Liquid-gas phase transition in strange hadronic matter with relativistic models
aa r X i v : . [ nu c l - t h ] O c t Liquid-gas phase transition in strange hadronic matter with relativistic models
James R. Torres, F. Gulminelli, and D´ebora P. Menezes Departamento de F´ısica - CFM - Universidade Federal de Santa Catarina,Florian´opolis - SC - CP. 476 - CEP 88.040 - 900 - Brazilemail: [email protected] CNRS and ENSICAEN, UMR6534, LPC,14050 Caen c´edex, Franceemail: [email protected] Departamento de F´ısica - CFM - Universidade Federal de Santa Catarina,Florian´opolis - SC - CP. 476 - CEP 88.040 - 900 - Brazilemail: [email protected]
Background
The advent of new dedicated experimental programs on hyperon physics is rapidly boosting the field, and the possibility ofsynthetizing multiple strange hypernuclei requires the addition of the strangeness degree of freedom to the models dedicated to nuclearstructure and nuclear matter studies at low energy.
Purpose
We want to settle the influence of strangeness on the nuclear liquid-gas phase transition. Because of the large uncertainties concerningthe hyperon sector, we do not aim at a quantitative estimation of the phase diagram but rather at a qualitative description of thephenomenology, as model independent as possible.
Method
We analyze the phase diagram of low density matter composed of neutrons, protons and L hyperons using a Relativistic Mean Field(RMF) model. We largely explore the parameter space to pin down generic features of the phase transition, and compare the results toab-initio quantum Monte Carlo calculations. Results
We show that the liquid-gas phase transition is only slightly quenched by the addition of hyperons. Strangeness is seen to be anorder parameter of the phase transition, meaning that dilute strange matter is expected to be unstable with respect to the formation ofhyper-clusters.
Conclusions
More quantitative results within the RMF model need improved functionals at low density, possibly fitted to ab-initio calculationsof nuclear and L matter. PACS numbers: 05.70.Ce, 21.65.Cd, 95.30.Tg
I. INTRODUCTION
It is well known that nuclear matter below saturation ex-hibits a first-order phase transition belonging to the Liquid-Gas (LG) universality class [1–9]. The study of the associatedphase diagram is not only a playground for many-body theo-rists, but it is also of clear relevance for nuclear phenomenol-ogy, since the very existence of atomic nuclei can be under-stood as a finite size manifestation of that phase transition. Ina similar way, one can ask whether the existence of hypernu-clei as bound systems implies the presence of a similar phasetransition in the extended phase diagram where strangenessrepresents an extra dimension.Since the first synthesis of L -hypernuclei in the 80’s, nu-merous nuclear matter studies including hyperons have beenperformed [10–14]. These early studies assumed very attrac-tive couplings in the strange sector in order to justify the ex-tra binding measurements of double L -hypernuclei [15, 16].As a consequence, it was predicted that multi-strange clustersand even strangelets could be stable and possibly accessible inheavy-ion collisions. In particular in Ref.[14], the occurrenceof a thermodynamic phase transition in strange compressedbaryonic matter was predicted, which would lead to a newfamily of neutron stars characterized by much smaller radiithan usually considered.However, more recent analysis [17, 18] of double L -hypernuclei tend to suggest a very small attraction in the L − L channel, and the stability of pure L -matter seems to be ruled out. Most hypernuclear matter studies are nowa-days essentially motivated by assessing the strange contentof neutron star cores, and therefore concentrate on matter in b -equilibrium [19]. At b -equilibrium, no hyperons appearbelow baryonic densities of the order of 3 r or more. Forthis reason, the influence of strangeness on the low densitynuclear matter phase diagram was never studied to our knowl-edge. Still, the existence of single and double L -hypernuclei,and the very active research on multiply strange nuclei withthe advent of new dedicated experimental programs such asJ-Parc in Japan or PANDA at FAIR [20–23] suggests that thenuclear liquid-gas phase transition should be preserved by theconsideration of the strangeness degree of freedom [24].In this paper, we explore the influence of strangeness onthe LG phase transition with popular Relativistic Mean-Field(RMF) models. Like in any other phenomenological effectivemodel, the couplings of the RMF are not fully known even atsubsaturation densities. In particular, neutron star physics hastaught us in the recent years that it is important to go beyonda simple SU(6) or even SU(3) symmetry, and extra attractive s ∗ and repulsive f mesons specifically coupled to the strangebaryons should be introduced [25–30], which leads to a poten-tially uncontrolled multiplication of parameters. However, ifwe limit ourselves to the simple system composed of neutrons,protons and L -hyperons, nuclear and hypernuclear structureprovide some constraints that can be used to limit the param-eter space of the model. In this paper, we consider the simplelinear and non-linear Walecka model for the np L system, anddiscuss the modification of the nuclear matter phase diagramunder a wide variation of coupling constants, in the acceptableparameter space constrained both from hypernuclear data andab-initio calculations of hypernuclear matter. We show thatin the whole parameter space the LG phase transition is pre-served by the addition of strangeness, even if the extension ofthe spinodal along the strange density direction is subject tolarge uncertainties. The instability zone is globally quenchedby strangeness, but the strange density is an order parame-ter of the transition. This means that from the thermodynamicpoint of view, the formation of hyperclusters with multiple L ’sshould be favored at low density [21–23], which has possibleimplications in relativistic heavy ion collisions [24].The paper is organized as follows: section II shortly recallsthe main equations of the Walecka model, both in its linearand non-linear version, for the np L system with inclusion ofstrange mesons. Section III defines the coupling parameterspace of the model, under the constraints of well defined val-ues for the L -potential as requested by the available hypernu-clear data. To further refine the domain of acceptable parame-ters, Section IV compares the RMF functionals with recent ab-initio predictions of n L -matter with the Auxiliary Field Diffu-sion Monte Carlo (AFDMC) technique [31]. In Section V thegeneral formalism for the analysis of spinodal instabilities inmulti-component systems is revisited. The main results of ourwork are presented in Section VI, which shows in detail theinstability properties of n L and np L matter with the differentchoices for the couplings. Finally Section VII summarizes thepaper. II. FORMALISM
In this section, we present the hadronic Equation Of State(EOS) used in this work. We describe matter within theframework of Relativistic Mean Field (RMF) models involv-ing the interaction of Dirac baryons mediated by the scalarand vector mesons which are independent degrees of free-dom [32, 33, 35–40]. The scalar-isoscalar s field medi-ates the medium-range attraction between baryons, the vector-isoscalar w field mediates the short-range repulsion betweenbaryons, the strange scalar s ∗ field mediates the medium-range attraction between hyperons, strange vector f field me-diates the short-range repulsion between hyperons and finallythe r meson field allows us to adjust isovector properties ofnuclear matter. In the present work, we used the Nonlin-ear Walecka Model (NLWM) and the Linear Walecka Model(LWM), which can be obtained by just turning off the nonlin-ear terms, in the presence of the mesons listed above. Non-linear means that there are also self interaction terms forthe scalar field s in the Lagrangian density, as proposed byBoguta and Bodmer [33, 34], what provides better results thanthe LWM [40]. The Lagrangian density reads: L = (cid:229) j y j (cid:2) g m (cid:0) i ¶ m − g w j w m − g f j f m − g r j ~ t · ~ r m (cid:1) − m ∗ j (cid:3) y j + (cid:0) ¶ m s¶ m s − m s s (cid:1) − bM N ( g s N s ) − c ( g s N s ) + (cid:0) ¶ m s ∗ ¶ m s ∗ − m s s ∗ (cid:1) − W mn W mn + m w w m w m − F mn F mn + m f f m f m , − ~ R mn · ~ R mn + m r ~ r m · ~ r m , (2.1)where m ∗ j = m j − g s j s − g s ∗ j s ∗ is the baryon effective massand m j is the bare mass of the baryon j . The terms W mn = ¶ m w n − ¶ n w m , F mn = ¶ m f n − ¶ n f m and ~ R mn = ¶ m ~ r n − ¶ n ~ r m − g r j (cid:0) ~ r m × ~ r n (cid:1) are the strength tensors, where the up arrowin the last term denotes the isospin vectorial space with the ~ t isospin operator. The coupling constants are g i j = c i j g iN ,with the mesons denoted by index i = s , w , r , s ∗ , f and thebaryons denoted by j . Note that c i j is a proportionality factorbetween g i j and the nucleon coupling constants g iN , with N = n , p . The couplings b and c are the weights of the nonlinearscalar terms. The sum over j can be extended over all baryonsof the octet (cid:0) n , p , L , S − , S , S + , X − , X (cid:1) .The values of the coupling constants of the nucleons withmesons s , w and r are obtained from the phenomenology.These constants are tuned to the bulk properties of nuclearmatter. Some of these properties are not known exactly, justwithin certain ranges, like the effective masses of the nucle-ons, therefore there are many sets of parameters that describethe bulk properties. The biggest uncertainties concern the hy-peron coupling constants, because the phenomenological in-formation from hypernuclei is not sufficient to completely pindown the interaction in the strange sector [41, 42]. The hy-peron couplings are chosen in different ways in the literature,either based on simple symmetry considerations [28, 43–47],or requiring an EOS in b -equilibrium sufficiently stiff to jus-tify the observation of very massive neutron stars [48, 49].Some different approaches, all affected by a certain degreeof arbitrariness, are listed here: 1) Some authors argue that c s j = c w j = c r j = p / c s L = c w L = c s S = c w S = / c s X = c w X = / c r L = c r S = c r X =
0; 3) Based on the ex-perimental analysis of L -hypernuclei data, an alternative con-straint is given by U L ( n N = n ) = c w L ( g w N ) − c s L ( g s N ) = −
28 MeV for the fixed c s L = .
75. This last case can beextended to the whole baryonic octet, indexed by j , setting c s j = . c w j is given by the above constraint and c r h = h is the hyperon index [48]; 4) Taking into account theresulting neutron star maximum mass [48, 49].In the case of the inclusion of the strange mesons, s ∗ and f [25–27], we have to ensure that the nuclear matter proper-ties are preserved when these new mesons are included. Newmesons mean new interactions and also new constants, there-fore the arbitrariness introduced by these constants must beeliminated by data whenever possible. In analogy with whathas been done with the g s L , when constrained by the hyper-nuclear potential U N L via hypernuclear data [48], we can try totie the strange constants to the U LL data available in literature[17, 18, 50–58]. In the next section we develop these ideas indetail.Applying the Euler-Lagrange equations to the lagrangiandensity Eq.(2.1) and using the mean-field approximation [37],( s → h s i = s ; w m → h w m i = d m w ; ~ r m → h ~ r m i = d m d i r ≡ d m d i r ; s ∗ → h s ∗ i = s ∗ ; f m → h f m i = d m f ), we obtain the following equations of motion for themeson fields at zero temperature: ( g s N s ) = D s (cid:229) j c s j r sj − bM n ( g s N s ) − c ( g s N s ) ! , ( g w N w ) = D w (cid:229) j c w j n j , (cid:0) g r N r (cid:1) = D r (cid:229) j t j c r j n j , ( g s N s ∗ ) = D ss ∗ (cid:229) j c s ∗ j r sj , ( g w N f ) = D wf (cid:229) j c f j n j , (2.2)where for simplicity we define the following factors: D s = (cid:16) g s N m s (cid:17) , D w = (cid:16) g w N m w (cid:17) , D r = (cid:16) g r N m r (cid:17) , D ss ∗ = (cid:16) g s N m s ∗ (cid:17) , D wf = (cid:16) g w N m f (cid:17) , c s j , c s ∗ j , c w j , c r j and c f j are ratios be-tween coupling constants and t j is the third component ofthe isospin projection of the j -baryon. The scalar and baryondensities are given respectively by r sj = g p Z k F j m ∗ j q p + m ∗ j p d p (2.3)and n j = g p Z k F j p d p . (2.4)The energy density of the baryons is given by e B = g p (cid:229) j Z k F j p q p + m ∗ j d p (2.5)and for the mesons e M = ( g s N s ) D s + ( g w N w ) D w + (cid:0) g r N r (cid:1) D r + ( g s N s ∗ ) D ss ∗ + ( g w N f ) D wf + bM n ( g s N s ) + c ( g s N s ) . (2.6)Finally the total energy density is the summation e = e B + e M . To obtain the chemical potential, one has to take the deriva-tives of the energy density with respect to the baryon den-sity [40]. Note the dependence of the Fermi momenta and thefields with the baryon density in the upper limit of the inte-grals in Eq.(2.5) and Eq.(2.6) respectively. Using the deriva-tive chain rule and the equation of motion for the s field, weobtain m ∗ j = m j − c s j ( g w N w ) − t j c r j (cid:0) g r N r (cid:1) − c w j (cid:0) g f N f (cid:1) . (2.7)The total pressure is p = p B + p M , where p B is the baryonic pressure given by p B = g p (cid:229) j Z k F j p q p + m ∗ j d p , (2.8)and p M is the pressure of the mesons: p M = − ( g s N s ) D s + ( g w N w ) D w + (cid:0) g r N r (cid:1) D r − ( g s N s ∗ ) D ss ∗ + ( g w N f ) D wf − bM n ( g s N s ) − c ( g s N s ) . (2.9) III. LAMBDAS IN (HYPER)NUCLEAR MATTER
Inspired by the pioneer works on the role of the isospinin the liquid-gas phase transition [3, 5, 7, 59–63], along withmore recent works on the role of the strangeness in the phasetransition of dense neutron star matter [64–69], in this workwe want to study the role of strangeness in the low densityand zero temperature LG phase transition, which can be phe-nomenologically associated to multiple strange bound hyper-nuclei [23, 24]. Because of the huge uncertainties in the strange sector wedo not aim at having quantitative predictions on that phasetransition, but would like to get qualitative statements andavoid as much as possible the model dependence of the re-sults. For this reason we shall explore as widely as possiblethe largely unconstrained parameter space of the hyperon cou-plings. In this section we detail the criteria employed to fix thesize of the parameter space.Concerning the nucleon sector, we used the GM1 parame-terization for the NLWM [44] and the original Walecka [37]parametrization for the LWM.The two sets of parameters are denoted by NLWM andLWM respectively shown in Tab.I with the fitted nuclear bulkproperties.
NLWM LWM n (fm − ) 0.153 0.17 K (MeV) 300 554 m ∗ / m B / A (MeV) -16.3 -15.95 E sym (MeV) 32.5 39.22 L (MeV) 94 127.22 D s (fm ) 11.785 13.670 D w (fm ) 7.148 10.250 D r (fm ) 4.410 4.410 D ss ∗ (fm ) 3.216 3.769 D wf (fm ) 4.212 6.040 b c -0.001070 0.000TABLE I: Sets of parameters used in this work and correspondingsaturation properties. It is well known that the value of the symmetric nuclearmatter incompressibility does not qualitatively influence thephase diagram, nor do the uncertainties on the other param-eters. We therefore consider the NLWM couplings as suffi-ciently well settled and do not play with them in the follow-ing. To fully explore the phenomenology of the model in thestrange sector, the hyperon couplings are considered as freeparameters, which however have to fulfill minimal require-ments in terms of the potential and the hypernuclei data. Tobe clear with the notation in the following, the general func-tion associated to the L -potential is the three variable func-tion: U L ( n n , n p , n L ) . For symmetric matter n n = n p we havea two variable function U L ( n N , n L ) . The one variable L N po-tential is denoted by U N L ( n N ) ≡ U L ( n N , n L = ) and finallyfor LL potential we have U LL ( n L ) ≡ U L ( n N = , n L ) , where n N = n n + n p is the nucleon density. For simplicity some-times we omit the dependence of the potential function withrespect to the density variables. The c L couplings tell ushow attractive or repulsive the U L can be. For the hyperoncoupling constants, it is difficult to fix these phenomenolog-ical parameters due to the scarcity of data available in spe-cial for the multi-hyperon nuclei. Hence when the s ∗ and f are taken into consideration we need some data from single- L and double- L nuclei. Based on data on single- L producedin ( p + , K + ) reactions, the presently accepted value of thesingle- L in symmetric nuclear matter at saturation density, U N L ( n ) is ≈ −
30 MeV [54, 55]. For multi-hyperon there areavailable data just for the double- L light nuclei, like LL Be, LL Be and LL He, and the measurements are related to the LL bond energy. This energy can be estimated from the bindingenergy difference between double- L and single- L hypernu-clei denoted by D B LL . In this work we consider the followingvalue D B LL = .
67 MeV [17, 18, 56–58]. The D B LL can beinterpreted as a rough estimation of the U LL potential at theaverage L density h n L i ∼ n / n is the saturation point of symmetric nuclear matterin Table I. Hence, the U N L ( n ) = −
28 MeV potential data canbe used to tie the c w L to the c s L . For strange mesons, us-ing U LL ( n / ) = − .
647 MeV we intend to link the c f L tothe c s ∗ L . The general form of the L -potential U L in the RMFmodels considered is given by U L ( n N , n L ) = c w L ( g w N w ) + c f L ( g w N f ) − c s L ( g s N s ) − c s ∗ L ( g s N s ∗ ) , (3.1)where the dependence on the densities is given by the equa-tions of motion of the meson fields, and n N is symmetric nu-clear matter density. Nucleons and L ’s exchange s and w mesons with each other, the first one being attractive while thesecond acts repulsively. These two mesons have no strangequantum number. The additional strange mesons are similarto the ordinary s and w but they see just the strange baryons,namely hyperons. The attractive force is due to the scalar me-son s ∗ and the repulsive is due to the strange vector meson f . For simplicity we can define w = ( g w N w ) , f = ( g w N f ) , s = ( g s N s ) and s ∗ = ( g s N s ∗ ) to rewrite Eq.(3.1) in termsof the densities instead of the fields U L ( n N , n L ) = c w L (cid:18) g w n m w (cid:19) n N − c s L (cid:18) g s n m s (cid:19) r sN ( s )+ " + (cid:18) c f L c w L (cid:19) (cid:18) m w m f (cid:19) × (cid:18) g w n m w (cid:19) ( c w L ) n L − " + (cid:18) c s ∗ L c s L (cid:19) (cid:18) m s m s ∗ (cid:19) × (cid:18) g s n m s (cid:19) ( c s L ) r s L ( s , s ∗ ) − c s L (cid:18) g s n m s (cid:19) (cid:2) − bm n s − c s (cid:3) . (3.2)We can look for one dimensional potential U N L ( n N ) , whichis just the L - potential for a L in nuclear symmetric matter andthis potential is a single variable function in nucleon density.It reads U N L ( n N ) = c w L (cid:18) g w N m w (cid:19) n N − c s L (cid:18) g s N m s (cid:19) × (cid:2) r sN ( s ) − bm n s − c s (cid:3) . (3.3)Here we can use the data for U N L ( n ) = −
28 MeV. Solvingthe above expression to c w L and using the equation of motionof the fields we have c w L = c s L s | N = n −
28 MeV w | N = n . (3.4) FIG. 1: (Color online). Relations between parameters in RMF.
The c s L is left to be a free parameter in the RMF mod-els. Fig.1 shows the relation between c s L and c w L when weconsider Eq.(3.4). For each choice of the c s L , a particularpotential is obtained in such a way that it is constrained to U N L ( n ; c s L ) = −
28 MeV. The linear dependence obtainedmeans that in the framework of the (N)LWM a strong attrac-tion at low densities is always correlated to a strong repulsionat high densities. It is interesting to remark that the same istrue in non-relativistic models [64–68].Fig.2 shows the family of potentials constrained by Eq.(3.4)in LWM and NLWM. We can see that a very wide varietyof behaviors is compatible with the hypernuclei constraint,which explains why dedicated RMF works to hypernuclearstructure have been able in the literature to reasonably fit theavailable single-particle levels with a large variety of choicesfor the couplings. We can also observe that the LWM andNLWM models produce very similar behaviors for this poten-tial. The main difference between the two models, for large c s L , is that the U N L potential in NLWM is deeper at low den-sities than the LWM due to the nonlinear terms and the pa-rameterization chosen.Now, we turn our attention to the U LL ( n L ) potential: U LL ( n L ) = " + (cid:18) c f L c w L (cid:19) (cid:18) m w m f (cid:19) ( c w L ) w − a ( c s L ) S , (3.5)where we have defined: a = + (cid:16) c s ∗ L c s L (cid:17) (cid:16) m s m s ∗ (cid:17) and S = s − (cid:18) g s N m s (cid:19) (cid:18) a − a (cid:19) (cid:0) − bm n s − c s (cid:1) . (3.6) (a)(b) FIG. 2: (Color online). U N L curves (constrained by data U N L ( n ) = −
28 MeV denoted by the black point) for some values of c s L in (a)LWM and (b) NLWM. (a) FIG. 3: (Color online). Relations between parameters in the NLWM.3D parameter space for for c f L constrained by U LL potential. Graypoints refer to parameters with which there is no numerical conver-gence in hyperonic stellar matter. The other chosen data is U LL (cid:0) n (cid:1) = − .
67 MeV. Therefore,solving Eq.(3.5) for c f L , we obtain: c f L = (cid:18) m f m w (cid:19) × vuut U LL (cid:0) n (cid:1) + ac s L S | n L = n − c w L w | n L = n c w L w | n L = n c w L . (3.7)The above expressions are valid for the NLWM, and theLWM expression is obtained for b = c =
0, when the S isreduced to the s field. Fig.3 shows the 3D parameter space c s L × c s ∗ L × c f L where we consider Eq.(3.7). Note that in Eq.(3.7) there are combinations of c s L and c s ∗ L that do notresult in real solutions. This gives a first trivial limitation forthe parameter values. The residual parameter space, shown inFig.3 for the NLWM, is still extremely large. Minimal con-straints can be added requiring that convergent solutions areobtained in hyperonic stellar matter (with all the baryon octet,electrons and muons included) in b equilibrium. c r = . L ’s are the first hyperons toappear and the (unconstrained) couplings to S do not play amajor role. The gray points in Fig.3 are related to divergentsolutions, where the L effective mass goes to zero at some fi-nite density. The red points yield possible solutions and, insome cases, the maximum masses can reach two solar masseswith a finite Y L [28, 29]. (a) (b)(c) (d) FIG. 4: (Color online). The black points in each of these figure denote U LL ( n / ) = − .
67 MeV. (a) and (b) show U LL potential without strangemesons for some values of c s L in LWM and NLWM respectively. (c) and (d) show U LL potential constrained to pass through the black pointfor some pairs of values of c s L and c s ∗ L in LWM and NLWM respectively. This study is only done with the NLWM because it is wellknown that the LWM leads to irrealistic results for high den-sity matter. Of course the LWM and NLWM gives differentEOS even at low density, but for now it is enough to restrictour parameter space substantially to start the study of possi-ble instabilities in hypernuclear matter at low density. Later, we see how drastic our choice for the c s L parameter is whenwe discuss the instabilities. A very similar reasoning with-out strange mesons was proposed in Ref.[36], where exper-imental values of the U N L were used to restrict the hyperon-meson coupling constants. In that paper, the resulting maxi-mum stellar masses were also analysed. Adding this conditionstill leaves us with a wide two-dimensional parameter space,which corresponds to an almost unconstrained model. A ma-jor simplification would be obtained if we do not introduceextra strange mesons. Indeed if we put c f L = c s ∗ L = L - L channel based on an analysis of hypernuclear datawhich nowadays appears questionable [30].The families of U LL potential curves without strange mesonsobtained with the LWM and the NLWM are shown in Figs.4(a) and (b). We can see that the only possibility of having thevery small extra binding suggested by experimental data, atthe low densities explored in hypernuclei, is to have a poten-tial which is unrealistically attractive at higher densities. Thisis due to the linear correlation between c s L and c w L observedin Fig.1. Consequently, the resulting EOS of stellar matter isclearly too soft. One can object that summarizing hypernu-clear data to two values for the L potential in infinite matter isa very crude approximation, which is certainly true. Howeverit is well known from very different approaches that dedicatedfits of hypernuclear data require some extra repulsion at higherdensity [10, 70], in qualitative agreement with our oversimpli-fied nuclear matter reasoning. This discussion implies that arealistic RMF model should probably include strange mesons,or alternatively more complex non-linear couplings, even ifthis is done at the price of considerably enlarging the param-eter space. In particular in this paper, our motivation beingto extract a phase diagram as general as possible, we preferconsidering a parameter space which is too large to one whichis too narrow. We will therefore stick to the parameter spacedefined by Fig.3.Figs.4 (c) and (d) display the LWM and NLWM U LL poten-tial with the inclusion of strange mesons, and with the extra re-quirement of fulfilling Eq.(3.7). We can see that a wide rangeof behaviors is still possible. Figs.4 (c) and (d) are globallysimilar, although in (d) the potential is slightly deeper than in(c) at very low density, i.e. n L < n /
5. For high densities, wecan clearly see that all curves in Fig.4 (d) are steeper than in(c), ie, for the parameters chosen, the U LL is more attractivewith the NLWM than with the LWM. If one observes the val-ues of the coupling constants, it is obvious that as the c s L and c s ∗ L values related to the attractive interactions increase, sodo the c w L e c f L values, related to the repulsive interaction. IV. RESULTS FROM AFDMC
In the recent years, ab-initio models based on the Brueckneror Dirac-Brueckner theory [71–73] or on different quantumMonte Carlo simulation techniques [31, 74–78] have been ap-plied to (hyper)-nuclear matter. Such models provide in thepure neutron sector, in the low density regime where the un-derlying interactions are well known from scattering data andthree-body effects are not expected to be important, a veryessential constraint to phenomenological mean field models,which starts to be routinely applied in order to fix some of the unknown couplings. Calculations including hyperons are stillvery scarce [31, 71–73]. We here compare our results to thevery recent AFDMC model [31], which has been satisfacto-rily compared to hypernuclear data [70] and allows producingvery massive neutron stars in agreement with the observations[31], though with negligible strangeness fraction. This modelis based on a phenomenological bare interaction inspired bythe Argonne-Urbana forces [79], with the addition of a purelyphenomenological three-body term. One of the advantages ofthe model is that the authors provide simple parametrizationsof their numerical results for the neutron- L energy function-als, allowing both an easier comparison with our RMF results,and a straigthforward calculation of the instability propertiesof hyper-matter as predicted by an ab-initio model. This latterpoint is discussed in the next section. The fit of the energydensity of the neutron- L mixture is given by [31]: e total ( n n , n L ) = " a (cid:18) n n n (cid:19) a + b (cid:18) n n n (cid:19) b n n + m L n L (cid:0) p n L (cid:1) / + ( m n n n + m L n L )+ c ′ n L n n + c ′ n L n n . (4.1)In this expression, the first term represents the energy densityof pure neutron matter, where the parameters a , a , b and b arelisted in Tab.II and n is saturation point of symmetric nuclearmatter. The second term highlights the kinetic energy densityof pure L -matter, and the last two terms, obtained from the fit-ting of the Monte Carlo results for different Y L = n L / n B frac-tions, provide an analytical parametrization for the differencebetween Monte Carlo energies of pure L and pure neutronmatter. Notice that L - L interactions are neglected in Ref.[31],which explains why pure L matter ( n n =
0) behaves as a Fermigas of noninteracting particles. This means that the extrapola-tions to high L densities have to be considered with a criticaleye. The constants c ′ ≡ c / n and c ′ ≡ c / n with c and c are given in Tab.III. Using Eq.(4.1), the chemical potentialsbecome: m n ( n n ) = a ( a + ) (cid:18) n n n (cid:19) a + b ( b + ) (cid:18) n n n (cid:19) b + m n + c ′ n L + c ′ n L n n , (4.2)and m L ( n L ) = m L (cid:0) p n L (cid:1) / + m L + c ′ n n + c ′ n n . (4.3)From thermodynamics we can also write the total pressure asfollows: p total ( n n , n L ) = ( a a (cid:18) n n n (cid:19) a + b b (cid:18) n n n (cid:19) b ) n n + m L n L (cid:18) p n L s L + (cid:19) / + c ′ n n n L + c ′ n L n n . (4.4) In Tab.III we show the sets of parameters proposed by theauthors of Ref.[31] when only two-body forces are taken intoaccount ( L N ), and also with the consideration of three bodyforces that yield two different parameterizations L NN (I) and L NN (II). In the case of pure neutron matter, in the AFDMCapproach the binding energy has no free parameters and wecan compare this result with the binding energy coming fromour phenomenological RMF models. When we include a L -fraction in the system, the ab-initio model itself needs phe- nomenological inputs and is associated to theoretical errorbars. This is due to the need of three-body forces in orderto properly reproduce hypernuclear data [70]. The interval ofpredictions between L NN (I) and L NN (II), obtained usingtwo different prescriptions for the three-body force, will beinterpreted in the following as the present theoretical error baron ab-initio models, such that a phenomenological model likeour RMF should lay between these two extreme cases. (a) (b)(c)
56 NLWMLWM0 0.05 0.1 0.15 0.2n B (fm -3 ) B i nd i ng E n e r gy ( M e V ) Y (cid:1) =0.269230769 ( (cid:0) N) (I)(II)= 1.0= 0.2 χ σΛ χ σ∗Λ = 1.0= 1.0 χ σΛ χ σ∗Λ = 1.0= 0.5 χ σΛ χ σ∗Λ (d) FIG. 5: (Color online). Binding energy obtained with three different models AFMDC, LWM and NLWM, for different L -fractions show in(a), (b), (c) e (d) figures. In Fig.5 we plot the binding energy for different values ofthe L -fraction present in Ref.[31] for AFDMC and for repre-sentative RMF models. Fig.5 (a) shows the binding energyfor pure neutron matter. It is known since a long time thatRMF models are systematically too stiff at high neutron den-sity in comparison to ab-initio models. However we can seethat for the sub-saturation densities of interest for the presentpaper, the LWM agrees very well with the AFDMC, betterthan the NLWM, which in principle should be more sophis-ticated. This remains true for finite L -fraction, as shown inFigs.5 (b) and (c), if this latter is small enough. In this regime,the values of the L coupling do not play an important role, PNM n (fm − ) 0.16 a (MeV) 13.4 a b (MeV) 5.62 b and the same level of reproduction is obtained for different L Nc (MeV) -70.1 c (MeV) 3.4 L N + L NN ( I ) c (MeV) -77.0 c (MeV) 31.3 L N + L NN ( II ) c (MeV) -70.0 c (MeV) 45.3TABLE III: Set of parameters used in the ab-initio AFDMC modelincluding two and three body forces, from [31]. choices of c s L , c s ∗ L .The effect of three-body forces increases with increasing L -fraction, and consequently the three versions of the AFDMCcalculation start to considerably deviating from each otherat the highest L -fraction considered by the authors of [31](Fig.5 (d)). In this condition, the AFDMC ( L N ) becomes verybound, due to the attractive feature of the L N potential, whilethe three-body force in AFMDC (I) and (II) insures the nec-essary repulsion to sustain massive neutron stars. We can seethat at high L fraction NLWM better reproduces the ab-initioresults, and the best reproduction is obtained for c s L ≪ c s ∗ L .We have observed that U LL is more sensitive to changes in c s L than c s ∗ L as seen in Figs.4 (c) and (d). No matter how muchwe change these parameters, we do not notably change the de-gree of agreement between the RMF models and the AFDMC.In this sense the orange and green curves in Fig.5 (d) repre-sent extreme choices for the RMF couplings in the two ver-sions LWM (full lines) and NLWM (dashed lines). To con-clude, the inclusion of strange mesons is necessary to producea RMF energy functional compatible with ab-initio results atlow baryonic density. For very low L -fractions, as it is thecase in hypernuclei, the sensistivity to the L couplings is verysmall, and the LWM surprisingly leads to a very good agree-ment to the AFDMC parametrization. However neither thelinear nor the non-linear version of the WM are satisfactory, ifone wants to describe matter with a non-negligible proportionof L ’s, and a dedicated fit with density dependent couplingsshould be done to reduce the parameter space. For the purposeof the present paper we will continue with both models in ourfurther analysis, keeping in mind that LWM results well re-produce ab-initio pure neutron matter, while NLWM with lowvalues of c s L ≈ . − . V. SPINODAL AND CURVATURE MATRIX
In the present section we focus on the calculation of theinstabilities in a system with neutrons, protons and L ’s at T = C associated to the scalar function e at a point de-noted by P ∈ ( n n × n p × n L ) . Since our benchmark ab-initiomodel only contains neutrons and L ’s, we consider first a two-component system case, where P ∈ ( n n × n L ) , and later wecomment about three-component systems [66, 69] which aremore relevant for hypernuclear physics. If e is smooth, or atleast twice continuously differentiable, C is symmetric. Thecurvature matrix elements are just second derivatives of the to-tal energy density with respect to each independent variable.In our case the curvature matrix is just 2 × C i j = (cid:18) ¶ e ( n i , n j ) ¶ n i ¶ n j (cid:19) , (5.1)where i , j = n , L . As this matrix is self-adjoint we can asso-ciate with it one bilinear form and one quadratic form at point P . So, the characteristic equation is Det ( C − l ) = , (5.2)where is 2 × l − Tr ( C ) l + Det ( C ) = . (5.3)The eigenvalues and eigenvectors of C have geometric mean-ing if P is a critical point. We can solve their roots explicitly l = (cid:18) Tr ( C ) + q Tr ( C ) − Det ( C ) (cid:19) (5.4)and l = (cid:18) Tr ( C ) − q Tr ( C ) − Det ( C ) (cid:19) , (5.5)where Det ( C ) = l l and Tr ( C ) = l + l . The uni-tary eigenvectors are given by b n = (cid:0) d n n , d n L (cid:1) and b n = (cid:0) d n n , d n L (cid:1) . For further analysis we define the direction bythe ratiostan q = d n L d n n = l − C nn C n L and tan q = d n L d n n = l − C nn C n L , (5.6)where q and q are an angle measured counterclockwise fromthe positive n n axis. If P is a critical point and hence C is justa Hessian matrix so the determinant term is exactly the Gausscurvature and the trace is twice the mean curvature [80] K = l l and H = ( l + l ) . (5.7)The stability properties of the system depend on the signs ofthe curvatures, K and H , at each point P ∈ ( n n × n L ) [5, 7, 64,65]:0 1. If K > H >
0, the system is stable.2. If K > H <
0, the system is unstable, both eigen-values are negative and two independent order parame-ters should be considered meaning that more than twophases can coexist.3. If K <
0, the system is unstable, meaning that the orderparameter of the transition is always one-dimensional,similar to the nuclear liquid-gas phase transition at sub-saturation densities.4. If K = H >
0, the system is stable.5. If K = H <
0, the system is unstable.In geometric terms the first and second condition tell us that P represents an elliptic point, third a hyperbolic point and fourthand fifth a parabolic. For a three-component system we haveto calculate numerically the following equation Det ( C − l ) = , (5.8)where is a 3 × l − Tr ( C ) l + h Tr ( C ) − Tr (cid:0) C (cid:1)i l − Det ( C ) = , (5.9)where we have to analyse the signs of three eigenvalues. Theremarkable feature of the liquid-gas phase transition is thatone of all eigenvalues is negative and the associated eigen-vector gives the instability direction [69], what means that theenergy surface is of a hyperbolic kind. Therefore, in the caseof the simpler n L system for a negative eigenvalue the ratio(5.6) became d n − L d n − n = l − − C nn C n L . (5.10) The next equation will be useful in the discussion about theratio for np L system with symmetric condition n n = n p . d n − L d n − N = l − − C NN C N L . (5.11)In the next section we comment our results. VI. RESULTS
In order to understand the instabilities possibly present inthe models discussed in sections III and IV, we need the anal-ysis done in the last section. We have calculated the curvaturematrix with the ab-initio and RMF models. The whole densityspace is a three-dimensional space and the spinodal region,when it occurs, is a three-dimensional volume that representsa geometric locus associated with the presence, at least, of onenegative eigenvalue. It is well known that in two-componentsystems with neutrons and protons the liquid-gas phase tran-sition occurs. The corresponding two-dimensional spinodalzone appears below the saturation density. So, in this system,one of the eingenvalues is negative. For a more complex sys-tem, with neutrons, protons and L ’s for example, we can fixthe L -fraction to see how the two-dimensional spinodal regionin the neutron-proton plane changes when lambdas are added.In all the models analyzed, for any proton fraction, and withall the different choices of couplings, we have systematicallyfound one and only one negative eigenvalue in a finite den-sity space defining a spinodal region. The only exception isgiven by the n L system studied with the LWM, which doesnot present any instability. However the instability is there inthe ab-initio model, and it appears in the LWM as soon as anon-zero proton fraction is added to the system, meaning thatthe result of the LWM n L mixture appears rather marginal. (a) (b) (c) FIG. 6: (Color online). Spinodals in the neutron-lambda plane for AFMDC with different parameterizations and NLWM with and withoutstrange mesons.
Therefore we can conclude that a transition exists in thesubsaturation nuclear matter including L hyperons, and thistransition belongs to the Liquid-Gas universality class. In the following, we turn to study the characteristics of this transitionin further details.1In Fig.6 we plot the spinodal areas in a system containingonly neutrons and L ’s. In Fig.6 (a) two spinodal zones forthe two different parameterizations of the ab-initio model in-cluding three-body forces are shown. The behavior at high L -density should be considered with caution, since the AFDMCcalculations were only done for Y L < . L matter with-out strange mesons are also displayed in Fig.6 (c). Note thatnone of these shapes touch the horizontal axis nor the verti- cal one, even if they look very close to the n n axis for someof the models. This result is due to the fact that pure neutronand pure L -matter are unbound. Indeed the spinodal instabil-ity at zero temperature leads to a phase transition where thesystem splits into two phases, the dense one representing thebound ground state. In the absence of a bound ground state,it is thus normal that the instability disappears. In the follow-ing, whenever the spinodal zone does not touch the axis it isclear that the reason underlying this behavior is an unboundsystem. It is interesting to observe that the widest extensionof the instability is obtained with the most repulsive model. (a) (b) (c) FIG. 7: (Color online) Spinodals in neutron- L -plane with eigenvectors in NLWM. From (a) to (b) we vary c s L with fixed c s ∗ L = .
0. (c)Ratio d n − L / d n − n as a function of the Y L for some couplings and baryon densities for Y p = . (a) (b) FIG. 8: (Color online). Three dimensional spinodal surfaces in the NLWM for a particular choice of coupling constrained parameters. In (a)the numbers denote cuts on the surface, (1) neutron-lambda spinodal area, (2) proton-lambda spinodal area, (3) neutron-proton spinodal areaand (4) the frontier of the spinodal area when we cut the three-dimensional spinodal volume by a vertical plane passing by n n = n p . (b) showthe slices when we fix Y L and the red shaded one is a special case. This counterintuitive result probably stems from the fact thatthe highest repulsion at high density is correlated to a strongerattraction at low density also in the ab-initio model. The be-havior of the unstable eigenvector, shown in Figs.7 (a) and (b)for the two RMF parameter sets that better reproduce the ab-initio EOS, is also interesting. We can see that it is close to the isoscalar direction n n + n L as it is in the standard LG [7]. Thissimply means that the transition is between a dense and a di-luted phase. In finite systems, the dense phase corresponds toan hypernucleus, and the dilute phase to a (hyper)-gas (whichat T = d n − L / d n − n as a function ofthe Y L for some couplings and baryon densities for Y p = . L fractions, the direction of phaseseparation is steeper than the constant L -fraction line. Thismeans that the dense phase is more symmetric than the dilutephase. We also depict the line that represents n L / n n , so that itbecomes visually easy to compare it with the direction of theeigenvectors.The instability direction can be better spotted from Fig.7(c), which displays the unstable eigenvector as a function ofthe L fraction. We can see that the unstable eigenvectors arealmost independent of the baryonic density. This means thatthe proportion of L in the dense phase following the spinodaldecomposition is the same whatever the timescales and dy-namics in the spinodal zone, and is well defined by the direc-tion of the unstable eigenvectors. This proportion monotoni-cally increase with the L fraction, but never reach the equalitybetween neutrons and L . This feature is due to the mass dif-ference between the two baryonic species, as well as to thereduced attraction in the L channel. It is at variance with theordinary nuclear liquid-gas which is associated to the fraction-ation or distillation phenomenon [3, 7], with the dense phasebeing systematically more symmetric than the dilute phase(see Fig.9). The optimal proportion of L increases with in-creasing the scalar coupling, as it can be intuitively expected.Now we would like to see how this affects the spinodalzone calculations in the three-component system, which ismore relevant for nuclear physics applications. Fig.8 showsthe three-dimensional spinodal volumes for particular cases: c s L = . c s ∗ L = . n n = n p . Fig.8 (b) is similarto (a) but in this case the black dashed lines represent constant Y L cuts. Y L = . L fraction, because itleads to the same representation as for the usual LG phasetransition, which is obtained in the limit Y L =
0. This is donein Fig.9, which shows the spinodal region in the neutron-proton plane obtained with the NLWM model for a largechoice of coupling parameters. It is important to remark thatonly NLWM gives reasonable properties for symmetric mat-ter in the absence of hyperons and for LWM we omitted thecorresponding results here. Fig.9 (a) shows the NLWM spin-odal for Y L = (a)(b) FIG. 9: (Color online) (a) Spinodal for neutron-proton matter witheigenvectors in NLWM. (b) The ratio d n − p / d n − n plotted as a functionof the proton fraction for Y L = (3) in Fig.8 (a). In Fig.9 (b) the ratios d n − p / d n − n are plottedas a function of the proton fraction for the same fixed baryondensities shown in Fig.9 (a).In Fig.10 (a) the gray curve is the frontier of the spinodal for Y L = . Y L = . Y L = . Y L = L -fractions(right side). In any case we can see that the phase diagrams ofthe two models are very similar, the NLWM instability zonebeing only slightly narrower.Panels (a) and (b) of Fig.9 recall the usual characteristicsof the nuclear liquid-gas phase transition [5, 7]. As it is wellknown, the instability covers a huge part of the sub-saturationregion and has an essentially isoscalar character. The unstableeigenvectors point towards a direction which is intermediate3between the isoscalar direction (observed only for symmet-ric matter n n = n p ), and the direction of constant isospin. Asa consequence, the dense phase is systematically more sym-metric than the dilute phase. Indeed, at zero temperature thedilute phase is a pure gas of neutrons (protons) if the sys-tem is neutron (proton) rich [7]. From panels (a) and (c) ofFig.10 we additionally learn that the LG instability is clearlypreserved by the addition of strangeness. However, the tran-sition is quenched for strongly coupled hyperons. Indeed, wecan clearly see that when c s L increases the spinodal area de-creases. Considering that the most realistic value lays around c s L ≈ . − .
5, this quenching is small. On the other side,when c s ∗ L increases, the modification on the spinodal is very small. This is expected, since the strange mesons are onlycoupled to strange baryons and are therefore expected to affectessentially the L density, which is not represented here. Dueto the weak effect of c s ∗ L in the spinodal frontier we selectthe value c s ∗ L = . d n − p / d n − n are plotted as a func-tion of the proton fraction. No difference can be seen withrespect to the normal LG: whatever the percentage of L ’s, theneutron-proton composition of the dense phase (i.e. the hy-pernucleus) is unmodified, even if the density is reduced. (a) (b) (c) FIG. 10: (Color online) (a) Spinodals in neutron-proton plane and Y L = . c s L = . c s ∗ L = .
0. (b) Spinodal forneutron-proton matter with eigenvectors in NLWM. (c) The ratio d n − p / d n − n plotted as a function of the proton fraction. This finding might seem in contradiction with recent studiesin multiply strange hypernuclei [21–23], where it is seen thatthe driplines are modified by the L -fraction. However thesemodifications are essentially due to shell and Coulomb effects,which are not accounted for in this infinite matter calculation.If we change our perspective from the neutron-proton planeto imagine the general three-dimensional spinodal locus andinstead of fixing Y L as before, we fix the symmetric mattercondition n N = n n = n p , the resulting plane slice crossingthis three-dimensional volume is similar to the curve denotedby number (4) in Fig.8 (a). The related spinodal areas forthe NLWM and many choices of the coupling parameters areshown in Fig.11.The comparison to the ab-initio model of Section IV sug-gests that the most realistic phase diagram should be be-tween the ones corresponding to c s L = . , .
5, which givesan energy functional intermediate between the two AFDMCparametrizations of three-body forces. We can see that thecoupling to the strange meson c s ∗ L has a bigger effect in thisplane as expected. Still, its influence on the spinodal is small.This means that the wide uncertainty on the strange mesonshas a negligible influence on the phase transition. The biggestuncertainty concerns the extension of the spinodal zone alongthe n L axis. It is however important to stress that this situa- (a) FIG. 11: (Color online) Spinodal in the nucleon- L -plane (keeping n n = n p ) (a) Colors contours are sliced shapes from 3d spinodal indensities space and varying c s L and c s ∗ L in NLWM. The gray dot-ted line represents the n N = n L line. tion n N < n L < n does not correspond to any known physicalsystem. Every shape shown touches the horizontal axis when Y p = .
5, as it should be considering that the np system is4bound. At this point we can report to Fig.3, to see that our re-striction of c s L does not affect much the spinodal zone analy-sis because when we increase c s L up to 1 . L -density direction. Even if thecalculation might be not realistic for very high L -fraction, wecan conclude that the LG phase transition is still present inmultistrange systems.Finally, Figs.12 (a) and (b) show the projections of the un-stable eigenvectors in the L -nucleon plane. We can see thata non negligible component of the order parameter lies alongthe n L direction, meaning that the L -density is an order pa-rameter of the phase transition, or in other words that thedense phase is also the phase with the higher strangeness con-tent. These eigenvectors are almost parallel to each other,and considerably deviate with respect to the direction of theconstant L -fraction lines as seen in Fig.12 (c). Interestingenough, the instability points towards an “optimal” compo-sition n L ≈ . n N for the dense phase, whatever the bary-onic density, coupling constants and L fraction. Only for verysmall and very high L fraction a deviation is observed. This is expected because by construction the instability must tend to-wards the non-strange direction in the absence of strangeness.It will be very interesting to verify if such an optimal com-position is obtained in calculations of multiple-strange hyper-nuclei. As in the case of the simpler n − L system, the factthat the instability always points towards L poor systems isat variance with the distillation phenomenon, characteristic ofthe LG phase transition with more than one component [3, 7],where the direction of phase separation tends to equal com-position. This symmetry breaking between nucleons and L ’scomes from the difference in the bare mass of the particles andthe less attractive couplings. Still, for very low L fractions,the direction of phase separation is steeper than the constant L -fraction line. This means that the dense phase is more sym-metric than the dilute phase. This thermodynamic finding iscompatible with the observation in Ref.[24] that the L ’s pro-duced in heavy-ion collisions should stick to the clusters (i.e.,the dense phase) rather than being emitted as free particles(i.e., the gas). (a) (b) (c) FIG. 12: (Color online) Spinodals in nucleon- L -plane ( n n = n p ) with eigenvectors in NLWM. From (a) to (b) we vary c s L with fixed c s ∗ L = .
0. (c) Ratio d n − L / d n − N . VII. SUMMARY AND CONCLUSIONS
We have investigated the thermodynamic phase diagram atsubsaturation density, for baryonic matter including neutrons,protons and L hyperons, within a RMF approach. For the nu-cleonic EOS, we have considered the GM1 parametrizationof NLWM, together with the simpler LWM. Strange mesonswere included to allow a wide exploration of the possible phe-nomenology for the (still largely unknown) hyperon-nucleonand hyperon-hyperon couplings, with minimal requirementson the potential depths extracted from hypernuclear data. Im-posing these requirements leads to a strong linear correlationbetween the attractive and the repulsive couplings, both forthe normal and the strange mesons. These constraints leave uswith a two-dimensional parameter space, which we have var-ied widely in order to pin down generic features of the phasediagram. Our main focus was the understanding of the instabilitiesin the hypernuclear matter, and specifically the influence of L ’s in the well known Liquid-Gas phase transition of nuclearmatter. The existence of an instability as a signature of a firstorder phase transition was identified by analyzing the curva-ture of the thermodynamic potential with respect to the nucle-onic and strange densities. In all our studies one and only onenegative eigenvalue has been found, showing that the phasetransition still exists in the presence of strangeness and is stillof LG type, even if its extension in the density space shrinkswith increasing strangeness. The negative eigenvalue corre-sponds to the direction in density space, in which density fluc-tuations get spontaneously and exponentially amplified in or-der to achieve phase separation. This eigenvalue is seen tosystematically have a non-negligible component in the direc-tion of the strange density. This means that strangeness canbe viewed as an order parameter of the transition.5Less expected is the fact that the instability direction sys-tematically points to a fixed proportion of L ’s in the densephase, at variance with the phenomenon of distillation typicalof binary mixtures. This proportion being of the order of 30%in the models we considered, this means that in a dilute sys-tem with a small contribution of L ’s, these latter will prefer-entially belong to the dense clusterized phase. These conclu-sions are general and appear largely model independent. Onthe contrary, the specific shape of the phase diagram wouldobviously depend on the choice of the free c s ∗ L and c f L cou-plings. Some hints on a more quantitative estimation of thethermodynamics were obtained from the analysis of the sim-pler n L phase diagram extracted from the ab-initio AFDMCcalculation of Ref.[31]. The characteristics of the phase tran-sition are confirmed in the ab-initio model, even if the phasediagram extension depends on the three-body force model inan important way.The comparison of the RMF with the AFDMC also revealssome limitations of the phenomenological model at low den-sity. Indeed the popular GM1 model is shown to comparevery poorely to the ab-initio calculation of pure neutron mat-ter even at the low densities considered in the present study.Unexpectedly, the simpler LWM is in very good agreementwith the ab-initio predictions at low density. Concerning the n L mixture, the energy functional is within the theoretical er-ror bars if 0 . ≤ c s L < . d meson also plays an important role in satisfyingboth nuclear bulk and stellar properties constraints. The useof another parameter set and/or the inclusion of this new de-gree of freedom requires a complete calculation from the verybeginning because the nucleon-lambda potential, Eq.(3.3), hasto be readjusted. However, the qualitative results will be cer-tainly similar, since the whole 3D parameter space associatedto strangeness was spanned. As a perspective for future work,it will be very interesting to analyze the instability behaviorof a density dependent coupling RMF model, directly fitted tothe ab-initio calculation. Acknowledgments
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