You are an expert at summarizing long articles. Proceed to summarize the following text: 
relativistic nucleus - nucleus ( a+a ) collisions provide us with experimental information about the phase diagram of quantum chromodynamics ( qcd ) and the strongly interacting matter properties . the last stage of such collisions is traditionally analyzed within the statistical approach which gives us an excellent opportunity to reveal the parameters of chemical freeze - out . this approach is based on the assumption of the thermal equilibrium existence during the last stage of a+a reaction . such an equilibrium can be reached due to intensive particle scattering . the stage of the system evolution when the inelastic reactions between hadrons stop is referred to as a chemical freeze - out ( fo ) . particle yields are determined by the parameters of fo , namely by chemical potentials and temperature . this general picture is a basis of the hadron resonance gas model ( hrgm ) @xcite which is the most successful one in describing the hadronic yields measured in heavy - ion experiments for energies from ags to lhc . despite a significant success of the hrgm , in the experimental data analysis there are a few unresolved problems . in general they are related to the description of hadron yields which contain ( anti)strange quarks . thus , the energy dependence of @xmath4 and @xmath5 ratios remained out of high quality description for almost a decade . excess of strange hadrons yields within the hrgm led physical community to ponder over strangeness suppression in heavy ion collisions . the first receipt to resolve this problem was to introduce the strangeness suppression factor @xmath0 which should be fitted in order to describe the experimental data @xcite . however , such an approach is not supported by any underlying physical model and the physical meaning of @xmath0 remains unclear @xcite . in addition the strangeness suppression approach in its original form does not contain a hard - core repulsion between hadrons , while the latter is an important feature of the hrgm . a significant role of the hard - core repulsion was demonstrated once more in ref . @xcite where the global fit of hadron yield ratios was essentially improved ( to @xmath6 1.16 ) compared to all previous analyses . the most advanced way to account for the hard - core repulsion between hadrons is to consider a hadron gas as a multi - component mixture of particles with different hard - core radii @xcite . within this approach all baryons and mesons except for the kaons and the pions are endowed by the common hard - core radii @xmath7 and @xmath8 , respectively . at the same time the kaon and the pion radii @xmath9 and @xmath10 are fitted independently in order to provide the best description of @xmath4 ratio @xcite . this is an important finding since the non - monotonic energy dependence of @xmath4 ratio may indicate some qualitative changes of the system properties and may serve as a signal of the deconfinement onset . this is a reason why such a ratio known as the strangeness horn is of a special interest . note , that the multi - component approach substantially increased the strangeness horn description quality , without spoiling the other ratios including @xmath5 one . however , even this advanced approach does not reproduce the topmost point of the strangeness horn indicating that the data description is still not ideal . in order to resolve this problem in ref . @xcite the @xmath0 factor was considered as a free parameter within the hrgm with multi - component repulsion . although the @xmath0 data fit sizably improves the quality of strangeness horn description , it does not seem to be useful for the description of other hadron multiplicities @xcite . furthermore , in contrast to the claims established on the low - quality fit @xcite , at low energies it was found @xcite that within the error bars in heavy ion collisions there is an enhancement of strangeness , i.e. @xmath11 , and not a suppression . the strangeness enhancement was confirmed very recently @xcite by the high quality fit of the available data within the multicomponent hrgm in which the hard - core radius of @xmath12 ( anti)hyperon was considered as a global fitting parameter in addition to the set of hard - core radii used in @xcite . however , the effect of apparent strangeness non - equilibration can be more successfully explained by the hypothesis of separate chemical fo for all strange hadrons . since all the hadrons made of @xmath13 and @xmath14 quarks are under thermal equilibration whereas the hadrons containing @xmath15 quark are not , then it is reasonable to assume two different fos for these two kinds of particles . following this conclusion in ref . @xcite a separate strangeness fo ( sfo ) was introduced . note , that according to @xcite both fo and sfo parameters are connected by the conservation laws of entropy , baryonic charge and isospin projection , while the net strangeness is explicitly set to zero at fo and at sfo . these conservation laws are crucial elements of the concept of separate sfo developed in @xcite which allows one to essentially reduce the number of independent fitting parameters . another principal element that differs the hrgm of @xcite from the ideal gas treatment used in @xcite is the presence of multi - component hard - core repulsion .    using the hrgm of @xcite it was possible to successfully describe all hadron multiplicities measured in a+a collisions at ags , sps and rhic energies with @xmath6 1.06 . the concept of separate sfo led to a systematic improvement of all experimental data description . however , the topmost point of the strangeness horn again was not fitted even within the experimental error , although the general description of @xmath16 ratio energy dependence was rather good except for the upper point . since an introduction of the @xmath17 factor demonstrated a remarkable description of all points of the strangeness horn , whereas the separate sfo led to a systematic improvement of all hadron yields description , we decided to combine these elements in order to describe an experimental data with the highest possible quality . in this way we would like to examine the problem whether the concept of separate sfo is able to completely explain a possible non - equilibrium of strange charge and whether on top of the sfo there exist a necessity to employ the @xmath0 factor in statistical approach . note that from the academic point of view the problem of residual strangeness non - equilibration , i.e. the question whether the strange charge is or is not in full chemical equilibrium , is of principal importance . this ambitious task is the main goal of the present paper . evidently , the best tool for such a purpose is the most successful version of the hrgm , i.e. the hrgm with the multi - component hadronic repulsion and a separate sfo . as it will be shown below , such an approach allows us to describe 111 independent hadron yield ratios measured for 14 values of the center of mass collision energy @xmath18 in the interval from 2.7 gev to 200 gev with very high quality . the paper is organized as follows . the basic features of the developed model are outlined in section 2 , while the fitting procedure of the present model is outlined in section 3 . the main obtained results are compared with the other models in section 4 . in section 5 we discuss in detail the new fit of hadronic multiplicity ratios with two chemical freeze - outs and @xmath0 factor , while section 6 contains our conclusions . in what follows we treat a hadronic system as a multi - component boltzmann gas of hard spheres . the effects of quantum statistics are negligible for typical temperatures of the hadronic gas whereas the hard - core repulsion between the particles significantly affects a corresponding equation of state @xcite . the present model is dealing with the grand canonical treatment . hence a thermodynamic state of system under consideration is fixed by the volume @xmath19 , the temperature @xmath20 , the baryonic chemical potential @xmath21 , the strange chemical potential @xmath22 and the chemical potential of the isospin third component @xmath23 . these parameters control the pressure @xmath24 of the system . in addition they define the densities @xmath25 of corresponding charges @xmath26 ( @xmath27 ) . introducing the symmetric matrix of the second virial coefficients @xmath28 , we can obtain the parametric equation of state of the present model in a compact form @xmath29 using the partial pressure @xmath30 of @xmath31-th sort of particles . the equation of state is written in terms of the solutions @xmath30 of the following system @xmath32   \ , , \quad \quad \\ & & \hspace*{-4mm}\phi_i ( t )   = \frac{g_i}{(2\pi)^3}\int \exp\left(-\frac{\sqrt{k^2+m_i^2}}{t } \right)d^3k   \ , . % \label{eqiv } % \end{aligned}\ ] ] each @xmath33 sort is characterized by its full chemical potential @xmath34 , mass @xmath35 and degeneracy @xmath36 . the function @xmath37 denotes the corresponding particle thermal density in case of ideal gas . the obtained model parameters for two freeze - outs and their dependence on the collision energy are discussed in the next section . they are obtained for the following values of hard - core radii which were determined earlier in @xcite : @xmath38 fm for pions , @xmath39 fm for kaons , @xmath40 for all other mesons and @xmath41 fm for all baryons .    in order to account for the possible strangeness non - equilibration we introduce the @xmath17 factor in a conventional way by replacing @xmath42 in eq . ( [ eqii ] ) as @xmath43 where @xmath44 is a number of strange valence quarks plus number of strange valence anti - quarks . the principal difference of the present model from the traditional approaches is that we employ an independent chemical fo of strange particles . let us consider this in some detail . the independent freeze - out of strangeness means that inelastic reactions ( except for the decays ) with hadrons made of s quarks are switched off at the temperature @xmath45 , the baryonic chemical potential @xmath46 , the strange chemical potential @xmath47 , the isospin third projection chemical potential @xmath48 and the three - dimensional emission volume @xmath49 . in general case these parameters of sfo do not coincide with the temperature @xmath50 , the chemical potentials @xmath51 , @xmath52 , @xmath53 and the volume @xmath54 which characterize the freeze - out of non - strange hadrons . the particle yields are given by the charge density @xmath55 in ( 1 ) and the corresponding volume at fo and at sfo . at the first glance a model with independent sfo contains four extra fitting parameters for each energy value compared to the traditional approach ( temperature , three chemical potentials and the volume at sfo instead of strangeness suppression / enhancement factor @xmath17 ) . however , this is not the case due to the conservation laws . indeed , since the entropy , the baryonic charge and the isospin third projection are conserved , then the parameters of fo and sfo are connected by the following equations @xmath56 where the entropy density @xmath57 , the density of baryonic charge @xmath58 and the density of the isospin third projection @xmath59 are found from the usual thermodynamic identities at the sfo ( a = sfo ) or at fo ( a = fo ) .    the effective volumes can be excluded , if these equations are rewritten as @xmath60 thus , the baryonic @xmath46 and the isospin third projection @xmath48 chemical potentials at sfo are now defined by eqs . ( [ eq : fo_sfo1 ] ) . note , that the strange chemical potentials @xmath52 and @xmath47 are found from the condition of vanishing net strangeness at fo and sfo , respectively . therefore , the concept of independent sfo leads to an appearance of one independently fitting parameter @xmath45 . hence , the independent fitting parameters are the following : the baryonic chemical potential @xmath21 , the chemical potential of the third projection of isospin @xmath23 , the chemical freeze - out temperature for strange hadrons @xmath45 , the chemical freeze - out temperature for all non - strange hadrons @xmath50 and the @xmath17 factor ( i.e. 5 fitting parameters for each collision energy ) .    an inclusion of the width @xmath61 of hadronic states is an important element of the present model . it is due to the fact that the thermodynamic properties of the hadronic system are sensitive to the width @xcite . in order to account for the finite width of resonances we perform the usual modification of the thermal particle density @xmath42 . namely , we convolute the boltzmann exponent under the integral over momentum with the normalized breit - wigner mass distribution . as a result , the modified thermal particle density of @xmath33 sort hadron acquires the form @xmath62 here @xmath35 denotes the mean mass of hadron and @xmath63 stands for the threshold in the dominant decay channel . the main advantages of this approximation is a simplicity of its realization and a clear way to account for the finite width of hadrons . it is appropriate here to mention that one could use other prescriptions to account for the width of resonances . however , in a recent work @xcite it was shown the breit - wigner prescription ( [ eqiii ] ) can provide somewhat better ( about 20% ) quality of the fit than the gaussian attenuation of resonance mass and essentially better one compared to the case without accounting for the width . at the same time in @xcite it was found that within the error bars such fo parameters as temperature , chemical potentials and @xmath17 factor are the same for different prescriptions of resonance width accounting . clearly , it is reasonable to expect that other physically motivated ways to account for the resonance width should give similar results . therefore , in this work we employed the prescription ( [ eqiii ] ) , which provides the better description of the data . the observed hadronic multiplicities contain the thermal and decay contributions . for example , a large part of pions is produced by the decays of heavier hadrons . therefore , the total multiplicity is obtained as a sum of thermal and decay multiplicities , exactly as it is done in a conventional model . however , writing the formula for final particle densities , we have to take into account that the fo and sfo volumes can be different : @xmath64 here the first term on the right hand side is due to decays after fo whereas the second one accounts for the strange resonances decayed after sfo . the factor @xmath65 can be replaced by @xmath66 due to baryonic charge conservation . @xmath67 denotes the branching ratio of the y - th hadron decay into the x - th hadron , with the definition @xmath68 = 1 used for the sake of convenience . the input parameters of the present model ( masses @xmath35 , widths @xmath61 , degeneracies @xmath36 and branching ratios of all strong decays ) were taken from the particle tables of the thermodynamical code thermus @xcite . * data sets*. the present model is applied to fit the data . we take the ratios of particle multiplicities at midrapidity as the data points . in contrast to fitting multiplicities themselves such an approach allows us to cancel the possible experimental biases . in this paper we use the data set almost identical to ref . @xcite . at the ags energies ( @xmath69 agev or @xmath70 agev ) the data are available with a good energy resolution above 2 agev . however , for the beam energies 2 , 4 , 6 and 8 agev only a few data points are available . they corresponds to the yields for pions @xcite , for protons @xcite , for kaons @xcite ( except for 2 agev ) . the integrated over @xmath71 data are also available for @xmath12 hyperons @xcite and for @xmath72 hyperons ( for 6 agev only ) @xcite . however , as was argued in ref . @xcite , the data for @xmath12 and @xmath72 should be recalculated for midrapidity . therefore , instead of raw experimental data we used the corrected values from @xcite . next comes the data set at the highest ags energy ( @xmath73 agev or @xmath74 agev ) . similarly to @xcite , here we analyzed only the na49 mid - rapidity data @xcite . since the rhic high energy data of different collaborations agree with each other , we analyzed the star results for @xmath75 gev @xcite , @xmath76 gev @xcite , @xmath77 gev @xcite and @xmath78 200 gev @xcite . .the fit results of different versions of the hrgm are compared for 14 values of the center of mass collision energies : the column @xmath79 corresponds to a single fo model of @xcite ; the column @xmath80 is found for the sfo with @xmath81 @xcite ; the column @xmath82 corresponds to the sfo+@xmath17 fit with added data points for @xmath83 while the column @xmath84 is obtained by the direct sfo@xmath85 fit . @xmath86 indicates the available number of independent hadronic ratios at given center of mass collision energy @xmath18 . in the row sum we list the sum of @xmath31-th column , while in the bottom row the number of degrees of freedom of each hrgm version is shown ( for more details see the text ) . [ cols="^,^,^,^,^,^ " , ]      in fig . [ fig : saguniiib ] we show the ratio of baryonic charge densities at sfo and at fo within the sfo@xmath87 model , which coincides with the inverse ratio of corresponding freeze - out volumes @xmath88 due to the baryonic charge conservation . from this figure one can see that visually small difference of temperatures and baryonic chemical potentials at sfo and fo leads , nevertheless , to quite sizable difference of other thermodynamic quantities . as one can see from the upper panel of fig . [ fig : saguniv ] , this is also true for the pressure existing at sfo and at fo .    also we confirm the existence of irregularities in the fo pressure found earlier @xcite . similar irregularities we find for the sfo pressure and for the effective number of degrees of freedom for sfo , @xmath89 , i.e. for the ratio of sfo pressure to the fourth power of the sfo temperature . from the upper panel of fig . [ fig : saguniv ] one can see that the largest increase of sfo pressure per increase of the center of mass energy of collision occurs at @xmath90 gev . in other words , for about 14% increase of @xmath18 the sfo pressure increases in 5.3 times and the ratio @xmath89 increases on about 65 % . according to the recent work @xcite these and other irregularities observed at chemical fo @xcite are signaling about the formation of the mixed quark - gluon - hadron phase and such an explanation can be experimentally verified in a few years at fair and nica . one more important finding of the present work can be seen from a comparison of the upper and lower panels of fig . [ fig : saguniv ] . note that in contrast to the temperature or baryonic chemical potential , the pressure allows one easier to distinguish the sfo from fo . moreover , comparing the squares in the upper and lower panels of fig . [ fig : saguniv ] one immediately concludes that the model of a single chemical fo @xcite with the same value of hard - core radius for all hadrons describes just the sfo for all values of @xmath18 below 62.4 gev . the same is true for a single fo model @xcite with different hard - core radii discussed above . this peculiar result can be easily understood , if one recalls that for the most values of collision energy the number of ratios involving strange hadrons is essentially larger than the number of ratios with non - strange particles .    at the same time a single chemical fo model reproduces the fo pressure of the sfo+@xmath17 model only at @xmath91 gev . the corresponding reasons we will discuss in the next section , while here we mention that at high rhic energies the fit quality of all models , including the single fo one , is rather high as one can see from table i. the main part of @xmath92 at these energies is formed by poor description of @xmath93-mesons at @xmath94 gev , @xmath12 hyperon at @xmath95 gev and @xmath96 hyperons at @xmath97 gev . such a comparison of the single fo model and the sfo+@xmath17 model pressures allows us to explain a cause of why in previous thorough analysis of the particle ratios within the realistic single fo model with the same hard - core radius of hadrons the main conclusion was that there is no deviation of strange particles from chemical equilibrium for the mid - rapidity data . in other words , it is possible to naturally explain the reason of why in @xcite it was found that within the error bars @xmath98 . our direct comparison of the sfo+@xmath17 model for the mid - rapidity data shows that the single fo models with the same or different hard - core hadronic radii reproduce the sfo pressure with @xmath98 almost at all collision energies , except for @xmath99 gev and for @xmath100 gev . in the former case one finds that @xmath101 , while in the latter case @xmath98 , but , as we discussed above , for high collision energies the pressure at fo found within the sfo+@xmath17 model reproduces the single fo model pressure . therefore , the main reason of why the chemical non - equilibrium of strange charge was not found within the single fo models is that the fitting procedure mainly described the ratios involving the strange particles and , hence , it would have been more appropriate to consider the chemical non - equilibrium of non - strange hadrons . the findings discussed above motivate us to study in some details what ratios and at what energies are improved . the most significant improvements correspond to the collision energies @xmath102 6.3 , 7.6 , and 12 gev , that are plotted in figs . [ fig : sagunv ] and [ fig : sagunvb ] . figs . [ fig : sagunv ] and [ fig : sagunvb ] demonstrate very high fit quality , especially for such traditionally problematic ratios as @xmath16 , @xmath103 , @xmath104 and @xmath105 , which is achieved within the sfo@xmath87 model compared to the single fo model and the sfo one . for instance , for @xmath102 7.6 gev seven ratios out of ten are improved , while for other energies the improvements are less significant . on the contrary , the particle ratios measured at @xmath102 17 gev ( see fig . [ fig : sagunvb ] ) are improved within the sfo model , while the sfo@xmath87 fit practically does not lead to any significant improvement compared to the sfo model . also we found that the sfo@xmath87 fit leads to a selective improvement and to a certain degradation of the fit quality of various ratios for different collision energies . for instance , the @xmath103 ratio is slightly increased for @xmath102 6.3 and 7.6 gev , but the situation drastically changes for @xmath102 12 gev . the same tendency is typical for @xmath106 . on the contrary , for @xmath107 ratio there is a noticeably worse data description within sfo+@xmath17 approach at @xmath102 6.3 , 7.6 gev , but for @xmath102 12 gev the fit quality is sizably better compared to all previous approaches . thus , within the present model we reveal a noticeable change in the trend of some ratios at @xmath102 7.6 - 12 gev , while at @xmath102 12 gev we do not observe any sizable improvement compared to the sfo model . a special attention in our consideration was paid to the strangeness horn , i.e. to the @xmath16 ratio , because such a ratio is traditionally the most problematic one for the hrgm to fit it . as one can see from fig . [ fig : sagunvi ] , the remarkable @xmath16 fit improvement for @xmath102 2.7 , 3.3 , 4.3 , 4.9 , 6.3 , 7.6 , 12 gev justifies the usage of the present model . quantitatively , we found that the @xmath108 improvement for the sfo+@xmath17 model is @xmath108=1.5/14 , i.e. even better than it was achieved in @xcite with @xmath108=3.3/14 for the @xmath17 fitting approach and with @xmath108=6.3/14 for the sfo model with @xmath81 .    from fig . [ fig : sagunvii ] one can see that at two highest rhic energies the description of ratios within the single fo model is very good for all ratios except for @xmath104 at @xmath95 gev and for @xmath109 at @xmath97 gev . the main reason is that at these energies all chemical potentials except for the strange one are almost zero and hence the number of particles and antiparticles is almost the same . as one can see from the upper panel of fig . [ fig : sagunvii ] , the sfo@xmath87 fit significantly improves only the @xmath110 and @xmath111 ratios compared to the single fo model , i.e. only two ratios of strange hadrons responded to the variation of two additional parameters . for @xmath97 gev the sfo@xmath87 fit significantly improves only the @xmath109 ratio and worsens the @xmath112 ratio and @xmath113 ( less ) , i.e. only three ratios of strange hadrons responded to such a sophisticated fit . for @xmath94 gev only two ratios out of five include kaons and , hence , using @xmath45 and @xmath17 one can perfectly reproduce the strange particle ratios without affecting the non - strange ones . treating the @xmath45 values found within the sfo model as an additional datum for the sfo@xmath87 fit , we obtain the same result . therefore , in contrast to low collision energies at high collision energies only a few ratios with strange particles can be improved by simultaneous variation of @xmath45 and @xmath17 and , hence , the sfo can not represent the majority of fitted ratios which are well reproduced even within the single fo model with a single or with several hard - core radii . the lower panel of fig . [ fig : saguniv ] evidently supports such a conclusion for the hrgm with the same value of hadronic hard - core radius .    within the sfo@xmath87 model the @xmath114 and @xmath115 ratios demonstrate some worsening compared to less sophisticated models . in fig . [ fig : sagunviii ] we show that the sfo@xmath87 model still does not improve these ratios . the @xmath114 fit quality , for instance , is @xmath108=10/8 . hence , up to now the best fit of the @xmath114 ratio was obtained within the sfo approach with @xmath81 . as it was mentioned in @xcite a too slow decrease of model results for @xmath114 ratio compared to the experimental data is typical for almost all statistical models . evidently , the too steep rise in @xmath114 behavior is a consequence of the @xmath116 anomaly @xcite . similar results are reported in refs . @xcite as the @xmath117 , @xmath2 and @xmath3 selective suppression . since even an introduction of the separate strangeness freeze - out with the strangeness enhancement factor does not allow us to better describe these ratios , we believe that there is a corresponding physical reason which is responsible for it . one of them could be a necessity to introduce the different hard - core radius @xmath118 for the @xmath119 ( anti)hyperons @xcite . we present a thorough investigation of the data measured at ags , sps and rhic energies within different versions of the multi - component hadron resonance gas model . the suggested approach to separately treat the freeze - outs of strange and non - strange hadrons with the simultaneous @xmath17 fitting gives rise for the top - notch strangeness horn description with @xmath108=1.5/14 . the developed model clearly demonstrates that the successful fit of hadronic multiplicities includes all the advantages of these two approaches discussed in @xcite . as a result for @xmath102 6.3 , 7.6 , 12 , 130 gev we found a significant data fit quality improvement .    at the same time the lack of available data at @xmath18 = 2.7 , 3.3 , 3.8 , 4.3 , 9.2 , 62.4 gev forced us to redefine the fitting procedure at these collision energies in order to avoid the mathematical inconsistency which in combination with the large experimental error bars led to rather large uncertainties of the fitting parameters . the suggested redefinition of the fitting procedure by including the @xmath45 temperatures obtained for these energies within the sfo model allowed us to avoid the mathematical problems and to get the safe answers on the values of the residual chemical non - equilibrium of strange particles . the developed sophisticated hrgm , i.e. the sfo+@xmath17 model , allowed us to describe the hadron multiplicity ratios with rather high quality @xmath120 0.93 . this very fact demonstrates that the suggested approach is a precise tool to elucidate the thermodynamics properties of hadron matter at two chemical freeze - outs . the fresh illustrations to this statement can be found in @xcite . the achieved total value of @xmath121 for the sfo+@xmath17 model is almost 50% lower than the @xmath92 value found for the single fo model and 30% lower than the sfo model @xmath92 value . the obtained @xmath17 values are consistent with the conclusion @xmath98 ( within the error bars ) . an evident exception is the topmost point of the strangeness horn ( located at @xmath102 7.6 gev ) , at which the mean value of the strangeness enhancement factor is @xmath122 . to reveal the physical reason for such a deviation we need more experimental data with an essentially higher accuracy . one of the main results obtained here is that the idea of separate chemical fo of strange hadrons provides a very high quality of the data description . further inclusion of the chemical non - equilibrium on top of the sfo is consistent with the result @xmath98 for all energies except for @xmath102 7.6 gev . thus , the found residual chemical non - equilibrium of strange particles is weak and , hence , it can be safely ignored for all energies except for @xmath102 7.6 gev . the strange charge enhancement of about 20% found at this collision energy allowed us to perfectly describe the topmost point of the strangeness horn , but at the expense of the worsening of @xmath114 and @xmath123 ratios . in addition , the description of ratios containing the non - strange particles , especially such as @xmath103 and @xmath106 , gets better compared to previously reported results @xcite . the remaining problem of the ratios involving the @xmath12 and @xmath124 ( anti)hyperons can be resolved by an inclusion of the different hard - core radius for @xmath12 ( anti)particles @xcite , but such a treatment is out of scope of the present work . the performed analysis of the sfo+@xmath17 model hadronic pressures existing at fo and at sfo allowed us to elucidate an important conclusion that the single fo models with the same hard - core radius @xcite or with different hard - core radii @xcite for all hadrons reproduce the sfo pressure for all collision energies below @xmath94 gev . the main reason for such a behavior is that the number of ratios involving strange hadrons is larger than the number of ratios with non - strange hadrons . also we report an existence of strong jumps in the sfo pressure , the sfo temperature and the corresponding effective number of degrees of freedom , when the center of mass collision energy changes from 4.3 to 4.9 gev . based on the concept of non - smooth chemical freeze - out introduced recently in @xcite , we parameterized the dependencies @xmath125 and @xmath126 which can be verified in the future experiments planned at fair ( gsi ) and nica ( jinr ) . we hope that the high precision data measured in these experiments will allow us to finally answer the question whether the residual non - equilibrium of strange charge is necessary to describe the topmost point of the strangeness horn or the concept of two separate chemical freeze - out for strange and non - strange hadrons can do this without introducing the @xmath17 factor . * acknowledgments . * we would like to thank a. andronic for providing an access to well - structured experimental data . the authors are thankful to i. n. mishustin , d. h. rischke and l. m. satarov for valuable comments . , a.i.i . and g.m.z . acknowledge a support of the fundamental research state fund of ukraine , project no f58/04 . k.a.b . acknowledges also a partial support provided by the helmholtz international center for fair within the framework of the loewe program launched by the state of hesse .
we present an elaborate version of the hadron resonance gas model with the combined treatment of separate chemical freeze - outs for strange and non - strange hadrons and with an additional @xmath0 factor which accounts for the remaining strange particle non - equilibration . within 
 suggested approach the parameters of two chemical freeze - outs are connected by the conservation laws of entropy , baryonic charge , third isospin projection and strangeness . 
 the developed model enables us to perform a high - quality fit of the hadron multiplicity ratios measured at ags , sps and rhic with @xmath1 . 
 a special attention is paid to a successful description of the strangeness horn . 
 the well - known problem of selective suppression of @xmath2 and @xmath3 hyperons is also discussed . 
 the main result is that for all collision energies the @xmath0 factor is about 1 within the error bars , except for the center of mass collision energy 7.6 gev at which we find about 20% enhancement of strangeness . 
 also we confirm an existence of strong jumps in pressure , temperature and effective number of degrees of freedom at the stage of strange particle chemical freeze - out , when the center of mass collision energy changes from 4.3 to 4.9 gev . 
 we argue that these irregularities may signal about the quark - hadron phase transition .