavy ion collisions provide the unique opportunity to explore
highly excited hadronic matter, i.e. the high density behavior of the nuclear
EoS, under controlled conditions (high baryon energy densities and tempera-
tures) in the laboratory [3]. Of particular recent interest is also the still poorly
known density dependence of the isovector channel of the EoS.
Suggested observables have been the nucleon collective flows [3,4] and the
distributions of produced particles such as pions and, in particular, particles
with strangeness (kaons) [5,6]. Because of the rather high energy threshold
(Elab = 1.56 GeV for Nucleon-Nucleon collisions), kaon production in HICs
at energies in the range 0.8− 1.8 AGeV is mainly due to secondary processes
involving ∆ resonances and pions (π). On the other hand, secondary processes
require high baryon density. This explains why the kaon production around
threshold is intimately connected to the high density stage of the nucleus-
nucleus collision. Furthermore, the relatively large mean free path of positive
charged (K+) and neutral (K0) kaons inside the hadronic environment causes
hadronic matter to be transparent for kaons [7]. Therefore kaon yields and
generally strangeness ratios have been proposed as important signals for the
investigation of the high density behavior of the nuclear EoS. This idea, as
firstly suggested by Aichelin and Ko [8], has been recently applied in HIC
at intermediate energies in terms of strangeness ratios, e.g. the ratio of the
kaon yields in Au+Au and C+C collisions [5,9]. In these studies it was found
that this ratio is very sensitive to the stiffness of the nuclear EoS. Indeed
comparisons with KaoS data [10] favored a soft behavior of the high density
nuclear EoS, a statement which is particularly consistent with elliptic flow
data of the FOPI collaboration [11].
The idea of studying particle ratios in HICs around the kinematical threshold
has been recently applied in the determination of the isovector channel of the
nuclear EoS, i.e. the high density dependence of the symmetry energy Esym. It
has turned out that particle ratios, such as (π−/π+) [12] or (K0/K+) [13–15],
are sensitive to the stiffness of the symmetry energy and, in particular to the
strength of the vector isovector field. However in medium effects on the kaon
propagation have been neglected so far. Here we will test the robustness of the
yield ratio against the inclusion and the variation of the corresponding kaon
potentials. At the same time in Ref. [16] the role of the in-medium modifica-
tions of NN cross sections has been studied in terms of baryon and strangeness
dynamics. It was found that the pion and kaon yields are sensitively influenced
by the reduced effective NN cross sections for inelastic processes. Here we will
see that the kaon yield ratio appears robust even with respect to the density
dependence of the in-medium inelastic NN cross sections, while at variance
the pion ratio seems to be more sensitive.
The collision dynamics is rather complex and involves the nuclear mean field
(EoS) and binary 2-body collisions. In the presence of a nuclear medium the
treatment of binary collisions represents a non-trivial problem. The NN cross
sections for elastic and inelastic processes, which are the crucial physical pa-
rameters here, are experimentally accessible only in free space and not for
2-body scattering at finite baryon density. Recent microscopic studies, based
on the G-matrix approach, have shown a strong decrease of the elastic NN
cross section [17,18] in a hadronic medium. These in-medium effects of the
elastic NN cross section considerably influence the hadronic reaction dynam-
ics [19]. Obviously the question arises whether similar in-medium effects of the
inelastic NN cross sections may affect the reaction dynamics and, in particular,
the production of particles (pions and kaons).
Furthermore, the strangeness propagation inside the nuclear medium is even
more complex and involves the additional consideration of kaon mean field
potentials in the dynamical description. This is an important issue when com-
paring with experimental kaon data [10]. In a Chiral Perturbation approach
at the lowest order (ChPT Potentials), the kaon (antikaon) potential has an
attractive scalar and a repulsive (attractive) vector part [20]. This leads to
weakly repulsive (strongly attractive) potentials for kaons (antikaons) with
corresponding scalar and vector kaon-nucleon coupling constants depending
on the parametrization [20,21] accounted for. Similar results can be obtained
in an effective meson-coupling model (OBE Potentials, in the RMF spirit),
where the K-meson couplings are simply related to the nucleon-meson ones,
in the spirit of ref. [22]. The latter approach has the advantage of being fully
consistent with the covariant transport equations used to simulate the reaction
dynamics [14,15]. We remind that the high density dependence of the kaon
self energies is still an object of current debate, e.g. see Refs. [23,7] in which
the role of the kaon potential has been investigated in terms of kaon in-plane
and out-of-plane flows. Moreover for studies aimed to the determination of
the symmetry energy from strangeness production one has to consider with
particular care the isospin dependence of the kaon mean field potential.
The main focus of the present work is on a detailed study of the robustness of
the pionic (π−/π+) and, in particular, the strangeness ratio (K0/K+) with re-
spect to the in-medium modifications of the imaginary part of the nucleon self
energy, i.e. the NN cross sections, and to the in-medium variations of the kaon
self energy, i.e. the density dependence of the kaon potential. This analysis,
which goes beyond our previous investigations of [14,15], is also motivated by
new measurements of the FOPI collaboration [24] by means of the strangeness
ratios.
The paper is organized as follows: The next Section describes the theoret-
ical treatment of the reaction dynamics within the Relativistic Boltzmann-
Uheling-Uhlenbeck (RBUU) transport equation. A detailed discussion on the
in-medium modifications of the inelastic NN cross sections is presented. In Sec-
tion 3 we discuss the kaon mean field potentials (in both ChPT and OBE/RMF
schemes) and their expected isospin dependence. Section 4 is devoted to a short
introduction to the dynamical calculations. Results are then shown in Section
5, mostly for central 197Au+197 Au collisions at 1AGeV , in terms of pion and
kaon yields. The initial presentation of the absolute yields is relevant for a de-
tailed discussion as well as for a comparison with theoretical results of other
groups and with experimental data of the KaoS and FOPI collaborations.
All together this intermediate step is important for testing the reliability of
the calculations, since ratios do not do it. Finally we present the pion and
strangeness ratios and discuss their dependence on the in-medium modifica-
tions of the cross NN cross sections and of the kaon potentials, including the
isospin effects. In Section 6 we conclude with a summary and some general
comments and perspectives.
2 Theoretical description of the collision dynamics
In this chapter we briefly discuss the transport equation focusing on the treat-
ment of two features important for kaon dynamics: (a) the collision integral by
means of the cross sections; (b) the kaon mean field potential and its isospin
dependence.
2.1 The RBUU equation
The theoretical description of HICs is based on the semiclassical kinetic the-
ory of statistical mechanics, i.e. the Boltzmann Equation with the Uehling-
Uhlenbeck modification of the collision integral [25]. The relativistic analog of
this equation is the Relativistic Boltzmann-Uehling-Uhlenbeck (RBUU) equa-
tion [26]
k∗µ∂xµ + (k
µν +M∗∂µxM
∗) ∂k
f(x, k∗) =
2(2π)9
W (kk2|k3k4)
f3f4f̃ f̃2 − ff2f̃3f̃4
, (1)
where f(x, k∗) is the single particle distribution function. In the collision term
the short-hand notations fi ≡ f(x, k∗i ) for the particle and f̃i ≡ (1− f(x, k∗i ))
0 50 100 150 200 250 300 350
 [MeV]
=1.1 fm
=1.34 fm
=1.7 fm
pn-data
Fig. 1. Elastic in-medium neutron-proton cross section σel at various Fermi momenta
kF as a function of the laboratory energy Elab. The free cross section (kF = 0) is
compared to the experimental total np cross section [17].
for the hole distributions are used, with E∗
M∗2 + k2. The collision in-
tegral explicitly exhibits the final state Pauli-blocking while the in-medium
scattering amplitude includes the Pauli-blocking of intermediate states.
The dynamics of the drift term, i.e. the lhs of eq.(1), is determined by the
mean field. Here the attractive scalar field Σs enters via the effective mass
M∗ = M − Σs and the repulsive vector field Σµ via the kinetic momenta
k∗µ = kµ − Σµ and via the field tensor F µν = ∂µΣν − ∂νΣµ. The dynamical
description according to Eq.(1) involves the strangeness propagation in the
nuclear medium. This topic will be discussed in more detail at the end of this
section.
2.2 In-medium effects on NN cross sections
The in-medium cross sections for 2-body processes (see below) enter in the
collision integral via the transition amplitude
W = (2π)4δ4 (k + k2 − k3 − k4) (M∗)4|T |2 (2)
with T the in-medium scattering matrix element.
In the kinetic equation (1) both physical input quantities, the mean field (EoS)
and the collision integral (cross sections) should be derived from the same
underlying effective two-body interaction in the medium, i.e. the in-medium
T-matrix; Σ ∼ ℜTρB, σ ∼ ℑT , respectively. However, in most practical
applications phenomenological mean fields and cross sections have been used.
In such approach the strategy is to adjust to the known bulk properties of
nuclear matter around the saturation point, and to try to constrain the models
at supra-normal densities with the help of heavy ion reactions [27,28]. Medium
modifications of the NN cross sections are usually not taken into account.
In spite of that for several observables the comparison to experimental data
appears to work astonishingly well [27–30]. However, in particular kinematical
regimes a sensitivity to the elastic NN cross sections of dynamical observables,
such as collective flows and stopping [19,31] or transverse energy transfer [32],
has been observed.
Microscopic Dirac-Brueckner-Hartree-Fock (DBHF) studies for nuclear matter
above the Fermi energy regime show a strong density dependence of the elastic
[17] and inelastic [18,33] NN cross sections. In such studies one starts from the
bare NN-interaction in the spirit of the One-Boson-Exchange (OBE) model by
fitting the parameters to empirical nucleon-nucleus scattering and solves then
the equations of the nuclear matter many body problem in the T -matrix or
ladder approximation. It is not the aim of the present work to go into further
details on this topic. An important feature of such microscopic calculations is
the inclusion of the Pauli-blocking effect in the intermediate scattering states
of the T -matrix elements and their in-medium modifications, i.e. the density
dependence of the nucleon mass and momenta. Here of particular interest are
the in-medium modifications of the inelastic NN cross sections since they di-
rectly influence the production mechanism of resonances and thus the creation
of pions and kaons according to the channels listed later (see Sect.3). DBHF
studies on inelastic NN cross sections are rare and in limited regions of density
and momentum [18]. For this reason we will first discuss in the following the
in-medium dependence of the elastic NN cross sections, which will be then
used as a starting basis for a detailed analysis of the density dependence of
the inelastic NN cross sections.
The microscopic in-medium dependence of the elastic cross sections can be
seen in Fig. 1, where the energy dependence of the in-medium neutron-proton
(np) cross section at Fermi momenta kF = 0.0, 1.1, 1.34, 1.7fm
−1, correspond-
ing to ρB ∼ 0, 0.5, 1, 2ρ0 (ρ0 = 0.16fm−3 is the nuclear matter saturation den-
sity) is shown. These results are obtained from relativistic Dirac-Brueckner
calculations [17]. The presence of the medium leads to a substantial suppres-
sion of the cross section which is most pronounced at low laboratory energy
Elab and high densities where the Pauli-blocking of intermediate states is most
efficient. At larger Elab asymptotic values of 15-20 mb are reached. Also the
angular distributions are affected by the presence of the medium. E.g. the ini-
tially strongly forward-backward peaked np cross sections become much more
isotropic at finite densities, mainly due to the Pauli suppression of intermedi-
ate soft modes (π-exchange) [17]. As a consequence a larger transverse energy
transfer can be expected.
The case of the inelastic NN cross sections is similar, but more complicated.
The presence of the medium influences not only the matrix elements, but also
the threshold energy Etr, which is an important quantity at beam energies be-
low or near the threshold of particle production. In free space it is calculated
from the invariant quantity s = (p
1 + p
2 )(p1µ + p2µ) with p
i , (i = 1, 2) the
4-momenta of the two particles in the ingoing collision channel, e.g. NN −→
N∆. This quantity is conserved in binary collisions in free space, from which
one determines the modulus of the momenta of the particles in the outgoing
channel. The threshold condition reads Etr ≡
s ≥ M1 +M2. Cross sections
in free space are usually parametrized in terms of
s or the corresponding mo-
mentum in the laboratory system plab within the One-Boson-Exchange (OBE)
model, see e.g. [34] for details.
At finite density, however, particles carry kinetic momenta and effective masses
and obey a dispersion relation p∗µp
∗µ = m∗2 modified with respect to the free
case. These in-medium effects shift the threshold energy in the free space
according to s∗ = (p
2 )(p
2µ) and the threshold condition for inelastic
processes inside the medium reads now E∗
s∗ ≥ m∗
. The requirement
of energy-momentum conservation can be carried out in terms of the quantity
s∗ or s, only as long as the in-medium mean fields or the corresponding self
energies do not change between ingoing and outgoing channels.
The application of free parametrizations of cross sections for inelastic processes
in dynamical situations of HICs at finite density leads thus to an inconsistency,
since the threshold condition is performed in terms of effective quantities,
but the matrix elements are carried out in free space, e.g. by fitting their
parameters to free empirical NN scattering. This effect can be seen in Fig.
2 (left panel) where the free inelastic NN −→ N∆ cross section σinel as a
function of the laboratory energy Elab is displayed, at various baryon densities
ρB. The threshold energy in the free space is Etr =
s = 2.014 GeV (for
M = 0.939 GeV and Mmin = 1.076 for the nucleon and the lower limit mass of
the ∆ resonance). The corresponding threshold value of the laboratory energy
Elab = (E
− 4M2)/2M is 0.32 GeV. However, at finite density the threshold
is shifted towards lower energies, i.e. the free cross section increases, due to
the reduction of the free masses of the outgoing particles in the threshold
condition E∗
. Obviously at higher energies far from threshold the
free cross section does not depend on the density.
A more consistent approach is the determination of the inelastic cross section
under the consideration of in-medium effects, i.e. the Pauli-blocking of inter-
mediate scattering states and in-medium modified spinors in the determina-
tion of the matrix elements within the OBE model. A simultaneous treatment
of the transport equation and the structure equations of DBHF for actual
anisotropic momentum configurations is not possible, due to its high com-
plexity. For this reason we have applied the same method as for the case
0 0.5 1 1.5 2
 (GeV)
0 0.5 1 1.5 2
 (GeV)
=0.5ρ0
free effective
Fig. 2. Inelastic NN −→ N∆ cross section σinel at various baryon densities ρB (in
units of the saturation density ρ0 = 0.16 fm
−3) as a function of the laboratory
energy Elab using the free parametrizations (left) and the in-medium modified ones
from DBHF [18] (right).
of elastic binary processes, i.e. in-medium parametrizations of the inelastic
cross sections of the type NN −→ N∆ within the same underlying DBHF
approach as already used for the elastic processes. Haar and Malfliet [18]
investigated this topic for infinite nuclear matter with the result of a strong
in-medium modification of the inelastic cross sections due to the reasons given
above. However, these studies were performed at various densities but only in
a limited region of momenta. For a practical application in HICs we have
thus extended these DBHF calculations using an extrapolation technique. We
have imposed an exponential decay law of the form ae−bplab on the values of
the in-medium cross sections of the channel NN → N∆ given in ref. [18].
The parameter a normalizes to the last value of the extrapolated cross sec-
tion and b is defined by fitting the slope of the free cross section, since it
does not change with density. For the density dependence we have enforced
a correction of the form f(ρB) = 1 + a0(ρB/ρ0) + a1(ρB/ρ0)
2 + a2(ρB/ρ0)
where a0 = −0.601, a1 = 0223, a2 = −0.0035, with ρ0 saturation density,
are extracted from the results of ref.[18]. The same modification is imposed
on the cross sections of all the inelastic channels, in a form of the type
σeff = σfree(Elab)f(ρB), with σfree taken from the standard free parametriza-
tions of Ref. [34]. Such a procedure is well appropriate at low energies but at
higher momenta can be less accurate. This, however, should not be a problem
at the reaction energies below the kaon production threshold considered in
this work.
Fig. 2 (right panel) shows the energy dependence of the inelastic NN cross
section at various densities as obtained from DBHF calculations [17] for sym-
metric nuclear matter. As in the case of elastic processes (see Fig. 1), the
inelastic one drops with increasing baryon density ρB mainly due to the Pauli
blocking of intermediate scattering states and the in-medium modification of
the effective Dirac mass [17]. There are also phenomenological studies [16,33]
which give similar medium effects on the inelastic cross sections, within the
limitation to isospin symmetric nuclear matter. More suitable results would
come from a DBHF approach to isospin asymmetric nuclear matter. Only re-
cently such studies have been started [35], however, limiting to low momenta
regions, below the threshold energy of inelastic channels.
3 Kaon Potentials
Before starting with the presentation of the results, it is important to analyse
the in-medium kaon potential, since it could be relevant when theoretical re-
sults will be compared with experiments. In fact it has been widely discussed
whether the kaon potential plays a crucial role in describing kaon production
and their dynamics [23,7,9]. Kaplan and Nelson [20] found that the explicit chi-
ral symmetry breaking is not so small forK mesons and this leads to significant
corrections to the free kaon mass at finite baryon density. There are different
models for the description of kaon properties in the nuclear medium. Here we
will briefly discuss two main approaches, one based on Chiral Perturbation
Theory (ChPT ) and a second on effective meson couplings (OBE/RMF ),
more consistent with the general frame of our covariant reaction dynamics.
The results are in good agreement and this is not surprising on the basis of
a simple physics argument. It is well established [7] that kaons (K0,+) feel a
weak repulsive potential in nuclear matter, of the order of 20 − 30 MeV at
normal density. This can be described as the net result of the cancellation of an
attractive scalar and a repulsive vector interaction terms. Such a mechanism
can be reproduced in the ChPT approach through the competition between
an attractive scalar Kaplan-Nelson term [20] and a repulsive vector Weinberg-
Tomozawa [36] term. The same effect can be obtained in an effective meson
field scheme just via a coupling to the attractive σ-scalar and to the repulsive
ω-vector fields.
In this paper antikaons K− and their strong attractive potential will be not
discussed, since for the higher threshold they have been not considered in the
energy range of interest here.
Finally, for studies aimed to the determination of the symmetry energy from
strangeness production one has to treat with particular care the isospin de-
pendence of the kaon mean field potential.
3.1 Chiral Perturbative Results
Starting from an effective chiral Lagrangian for the K mesons one obtains a
density and isospin dependence for the effective kaon (K0,+) masses [7]. In
isospin asymmetric matter we finally get
m∗K =
m2K −
ρs3 + VµV µ (upper sign, K
+), (3)
where ρs, ρs3 are total and isospin scalar densities, with mK = 494MeV the
free kaon mass, fπ = 93MeV the pion decay constant, and ΣKN the kaon-
nucleon sigma term (attractive scalar), here chosen as 450 MeV. The vector
potential is given by:
8f ∗2π
8f ∗2π
jµ3 (upper sign, K
+), (4)
with jµ, jµ3 baryon and isospin currents. The f
π is an in-medium reduced pion
decay constant. It is expected to scale with density in a way similar to the chiral
condensate [37]. This leads to a reduction around normal density f ∗2π ≃ 0.6f 2π .
Such a reduction is compensated in one-loop ChPT by other contributions
in the scalar attractive term so we will use f ∗π only for the vector potential,
with an enhanced repulsive effect [7]. The constant C has been fixed from the
Gell-Mann-Okubo mass formula (i.e. in free space) to a value of 33.5MeV
[22]. In Eqs. (3-4) upper signs hold for K+ and lower signs for K0. As can be
seen, the vector term, which dominates over the scalar one at high density, is
more repulsive for K0 than for K+. This leads to a higher (lower) K0 (K+)
kaon in-medium energy given by the dispersion relation
EK(k) = k0 =
k2 +m∗2K + V0 (5)
The density dependence, evaluated in the chiral approach, of the quantity
EK(k)k=0 = m
K + V0 for K
0,+, that directly influences the in-medium pro-
duction thresholds is shown by the upper curves in Fig. 3 (left panel). In
particular, it can be noted that that K0 and K+ in medium-energy differs by
≈ 5% at ρB = 2ρ0 (with EK0 > EK+), at a fixed isospin asymmetry around
0.2. Therefore, the inclusion of isovector terms favors K+ over K0 production,
with a consequent reduction of the K0/K+ strangeness ratio.
0 1 2 3
0 1 2 3
0 - E
Fig. 3. Density dependence (ρ0 is the saturation density) of in medium kaon energy
(left panel) in unit of the free kaon mass (mK = 0.494GeV ). Upper curves refer
to ChPT model calculations: the central line corresponds to symmetric matter, the
other two give the isospin effect (up K0, down K+). Bottom curves are obtained
in the OBE/RMF approach, the solid one is for symmetric matter. The isospin
splitting is given by the dashed (NLρ) and dotted (NLρδ) lines, again up K0, down
K+. Right panel: relative weight of the isospin splitting, see text. All the curves are
obtained considering an asymmetry parameter α = 0.2.
3.2 Relativistic Mean Field Results
Kaon potentials can be also derived within an effective meson field OBE ap-
proach, fully consistent with the RMF transport scheme used to simulate the
reaction dynamics, see Eq.(1). We will use a simple constituent quark-counting
prescription to relate the kaon-meson couplings to the nucleon-meson cou-
plings, i.e. just a factor 3 reduction. Following the chiral argument discussed
before, only for the scalar vector case we have further increased the kaon
coupling to gωK ≃ 1.4/3gωN . This will ensure the required repulsion around
normal densities for K+s. Consistently the isospin dependence will be directly
derived from the coupling between the kaon fields and the ρ and δ isovector
mesons [22].
The in-medium energy carried by kaons will have the same form as in Eq.(5)
but with effective masses and vector potentials given by
m∗K =
m2K −mK(gσKσ ±
gδKgδN
m2K −mK(gσKσ ±
fδρs3) (6)
gωKgωN
gρKgρN
∗ρB ± fρρB3) (7)
where upper signs are forK+s. The fi ≡ g2iN/m2i , i = σ, ω, ρ, δ are the nucleon-
meson coupling constants used in our RMF Lagrangians and f ∗ω = 1.4fω due
to the enhanced kaon-scalar/vector coupling. σ represents the solution of the
non linear equation for the scalar/isoscalar field which gives the reduction of
the nucleon mass in symmetric matter, therefore we can directly evaluate the
kaon-σ coupling using
gσKσ =
(M −M∗)
where M∗ is the nucleon effective mass at the fixed baryon density.
In this RMF approach we can derive an almost analytical expression for
the isospin effects on the kaon in-medium energy Eq.(5) at k = 0. Using
the approximate form ρs ≃ M∗/E∗FρB for the scalar density, we get a rel-
ative weight of the isospin splitting of the kaon potentials ∆EK(k)k=0 ≡
EK0(k)k=0 −EK+(k)k=0 given by
2α(fρ − M
f ∗ω +
(mK − 16(M −M∗))
with α ≡ ρB3/ρB the asymmetry parameter.
We can now easily estimate the isospin splitting of K0 vs. K+ for the two
isovector mean field Lagrangians used here, NLρ and NLρδ. The effect will
be clearly larger when the δ coupling is included since we have to increase
the ρ-coupling fρ, see [14,15], but still the expected weight is relatively small,
going from about 1.5% (NLρ) to about 3.0% (NLρδ) at ρB = 2ρ0, for a fixed
isospin asymmetry around 0.2. The complete results are also shown in Fig. 3
(right panel). The agreement with the ChPT estimations is rather good, but
in the RMF scheme we see an overall reduced repulsion and a smaller isospin
splitting. Both effects are of interest for our discussion, the first affecting the
K0,+ absolute yields, the second important for the K0/K+ yield ratios.
4 Numerical realization and notations
The Vlasov term of the RBUU equation (1) is treated within the Relativistic
Landau-Vlasov method, in which the phase space distribution function f(x, p∗)
is represented by covariant Gaussians in coordinate and momentum space [38].
For the nuclear mean field or the corresponding EoS in symmetric matter the
fσ (fm
2) fω (fm
2) fρ (fm
2) fδ (fm
2) A (fm−1) B
NLρ 9.3 3.6 1.22 0.0 0.015 -0.004
NLρδ 9.3 3.6 3.4 2.4 0.015 -0.004
Table 1
Coupling parameters in terms of fi ≡ ( gimi )
2 for i = σ, ω, ρ, δ, A ≡ a
and B ≡ b
for the non-linear NL models [14] using the ρ (NLρ) and both, the ρ and δ mesons
(NLρδ) for the description of the isovector mean field.
NL2 parametrization [26] of the non-linear Walecka model [39] is adopted with
a compression modulus of 200 MeV and a Dirac effective mass of m∗ = 0.82 M
(M is the bare nucleon mass) at saturation. The momentum dependence enters
via the relativistic treatment in terms of the vector component of the baryon
self energy. The isovector components in the mean fields are introduced in the
NLρ,NLρδ Lagrangians as in the recent Refs. [14,15]. In Table 1 we report
all the coupling constants and the coefficients of the non-linear σ-terms.
The collision integral is treated within the standard parallel ensemble algo-
rithm imposing energy-momentum conservation. For the elastic NN cross sec-
tions the DBHF calculations of Ref. [17] have been used throughout this work.
At intermediate relativistic energies up to the threshold of kaon (K0,+) pro-
duction, i.e. Elab = 1.56 GeV, the major inelastic channels are (B, Y,K stand
for a baryon (nucleons N or a ∆-resonance), hyperon and kaon, respectively)
– NN ←→ N∆ (∆-production and absorption)
– ∆←→ πN (π-production and absorption)
– BB −→ BYK, Bπ −→ Y K (K-production from BB and Bπ-channels)
The produced resonances propagate in the same mean field as the nucleons,
and their decay is characterized by the energy dependent lifetime Γ which is
taken from Ref. [34]. The produced pions propagate under the influence of the
Coulomb interaction with the charged hadrons. Kaon production is treated
hereby perturbatively due to the low cross sections, taken from Refs. [40].
Kaons undergo elastic scattering and their phase space trajectories are deter-
mined by relativistic equations of motion, if the kaon potential is accounted
In the next section the results of transport calculations in terms of pion and
kaon yields and their rapidity distributions will be presented. The following
cases for the inelastic NN cross sections σinel and the kaon potential ΣK (scalar
and vector) will be particularly discussed:
– free σinel, without ΣK (w/o K-pot σfree)
– free σinel, with ΣK (w K-pot σfree)
– free σinel, with isospin dependent ΣK (w ID K-pot σfree)
– effective σinel, without ΣK (w/o K-pot σeff )
– effective σinel, with ΣK (w K-pot σeff )
– effective σinel, with isospin dependent ΣK (w ID K-pot σeff )
For pions only the different cases of σinel will be labelled, since they do not ex-
perience any potential, apart coulomb. One should note that in all calculations
only inelastic processes including the lowest mass resonance ∆(1232MeV )
have been considered, without accounting for the N∗(1440) resonance. This
will have not appreciable consequences for pions yields, but it slightly reduces
the kaon multiplicities.
5 Results
As mentioned in the introduction, the main topic of the present work is to
study the sensitivity of particle ratios to physical parameters such as in-
medium effects of cross sections and the isospin dependence of the kaon po-
tential. This is an important issue to clarify since there is some evidence sug-
gesting the yield ratios as good observables in determining the high density
behavior of the symmetry energy. In a near future these data will be exper-
imentally accessible with the help of reactions with radioactive ion beams.
However, a comparison of absolute values with experimental data, although it
is not the aim of this work, is essential and it has to be included in order to
show the consistency of our approach. Thus we will start the presentation of
the results first in terms of absolute yields, and comparison with data, before
passing to the main section on the particle ratios. Most calculations refer to
central 197Au+197 Au collisions at 1 AGeV .
5.1 Effects of in-medium inelastic NN cross sections on particle yields
5.1.1 Resonance and Pion Production
Here we study the role of the density dependence of the effective inelastic NN
cross sections on particle yields (pions and kaons). We start with the temporal
evolution of the ∆ resonances and the produced pions, as shown in Fig. 4.
The maximum of the multiplicity of produced ∆-resonances occurs around 15
fm/c which corresponds to the time of maximum compression. Due to their
finite lifetimes these resonances decay into pions (and nucleons) as ∆ −→
πN . Some of these pions are re-absorbed in the inverse process, i.e. πN −→
∆ but chemical equilibrium is never reached, as pointed out in [15]. This
0 10 20 30 40 50 60
time (fm/c)
0 10 20 30 40 50 60
time (fm/c)
Fig. 4. Time evolution of the ∆-resonances (left panel) and total pion yield (right
panel) for a central (b = 0 fm) Au+Au reaction at 1 AGeV incident energy. Cal-
culations with free (solid lines) and effective (DBHF, dashed lines) σinel are shown.
mechanism continues until all resonances have decayed leading to a saturation
of the pion yield for times t ≥ 50 fm/c (the so-called freeze-out time). The
resonance production takes place during the high density phase, where the
in-medium effects of the effective cross sections are expected to dominate.
In fact, the transport results with the in-medium modified σinel reduce the
multiplicity of inelastic processes, and thus the yields of ∆ resonances and
pions. However, the in-medium effect is not so pronounced here with respect
to similar phenomenological studies of Ref. [16,33], which should come from
the moderate density dependence of the effective cross sections, see also again
Fig. 2.
Fig. 5 shows the centrality dependence of the charged pion yields for Au+Au
collisions at 1.0 AGeV incident energy. The degree of centrality is characterized
by the observable Apart, which gives the number of participant nucleons and
can be calculated within a geometrical picture using smooth density profiles for
the nucleus [41]. Obviously Apart increases with decreasing impact parameter
b and its value approaches the total mass number of the two colliding nuclei
in the limiting case of b = 0 fm. As can be seen in Fig. 5, the charged pion
yields are enhanced with increasing Apart, particularly in a non-linear Apart-
dependence. As pointed out in [41], the charged pion multiplicities show a
similar non-linear increase also in the data. However, by directly comparing
the theoretical charged pion yields with the experiments [41] we observe that
our calculations overpredict the data, even when the in-medium reductions in
σinel are accounted for.
This discrepancy is a general feature of the transport models and may lie on
the role of the rescattering processes that take place in the spectator region,
0.0 100.0 200.0 300.0 400.0
 0  0
 5  5
 10  10
 15  15
 20  20
 (FOPI)
 (FOPI)
E=1.0 AGeV
Au+Au
Fig. 5. Centrality dependence (in terms of Apart) of the negative (π
−) and positive
(π+) charged pions for Au+Au collisions at 1 AGeV incident energy. Calculations
with free (solid lines, filled circles) and effective (dashed lines, filled squares) cross
sections are shown as indicated. Experimental data, taken from FOPI collaboration
[41], are also displayed for comparison.
where nuclear surface effects can play a crucial role. In order to check this
point we have performed a selection on pions produced at central rapidity,
where data are also available [41].
In Fig. 6 we present the inclusive (all centralities) pion rapidity distributions
vs. the FOPI data for charged pions. We see that the agreement is rather good
at mid-rapidity while we see a definite overcounting in the spectator sources.
Such a good evaluation of the pion production ad mid-rapidity is confirmed
by the results shown in Fig. 7, where we present the inclusive (all centralities)
pion transverse spectrum at midrapidity (−0.2 < y0 < 0.2). We first note that
this is also not much affected by the inclusion of the in-medium inelastic cross
sections. Moreover we see again that our results are in good agreement with
the experimental values from the FOPI collaboration [41], in the same rapidity
selection. The overestimation of the pion yields shown in Fig. 5 probably
results from other rapidity regions where the role of the spectator sources is
more evident. We have also to say that we are not imposing any experimental
filter to our results. The point is rather delicate since the main discrepancies
appear in high rapidity regions. In any case such a fine agreement at mid-
-2 -1 0 1 2
-2 -1 0 1 2
Fig. 6. Inclusive (all centralities) pion rapidity distributions for a Au+Au reaction
at Ebeam = 1 AGeV incident energy. Comparison with the experimental values given
by FOPI collaboration [41]; as in the data we have used a transverse momentum
cut to pt > 0.1GeV/c.
0 200 400 600 800
 (MeV)
 FOPI
 FOPIπ
0.1*π
Fig. 7. Inclusive transverse spectrum at midrapidity of π−, π+ for a Au+Au reaction
at Ebeam = 1 AGeV incident energy. Comparison with the experimental values given
by FOPI collaboration [41]. The cross sections are normalized to a rapidity interval
dy = 1.
rapidity is very important for the reliability of our results on kaon production,
mostly produced in that rapidity range via secondary πN,∆N channels, see
[15].
The pion reaction dynamics is furthermore not sensitively affected by the
in-medium inelastic cross sections. We restrict here the analysis to central
Au+Au collisions at 1 AGeV. In Fig. 8 we show cross section effects on the
-2 -1 0 1 2
-2 -1 0 1 2
Fig. 8. Rapidity distributions of negative and positive charged pions (left and right
panels, respectively) for a central (b = 0 fm) Au+Au reaction at Ebeam = 1 AGeV
incident energy.
rapidity distributions (normalized to the projectile rapidity in the cm sys-
tem) for π±, an observable which characterizes the degree of stopping or the
transparency of the colliding system. This is due to the fact that the global
dynamics is mainly governed by the total NN cross sections, in which its elas-
tic contribution is the same for all the cases. In previous studies [19,31] the
in-medium effects of the elastic NN cross sections gave important contribu-
tions to the degree of transparency or stopping. It was found that a reduction
of the effective NN cross section particularly at high densities is essential in
describing the experimental data [19], as confirmed by various other analy-
ses [31]. The density effects on the inelastic NN cross section influence only
those nucleons associated with resonance production, and therefore they do
not affect the global baryon dynamics significantly.
5.1.2 Kaon Production
The situation is different for kaon production, see Fig. 9. The influence of the
in-medium dependence of σinel is important, and reduces the kaon abundancies
by a factor of ≈ 30%. This is due to the fact that the leading channels for
kaon production are N∆ −→ BY K and Nπ −→ ΛK. Thus kaon production is
essentially a twostep process and the medium-modified inelastic cross sections
enter twice, leading to an increased sensitivity.
Fig. 10 shows the rapidity distributions of kaons, where the in-medium effect
is more visible with respect to the corresponding pion rapidity distributions
(see Fig. 8). These results seem to show that kaon production could be used
to determine the in-medium dependence of the NN cross section for inelas-
0 10 20 30 40 50 60
time (fm/c)
0 10 20 30 40 50 60
time (fm/c)
w/o K-pot σ
w/o K-pot σ
Fig. 9. Time evolution of the K0 (left panel) and K+ (right panel) multiplicities,
for the same reaction and models as in Fig. 4, with free and in-medium inelastic
cross sections, without the inclusion of the kaon potentials.
-2 -1 0 1 2
w/o K-pot σ
w/o K-pot σ
-2 -1 0 1 2
Fig. 10. Same as in Fig. 9, but for the normalized rapidity distributions.
tic processes. Similar phenomenological studies based on the BUU approach
[16,33] strongly support in-medium modifications of the free cross sections. It
is of great interest to perform an extensive comparison with experimental data
on kaon production, in order to have a more clear image of the effect of the
in-medium cross sections on their production. The point is that kaon absolute
yields are also largely affected by the kaon potentials, see the following, as
expected from the general discussion of the previous section. However since
kaons are mainly produced in more uniform high density regions the effects
of the medium on cross sections tend to disappear in the yield ratios. In the
next section we will show that the same holds true for the K0, K+ potentials.
Our conclusion is that the kaon yield ratios might finally be a rather robust
0 10 20 30 40 50 60
time (fm/c)
0 10 20 30 40 50 60
time (fm/c)
w/o K-pot
w K-pot
w ID K-pot
Fig. 11. Time evolution of the K0 (left panel) and K+ (right panel) multiplicities
for the same reaction as in Fig. 4. Calculations without (w/o K-pot, solid), with (w
K-pot, dashed) and with the isospin dependent (w ID K-pot,dotted-dashed) kaon
potential are shown. In all the cases the free choice for σinel is adopted.
observable to probe the nuclear EoS at high baryon densities.
5.2 The role of the kaon potential
As discussed in the previous sections, the important quantity which influences
the kaon production threshold is the in-medium energy at zero momentum [7].
This quantity rises with increasing baryon density and in the general case of
isospin asymmetric matter shows a splitting between K0 and K+, see Fig. 3.
We are presenting here several K-production results in ab initio collision sim-
ulations using the Chiral determination of the K-potentials, ChPT . Fig. 11
shows the time dependence of the two isospin states of the kaon with respect
to the role of the kaon potential and its isospin dependence. First of all, the
repulsive kaon potential considerably reduces the kaon yields, at least in this
ChPT evaluation.
The inclusion of the isospin dependent part of the kaon potential slightly
modifies the kaon yields, towards a larger K+ production in neutron-excess
matter. However by comparing to the corresponding isospin dependence of
the in-medium kaon energy, see Fig. 3, the effect is less pronounced in the
dynamical situation. This is due to the fact that in heavy ion collisions the local
asymmetry in the interacting region varies with time, see [15]. In particular,
it decreases with respect to the initial asymmetry because of partial isospin
equilibration due to stopping and inelastic processes with associated isospin
exchange. This is reflected also in the kaon rapidity distributions, see Fig. 12,
-2 -1 0 1 2
-2 -1 0 1 2
w/o K-pot
w K-pot
w ID K-pot
Fig. 12. Same as in Fig. 11, but for the rapidity distributions.
-1 0 1 2
normalized rapidity y
-1 0 1 2
normalized rapidity y
-1 0 1 2
normalized rapidity y
w/o K-pot w K-pot w ID K-pot
Fig. 13. K+ rapidity distributions for semi-central(b < 4 fm) Ni+Ni reactions at
1.93 AGeV. Theoretical calculations (as indicated) are compared with the exper-
imental data of FOPI (open triangles) and KaoS (open diamonds) collaborations
[42,43].
where the role of the kaon potential is crucial, but not its isospin dependence.
As we have already seen even in-medium modifications of inelastic cross sec-
tions are affecting the kaon absolute yields, so it appears of interest to look at
the combined effects. For that purpose we have performed calculations for a
semi-central (b < 4fm) Ni+Ni system at 1.93AGeV , where data are existing
from the FOPI [42] and KaoS [43] collaborations. The results for K+ rapidity
distributions, compared to experimental data, are shown in Fig.13.
We observe that although the kaon yields are reduced when using the in-
medium inelastic cross section, we are still rather far away from the data, left
panel of Fig. 13. We note that the reduction due to the density dependence
of the effective inelastic cross sections is rather moderate here with respect
to that of the heavier Au-system (see Fig. 10). for kaons). This is due to the
less compression achieved for the lighter Ni-systems. The inclusion of the kaon
potential, without (central panel) and with (right panel) isospin dependence,
is further suppressing the K+ yield, towards a better agreement with data, as
expected for the repulsive behavior at high density.
In fact the results obtained with kaon potentials and effective cross sections
seem to underestimate the data. This could be an indication that the ChPT
K-potentials are too repulsive at densities around 2ρ0 where kaons are pro-
duced, see [15]. We like to remind that the parameters of ChPT potentials
are essentially derived from free space considerations. When we follow a more
consistent RMF approach, directly linked to the effective Lagrangians used to
describe bulk properties of the nuclear matter as well as the relativistic trans-
port dynamics, we see less repulsion, bottom curves in Fig. 3 (left panel). This
is valid also for the isospin dependent part of the K-potentials, that more di-
rectly will affect the K0/K+ yield ratio. We see from the same Fig. 3 (right
panel) that in the RMF frame this splitting is reduced to a few percent for
all the different isovector interactions. The conclusion is that when kaon po-
tentials are evaluated within a consistent effective field approach we have a
better agreement with data for absolute yields, with a very similar reduction
of the K0 and K+ rates. This is important for the yield ratio, that then should
be not much sensitive to the in-medium effects on kaon propagation.
A similar conclusion on K-potential effects, obtained within the ChPT ap-
proach, can be drawn from the centrality dependence of the K+ yields shown
in Fig. 14 in the case of Au+Au collisions at 1 AGeV beam energy and com-
pared with KaoS data [44]. The trend in centrality can be reproduced by all
theoretical calculations (with different cross sections), however, all of them
seem to underestimate the experimental yields.
In fact we have to mention that another possible source of the discrepancy
with data can be that in all our simulations only the lowest mass resonance
∆(1232MeV ) has been dynamically included. Transport calculations from
other groups, that take care also of the N∗(1440MeV ) resonance, are getting
an enhancement of the K+ yield for Au+Au collisions at 1 AGeV incident
energy [7]. This significant dependence of the kaon yields on the N∗ resonance
comes from the 2-pionic N∗-decay channel, i.e. N∗ −→ ππN . Therefore, since
the most important channels of kaon production are the pionic ones, we can
expect some underestimation of the absolute yields in our calculations. Just
to confirm this point, in Fig. 14 we report also transport results from the
Tübingen group, in which all resonances are accounted for [7]. We finally re-
0,0 0,2 0,4 0,6 0,8
 5,0×10
 1,0×10
 1,5×10
 2,0×10
0,0 0,2 0,4 0,6 0,8 1,0
w K-pot w ID K-pot
Fig. 14. K+ centrality dependence in Au+Au reactions at 1 AGeV incident energy.
Our theoretical calculations (as indicated) are compared with KaoS data from [44]
(open diamonds) and with results of the Tübingen group (open squares).
mark that the inclusion of other nucleon resonances in neutron-rich matter
will further contribute to increase the K0 yield through a larger intermedi-
ate π− production. This can contribute to compensate the opposite effect of
isospin dependent part of the K-potentials on the K0/K+ yield ratios.
5.3 Pionic and Strangeness Ratios
A crucial question is whether particle yield ratios are influenced by in-medium
effects both on inelastic cross section and kaon potentials. This point is of ma-
jor importance particularly for kaons, since ratios of particles with strangeness
have been widely used in determining the nuclear EoS at supra-normal density.
Relative ratios of kaons between different colliding systems have been utilized
in determining the isoscalar sector of the nuclear EoS [5]. More recently, the
(π−/π+)- and (K0/K+)-ratios have been proposed in order to explore the high
density behavior of the symmetry energy, i.e. the isovector part of the nuclear
mean field [14,15,13,12].
Fig. 15 shows the incident energy dependence of the pionic (π−/π+, left panel)
and strangeness (K0/K+, right panel) ratios for the different choices of in-
elastic cross sections and kaon potentials, as widely discussed in the previous
sections.
First of all, a rapid decrease of the pionic ratio with increasing beam energy
0.8 1 1.2 1.4 1.6 1.8 2
 (AGeV)
0.8 1 1.2 1.4 1.6 1.8 2
 (AGeV)
free w/o K-pot
eff w/o K-pot
0.8 1 1.2 1.4 1.6 1.8 2
 (AGeV)
free w K-pot
eff w K-pot
0.8 1 1.2 1.4 1.6 1.8 2
 (AGeV)
tio σ
free w ID K-pot
eff w ID K-pot
Fig. 15. Energy dependence of the π−/π+ (left panel) and K0/K+ (right panel)
ratios for central (b = 0 fm) Au+Au reactions.
is observed, related to the opening of secondary rescattering processes (reab-
sorption/recreation of pions with associated isospin exchange) channels. The
corresponding strangeness ratio depends only moderately on beam energy due
to the absence of secondary interactions with the hadronic environment.
The pionic ratio is partially affected by the in-medium effects of σinel, as it can
be seen in the left panel of Fig.15. Its slope is slightly changing with respect to
beam energy. The situation is similar for the strangeness ratio, which actually
appears even more robust vs. in-medium modifications, even with the kaon
potentials. This can be seen in the right panel of Fig.15, where for all the
considered beam energies the ratio remains almost unchanged. Such a result
is consistent with those of the previous sections, where it was found that the
absolute kaon yields decrease in the same way when the effective σinel are
applied and when the K-potentials are included.
The different sensitivity to variations in the inelastic cross sections of pionic
vs. strangeness ratios can be easily understood. For the large rescattering and
lower masses pions can be produced at different times during the collision,
in different density regions. At variance kaons are mainly produced at early
times in a rather well definite compression stage, i.e. in a source with a more
uniform high density, and so the density dependence of the inelastic cross
sections will affect in the same way neutral and charged kaon yields, leaving
the ratio unchanged. At this level of investigation one could argue that the
strangeness ratio is a very promising observable in determining the nuclear
EoS and particularly its isospin dependent part. This has been also the main
conclusion in Ref. [15].
However, a strong isospin dependence of the kaon potentials could directly
NL NLρ NLρδ
K0/K+ (w/o K-pot) 1.24 (± 0.02) 1.35 (± 0.01) 1.43 (± 0.02)
K0/K+ (w ID K-pot) 1.02 (± 0.03) 1.22 (± 0.04) 1.34 (± 0.05)
Table 2
Sensitivity of the strangeness ratio K0/K+ to the isospin dependent kaon potential
and to the isovector mean field (NL, no isovector fields, NLρ and NLρδ). The
considered reaction is a central (b = 0 fm) Au+Au collision at 1 AGeV incident
energy.
affect the ratio, since the K0 and K+ rates will be modified in opposite ways.
This is shown by the two triangle points at 1 AGeV in the right panel of Fig.
15. As already discussed this large isospin dependence of the kaon potential,
clearly present in the ChPT evaluation, is greatly reduced in a consistent
mean field approach, see Fig. 3 and the arguments presented in Section 3.
In any case this point deserves more detailed studies. We plan to perform
ab initio kaon-production simulations within the OBE/RMF evaluation of
kaon potentials, with an isospin part fully consistent with the isovector fields
of the Hadronic Lagrangians used for the reaction dynamics.
An interesting final comment is that the sensitivity of the strangeness ratio
to the isovector part of the nuclear EoS remains even when strong isospin
dependence of the kaon potentials is inserted, as in the ChPT case.
In order to check this, we have repeated for Au+Au at 1 AGeV incident energy
the calculations by varying the isovector part of the nuclear mean field. As
in Refs. [14,15], three options for the isovector mean field have been applied:
the NL (no isovector fields), NLρ and NLρδ parametrizations, but now in-
cluding the isospin effect in the kaon potential in the ChPT evaluation. The
options of the symmetry energy differ from each other in the high density stiff-
ness. NL gives a relatively soft Esym, NLρδ a relatively stiff one, and NLρ
lies in the middle between the other limiting cases [14]. Table 2 shows the
strangeness ratio as function of these different cases for the isovector mean
field, keeping now constant the other parameters (free σinel, isospin dependent
kaon potential). The ratio, indeed, strongly decreases when the isospin part in
the kaon potential is accounted for. The more interesting result is, however,
that the relative difference between the different choices of the symmetry en-
ergy remains stable. This can be understood from the fact that in the kaon
self energies the isospin sector contains only the isospin densities and currents
without additional parameters such meson-nucleon coupling constants. Since
the local asymmetry does not strongly vary from one case to the other (NL,
NLρ, NLρδ), one would expect a robustness of the EoS dependence. Thus
we conclude that the strangeness ratio appears to be well suited in determin-
ing the isovector EoS, however, a fully consistent mean field approach is still
missing.
6 Conclusions
We have investigated the role of the in-medium modifications of the inelastic
cross section and of the kaon mean field potentials on particle production in
intermediate energy heavy ion collisions within a covariant transport equation
of a Boltzmann type. We have used for both, the elastic and inelastic NN cross
sections the same DBHF approach which provide in a parameter free manner
the in-medium modifications of the imaginary part of the self energy in nu-
clear matter. The kaon potential has been evaluated in two ways, following a
Chiral Perturbative approach and an Effective Field scheme, considering va-
lence quark-meson couplings. We have applied these modifications of the cross
sections and kaon potentials to the collision integral of the transport equation
and analyzed Au+Au and Ni+Ni collisions at intermediate relativistic energies
around the kaon threshold energy.
Our studies have shown a good sensitivity of the particle multiplicities and
rapidity distributions of pions and kaons. In particular, a moderate reduction
for pions has been seen when the in-medium effects in the inelastic cross section
are accounted for. The pion yields are still overestimating the inclusive data
while we have a very nice agreement with the pion spectra and multiplicities at
mid-rapidity. The latter point is important for trusting the kaon production,
mainly due to secondary pion collisions at mid-rapidity.
At variance the kaon (K0,+) yields show a larger sensitivity to the reduction
of the inelastic cross sections, with a decrease of about 30 %. However we
see that the introduction of a repulsive kaon potential is essential in order to
reproduce even inclusive data.
We have then focused our attention on π−/π+ and K0/K+ yield ratios, re-
cently suggested as good probes of the isovector part of the EoS at high den-
sities. The pionic ratios, due to their strong secondary interaction processes
with the hadronic environment, show a dependence on the density behavior
of the inelastic cross sections. A further selection of the production source, i.e.
a transverse momentum discrimination, could be required in order to have a
more reliable probe of the nuclear EoS.
The situation appears more favorable for the kaon ratios. In fact we find that
the multiplicities of K0 and K+ are influenced in such a way that their ratio
is not affected by the density dependence of the inelastic cross sections. This
is due to the long mean free path of the K0,+ that are produced only in the
compression stage of the collision [15]. The effects of the in medium kaon
potentials are also largely compensating in the K0/K+ yield ratio, due to
the similar repulsive field seen by K0 and K+ mesons. Such a result can be
modified by the isospin dependence of the kaon potentials which is expected to
act in opposite directions for neutral and charged kaons rates. Actually this is a
rather stimulating open problem. In our analysis with two completely different
approaches, ChPT vs. RMF , we get a good agreement for the isoscalar kaon
potential but a rather different prediction for the isovector part.
However, a study in terms of the different choices of the isovector kaon field
has shown that the relative dependence of the strangeness ratios on the stiff-
ness of the isovector nuclear EoS remains a well robust observable. This is
an important issue in determining the high density behavior of the symmetry
energy in more systematic analyses in the future, when more experimental
data will be available.
Acknowledgments. This work is supported by BMBF, grant 06LM189 and
the State Scholarships Foundation (I.K.Y.). It is also co-funded by European
Union Social Fund and National funded Pythagoras II - EPEAEK II, under
project 80861. One of the authors (V.P.) would like to thank H.H. Wolter and
M. Di Toro for the warm hospitality during her short stays at their institutes.
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