EEur. Phys. J. A (2020) 56:258 https://doi.org/10.1140/epja/s10050-020-00262-1
Light clusters in dilute heavy-baryon admixed nuclear matter
Armen Sedrakian , Frankfurt Institute for Advanced Studies, Ruth-Moufang str. 1, D-60438 Frankfurt am Main, Germany Institute of Theoretical Physics, University of Wroc(cid:32)law, pl. M. Borna 9, 50-204 Wroc(cid:32)law, PolandReceived: 1 September 2020 / Accepted: 26 September 2020 © Author(s) 2020
Abstract.
We study the composition of nuclear matter at sub-saturation densities, non-zero temperatures,and isospin asymmetry, under the conditions characteristic of binary neutron star mergers, stellar collapse,and low-energy heavy-ion collisions. The composition includes light clusters with mass number A ≤
4, aheavy nucleus ( Fe), the ∆ -resonances, the isotriplet of pions, as well as the Λ hyperon. The nucleonicmean-fields are computed from a zero-range density functional, whereas the pion-nucleon interactions aretreated to leading order in chiral perturbation theory. We show that with increasing temperature and/ordensity the composition of matter shifts from light-cluster to heavy baryon dominated one, the transitiontaking place nearly independent of the magnitude of the isospin. Our findings highlight the importanceof simultaneous treatment of light clusters and heavy baryons in the astrophysical and heavy-ion physicscontexts. PACS.
The formation of light clusters in dilute, warm nuclearmatter is of interest in astrophysics of binary neutron starmergers, stellar collapse, as well as in heavy-ion physics.The details of the matter composition are important forthe accurate determination of transport coefficients ap-pearing in dissipative relativistic fluid dynamics as wellas the neutrino Boltzmann transport in various astro-physical scenarios. The clustering phenomenon is also ofgreat interest in nuclear structure calculations (e.g. alpha-clustering) and heavy ion collisions in laboratory experi-ments.A great deal of effort during the last decade was fo-cused on the accurate determination of the compositionof dilute nuclear matter at finite temperatures and isospinasymmetry within a range of methods based on the ideasof nuclear statistical equilibrium [1–36] and virial expan-sion for quantum gases [37–40]. The appearance of clustersleads to a range of interesting phenomena, in particular α -condensation at low temperatures [25, 26, 30, 41–44].In astrophysics, light clusters and their weak interactionswith neutrinos were studied in detail in the context ofstellar collapse and supernova physics [45–48]. The elec-troweak interactions of leptons with baryonic matter arealso of interest in describing the transport in binary neu-tron star mergers, in particular the bulk viscosity [49–52]and electrical conductivity [53, 54]. The formation of the heavy baryons in dense and coldnuclear matter, in particular hyperonic members of the J / baryonic octet in combinations with the non-strangemembers of baryon J / decouplet ( ∆ -resonances) hasattracted attention in recent years [55–64]. The relativis-tic density functionals were successfully tuned to removethe tension between the softening of the equation of stateof dense matter associated with the onset of the baryonsand the astrophysical observations of the massive neutronstars with masses 2 M (cid:12) [58–60].The motivation of this work is to explore the inter-play between the clustering and heavy-baryon degrees offreedom in dilute, finite-temperature nuclear matter. Forthis purpose we set-up a model which includes both lightnuclear clusters with mass number A ≤
4, a representa-tive heavy nucleus ( Fe) as well as the Λ -hyperon, thequartet of ∆ -resonances, and the isotriplet of pions π ± , .Previously, hyperons were included in the finite tempera-ture composition of matter in stellar collapse and proto-neutron star studies [63, 65, 66]. Pions and pion conden-sation has been studied recently in the stellar context inRefs. [66–69]. While the light nuclear clusters have beenaccounted for in the low-density envelops used in somemodels, a combined study of the clustering, heavy baryonsand pions is missing so far.In this work, we extend the approach of Ref. [25] to in-clude heavy baryons and pions in the composition and theequation of state of isospin asymmetrical nuclear matter. a r X i v : . [ nu c l - t h ] O c t Armen Sedrakian: Light clusters in dilute heavy-baryon admixed nuclear matter
In addition to the mean-field effects included in the pre-vious study, we will treat also the Pauli-blocking effectson the binding energies of the light clusters in an approxi-mate manner. We will focus on temperatures T ≥
10 MeV,which is above the critical temperature of Bose-Einsteincondensation of α particles in the clustered environment,see for further details [25, 26, 30, 42–44]. Indeed, lowtemperatures disfavor the heavy baryons in low-densitynuclear matter and the problem of α condensation is un-affected by their nucleation. While we include in our com-position a heavy nucleus, its effect will turn out to beminor in the parameter range studied in this work.The paper is organized as follows. Section 2 extendsthe formalism of the quasiparticle gas model [25] to in-clude heavy baryons and pions. In Sec. 3 we present thenumerical results for the composition and equation of stateof matter. Section 4 provides a summary and an outlook. We consider matter composed of unbound nucleons, heavybaryons, light nuclei ( A ≤ Fe and pions at temper-ature T and baryon number density n B . We assume thatthe charge fraction is fixed to a value Y Q = n L /n B , where n L = ( n e − n e + ) + ( n µ − n µ + ) where n e , n e + , n µ and n µ + are the number densities of electrons, positrons, muonsand anti-muons. The thermodynamical potential of thesystem can be expanded into a sum of contributions ofconstituents Ω ( µ n , µ p , T ) = (cid:88) j Ω j ( µ j , T ) , (1)where j runs over the all elements of the composition ofmatter, specifically, j = A, Z for nuclei with mass number A and charge Z , j = n, p for neutrons and protons, j = { ∆ , ∆ + , ∆ ++ , ∆ − } for ∆ -resonances, j = Λ for the Λ -hyperon, and π , π ± for the isotriplet of pions. Here thechemical potentials of the species µ j are functions of thechemical potentials of neutrons and protons µ n and µ p in“chemical” equilibrium with respect to weak and stronginteractions.If a nucleus is characterized by mass number A andcharge Z its chemical potential is expressed as µ A,Z = ( A − Z ) µ n + Zµ p . (2)For the chemical potentials of heavy baryons the followingrelations hold µ Λ = µ ∆ = µ n = µ B , (3) µ ∆ − = 2 µ n − µ p = µ B − µ Q , (4) µ ∆ + = µ p = µ B + µ Q , (5) µ ∆ ++ = 2 µ p − µ n = µ B + 2 µ Q , (6)where we introduced the baryon number chemical poten-tial µ B and the charge chemical potential µ Q = µ p − µ n . The chemical potentials of the pions obey the followingrelations µ π = 0 , (7) µ π + = µ p − µ n , (8) µ π − = µ n − µ p . (9)The baryon number density and the charge neutrality con-ditions are given by the relations n B = n n + n p + (cid:88) c A c n c + n ∆ ++ + n ∆ + + n ∆ − + n ∆ + n Λ , (10) n B Y Q = n p + (cid:88) c Z c n c + 2 n ∆ ++ + n ∆ + − n ∆ − + n π + − n π − , (11)where the c -summation goes over the densities of deuteron( d ), triton ( t ), He ( h ), α -particle and Fe nucleus. Thelatter nucleus is considered below in its ground state, i.e.,the states that are excited at finite temperatures are ne-glected. The inclusion of these states will act to enhancethe fraction of this particular nucleus or other heavier nu-clei in matter, should they be included in the composi-tion. Equations (10) and (11) determine the two unknownchemical potentials µ n and µ p at any temperature T forfixed values of n B and Y Q .The thermodynamical potential for each species canbe expressed through the densities Ω j ( µ j , T ) = − V (cid:90) µ j −∞ dµ (cid:48) j n j ( µ (cid:48) j , T ) , (12)where n j ( µ (cid:48) j , T ) is the number density of species j , V isthe volume.In the stellar context, the matter is charge neutral,the positive charge of baryons being neutralized by lep-tons (electrons and muons). The lepton thermodynamicpotential is given by Ω L = − (cid:88) l = e,µ g l T (cid:90) d k (2 π ) ln (cid:2) f − l ( − E l ( k ) + µ l ) (cid:3) , (13)where the index l sumes of electrons e and muons µ ( τ -leptons can be neglected), g l = 2 is the degeneracy factor,the lepton energy is given by E l = (cid:112) k + m l , where m l is the lepton mass and µ l their chemical potential and f l stands for the lepton Fermi distribution function. Thelepton density is obtained then as n l = ∂Ω L /∂µ l . At fi-nite temperatures a small fraction of positrons may ap-pear: their thermodynamical potential is obtained fromEq. (13) by interchanging the sign of the electron chemi-cal potential. To obtain the full thermodynamical poten-tial of matter in astrophysical contexts one needs to takeinto account, in addition, the thermodynamical potentialof neutrinos and anti-neutrinos. For any fixed flavor it hasthe same form as Eq. (13), the only difference being thedegeneracy factor g ν = 1 (as implied by the StandardModel) and vanishingly small neutrino mass. rmen Sedrakian: Light clusters in dilute heavy-baryon admixed nuclear matter 3 Having computed partial contributions Ω j , the ther-modynamic quantities can be obtained from the thermo-dynamic potential Eq. (1) for nuclear systems and fromthe sum of Eq. (1) and Eq. (12) in the charge neutral stel-lar systems. In particular, we recall that the pressure andthe entropy are given by P = − ΩV , S = − ∂Ω∂T . (14) We now turn to the computation of the partial densitiesof constituents. This can be done in a unified manner forquasiparticles, resonances, and clusters using the real-timefinite temperature Green’s function (hereafter GF) for-malism. The density of species j are directly related tothe following GFs iG Fig. 1. Dependence of the mass fractions of the particlesin dilute nuclear matter on temperature at constant density n B /n = 10 − . The top and lower panels correspond to chargefractions Y Q = 0 . α -particles (dash-dotted), ∆ resonances (dash-double-dot), Λ -hyperon (dash-triple-dot), and pions (double-dash-dot). The mass fraction of Fe is not visible on the figure’s scale. The binding energies of clusters are functions of den-sity and temperature in general. The nuclear environmentinfluences the binding energies through phase space oc-cupation (Pauli-blocking). To take this into account, weuse the results of the solutions of in-medium two-bodyBethe-Salpeter and three-body Faddeev equations in di-lute nuclear matter given in Ref. [72]. These solutionsare fitted by the following procedure: (a) first we deter-mine the critical value of the inverse temperature β forwhich a cluster disappears via the formula: β cr [MeV − ] =0 . . n /n B ), where n = 0 . 16 fm − , whichis assumed to be universally independent of A and Y Q ,and (b) the in-medium binding energies B j ( n B , T ) are ob-tained via a linear fit given by B j ( n B , T ) = E j (cid:20) − ββ cr ( n /n B ) (cid:21) . (28)Then the spectral function (25) takes the form S j ( ω, p ) (cid:39) πδ (cid:18) ω − p M − B j + µ ∗ j (cid:19) , (29)where any contribution to the self-energy beyond the mod-ifications of the binding energy is energy and momentumindependent and, thus, can be absorbed in the chemicalpotential µ ∗ j . The system of Eqs. (10) and (11) was solved simultane-ously for unknown chemical potentials µ n and µ p at fixed temperature T , baryon number density n B and chargefraction Y Q . We consider two values of the latter param-eter Y Q = 0 . 1, which is characteristic to binary neutronstar mergers, and Y Q = 0 . mass fraction X j = A j n j /n B ,where A j is the mass number of a constituent, as a func-tion of temperature in cases (a) nucleons and clusters onlyand (b) nucleons, clusters, heavy baryons and pions, for Y Q = 0 . n B /n = 10 − , where n = 0 . − is the nuclear saturation density. The mass fractionof Fe is not visible on figure’s scale. It is seen that nu-cleons are the dominant component at all temperatures,but there is a change in the composition of matter withrespect to the remaining constituents with increasing tem-perature. For temperatures T ≥ 30 MeV the dominantmass fraction is in the heavy baryons, whereas at lowertemperatures the clusters are the dominant component.Note also that the inclusion of heavy baryons and pionsreduces the isospin asymmetry in the neutron and pro-ton components and, as a consequence, the helion andtriton abundances are much closer to each other in thiscase. A previous study of hyperon abundances at finitetemperatures in Ref. [28] finds that the hyperon fractionexceeds 10 − at density n B /n = 10 − for temperatures T ≥ 40 MeV. According to Fig. 1 this occurs in our modelfor T ≥ 20 MeV. This difference may be a consequence ofdifferent treatment of nuclear interactions and differentcompositions allowed in the models. Ref. [27] finds that Λ hyperon fraction stays below 10 − for temperatures upto 14 MeV in the inhomogeneous “pasta” phases of su-pernova matter independent of the value of Y Q , which isconsistent with present results.Figures 2 and 3 show the mass fractions mass frac-tion X j at two fixed temperatures T = 30 MeV and T = 10 MeV and varying density. It is seen that theabundances of the nucleons, heavy baryons, and pionsare insensitive to the density, whereas the cluster abun-dances increase as the density increases. In other words,the increase in the nucleonic density at a fixed temper-ature is accommodated by the system by increasing thenumber of the light clusters, whereas the fractions of neu-trons and protons remain constant in a wide density range.Since the heavy baryon fraction are determined by their“chemical” equilibrium with respect to neutrons and pro-tons via the relations (3)-(6), their fractions stay constantwith the density as well. The same applies also to pionfractions, which are likewise related to proton and neu-tron concentrations via Eqs. (8) and (9). The reductionof isospin asymmetry among neutrons and protons men-tioned above is seen here as well. Note that the Pauli-blocking at T = 30 MeV is ineffective within the densityrange considered, but its effect is seen in the right panelsof Fig. 3 corresponding to T = 10 MeV. It is seen that n B /n (cid:39) . B j ( n B , T ) → rmen Sedrakian: Light clusters in dilute heavy-baryon admixed nuclear matter 5 Fig. 2. Dependence of the mass fractions of the particlesin dilute nuclear matter on density for T = 30 MeV. Thetop and lower panels correspond to charge fraction Y Q =0 . α -particles (dash-dotted), ∆ reso-nances (dash-double-dot), Λ -hyperon (dash-triple-dot), and pi-ons (double-dash-dot). In the right figure, the clusters disap-pear for n B /n ≥ × − (shaded area) due to the Pauli-blocking of the phase-space. The mass fraction of Fe is notvisible on the figure’s scale. Fig. 3. Same as in Fig. 2 but for T = 10 MeV. the phase space vanishes with increasing the density moresmoothly: the clusters with the lower-momenta are elimi-nated first, while those with high-momenta remain intact.It is also seen that the pion mass fraction undergoes at thesame point an abrupt change, clearly visibly for Y Q = 0 . Fig. 4. Pressure as a function of normalized density n B /n for temperature values (in MeV) T = 20 (solid lines), 30 (long-dashed), 40 (short-dashed), and 50 (dash-doted). The upperpanels correspond to Y Q = 0 . Y Q = 0 . ing energies of clusters, the transition is found to be lessabrupt.Figure 4 shows the pressure as a function of the nor-malized density for temperature values T = 20 , 30, 40,and 50 MeV for two values of charge fraction Y Q = 0 . Y Q = 0 . 1, already observedin Fig. 2, which leads to pressure values that are similarto those for the case Y Q = 0 . The composition of warm dilute nuclear matter was com-puted including simultaneously light clusters with A ≤ Fe), heavy baryons ( Λ ’sand ∆ ’s) and pions. We find that with increasing temper-ature the mass fraction shifts from light clusters to heavybaryons, whereby the nucleons remain the dominant com-ponent within the parameter range considered. The heavynucleus Fe does not play a significant role at tempera-tures T ≥ 10 MeV, but is known to suppress stronglythe abundances of light clusters at low temperatures ofthe order 1 MeV [25, 44]. The addition of heavy baryonsand pions makes the nucleonic component more isospinsymmetric and, as a consequence, the cluster abundancesbecome less sensitive to the value of the isospin asymme-try. At low temperatures T (cid:39) 10 MeV, the phase-spaceoccupation strongly suppresses the cluster abundances fordensities n B /n ≥ . 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