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the knowledge of the in - medium hadronic properties at supra - normal densities is of major importance for the understanding of ( nuclear ) astrophysics such as the physical mechanism of supernovae explosions and the physics of neutron stars @xcite . strangeness production at intermediate energy heavy - ion collisions has been a very helpful tool in studying the hadronic equation of state ( eos ) and the properties of strangeness matter under extreme conditions of baryon density and temperature @xcite . it is well established that ( @xmath0 ) kaons feel a weak repulsive potential of the order of @xmath1 mev at saturation density @xmath2 , as it has been verified by studying strangeness flows in heavy - ion collisions @xcite . the determination of the high density behavior of the nuclear eos has been successfully investigated in this context @xcite . it has been realized that strangeness production could also serve as an important messenger for the isovector part of the nuclear eos at high densities , i.e. for the symmetry energy @xcite . a reliable determination of the symmetry energy at supra - normal densities from heavy - ion collisions is still an object of current debates and investigations @xcite . however , a reliable study of the high density nuclear eos and , in particular , its isovector part involves a detailed analysis of the role of the isospin dependent part of the kaon - nucleon interaction on strangeness production in intermediate energy nucleus - nucleus collisions , which has not been studied so far . theoretical studies on static hadronic matter based on the chiral perturbation theory ( chpt ) @xcite and on the one - boson - exchange approach ( obe ) @xcite predict a splitting of self energies between positive charged ( @xmath3 ) and neutral ( @xmath4 ) kaons . this isospin effect increases with baryon density and isospin asymmetry , and it may thus influence the production and propagation of particles with strangeness in dense hadronic matter during a nucleus - nucleus collision . it may thus be a sensitive observable to determine the symmetry energy at high densities . this work is an extension of former theoretical investigations of kaon medium effects in heavy - ion collisions @xcite to asymmetric hadronic matter . the main topic of our study is therefore the influence of the isovector sector of the kaon - nucleon interaction on kaon dynamics at incident energies near the strangeness production threshold . in extension to our previous work @xcite , two different models for the kaon - nucleon interaction and their particular isovector properties are now discussed in detail : one based on the chiral perturbation theory ( chpt ) @xcite and a second one based on the one - boson - exchange ( obe ) approach for the effective meson - nucleon field theory @xcite . we use a fully relativistic transport model , in which isospin effects are included in the mean - field and in the collision integral @xcite , and also in the different realizations of the kaon potential . these models are applied in heavy - ion collisions at intermediate energy below the threshold for strangeness production . the production of the @xmath5 isospin states and , in particular , their ratio is extensively investigated within the different scenarios for the isovector sector of the nuclear mean - field and the kaon - nucleon potential . the antikaon field @xmath6 is not considered here . a discussion of the stronger and more complex @xmath7-nucleon interaction would go beyond the scope of this work , e.g. , we refer to refs . in this section we describe the theoretical model used in this work . after a brief introduction of the transport equation we focus on the different treatments of the kaon - nucleon potential in isospin - asymmetric hadronic matter , which is the major focus of the present work . the theoretical description of a heavy ion collision is based on the relativistic boltzmann - uehling - uhlenbeck ( rbuu ) dynamical approach , as introduced in refs . @xcite and then extended for isospin and strangeness degrees of freedom @xcite : @xmath8 f_i(x , k^ { * } ) = { \cal i}_{\rm coll } \label{rbuu } \quad .\end{aligned}\ ] ] the rbuu transport equation describes the phase space evolution of the one - particle density distribution functions @xmath9 for the different particle species under consideration under the influence of a mean - field ( lhs of eq.([rbuu ] ) ) and binary collisions ( rhs of eq.([rbuu ] ) ) which couple the different fields . here we consider the propagation of all baryons up to the @xmath10-resonance and hyperons ( @xmath11 ) . the production of pions results through the resonance decay and kaons ( @xmath0 ) are produced in baryon - baryon and baryon - pion binary collisions ( see for details ref . @xcite ) . the mean - field or the nuclear eos enters via the effective mass @xmath12 , the kinetic momenta @xmath13 and the field tensor @xmath14 , where @xmath15 and @xmath16 are the lorentz - scalar and lorentz - vector components of the baryon self energy . pions are propagated under the influence of the coulomb field only . mesons with strangeness obey the same transport equation ( [ rbuu ] ) , however , with different self energies with respect to the baryon ones , as it will be discussed below . the collision integral is determined by the total elastic and inelastic in - medium cross sections and includes all important channels on pion and kaon production , without the inclusion of antikaons . isospin effects on the threshold conditions are explicitly accounted for to respect energy conservation @xcite . more details can be found in ref . furthermore , the quasi - elastic @xmath17 channel with isospin exchange is included , however , it does not essentially influence the strangeness ratios . kaplan and nelson were the first to apply the chiral lagrangian approach to the properties of kaons in dense nuclear matter @xcite . starting from the effective chiral lagrangian an in - medium klein - gordon equations for kaons at the mean field level follow can be derived @xmath19 \phi_{k}(x)= 0 \label{kaonmean } \quad , \ ] ] where @xmath20 and @xmath21 are the baryon current and the scalar density , respectively . ( [ kaonmean ] ) can be rewritten in a more compact form as @xmath22\phi_{k}(x)= 0 \quad . \label{kaonmean2}\ ] ] in isospin asymmetric matter the kaon effective mass is given by @xcite @xmath23 where @xmath24 are the total and isospin scalar baryon densities . the vector potential is defined as @xmath25 with @xmath26 isoscalar and isovector baryon currents , respectively , and @xmath27 for @xmath3 and @xmath4 . @xmath28 @xmath29 is the free kaon mass , @xmath30 @xmath29 the pion decay constant and @xmath31 the kaon - nucleon sigma term , usually chosen in the range @xmath32 @xmath29 . @xmath33 is the in - medium pion decay constant @xcite which appears only in the vector field . according to brown and rho @xcite this accounts for the fact that the enhancement of the the scalar part is compensated by higher order corrections in the chiral expansion . finally , the parameter @xmath34 is fixed to the value of @xmath35 @xcite .      in an alternative approach kaon potentials can also be derived within an effective meson field @xmath36 approach @xcite . in this picture the kaons interact through the exchange of mesons with different lorentz properties ( @xmath37 ) . these mesons mediate the isoscalar - scalar , isoscalar - vector , isovector - vector and isovector - scalar parts of the kaon - kaon interaction . in the mean - field approximation the kaon equation of motion reads @xmath38\phi_k(x)=0 \label{kaoneq2 } \quad .\end{aligned}\ ] ] as in the chpt case ( see eq . ( [ kaonmean2 ] ) ) eq . ( [ kaoneq2 ] ) can be written more compact as : @xmath39 \phi_k(x)=0 \label{kaoneq}\ ] ] the vector potential @xmath40 and the effective mass @xmath41 are now defined differently relative to the chpt approach @xmath42 where @xmath43 and @xmath44 , @xmath45 and @xmath46 are the isoscalar , isovector baryon currents , the isoscalar - scalar field and isovector - scalar baryon density , respectively . we will use a simple constituent quark - counting prescription to relate the kaon - meson couplings to the nucleon - meson couplings , i.e. a reduction by a factor @xmath47 ; @xmath48 ( @xmath49 ) . thus , in the isoscalar - vector sector we have @xmath50 due to the enhanced kaon - scalar / vector coupling . in the isovector - vector sector one obtains @xmath51 . the scalar kaon field finally includes the isoscalar - scalar and isovector - scalar couplings @xmath52 and @xmath53 , respectively . for the parameters of the nucleon - meson couplings we refer to ref . we note that the picture of a simple constituent quark - counting may not be consistent with @xmath54 scattering in free space . however , this is not of importance here , since @xmath54 scattering may influence kaon spectra , but not absolute kaon yields and their ratio , which is the main topic of this work . ( 0.5,0.0 )    comparing both models for the kaon - nucleon interaction , eqs . ( [ vpot2],[emass ] ) versus ( [ kaoneffiso],[kapotiso ] ) , one sees that in the @xmath36 model the nucleon - meson couplings enter explicitly in the kaon self energies , while in the @xmath18 approach the kaon mean field is determined by the parameters @xmath31 and @xmath34 . it is therefore helpful to study the isospin dependence of the @xmath36 model for various parametrizations of the isovector eos , i.e. , the symmetry energy . as in previous works @xcite , we use the @xmath55 , @xmath56 and @xmath57 prescriptions for the isovector mean - field . in the first one ( @xmath55 ) no isospin potential is included . @xmath56 contains only the @xmath58 meson , while @xmath57 accounts for both , the @xmath58 and the @xmath59 mesons in the isovector potential . the symmetry energy becomes stiffer in the direction @xmath60 , since one has to increase the @xmath58-meson coupling when the @xmath59 meson field is taken into account in the potential part of the symmetry energy . the crucial quantity for the interpretation of the transport results shown later is the kaon in - medium energy , given by the dispersion relation : @xmath61 the density dependence of the in - medium kaon energy @xmath62 for @xmath0 is shown in fig . [ kmass ] for both models for the kaon - nucleon potential . in general the @xmath18 approach leads to a stiffer density dependence of the in - medium kaon energy relative to the @xmath36 model . this feature is more pronounced at high densities @xmath63 , which are reached in intermediate energy heavy - ion collisions near the kaon production threshold . thus , one expects observable effects of the model of the symmetry energy in the produced kaons ( @xmath0 ) from heavy - ion collisions .    the isospin effect for the considered asymmetry parameter of @xmath64 ( @xmath65 are the neutron and proton numbers ) , which corresponds approximately to @xmath66-nuclei , is pronounced at the baryon densities of @xmath63 , see again fig . [ kmass ] . according to eqs . ( [ kaoneffiso],[emass ] ) the isovector contributions to the kaon - nucleon @xmath18 and @xmath36 scalar self energies lead to an effective mass splitting between positive ( @xmath3 ) and neutral ( @xmath4 ) kaons . also the vector self energies between @xmath3 and @xmath4 differ according eqs . ( [ kapotiso],[vpot2 ] ) . these isospin effects finally lead to the pronounced isospin dependence of the in - medium kaon energy . we should note here the different origin of the isospin part of the kaon - nucleon potential between the two approaches . in the @xmath18 case the isospin dependence enters only through the isospin - currents , while in the @xmath36 model the effective isovector meson - nucleon couplings are explicitly included . this leads to an isospin splitting between @xmath4 and @xmath3 , which additionally depends on the model for the symmetry energy .    the non - trivial isospin dependence of the two models for the kaon - nucleon potential may have interesting effects on the absolute @xmath4 and @xmath3 yields and their collective flows and , in particular , on the strangeness ratio @xmath67 in heavy - ion collisions at intermediate energies near the kaon production threshold , which is the topic of the next section . the results of the following dynamical calculations are based on the transport equation ( [ rbuu ] ) , in which the isospin effects of the kaon - nucleon potentials are of particular interest , since they are investigated for the first time and may affect absolute kaon yields and strangeness ratios . we first discuss in subsection @xmath68 the qualitative features of the two models of the kaon - nucleon potential and compare with experiments . for a transparent discussion on the the isospin effects the following strategy has been chosen : the isoscalar part of the nuclear mean - field is the same for all different cases of transport calculations , while the isovector channel includes only the @xmath58 meson . in the strangeness sector following cases of different transport calculations will be discussed :    1 . @xmath18 refers to transport results with only the isospin independent part of the chiral model for the kaon potential . 2 .   @xmath36 refers to transport results with only the isospin independent part of the effective mean - field model for the kaon potential . 3 .   @xmath69 refers to transport results with the full isospin dependent chiral model for the kaon potential . 4 .   @xmath70 refers to transport results with the full isospin dependent effective mean - field model for the kaon potential . thus in the first two cases isospin effects are included only in the baryonic sector ( @xmath58 meson ) , while in the last two ones they are taken into account also in the strangeness sector . we then investigate in subsection @xmath71 in detail the role of the different isospin - dependent kaon - nucleon potentials on the strangeness @xmath72-ratio in @xmath73 central collisions . here we particularly distinguish for each case listed above between the @xmath55 , @xmath56 and @xmath57 models , in which the symmetry energy becomes more stiffer in the sequence @xmath74 . we start the discussion with the influence of the different kaon potentials on absolute @xmath75 meson yields . this is shown in fig . [ kpotrap ] for heavy - ion collisions between heavy systems ( au+au ) , in terms of their rapidity distributions for the four cases . consistent with fig . [ kmass ] the transport calculations with the @xmath18 model with the larger repulsion leads to overall smaller kaon yields with respect to those in the @xmath36 approach . the inclusion of the isospin dependent part in the kaon potentials ( compare the curves @xmath69 and @xmath18 as well as those between @xmath70 and @xmath36 in fig . [ kpotrap ] ) changes only very little the absolute yields . this is again consistent with fig . [ kmass ] , where the isospin effects on the two isovector states are small at baryon densities around @xmath76 . furthermore , in fig . [ kmass ] the asymmetry parameter was fixed to a value of @xmath77 corresponding to the asymmetry of a @xmath66-nucleus ) , while in heavy - ion collisions the local isospin - asymmetry , in the stage of kaon production , is reduced with respect to that of the initial situation @xcite . this is because of partial isospin equilibration due to stopping and inelastic processes with associated isospin exchange . ( 2,0.0 )    it is interesting to study the role of the different kaon potentials in lighter colliding systems , such as @xmath78 , in which also experimental information is available @xcite . [ knini ] shows the comparison between the data and the transport calculations obtained with both sets of the kaon potential without and with the isospin dependence ( left and right panel , respectively ) . the isospin dependence is again not of relevance here . the different kaon potentials influence the kaon yields in a similar manner as in the previous figure . these in - medium dependencies are less pronounced now , since less compression is achieved in the stage of kaon production for the lighter @xmath78-system . both sets of calculations are close to the experimental data , with the tendency of an improvement in the @xmath36 and @xmath70 cases . however , this result does not neccesarily mean that the @xmath36 and @xmath70 kaon - nucleon potentials are the most realistic ones , because other physical quantities may influence the transport calculations . a general and non - trivial feature in heavy - ion collisions concerns the treatment of the in - medium effects on the elastic and inelastic cross sections , which are usually parametrized to empirical free nn - scattering . microscopic studies on elastic nn - scattering at finite baryon density exist @xcite , but for inelastic processes , e.g. , @xmath79 , theoretical calculations are very rare and often limited in baryon density and momentum @xcite . attempts have been recently done in studying this non - trivial problem in heavy - ion collisions @xcite using a simple parametrization from microscopic dirac - brueckner calculations @xcite . for the strangeness sector , e.g. , @xmath80 and @xmath81 ( @xmath82 , @xmath83 and @xmath84 stand for nucleons or higher resonances , hyperons and kaons , respectively ) , the in - medium effects are trivially included only in the phase - space factors and the threshold conditions , but not in the matrix elements , i.e. , here free parametrizations are used @xcite . a selfconsistent treatment of kaon - nucleon scattering in the hadronic environment would be desirable in this context , as it has been already investigated for the antikaonic sector @xcite . furthermore , a more realistic density and , in particular , energy dependence of the hadronic relativistic mean field would be important @xcite . however , in order to keep the discussions transparent , we do not expand on these topics here , and use the same parametrization as in @xcite . ( 2.,0.0 )    in fig . [ kpotcentr ] the centrality dependence of the @xmath3 yields for @xmath85-collisions is shown . first of all , all sets of data understimate the data due to the strong density dependence of the inelastic cross section for @xmath10-production , see also ref . @xcite and the discussion above . the in - medium dependence of the kaon potentials is now more pronounced for the most central @xmath85-collisions , see left panel in fig . [ kpotcentr ] . isospin effects of both kaon potentials appear due to the asymmetry of the colliding @xmath86-systems , see right panel of the same figure . the effect of the inclusion of the isovector potential turns out to be stronger in the @xmath69 case compared to the @xmath70 one , since the in - medium energy for @xmath3 is stronger affected in the @xmath18 model , see again fig . [ kmass ] . thus , the kaon yield is enhanced more strongly in the @xmath69 transport calculations , and comes closer to the @xmath70 calculations , which are less sensitive to isospin effects . collective kaon ( @xmath3 ) flow ( not shown here ) has been also studied in the framework of the different kaon potentials . all the calculations reproduce the experimental flow @xcite reasonable well , however , the isospin effects were only moderate . it would be more helpful to study _ isospin _ strangeness flows , i.e. , the difference of collective flows between @xmath4 and @xmath3 , since the isovector part of the kaon potential affects @xmath3 and @xmath4 differently . in conclusion , the evaluation scheme of the kaon - nucleon potential within the effective @xmath36 approach seems to lead to a better description of the experimental data . however , a definitive conclusion on the question , which potential would be more realistic , can not be drawn at this level of study . a more systematic study on in - medium effects also on the level of @xmath87-scattering is in this context neccesary .    ( 8.,7.0 ) ( 2.0,0.0 )    thus at this level of the comparison between the theoretical calculations and the data the @xmath36 framework describes the experiment better than the @xmath18 model . however , for a transparent discussion specifically of the isospin effects on strangeness ratios both models of the kaon - nucleon interaction will be investigated . a more promising observable to study isovector effects may be given by the ratio of the two isospin kaon states , e.g. , the strangeness ratio @xmath67 , since ratios are expected to be independent on possible in - medium modifications of the isoscalar medium . e.g. , the influence of in - medium inelastic @xmath88 cross sections has been studied in @xcite with the conclusion of a very moderate dependence of the strangeness ratio on the density dependence of the inelastic cross section . a crucial question is whether particle yield _ ratios _ are influenced by the in - medium modifications of kaon potentials . this is important particularly for kaons , since ratios of particles with strangeness have been widely used to determine the nuclear eos at supra - normal density @xcite . more recently , the @xmath89- and @xmath72-ratios have been proposed as a good probe to determine the high density behaviour of the isovector part of the nuclear mean field @xcite . we thus extend our previous studies @xcite by explicitly including both schemes of the kaon potential in the transport calculations ( @xmath18 and @xmath36 models ) . in particular , we distinguish between the different scenarios for the symmetry energy in the baryon sector , i.e. , by neglecting the isovector potential ( @xmath55 case ) , by including only the @xmath58-meson ( @xmath56 case ) , and both , the @xmath58 and @xmath59 mesons ( @xmath57 ) in the isovector baryon and kaon potentials ( @xmath69 and @xmath70 cases ) . ( 8.,7.0 ) ( 2.0,0.0 )    fig . [ ratios ] shows the strangeness ratio for the different parametrisations of the isovector part of the nuclear @xmath90 . calculations are also performed for the two approaches of the kaon potential ( @xmath69 and @xmath70 model ) . first of all , an overall linear increase of the ratio with respect to the stiffness of the symmetry energy is observed . this feature is not trivial and reflects the interplay of isospin effects between mean - field and collision dynamics at the threshold . as discussed in detail in ref . @xcite , isovector mean - field and threshold effects act differently on the isospin kaon states @xmath3 and @xmath4 , where the later feature governs the strangeness dynamics at low intermediate energies near the kaon production threshold . the sensitivity of the strangeness ratio on the different evaluations of the isospin _ independent _ kaon potential ( @xmath18 and @xmath36 cases ) is very moderate , as expected . if the isovector part is neglected , the density behavior of the kaon potential is the same for both isospin @xmath4 and @xmath3 states . therefore the ratio should be rather independent on the evaluation scheme of the kaon potential , as it is the case in the transport calculations of fig . [ ratios ] . taking explicitly account of the isospin effects in the kaon potentials ( @xmath69 and @xmath70 cases ) the dynamical situation changes . the isospin effect in the @xmath69 model strongly decreases the strangeness ratio , however , the relative difference between the various choices of the symmetry energy ( @xmath55 , @xmath56 and @xmath57 ) remains stable . this is due to the fact that the isovector sector of the @xmath69 kaon self energies contain only the isospin lorentz - scalar and lorentz - vector densities without additional parameters , e.g. , meson - nucleon couplings @xcite . this feature is also shown in fig . [ kmass ] , in which the isospin splitting is independent on the stiffness choice of the symmetry energy . the decrease of the ratio in the @xmath69 calculations is related to the fact that the in - medium energy becomes more stiffer ( softer ) for @xmath4 ( @xmath3 ) . the situation in the @xmath70 calculations is similar , but the isospin effects are less pronounced . the suppression of the kaon ratio in the transport calculations due to the isospin effects is now weak , due to the softer character of the @xmath91 kaon potential , compared to the @xmath92 model . however , the relative difference between the different models of the symmetry energy is slightly modified . when isospin effects are accounted for , the @xmath70 self energies explicitly contain the meson - nucleon couplings , which are different for each model of the symmetry energy , see also again fig . [ kmass ] . this leads to the observed different slope of the kaon ratio in the @xmath70 transport calculations , however , the effect is rather moderate . it is interesting to note that the situation in a chemical and thermal equilibrated ( closed ) system , e.g. , in studying strangeness production in a box with periodic boundary conditions , is rather different . the isospin effects on the strangeness ratio are very pronounced in box calculations ( see fig . @xmath93 in ref . however , the isospin effect is considerably reduced in an open system in heavy - ion collisions due to , e.g. , isospin transparency which decreases the local asymmetry of the kaon production source .    in summary , the consideration of the isospin effects on the level of the kaon - nucleon interaction leads to modifications of the strangeness ratio on the absolute value , but not to significant changes with respect to the stiffness of the symmetry energy . in the @xmath69 and @xmath70 approaches a sensitivity to the symmetry energy on the level of @xmath94 and @xmath95 survives . we thus conclude that the strangeness ratio is a very promising observable to determine the isospin dependence of the underlying kaon - nucleon interaction and particularly the isospin part of the nuclear eos . we have investigated the role of different kaon potentials and , in particular , their explicit isospin dependence in asymmetric nuclear matter . the kaon potentials have been evaluated in two schemes , one following a chiral perturbation approach and a second one within the framework of an effective one - boson - exchange picture . we have studied in detail the effects of various isospin dependent kaon potentials on kaon dynamics in heavy - ion collisions within a covariant transport equation . the isospin effects turns out to be less sensitive to absolute kaon yields , but more pronounced to the strangeness @xmath67 ratio . we thus focused our attention to the @xmath67 ratio in collisions between heavy and isospin - asymmetric systems ( @xmath85 ) at incident energies below the kaon production threshold . different evaluation schemes of the kaon potential and its isovector part lead to different results on the strangeness ratio . the isospin dependence changes this ratio on an absolute level . as an interesting result , however , the relative dependence of the strangeness ratio to the stiffness of the isovector eos remains stable .    at this level of investigation we conclude that the strangeness ratio is an important observable to determine the high density behavior of the symmetry energy . future experiments with exotic radioactive beams , as they are planned in new experimental facilities at msu and at gsi ( fair project ) , will be very helpful to resolve the theoretical uncertainties of the symmetry energy at high baryon densities . * acknowledgements * + we acknowledge very fruitful discussions with a.b . larionov , m. kaskulov , h. lenske . we would like to particularly thank m. kaskulov and h. lenske for a carefully reading of the manuscript and further suggestions . this work is supported by bmbf and by the dfg cluster of excellence _ origin and structure of the universe_. one of the authors ( v.p . ) acknowledges support by the hellenic state scholarships foundation ( i.k.y . ) . v.  thorsson , m.  prakash , j.m . lattimer , nucl . phys . * a572 * ( 1994 ) 693 ; + r.  knorren , m.  prakash , p.j . ellis , phys . rev . * c52 * ( 1995 ) 3470 ; + g.e . brown , k.  kubodera , m.  rho , v.  thorsson , phys . lett . * b291 * ( 1992 ) 355 ; + v.  thorsson , j.p . ellis , phys . rev . * d55 * ( 1997 ) 5177 ; + t.  klhn et al . , phys . rev . * c74 * ( 2006 ) 035802 .
kaon properties are studied within the framework of a fully covariant transport approach . 
 the kaon - nucleon potential is evaluated in two schemes , a chiral perturbative approach and an effective one - boson - exchange model . 
 isospin effects are explicitly accounted for in both models . 
 the transport calculations indicate a significant sensitivity of momentum distributions and total yields of @xmath0 isospin states on the choice of the kaon - nucleon interaction . 
 furthermore , isospin effects are rather moderate on absolute kaon yields , but appear on strangeness ratios . 
 this is an important issue in determining the high density symmetry energy from studies of strangeness production in heavy - ion collisions . 
 asymmetric nuclear matter , symmetry energy , relativistic heavy ion collisions , kaon - nucleon potential , strangeness production . 
 + pacs numbers : * 25.75.-q * , * 21.65.+f * , 21.30.fe , 25.75.dw .