Light clusters in warm stellar matter: calibrating the cluster couplings
Tiago Custódio, Alexandre Falcão, Helena Pais, Constança Providência, Francesca Gulminelli, Gerd Röpke
aa r X i v : . [ nu c l - t h ] N ov EPJ manuscript No. (will be inserted by the editor)
Light clusters in warm stellar matter: calibrating the clustercouplings
Tiago Cust´odio , Alexandre Falc˜ao , Helena Pais , Constan¸ca Providˆencia , Francesca Gulminelli , and GerdR¨opke CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal Normandie Univ., ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, F-14000 Caen, France Institut f¨ur Physik, Universit¨at Rostock, D-18051 Rostock, Germany National Research Nuclear University (MEPhI), 115409 Moscow, RussiaReceived: date / Revised version: date
Abstract.
The abundances of light clusters within a formalism that considers in-medium effects are cal-culated using several relativistic mean-field models, with both density-dependent and density-independentcouplings. Clusters are introduced as new quasiparticles, with a modified coupling to the scalar mesonfield. A comparison with experimental data from heavy ion collisions allows settling the model dependenceof the results and the determination of the couplings of the light clusters to the meson fields. We findthat extra experimental constraints at higher density are needed to convincingly pin down the densityassociated to the melting of clusters in the dense nuclear medium. The role of neutron rich clusters, suchas He, in asymmetric matter is discussed.
PACS.
XX.XX.XX No PACS code given
Below saturation density, nuclear matter is supposed toundergo a liquid-gas phase transition [1,2,3]. Since in phys-ical systems nuclear matter is electrically charged, thephase separation will produce clusterized matter. This be-havior is directly reflected in several astrophysical sites,like core-collapse supernovae [4,5,6,7,8], neutron star (NS)mergers [9,10,11], and the inner crust of neutron stars[12,13,14]. The form of the clusterized matter dependson temperature and isospin asymmetry. In cold catalysedbeta-equilibrium matter, as the one occuring in neutronstars, spherical clusters are found in almost the whole in-ner crust region, and close to the crust-core transition,the competition between surface and Coulomb forces givesrise to cluster configurations of different geometries coined“pasta phases” [12]. These types of clusters may surviveeven at finite temperature [15,16,17,18,19,20]. Light clus-ters, i.e. light nuclei like deuterons or α − particles, willform in warm stellar matter as found in core-collapse su-pernova matter, proto-neutron stars or binary neutronstar mergers, and may also coexist with heavy clustersat densities above 10 − fm − , if the temperature does notovercome a few MeV. The presence of light clusters af-fects the rates of the reactions involving the weak force,and therefore, may impact the supernova dynamics [4,21].In the NS merger evolution, the α particles play an im- Correspondence to : Helena Pais, [email protected] portant role on the dissolution of the remnant torus ofaccreted matter that surrounds the central high-mass NSformed after the merging. Matter from this accretion diskis also contributing to the ejecta that originates the kilo-nova observation [10,11,22].Clusterized warm matter at low densities has been de-scribed within a generalization of the relativistic mean-field (RMF) approach. Within this framework, light clus-ters are included as independent degrees of freedom thatinteract with the medium through their coupling to themesonic fields [23,24,25,26,27,28]. In previous papers [29,30], some of the present authors introduced a new for-malism, that takes into account in-medium effects for thecalculation of the equation of state with light clusters forapplications in astrophysical systems. These effects are in-troduced via the scalar cluster-meson coupling, and alsovia an extra term that is added to the total binding energyof the clusters. This term not only avoids double countingof single particle continuum states, but also affects the dis-solution of clusters at high densities. In both references,the studies have been developed within the FSU model[31], which has been fitted to both static and dynamicproperties, and it is adequate to describe nuclear matterat saturation density and below. This model, however, istoo soft and does not describe neutron stars with two solarmasses. It is, therefore, necessary to generalize the previ-ous studies to other models currently used with success todescribe symmetric and asymmetric nuclear matter.
Tiago Cust´odio et al.: Light clusters in warm stellar matter: calibrating the cluster couplings
Some RMF models are frequently used in simulationsand in the study of astrophysical observations, such as theRMF models with non-linear mesonic terms TM1 [32,33]and its modifications [34,35,36,37], NL3 [38] and its mod-ifications [39,36], SFHo [40], FSU2R [41], and RMF withdensity dependent couplings, like DD2 [25] and DDME2[42]. Models such as TM1 and NL3 have been fitted tothe ground state properties of nuclei. However, they bothpresent a too large slope of the symmetry energy at satu-ration, far from what the experimental constraints [43,44,45], or ab-initio chiral effective field theory calcula-tions (CEFT) [46], indicate, and, therefore, they have beenmodified by a non-linear ωρ term that smoothens thedensity dependence of the symmetry energy [39,34,35,36,37]. The parametrization FSU2R [41] is based in the FSUmodel, which was modified in order to be able to describetwo solar mass stars and still to satisfy CEFT results.In [29,30], it was found that the equilibrium constantsdetermined from the NIMROD data [47] could be welldescribed taking a universal coupling of the σ -meson to allthe light clusters considered. Having in mind the inclusionof light clusters in other RMF models besides FSU, it isimportant to study their behavior at low densities whenmatter is clusterized and light clusters have an importantrole in the definition of the transport properties of matter.In this paper, we want to apply the generalized RMF(gRMF) formalism to low density clusterized matter, whereclusters are treated as new quasiparticles, but with a mod-ified coupling to the σ meson field developed in [29,30], us-ing different relativistic mean field models, to understandhow they behave. For that matter, we have chosen mod-els with both density-dependent and density-independentcouplings. We consider the four usual light clusters, thatis, H, H, He, and He, and we add, inspired by the workdone in Ref. [29], another cluster, He, to our calculations.This cluster has been included in the analysis of the IN-DRA collaboration [48], which has recently published anew set of equilibrium constants [48] with a controlledbayesian determination of the system density during theexpansion, including the possibility of in-medium modifi-cations [49,50]. Note that this gRMF approach has beenapplied recently also to the description of yields of clustersproduced at ternary fission [51].The aim of the work is to compare different approachesfor the RMF parametrization, the description of the cou-pling to the meson field, and the comparison with therecent INDRA data to investigate the influence of correla-tions on the equation of state. This comparison serves ascriterion to validate different models for the compositionof subsaturation nuclear matter.This paper is organized as follows: a brief summary ofthe formalism applied is given in the next Section, someresults are shown in Section 3, and, finally, in Section 4,some conclusions are drawn.
We briefly review the RMF models that will be consideredin the present work, in particular, FSU2R [41], NL3 ωρ Table 1.
A few symmetric nuclear matter properties for themodels used in this work, calculated at saturation density, n :the binding energy per particle B/A , the incompressibility K ,the symmetry energy E sym , the slope of the symmetry energy L , and the nucleon effective mass M ∗ . All quantities are inMeV, except for n that is given in fm − , and for the effectivenucleon mass that is normalized to the nucleon mass M .Model n B/A K E sym
L M ∗ /M FSU2R 0.15 -16.28 238 30.7 47 0.59NL3 ωρ ωρ [36], TM1 ωρ [36], SFHo [40], DDME2 [42], and DD2 [25].In Table 1, we show some symmetric nuclear matter prop-erties calculated at saturation density for these models.These properties are obtained from the Lagrangiandensity, that describes nucleons with vacuum mass M coupled to the scalar meson σ with mass m σ , the vectorisoscalar meson ω with mass m ω and the vector isovectormeson ρ with mass m ρ , L = X i = p,n L i + L σ + L ω + L ρ + L σωρ , (1)where L i is L i = ¯ ψ i [ γ µ iD µ − M ∗ ] ψ i , with iD µ = i∂ µ − g ω ω µ − g ρ τ · ρ µ , and the Dirac effectivemass, M ∗ = M − g σ σ . The mesonic Lagrangian densitiesare given by: L σ = 12 (cid:18) ∂ µ σ∂ µ σ − m σ σ − κσ − λσ (cid:19) , L ω = − Ω µν Ω µν + 12 m ω ω µ ω µ + ζ ζg ω ( ω µ ω µ ) , L ρ = − B µν · B µν + 12 m ρ ρ µ · ρ µ + ξ g ρ ( ρ µ ρ µ ) , (2)where Ω µν = ∂ µ ω ν − ∂ ν ω µ , B µν = ∂ µ ρ ν − ∂ ν ρ µ − g ρ ( ρ µ × ρ ν ), and τ are the SU(2) isospin matrices.The Lagrangian density of the models FSU2R, NL3 ωρ ,TM1 ωρ and SFHo includes non-linear mesonic terms, whichare either self-interaction terms, or terms that mix the σ, ω , and ρ mesons [40]: L σωρ = g ρ f ( σ, ω µ ω µ ) ρ µ · ρ µ , (3)where f is given by f ( σ, ω µ ω µ ) = X i =1 a i σ i + X j =1 b j ( ω µ ω µ ) j (4)for SFHo, and by f ( σ, ω µ ω µ ) = Λ v g v ω µ ω µ (5) iago Cust´odio et al.: Light clusters in warm stellar matter: calibrating the cluster couplings 3 for FSU2R, NL3 ωρ and TM1 ωρ .The models DD2, DDME2 have density-dependent cou-plings and no nonlinear mesonic terms. Their isoscalarcouplings of the mesons i to the baryons are given by g i ( n B ) = g i ( n ) a i b i ( x + d i ) c i ( x + d i ) , i = σ, ω, (6)and the isovector meson-nucleons coupling by g ρ ( n B ) = g ρ ( n ) exp [ − a ρ ( x − . (7)In the last expressions, n is the model-dependent sym-metric nuclear saturation density, see Tab. 1, and x = n B /n , with n B the baryonic density.In order to include explicitly the light clusters as con-stituents of nuclear matter, we consider them as point likeparticles, neglecting intrinsic structures. This approxima-tion is acceptable at low densities when the volume ac-cessible to each clusters is much larger than its volume.In RMF approach they will interact with the mediumthrough their coupling to the mesons. We add to the La-grangian density the cluster contributions, and for homo-geneous matter each one contributes to the total energydensity of the system with E i = 2 S i + 12 π R k i E i ( f i + ( k ) + f i − ( k )) dk i + g iω ω n i + g ρ ρ I i n i , (8)where E i = p k i + M ∗ i is the cluster single-particle en-ergy, and M ∗ i the effective mass of cluster i . The clusterspin, isospin and density are denoted, respectively, by S i , I i , and n i . The distribution functions for the particles andantiparticles f i ± are defined as f i ± = 1exp[( E i ∓ ν i ) /T ] + η , (9)with η = 1 for fermions and η = − T thetemperature, and ν i = µ i − g iω ω − g ρ I i ρ . The couplingof the clusters to the ω -meson is given by g iω = A i g ω ,with A i the cluster mass number.We define the binding energy of the cluster i in themedium as in [29] B i = A i m ∗ − M ∗ i , i = d, t, h, α, He , (10)with M ∗ i given by M ∗ i = A i m − g iσ σ − (cid:0) B i + δB i (cid:1) . (11)In this last expression, B i is the binding energy of thecluster in the vacuum, which will be fixed to the exper-imental values, the second term denotes the coupling ofthe cluster to the σ -meson, and the last term describes abinding energy shift. In the following, we will write thecoupling g iσ in terms of the coupling of the nucleon to the σ -meson, g σ , as g iσ = x s A i g σ , with x s its fraction, rangingfrom 0 to 1, to modify (decrease) the nuclear attraction.For the binding energy shift δB i , we consider [29,52,53] δB i = Z i n (cid:0) ǫ ∗ p − mn ∗ p (cid:1) + N i n ( ǫ ∗ n − mn ∗ n ) , (12) -3 -2 -1 n B [fm -3 ] -4 -3 -2 -1 X i np2H3H3He4He6He Fig. 1. (Color online) The abundances (mass fraction) of thestable isotopes n, p, d, t, h, α , and He considered as a functionof the density for T = 5 MeV and a fixed proton fraction of0.2. The NSE (dashed) is compared to a QS calculation (fulllines), see text. where the gas energy density ǫ ∗ j and nucleonic density n ∗ j , j = n, p , are given by [29] ǫ ∗ j = 1 π Z k Fj (gas)0 k E j ( f j + ( k ) + f j − ( k )) dk (13) n ∗ j = 1 π Z k Fj (gas)0 k ( f j + ( k ) + f j − ( k )) dk , (14)with k F j (gas) = (cid:0) π n j (cid:1) / defined using the zero tem-perature relation between density and Fermi momentum.This was proposed in [29] as an effective way of implement-ing Pauli blocking in a range of temperatures for whichthe Fermi distribution is close to a step function. Thesetwo quantities, ǫ ∗ j and n ∗ j , define the energy density andthe density of nucleon j associated to the levels below theFermi momentum, k F j (gas), of the T = 0 nucleonic den-sity n j . The term δB i can be identified as an excludedvolume mechanism in the Thomas-Fermi approximation.The gRMF model is able to describe the occurrence ofclusters in nuclear matter and density modifications. Theparameters are introduced in an empirical way, and it isour goal to discuss different choices for these parameters,in particular the coupling of clusters x s to the σ -meson.There are also alternatives to describe density effects suchas the excluded volume concept, see [6], which also usesempirical parameters. A microscopic, quantum statisticalapproach can be given, see [54] and references therein,which, however, is not simple to be used for practical ap-plications. Before comparing different gRMF models, we give a gen-eral discussion of the problem to include correlations and
Tiago Cust´odio et al.: Light clusters in warm stellar matter: calibrating the cluster couplings cluster formation in the EoS. In Fig. 1, the mass fraction X i = A i n i /n B for the stable isotopes i = n, p, d, t, h, α ,and He is shown for T = 5 MeV, y p = 0 . .
001 fm − < n B < . − ; y p isthe fixed global proton fraction, and n i the particle num-ber density. We have P i X i = 1. The calculation of thecomposition according to the simple model of nuclear sta-tistical equilibrium (NSE) neglecting all interactions be-tween the constituents, and considering only the groundstates of the stable isotopes, is compared to a quantumstatistical (QS) calculation taking in-medium effects intoaccount, in particular Pauli blocking and the quasiparticleshift taken as DD2-RMF, see [54]. In the low-density re-gion, the interaction between the constituents of nuclearmatter can be neglected so that this limiting region isappropriately described by the NSE. A small difference isobserved for the deuteron mass fraction owing to the viriallimit which is correctly described by the QS approach. Theaccount of excited states implements also the account ofscattering states in the virial EoS [55,56] which leads to asignificant contribution for the deuteron fraction becauseof its small binding energy. As seen from Fig. 1, in-mediumeffects become appreciable above n B = 10 − fm − .The most striking effect is the suppression of boundstate abundances because of Pauli blocking so that nearthe saturation density these clusters nearly disappear, andwe obtain a Fermi liquid of neutrons and protons. Theseare treated as quasiparticles containing a mean-field en-ergy shift. Different versions of these RMF approaches arepresented and compared in this work. We have to inter-polate between two limiting cases, the virial expansionin the low-density limit and the RMF approach near thesaturation density. In particular, the virial EoS dependsonly on the experimentally determined binding energiesand scattering phase shifts [55,56], and provides at finitetemperature the correct zero density limit. The exact mi-croscopic description of the intermediate region is a diffi-cult many-particle problem. Continuum correlations andhigher order clustering in a dense medium is hard to calcu-late within the QS approach. Interpolations are of interestwhich may be probed by laboratory experiments as shownin this paper.An interesting issue seen also in Fig. 1 is that withinthe NSE calculation with a given set of isotopes, the neutron-rich clusters become dominant at increasing density, be-cause the proton fraction is small so that exotic nucleilike He are most abundant near the saturation density.We can extend the NSE including resonant states such as H and He. In a recent publication by Yudin et al. [57], itwas claimed that these exotic nuclei may be of importancein stellar matter, in particular with respect to the neu-trino opacity. However, as shown in Refs. [58,21], withina more systematic approach the contribution of the reso-nant states should be expressed in terms of the scatteringphase shifts so that their contribution is strongly reduced.Therefore, in this paper, we restrict our calculations onlyto He, which is also measured in the INDRA experiment.Within a more exhaustive investigation one can also searchfor He and other clusters, but it is expected that their -3 -2 -1 T=5MeVy p =0.2FSU2R x s =0.85 X i He H He H He10 -3 -2 -1 x s =0.85DDME2 X i x s =0.92x s =0.9210 -4 -3 -2 -1 s =0.70 X i n B (fm -3 ) 0.02 0.04 0.06 0.10x s =0.85n B (fm -3 ) Fig. 2. (Color online) The abundances of all the clusters con-sidered as a function of the density for T = 5 MeV and a fixedproton fraction of 0.2 for FSU2R (top), DDME2 (middle), andSFHo (bottom) obtained with x s = 0 .
85 (left) and x s = 0 . mass fractions are very small. We will not go in more de-tails here, but discuss what can be learned from laboratoryexperiments to derive adequate interpolation expressionsto infer the composition in the whole subnuclear densityregion.To this aim, in the present section, we compare thedistribution of light clusters at low densities obtained withthe models FSU2R, NL3 ωρ , TM1 ωρ , SFHo, DDME2, andDD2, and we discuss how well these models describe theequilibrium constants determined from the INDRA [48,49,50] data.As referred before, in [29] it was shown that takingthe universal coupling x s = 0 . ± .
05 of the σ -meson tothe light clusters d, t, h, α would describe well the equilib-rium constants determined from the NIMROD data [47].These data, however, have been analyzed making the hy-pothesis that the system volume can be obtained assumingan ideal gas equation of state for the clusters. Recently,the INDRA collaboration [48] has performed a similar ex-periment with the heavier system Xe-Sn, where differenttin and xenon isotopes were used, and considered in theanalysis also the He cluster, besides the light clusters d, t, h, α . The Bayesian analysis of these data performedin [49,50], allowing possible in-medium corrections in thedetermination of the system volume, has shown that alarger value x s = 0 . ± .
02 should be taken for the clus-ter σ -meson coupling. However, this value is obtained witha specific version of the RMF model, namely FSU, and itcould be model dependent. For this reason, in the follow- iago Cust´odio et al.: Light clusters in warm stellar matter: calibrating the cluster couplings 5 ing figures, we will consider different values of x s for the σ meson-cluster coupling within the RMF models intro-duced above.In Fig. 2, we show the mass fractions of all the clus-ters considered as a function of the density for a temper-ature of 5 MeV and a fixed proton fraction y p = 0 . x s is necessary to reproduce the virial EoS at low densi-ties, and to fit the equilibrium constants deduced from theINDRA data as will be discussed later. We can see thatbelow n B ≈ × − fm − , the most abundant clustersare H, He, and He. This is because the clusters Heand H are the neutron-richest ones, and we are consider-ing asymmetric neutron rich matter, and He is the mostbound one. The heaviest cluster He is the first to dis-solve while the lightest one H is the last one. For FSU2Rand SFHo, He is the most abundant cluster in a shortrange of densities taking, respectively, x s = 0 .
85 and 0.7.It is striking that for x s = 0 .
92 (0.85) for FSU2R (SFHo), He becomes the most abundant in the range 0.01 to 0.04fm − . Within DDME2, the clusters dissolve at the small-est densities, and He is the most abundant cluster in ashort range of densities only when the larger value of x s is considered. For x s =0.85, the dissolution density rangefor all clusters is ∼ − × − fm − for FSU2R andDDME2. It is interesting to see that for the RMF withnon-linear terms, such as FSU2R and SFHo, the fractionof He clusters becomes quite large just before the bind-ing energy of the cluster goes to zero, followed by a steepdecrease of the abundance of this cluster. In the case ofSFHo and x s =0.85, we can even identify a first order phasetransition at this density.In Fig. 3, we compare the α -particle mass distributionsobtained for the models FSU2R, NL3 ωρ , TM1 ωρ , SFHo,DD2 and DDME2, considering a temperature of T = 10MeV, and proton fraction of y p = 0 .
41. For all the mod-els shown, the scalar cluster-meson coupling fraction isset to x s =0.85. For SFHo, we have also plotted the abun-dances with x s = 0 . ± .
05, represented by the hashedregion. The differences reflect the properties of these mod-els at subsaturation densities. Below 0.02 fm − , i.e. not farfrom the range of densities where the virial EoS is valid,all models give similar results. Above 0.03 fm − , the mod-els start to differ, with DD2 and DDME2 predicting thesmallest dissolution density, slightly below 0.05 fm − , andSFHo the largest one, ≈ . − . SFHo is, in fact, aspecial case because all the other models predict dissolu-tion densities in a narrow range of ∼ . − .
06 fm − .With SFHo, we get a similar result, if the x s is reducedto ≈ . − .
7. It is expectable that under the conditionswhere light clusters play an important role, the predictions -4 -3 -2 -1 p =0.41T=10MeV X H e n B (fm -3 )FSU2RNL3wrTM1wrDDME2DD2SFHoSFHo Fig. 3. (Color online) The mass fraction of the α cluster asa function of the density for all the models considered, and T = 10 MeV. The scalar cluster-meson coupling fraction is setto x s = 0 .
85, except for SFHo where the results are shownfor x s = 0 . ± .
05 (wide hashed band) and for x s = 0 . obtained with SFHo will differ from the ones obtainedwith any of the other five models. This could be expectedfrom the results of Ref. [59] where the spinodal sectionsobtained for SFHo extend to a much larger ( n p , n n ) phase-space region than FSU2R, TM1 ωρ or DDME2.In Fig. 4, we plot, for comparison, the mass distribu-tions of the five clusters considered, calculated within themodels FSU2R, NL3 ωρ , TM1 ωρ , SFHo, DD2 and DDME2,for two temperatures, T = 5, and 10 MeV, and two protonfractions y p = 0.2 and 0.41. The temperatures chosen aretypical in proto-neutron stars, and the proton fractions re-flect two different stages of the star evolution. Some gen-eral comments are in order: (i) In average, models withdensity-dependent couplings give different fractions fromthe others if the same value of x s is chosen. They gener-ally predict the cluster dissolution at smaller densities andsmaller particle fractions at low densities for the neutronrich clusters; (ii) In the low-density range shown, the neu-tron rich clusters, tritium and He, are the most abundantclusters for y p = 0 . T = 5 MeV. At T = 10 MeV,this is still true for the tritium, the one with the smallestmass; (iii) Models with non-linear mesonic terms show asteeper behavior close to the dissolution density.We now turn to examine how the differences observedin the models reflect in the predictions for the equilibriumconstants, which are the quantities determined from theexperimental data [47,48]. The equilibrium constants aredefined as the ratio K ci = n i n N i n n Z i p , where n i is the density of cluster i , n n and n p are, respec-tively, the density of free neutrons and protons, and Z i , N i are the number of protons and neutrons in cluster i .In Fig. 5, we plot the equilibrium constants obtainedwith the different models on typical ( n B − T ) trajectories Tiago Cust´odio et al.: Light clusters in warm stellar matter: calibrating the cluster couplings -4 -3 -2 -1 y p =0.41T=5MeVx s =0.85 X H DDME2DD2NL3wrTM1wrFSU2R10 -4 -3 -2 -1 X H -4 -3 -2 -1 X H e -4 -3 -2 -1 X H e -4 -3 -2 -1 X H e n B (fm -3 ) y p =0.2T=5MeVx s =0.850.02 0.03 0.06 0.10n B (fm -3 ) y p =0.2T=10MeVx s =0.850.02 0.03 0.06 0.10n B (fm -3 ) Fig. 4. (Color online) The mass abundances of all the clusters considered as a function of the density for the models FSU2R,NL3 ωρ , TM1 ωρ , SFHo, DD2 and DDME2 and T = 5 MeV and y p = 0 .
41 (left column), T = 5 MeV and y p = 0 . T = 10 MeV and y p = 0 . x s = 0 .
0 0.005 0.01 0.015 0.02 0.025 0.03(a) K c H e ( f m ) n B (fm -3 )DDME2, x s =0.85FSU2R, x s =0.85FSU, x s =0.85SFHo, x s =0.70SFHo, x s =0.85 10
0 0.005 0.01 0.015 0.02 0.025 0.03(a)(b) K c H e ( f m ) n B (fm -3 )
0 0.005 0.01 0.015 0.02 0.025 0.03(d) K c H ( f m ) n B (fm -3 )10
0 0.005 0.01 0.015 0.02 0.025 0.03(c) K c H ( f m ) n B (fm -3 ) Fig. 5. (Color online) The chemical equilibrium constants as a function of the density from a calculation where we considerhomogeneous matter with four light clusters for the DDME2 (red circles), FSU2R (crosses), and FSU (black triangles) models,calculated at the experimental values ( T , n B exp , y p exp = 0 .
41) proposed in Ref. [47], and considering the cluster coupling as x s = 0 .
85. Also shown are the results for SFHo with x s = 0 . x s = 0 .
85 (black diamonds). that can be explored in heavy ion collisions. The choice ofthe temperature value, at each density point, is the oneestimated in Ref. [47], and the proton fraction is fixedto y p = 0 .
41 at each point. The volume estimation inthat paper is not fully realistic, since it was made in thesimplifying assumption of an ideal cluster gas. However,many different theoretical calculations [60] were producedassuming the (
T, n B , y p ) correlation of Ref. [47], there-fore this particular choice is useful to assess the modeldependence of the calculations. In this Figure, the sensi-tivity of the chemical constants to the scalar cluster-mesoncoupling for one representative model, SFHo, is shown.We can see that the effect previously observed in Fig. 3,namely the positive correlation between the value of x s ,here reflects into higher values for the equilibrium con-stants when x s is increased. This effect is sizeable and po-tentially bigger than the experimental error bars on equi-librium constants, meaning that a comparison with exper-imental data, within a given model, allows predicting thedissolution density of clusters in dense matter. The resultsof three different models DDME2, FSU2R and SFHo, us-ing the same value for that coupling, fixed to x s = 0 .
85, asproposed in [29], are also displayed in this figure, and theyshow the model dependence of the equilibrium constantprediction. Models TM1 ωρ and NL3 ωρ have a behavior very close to FSU2R, and are not represented. The differ-ence between the predictions is a measure of the modeldependence of the calculation. We can see that the chem-ical constants are smaller for DDME2 reflecting the factdiscussed before that the clusters dissolve at smaller den-sities within this model. We can, however, say that thebehavior of FSU2R and DDME2 models is similar to theone obtained in [29] within the model FSU, probably dueto the fact that these models have similar properties atsubsaturation densities.As expected, the coupling x s is model-dependent, andit should be fitted to some experimental data or ab-initiocalculation. In the following, we determine for each modelthe range of x s values that best reproduces the INDRAequilibrium constants for the different clusters as calcu-lated in [49]. In Fig. 6, we show how the chemical equilib-rium constants calculated within models FSU2R, DDME2and SFHo with the best x s values compare with the onesextracted from the experimental data obtained by the IN-DRA collaboration. We have obtained for FSU2R x s =0 . ± .
02, for DDME2 x s = 0 . ± .
02, and for SFHo x s = 0 . ± .
03. It is observed that the quality of thefit depends slightly on the size of the clusters, and thisdeserves an investigation in a future work. Let us recallthat previously, in Refs. [49,50], the authors performed an
Tiago Cust´odio et al.: Light clusters in warm stellar matter: calibrating the cluster couplings He He H, He H Xe+
SnFSU2R K c ( A Z ) ± σ ( f m ( A - ) ) n B (fm -3 ) x s =0.91 ± He He H, He H Xe+
SnDDME2 K c ( A Z ) ± σ ( f m ( A - ) ) n B (fm -3 ) x s =0.93 ± He He H, He H Xe+
SnSFHo K c ( A Z ) ± σ ( f m ( A - ) ) n B (fm -3 ) x s =0.83 ± Fig. 6. (Color online) The equilibrium constants as a function of the density. The full lines represent the experimental resultsof the INDRA collaboration, with 1 σ uncertainty. The grey bands are the equilibrium constants from a calculation [30] wherewe consider homogeneous matter with five light clusters for the FSU2R EoS (left), the DDME2 EoS (middle) and SFHo EoS(right), calculated at the average value of ( T , n B exp , y p exp ), and considering cluster couplings in the range of x s = 0 . ± . x s = 0 . ± .
02 (DDME2) and x s = 0 . ± .
03 (SFHo). The color code represents the global proton fraction. -3 -2 -1 T=5MeVy p =0.41FSU2Rx s =0.91 X i He H He H He10 -3 -2 -1 x s =0.93DDME2 X i T=10MeV10 -3 -2 -1 s =0.83 X i n B (fm -3 ) 0.02 0.04 0.06 0.10 0.15n B (fm -3 ) Fig. 7. (Color online) The abundances of all the clusters con-sidered as a function of the density for T = 5 MeV (left) and T = 10 MeV (right) and a fixed proton fraction of 0.41 forFSU2R with x s = 0 .
91 (top), DDME2 with x s = 0 .
93 (mid-dle), and SFHo with x s = 0 .
83 (bottom). analysis of the experimental data within the FSU model,and in order to reproduce data, it was necessary to take x s = 0 . ± .
02, a result very close to the one obtained -4 -3 -2 -1 p =0.41 X H e n B (fm -3 )FSU2R, x s =0.91 ± s =0.93 ± s =0.83 ± Fig. 8. (Color online) The mass fraction of the α − particle asa function of the density for T = 10 MeV and a fixed protonfraction of 0.41 for FSU2R with x s = 0 . ± .
02, DDME2 with x s = 0 . ± .
02, and SFHo with x s = 0 . ± . with FSU2R and DDME2. For SFHo, and as we saw be-fore, we need a smaller coupling x s to fit this data.Choosing the scalar cluster-meson coupling ratio thatbest fits the INDRA data, we calculate the clusters abun-dances for FSU2R, DDME2 and SFHo, for y p = 0 .
41 and T = 5 and 10 MeV, see Fig. 7. All models predict similarabundances of all the clusters considered up to a density ≈ .
05 fm − for T = 5 MeV and ≈ .
06 fm − for T = 10MeV. This result is very interesting: in fact, as shown in[50], the INDRA data explore densities up to ≈ .
06 fm − ,however these larger densities are attained at a temper-ature ≈ T = 5 MeV corresponds to iago Cust´odio et al.: Light clusters in warm stellar matter: calibrating the cluster couplings 9 densities below ≈ .
02 fm − . We may, therefore, expectthat a fit to the INDRA data is giving information on theabundances of light clusters corresponding to a pair ( T , n B ). Although the proton fraction is also changing alongwith T and n B , it takes values in a very narrow range,0.39-0.42, very close to y p = 0 .
41 used to calculate thecluster abundances in Fig. 7.Models FSU2R and DDME2 show very similar frac-tions also above ≈ .
05 fm − , in particular, at the max-imum of the distributions and at the dissolution density,which we define as the density above which the fractionsare below 10 − . However, SFHo predicts dissolution den-sities ∼
30% larger than the other two models.Having this in mind, we plot in Fig. 8 the mass frac-tion of the α − particle as a function of the density forthe three models previously considered, and a temperatureof 10 MeV and a fixed proton fraction of 0.41. For eachmodel, we choose the range of the σ − coupling that bestfits the INDRA data. We confirm that the α -abundancespredicted by the three models coincide up to ≈ .
06 fm − .Moreover, for FSU2R and DDME2 we do have a completesuperposition of the bands, indicating a similar predictionfor the dissolution density. SFHo, however, shows a higherdissolution density, ∼
30% larger.The present results seem to indicate that a good re-production of the equilibrium constants obtained from theexperimental data could imply a unique prediction for thecluster abundances, and, in particular, of the dissolutiondensity only if we could have some extra experimentalconstraints at a slightly higher density.
We have analyzed the appearance of light clusters in warmnon-homogeneous matter at densities below saturation den-sity in the framework of RMF models. We used six modelsthat properly describe nuclear matter properties, and pre-dict stars with more than two solar masses, two of whichwith density-dependent couplings, and the other four withnon-linear mesonic terms. Light clusters were includedas point-like particles that are affected by the mediumthrough their couplings to the mesons. For these cou-plings, we have considered: (a) the results of [29], where,for the σ -meson coupling, a universal coupling propor-tional to A i x s g σ , with x s to be fixed on experimental data,was proposed; (b) the couplings determined in [49,50] ex-tracted from the INDRA [48] experimental data.Except for the model SFHo, we have found that differ-ent models predict similar abundances of clusters. Overall,for the density-dependent models we have obtained 15% to20% smaller dissolution densities, but far from the disso-lution density, the abundances are similar with respect tothe non-linear models. For SFHo, taking the same scalarcluster-meson coupling, the dissolution densities are ap-proximately the double, and the cluster abundances arelarger. It is, therefore, expectable that simulations thatuse SFHo to describe supernova explosions or binary NSmergers will have larger contributions of light clusters. Inorder to reproduce the equilibrium contants obtained from heavy ion collisions, a smaller coupling of the light clus-ters to the σ -meson has to be considered. We conclude thatthe clusterization effect, in particular the amount and thechemical composition of clusters, depends on the behaviorof the model in the corresponding density range. Takinguniversal couplings for the clusters highlights the differ-ences. The present heavy ion constraints are not enoughto distinguish between models like DDME2 and FSU2R,but clearly shows that SFHo requires a different treat-ment.In the present comparison, we have considered besidesthe lighter clusters d, t, h, and α , also the heavier cluster He. In asymmetric matter, it was shown that the con-tribution of this cluster is quite important in a range ofdensities not far from the dissolution density. A discussionof the role of heavier clusters at the densities and temper-atures studied in the present work has been presented in[30]. Moreover, we believe there is a need of experimentalmeasurements for heavier clusters in order to discriminatethe different models.The gRMF formalism presented here allows to takecluster formation into account for hot and dense nuclearmatter, in particular stellar matter. For the contributionof nucleon quasiparticles ( n, p ) different parametrizationswithin the RMF are possible. We considered several mod-els, and some of them were calibrated to the INDRA data,namely FSU2R, DDME2, and SFHo. The coupling param-eter x s for the interaction with the σ field can be intro-duced as a global quantity for all clusters. It determinesthe density where the respective clusters are dissolved. Wehave shown that if x s is fitted to equilibrium constants de-termined from experimental data, different models predictsimilar abundances up to the densities and temperaturesexplored by INDRA data. The dissolution densities, how-ever, differ: while two of the models, FSU2R and DDME2predict similar behavior at dissolution, the third model,SFHo, gives dissolution densities that are at least 30%larger. In the future, a more careful analysis will be un-dertaken using statistical methods to extract these quan-tities. Besides, a microscopic approach to this couplingparameter may show a dependence on the respective nu-cleus, as well as on thermodynamic parameters, like thetemperature. This may indicate that the model appliedin the present study needs to take these dependences intoaccount. This point is left for future developments. We thank R. Bougault for useful discussions. This work wassupported by the FCT (Portugal) Projects No. UID/FIS/04564/2019and UID/FIS/04564/2020, and POCI-01-0145-FEDER-029912,by PHAROS COST Action CA16214, and by the GermanResearch Foundation (DFG), Grant
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