Phenomenological quark-hadron equations of state with first-order phase transitions for astrophysical applications
PPhenomenological quark-hadron equations of state with first-order phase transitionsfor astrophysical applications
Niels-Uwe F. Bastian
Institute for Theoretical Physics, University of Wroclaw, pl. M. Borna 9, 50-204 Wroclaw, Poland (Dated: September 24, 2020)In the current work an equation of state model with a first-order phase transition for astrophysicalapplications is presented. The model is based on a two-phase approach for quark-hadron phasetransitions, which leads by construction to a first-order phase transition. The resulting model hasalready been successfully used in several astrophysical applications, such as cold neutron stars, core-collapse supernova explosions and binary neutron star mergers. Main goal of this work is to presentthe details of the model, discuss certain features and eventually publish it in a tabulated form forfurther use.
I. INTRODUCTION
The investigation of the equation of state (EoS) ofstrongly interacting matter is an ongoing problem ofnuclear and high-energy physics. Direct approaches tosolve the underlying theory of quantum chromodynamics(QCD) are only accessible at high temperatures and van-ishing densities or at asymptotically high densities. Forthe region, which is relevant in astrophysics, mainly ef-fective and phenomenological approaches are in use (e.g.see reviews [1, 2] for further reference).Of particular interest is the possible transition fromordinary hadronic matter at low densities/temperaturesto a phase of deconfined quarks. Numerical re-sults from Lattice QCD predict a crossover transitionwith a pseudo-critical temperature of T ( µ B = 0) =156 . ± . [3]. However, these calculations are lim-ited to small baryon chemical potentials and can notreach densities relevant for astrophysics. At asymptot-ically high densities perturbative QCD (pQCD) can beapplied, which predicts a phase of deconfined quark mat-ter [4, 5]. Unfortunately, it is not possible to reach densi-ties, where the transition from hadronic to quark matteroccurs, and therefore both the position and order of thistransition at low temperatures are currently speculative.The two possible scenarios are the existence of at leastone critical endpoint (CEP), changing the cross over tran-sition into a first-order transition, and the absence of anyCEP, which would result in the cross over transition tospan the entire phase diagram.In the presented model, the existence of a first-orderphase transition at high baryon densities is assumed asa working hypothesis, in order to explore possible impli-cations in astrophysics, which might lead to measurablesignals. In contrary to the original publication of themodel in [6], where neutron star configurations are stud-ied, here the applicability is extended to such dynamicphenomena as core-collapse supernova (CCSN) or binaryneutron star merger (BNSM), which reach not only highdensities, but also high temperatures.. This extensionbroadens the spectrum of predictable signals from mass-radius relations and tidal deformability (of neutron stars)to, e.g., gravitational waves, neutrino signals and possi- bly electromagnetic counter parts. Additionally the hy-pothetical case of multiple CEPs is covered, due to thepossibility to probe the existence of a first-order phasetransition at various temperatures.The material, presented here, was so far successfullyapplied to BNSM in [7], suggesting a possible signal ingravitational waves, which would unambiguously iden-tify the existence of a sudden softening in the EoS like afirst-order phase transition. Furthermore, it was appliedto CCSN simulations in [8], predicting a second shockwave, which leads to the successful explosion of a
50 M (cid:12) progenitor star and a measurable neutrino signal, origi-nating from the phase transition. The additional shockwave in the CCSN leads to an altered result of nucle-osynthesis analyses, bringing back supernovae as sourceof r -process elements, as shown in [9].The document is structured in the following way: Firstof all, section II presents the model in its current state,including the description of each component and the real-isation of phase transition. Afterwards, in section III thecreated parameter sets and their properties are shown.Finally, in section IV a discussion is laid out about theconsequences of certain aspects of the model and possiblealternative approaches.Note, that in this publication natural units ¯ h = c = k B = 1 and the unity volume V = 1 is applied. II. HYBRID EQUATION OF STATE MODEL
The particles involved in astrophysical systems can begrouped into 5 classes. Due to the different nature oftheir thermodynamic interactions it is reasonable to as-sume, that mixed terms in the thermodynamic potentialare small and their contributions can be addressed sepa-rately:
Ω = Ω nucleons + Ω nuclei + Ω quarks + Ω leptons + Ω photons . (1)Since photons do not carry any charge, they are only in-cluded as thermal excitation at high temperatures and donot affect the phase transitions. Included leptons, such as a r X i v : . [ nu c l - t h ] S e p electrons and neutrinos, are fully degenerate at our densi-ties and can be described as ideal gas. They carry electricand leptonic charge and are (dependent on the system) inchemical equilibrium with the strongly-interacting parti-cles. The separation of the strongly-interacting part intoquarks and hadrons is an assumption of the presentedmodel, which is commonly used in astrophysics (so calledtwo-phase approach).In section II A the relativistic density functional (RDF)formalism is derived which is used to describe homo-geneous quark and hadron matter in a mean field ap-proximation. Afterwards the particular models used for Ω nucleons (section II B) and Ω quarks (section II C) areshown. Section II D presents the model which is usedfor nuclear cluster formation Ω nuclei and section II E ex-plains the inclusion of leptons Ω leptons . A. Relativistic density functional derived fromfield theory
The RDF approach as introduced in [6] is a frameworkcapable of dealing with very complex interaction contri-butions, e.g., confinement. Here the derivation is donein a more thorough and complete manner, including iso-vector couplings, which have major effects to the iso-spinasymmetric phase diagram, relevant for astrophysics.The derivation is done self-consistently from the path-integral formalism, based on an effective Lagrangian oflow-energy QCD to obtain the partition function Z andhence the thermodynamic potential Ω = − T ln Z . (2)The RDF approach can be obtained using the path inte-gral approach to the partition function, which analogousto the treatment of the Walecka model of nuclear matterin [10] takes the form Z = (cid:90) D ¯ q D q exp (cid:26)(cid:90) d x (cid:2) L eff + ¯ qγ ˆ µq (cid:3)(cid:27) , (3)where in the case of a two-flavour quark model q = (cid:18) q u q d (cid:19) , (4)and ˆ µ = diag( µ u , µ d ) is the diagonal matrix of the chemi-cal potentials conjugate to the conserved numbers of cor-responding quarks. The effective Lagrangian density isgiven by L eff = L free − U , (5) L free = ¯ q ( ıγ µ ∂ µ − ˆ m ) q , (6)where ˆ m = diag( m u , m d ) is the matrix of current quarkmasses. The interaction is given by the potential en-ergy density U = U (¯ qq, ¯ q(cid:126)τ q, ¯ qγ q, ¯ q(cid:126)τ γ q ) which in generalis a non-linear functional of the field representations of the scalar, vector and corresponding isovector quark cur-rents. Note here, that I restrict myself to quark fields outof readability of the manuscript - the formalism can bewritten down for any fermionic many body system, whichneeds to be treated in a relativistic scheme by exchang-ing the respective quark fields q and ¯ q by, e.g., nucleonicfields ψ and ¯ ψ .In the isotropic case, the vector four-current reducesto its zeroth component and the gradients of the fieldsvanish, so that γ µ ∂ µ = γ ∂ . Due to charge conservation,only the third component of any isovector fields wouldremain, so that only the third component of the isospin–vector (cid:126)τ = ( τ , τ , τ ) T would remain, which is here short-handed written as τ = τ .To achieve a quasi-particle representation, the poten-tial energy density shall depend linearly on the Diracspinor bilinears representing the relevant currents of thesystem. The linearisation of the interaction is facilitatedby a Taylor expansion around the corresponding expec-tation values (cid:104) ¯ qq (cid:105) = n s , = (cid:88) i n s ,i = − (cid:88) i T ∂∂m i ln Z , (7a) (cid:104) ¯ qτ q (cid:105) = n si , = (cid:88) i τ i n s ,i = − (cid:88) i τ i T ∂∂m i ln Z , (7b) (cid:10) ¯ qγ q (cid:11) = n v , = (cid:88) i n v ,i = (cid:88) i T ∂∂µ i ln Z , (7c) (cid:10) ¯ qτ γ q (cid:11) = n vi , = (cid:88) i τ i n v ,i = (cid:88) i τ i T ∂∂µ i ln Z , (7d)of the scalar density n s , , the vector density n v , , thescalar–isovector density n si , and the vector–isovectordensity n vi , , respectively. Here τ i is the isospin quan-tum number of the particle species i . This expansionresults in U = ¯ U + (¯ qq − n s , )Σ s , + (¯ qτ q − n si , )Σ si , + (¯ qγ q − n v , )Σ v , + (¯ qτ γ q − n vi , )Σ vi , + . . . , (8)where the notation Σ s , = ∂U∂ (¯ qq ) (cid:12)(cid:12)(cid:12)(cid:12) ∗ = ∂ ¯ U∂n s , , (9a) Σ si , = ∂U∂ (¯ q(cid:126)τ q ) (cid:12)(cid:12)(cid:12)(cid:12) ∗ = ∂ ¯ U∂n si , , (9b) Σ v , = ∂U∂ (¯ qγ q ) (cid:12)(cid:12)(cid:12)(cid:12) ∗ = ∂ ¯ U∂n v , , (9c) Σ vi , = ∂U∂ (¯ q(cid:126)τ γ q ) (cid:12)(cid:12)(cid:12)(cid:12) ∗ = ∂ ¯ U∂n vi , , (9d)respectively for the different self energies, is introduced.The derivatives (marked by the asterix ∗ ) are taken at theexpectation values of the field bilinears ¯ qq = n s , , ¯ qτ q = n si , , ¯ qγ q = n v , , and ¯ qτ γ q = n vi , . The newly defined ¯ U = U ( n s , , n si , , n v , , n vi , ) is the potential energy densityat the expectation values. The expansion is truncated atthe second term, assuming the fluctuations around theexpectation values of the fields are small. Applying thisquasi-particle approximation to the effective Lagrangianof eq. (5) and reordering the terms results in L eff = L qu − Θ( n s , , n si , , n v , , n vi , ) , (10)with the quasi particle contribution L qu = ¯ q (cid:0) γ (cid:16) ı∂ + Σ v , + τ Σ vi , (cid:17) − (cid:16) ˆ m + Σ s , + τ Σ si , (cid:17) (cid:1) q , (11)and the effective potential energy density Θ = ¯ U − Σ s , n s , − Σ si , n si , − Σ v , n v , − Σ vi , n vi , . (12)Now, the partition function from eq. (3) takes the form Z = (cid:90) D ¯ q D q exp (cid:20) S qu [¯ q, q ] − T Θ( n s , , n si , , n v , , n vi , ) (cid:21) (13)where the quasi particle action in Fourier-Matsubara rep-resentation is given by [10] S qu [¯ q, q ] = 1 T (cid:88) n (cid:88) (cid:126)p ¯ q G − ( ω n , (cid:126)p ) q , (14) G − ( ω n , (cid:126)p ) = γ ( − iω n + ˆ˜ µ ) − (cid:126)γ · (cid:126)p − ˆ M , (15)with the Dirac effective masses M i = m i +Σ s , + τ i Σ si , andthe renormalised chemical potential ˜ µ i = µ i − Σ v , − τ i Σ vi , .Note here that the hat notation of ˆ˜ µ and ˆ M stands againfor diagonal matrix over all species. For convenience theshort-hand notations are introduced: S i = Σ s , + τ i Σ si , (16) V i = Σ v , + τ i Σ vi , (17)defining the scalar shift S i and the vector shift V i , respec-tively for each particle species i . The functional integralcan be performed in this quasi particle approximationwith the result Z qu = (cid:90) D ¯ q D q exp {S qu [¯ q, q ] } = det[ 1 T G − ] , (18)where the determinant operation acts in momentum–frequency space as well as on the Dirac, flavour, andcolour indices. Using the identity ln det A = Tr ln A andthe representation of the gamma matrices, one obtains forthe thermodynamic potential (for details see, e.g., [10]) Ω qu = − T ln Z qu = − T Tr ln[ 1
T G − ]= (cid:88) i g i (cid:90) d p (2 π ) T (cid:16) ln (cid:104) − T ( E ∗ i − ˜ µ i ) (cid:105) + ln (cid:104) − T ( E ∗ i +˜ µ i ) (cid:105)(cid:17) , (19) where the so-called “no sea” approximation (as is cus-tomary in the Walecka model) is tacitly used by remov-ing the vacuum energy term which corresponds to thephase space integral over the kinetic one-particle energy E ∗ i = (cid:112) p + M i . Self consistency
In order to evaluate the thermodynamics of the RDFapproach, one has to solve a self consistency problem,since the thermodynamic potential is a functional of thescalar and vector densities, which themselves are definedas derivatives of the thermodynamic potential by eqs. (7).The thermodynamic potential takes now the form
Ω = − (cid:88) i g i (cid:90) d p (2 π ) T (cid:16) ln (cid:104) − T ( E ∗ i − ˜ µ i ) (cid:105) + ln (cid:104) − T ( E ∗ i +˜ µ i ) (cid:105)(cid:17) + Θ , (20)while its derivatives are n s ,i ( T, { µ j } ) = (cid:18) ∂ Ω( T, { µ j } ) ∂m i (cid:19) T, { µ j } , { m j (cid:54) = i } = (cid:90) d p (2 π ) M i (cid:112) p + M i (cid:0) f i + ¯ f i (cid:1) , (21) n v ,i ( T, { µ j } ) = − (cid:18) ∂ Ω( T, { µ j } ) ∂µ i (cid:19) T, { µ j (cid:54) = i } = (cid:90) d p (2 π ) (cid:0) f i − ¯ f i (cid:1) , (22)with the Fermi distributions for particles and antiparti-cles f i = 1e ( √ p + M i − ˜ µ i ) /T + 1 , (23) ¯ f i = 1e ( √ p + M i +˜ µ i ) /T + 1 . (24)Once this set of equations is solved, one can compute allthermodynamic quantities explicitly.Based on this formalism, the specific interaction po-tential for the hadronic and quark models are introducedand the resulting self energies S i and V i are discussed inthe next section. B. Hadron model – Walecka model in the RDFformulation
Within this section it will be shown how the DD2 EoScan be written in the RDF formalism. While the rela-tivistic mean field (RMF) approach with density depen-dent couplings was already depicted in [11], the currentparametrization can be found in [12]. In this work, DD2is the hadronic EoS of choice, because it is well estab-lished in astrophysics and provides an excellent reproduc-tion of nuclear properties and withstands all constraints,which are important in astrophysical applications, ex-cept the flow-constraint, what is corrected in the DD2Fparametrization (see next section).Both DD2 and DD2F model can be expressed in theRDF formalism with the potential U = −
12 Γ σ m σ n , + 12 Γ ω m ω n , + 12 Γ ρ m ρ n , , (25)with the density-dependant coupling parameters Γ { σ,ω,ρ } ,the meson masses m { σ,ω,ρ } and the corresponding densi-ties. It results in the scalar self energy S = − Γ σ m σ n s , , (26)and the species dependent vector self energy V i = Γ ω m ω n v , + τ i Γ ρ m ρ n vi , − Γ σ Γ (cid:48) σ m σ n , + Γ ω Γ (cid:48) ω m ω n , + Γ ρ Γ (cid:48) ρ m ρ n , . (27)With these quantities, one can formulate all thermody-namic observables, as described in section II A.The form of the coupling terms are not altered in thisrepresentation and can be used from the original work.
1. Corrections due to flow constraint – DD2F
With regard to the flow constraint [13], the behaviourat supersaturation densities was altered in [14] by redefin-ing the coupling parameters as Γ DD2f i = F i Γ DD2 i , (28)with F i as new density-dependent function F i = k (1+ p ) y m k (1 − p ) y m for i = σ k (1 − p ) y m k (1+ p ) y m for i = ω for i = ρ , (29)and y = (cid:40) n/n ref − ∀ n > n ref ∀ n ≤ n ref . (30)The occurring parameters are set to k = 0 . , p = 0 . ,and m = 2 . . The reference density n ref = n sat =0 .
149 fm − is again set to the model’s saturation den-sity. One sees that, for n ≤ n ref → F i = 1 it returnsto original DD2, so the change does not touch the be-haviour below saturation density, where DD2 is alreadya very sophisticated model. The coupling to the ρ field isnot altered at all, resulting in an unchanged asymmetrybehaviour.
2. Nuclear properties
Both DD2 and DD2F models give the same nuclearproperties. The nuclear saturation density is n sat =0 .
149 fm − and the corresponding binding energy is E B = 16 .
02 MeV . The incompressibility and symmetryenergy at saturation can be found as K = 255 . and E Sym = 31 .
674 MeV . Furthermore they feature a criti-cal endpoint of the liquid-gas phase transition at baryondensity n crit = 0 .
044 8 fm − and temperature T crit =13 .
71 MeV . Important information for astrophysics arethe maximal mass for neutron stars M DD2max = 2 .
44 M (cid:12) , M DD2Fmax = 2 .
08 M (cid:12) and the direct-Urca threshold density n DU = ∞ .For a recent overview on this class of hadronic EoSmodels and how they are adjusted to nuclear observablesplease see [15]. C. Quark model – confinement as densityfunctional
The nature of quarks and gluons, which are stronglyinteracting particles, is yet little understood. The QCDas underlying theory of strong interaction, can only besolved numerically at vanishing baryon-chemical poten-tial, or perturbative in the asymptotic limit of infinitetemperature or density. For the region of dense, yet non-perturbative, matter, recently, most of the approachesemploy Nambu-Jona-Lasinio (NJL)-type models for thedescription of chiral symmetry breaking (restoration) butlack the deconfinement. The aim of the current work isto describe all phenomena, known for quark interaction,in a consistent way. To this end I adopt the followingdensity functional for the interaction energy: U ( n s , , n v , , n vi , ) = Cn / , + D ( n v , ) n / , + ω n , + ω ω (cid:48) n , n , + ρ n , . (31)Because of the number of terms occurring in eq. (31),each representing a physical effect, I will go through themin several subsections.
1. Quark confinement
The first terms in eq. (31) U SFM ( n s , , n v , ) = Cn / , + D ( n v , ) n / , , (32)represent the so-called string-flip model (SFM) as intro-duced in [6]. It captures aspects of (quark) confinementthrough its resulting density dependent scalar self energycontribution to the effective quark mass M : S SFM = 43 Cn / , + 23 D ( n v , ) n − / , . (33) B [fm -3 ]0100200300400500600 M [ M e V ] α = 0 α = 0.2 fm α = 0.4 fm FIG. 1. Effective quark mass due to confinement for differentcolour screening parameters α , see eq. (35). The effective con-finement manifests itself by a divergence of the quasi-particlemass at low densities. Here √ D = 240 MeV and C = 0 , seeeq. (32). Here the first term is motivated by the Coulomb-like one-gluon exchange, while the second term represents the lin-ear string potential between quarks. The effective massdiverges for densities approaching zero, see fig. 1, andthus suppresses the occurrence of the quasi-particle ex-citations corresponding to these degrees of freedom. Forcolour neutral hadrons this divergence of the self energyis entirely compensated by that of the confining interac-tion in the equation of motion [16]. For quark matter incompact stars, such a mechanism has been used in [17].Note that in its non-relativistic formulation with energyshifts [18], the SFM has already been applied successfullyto describe massive hybrid stars with quark-matter cores[19].The SFM modification takes into account the occupa-tion of the surrounding medium by colour fields, whichleads to an effective reduction of the in-medium stringtension. This is achieved by multiplying the vacuumstring tension parameter D with the available volumefraction Φ( n v , ) : D ( n v ) = D Φ( n v , ) . (34)This reduction of the string tension is understood asa consequence of a modification of the pressure on thecolour field lines by the dual Meissner effect since the re-duction of the available volume corresponds to a reduc-tion of the non-perturbative dual superconductor QCDvacuum that determines the strength of the confining po-tential between the quarks.A standard ansatz for the volume fraction is the Gaus-sian approach Φ( n v ) = (cid:40) exp (cid:104) − α ( n v , − n ref ) (cid:105) n v , > n ref n v , ≤ n ref (35)depending on the vector density n v , , a volume fractionparameter α , which scales the reduction and a reference density n ref from which the effect starts. The refencedensity is set to zero in the presented models.
2. Vector repulsion
The following contribution in eq. (31), as it was alreadyintroduced in [6] U v ( n v , ) = ω n , + ω ω (cid:48) n , n , (36)stands for the repulsion stemming from a four-fermion in-teraction in the Dirac vector channel and a higher order(8-fermion) repulsive interaction in the vector channel.Such higher order vector mean fields have been consid-ered already in the description of nuclear matter (see, e.g.[20]), and it is therefore natural to invoke them also in thedescription at the quark level. The higher order quark in-teractions have been introduced in [21] for the descriptionof hybrid stars in order to provide a sufficient stiffening athigh densities required to fulfill the M (cid:12) mass constraintfrom the precise mass measurement of [22] and [23]. Thisallows one to obtain a separate third family of high-masshybrid stars [24]. The denominator (1 + ω (cid:48) n , ) − in thehigher order term compensates its asymptotic behaviourto ensure the speed of sound c s = (cid:112) ∂P/∂ε does not ex-ceed the speed of light. It can be seen as a form factorof the 8-fermion interaction vertex.
3. Isospin mean field
The last contribution in eq. (31) U vi ( n vi , ) = ρ n , , (37)is representing an vector–isovector interaction, compa-rable to the ρ -meson interaction in Walecka-type mod-els. An isovector mean field is usually not considered inquark models, since the symmetry energy of quark mat-ter is not known. On the other hand, such models areoften designed to fit only neutron stars, in which case thecontribution of the isovector mean field can be absorbedin the usual vector mean field and its explicit contribu-tion is hidden. To achieve the goal of a unified EoS forboth astrophysics and heavy-ion collisions (HIC), the in-troduction of this term is crucial. Now it is possible toadjust the parameters for symmetric matter applicationsand constraints independently of the neutron rich scenar-ios, e.g., in neutron stars. Furthermore the shape of thedeconfinement phase transition can be adjusted by thiscontribution, hence is has shown that a model withoutisovector interaction gives unreasonable onset densitiesfor neutron matter, even below saturation density. Addi-tionally the composition of matter in neutron stars, e.g.,the baryonic charge fraction y c = 1 n B (cid:88) i C i n v ,i , (38) y C n B [fm -3 ] ρ = 0 MeV fm ρ = 1 MeV fm ρ = 10 MeV fm ρ = 100 MeV fm FIG. 2. Charge fraction as a function of baryon density atneutron star conditions. The different lines correspond to thestrength of the vector–isovector parameter ρ . E s y m [ M e V ] n B [fm -3 ]DD2 ρ = 0 MeV fm ρ = 10 MeV fm ρ = 80 MeV fm ρ = 100 MeV fm FIG. 3. Symmetry energy of quark and hadron matter ofhadronic EoS (DD2) and parametrizations of the quark modelwith different values of the vector-isovector parameter ρ de-pending on the baryon density. with C i being the charge number and n B the total baryondensity, strongly depends on this field, as it can be seenin fig. 2. Without any coupling to the vector–isovectormean field, the quark model does not even reach a chargefraction of .Still the question of parametrization is open. One pos-sibility is to utilise the coupling parameter of a baryonicmodel and scale by the factor (“quarks counting rule”).Another possibility to fit the parameter (which is used,e.g., in [7, 8]), is to demand an approximately smoothtransition of the symmetry energy E sym at the deconfine-ment phase transition, as it can be seen in fig. 3. Withouta sufficiently large coupling to the vector–isovector meanfield, the symmetry energy would have a significant jumpat the quark-hadron phase transition by a factor of up tofive. D. Nuclear Clusters at sub-saturated densities
In astrophysical applications the description of nuclearcluster formation is of special importance. It does notonly changes the EoS itself, but also significantly influ-ences aspects like neutrino response, for details see, e.g.,Furusawa et al. [25], Fischer et al. [26].The current work focuses on the description of highdensity matter and possible phase transitions to decon-fined quark matter. For the low density problem of nu-clear cluster formation, we use the established model ofHempel and Schaffner-Bielich [27]. Here a nuclear sta-tistical equilibrium (NSE) is assumed, where any boundstate is considered a new species (chemical picture). Themodel takes into account experimental values of Audi et al. [28] and theoretical values of Moller et al. [29].At nuclear saturation density, all clusters should havedissolved and the the system should form homogeneousmatter of neutrons and protons. In order to obtain such aMott transition, the model uses a classic excluded volumeapproach for the bound states which suppresses them athigher densities.
E. Leptonic degrees of freedom
In the current model, leptons are not taken into ac-count for the phase transition. In the published tabula-tions, one version is completely without leptons, becausesome applications need to add them during the evolutionand treat them out of equilibrium. The other version haselectrons added to fulfil electric charge neutrality afterthe phase transition is constructed. Muons and neutri-nos are not included.
F. Chemical equilibrium and phase transitions
In order to obtain a phase transition from hadronicin quark matter, the so-called two-phase approach is ap-plied. This approach, which is commonly used in astro-physics, both phases are derived independently and thenmerged on a thermodynamic level, applying Gibbs con-ditions of phase equilibrium. The fact, that hadrons arecomposite particles, made of quarks is not considered inthis description.In the case of chemical equilibrium, the distributionfunction of all present particles can be expressed by thechemical potentials of their charge numbers by µ i = (cid:80) j A ij µ j , for each particle i , the chemical potential µ j of its charges j and the associated charge number A ij .Considered here is baryon number, lepton number andelectric charge resulting in the definition of the particlechemical potentials asProton: µ p = µ B + µ C , (39a)Neutron: µ n = µ B , (39b)Up-Quark: µ u = 13 µ B + 23 µ C , (39c)Down-Quark: µ d = 13 µ B − µ C , (39d)Electron: µ e = µ l − µ C , (39e)Neutrino: µ ν e = µ l . (39f)Aspects of strangeness are not considered in the currentwork. Nevertheless, strangeness is not a conserved chargein astrophysical applications, due to weak equilibrium.Baryon and charge number are conserved charges. Lep-ton number is in astrophysical applications not a con-served charge, and the lepton chemical potential is dic-tated by charge neutrality. Furthermore, neutrinos cannot always be considered in chemical equilibrium.Gibbs conditions of phase equilibrium involve thermalequilibrium T H = T Q , mechanical equilibrium p H = p Q and chemical equilibrium µ HB , C = µ QB , C for baryon andcharge chemical potential simultaneously. Since leptons(particularly neutrinos) can not be assumed in equilib-rium, they are not taken into account for the phase tran-sition. At this point, the quark-hadron phase transitionis considered to be a phenomenon of strongly interactingparticles.In fig. 4 the black lines show a resulting phase dia-gram in the charge fraction over baryon density plane.For every given charge fraction of the hadronic side, acorresponding EoS point on the quark side was obtained,which is shown as blue lines. Generally, these pairs do nothave the same charge fraction, because they have equalcharge chemical potentials and the dependency of chargechemical potential to charge fraction is system/modeldependent. Note that in two-flavour quark matter thecharge fraction can go from y C = − (pure down quarkmatter) to y C = +2 (pure up quark matter)The resulting tabulation of the EoS needs to haveisothermal lines of constant charge fraction. To obtainthose, new pairs of points on the phase boundary whereselected, which do have the same charge fraction but arenot in Gibbs-equilibrium. With those points the mixedphase points where obtained via linear interpolation (indensity dimension).An alternative strategy would be to use the originalpairs (as shown in fig. 4) and interpolate until you get thewanted configuration of baryon density and charge frac-tion. But despite the additional numerical overhead anderrors due to additional interpolations, the effect shouldbe small, due to the small difference in charge fractionsin our models. This small differences are a result of thechoice of the parameter ρ , which controls the symmetryenergy of the quark model. Without this parameter, theeffect would be much more drastic. III. RESULTS
Main result of this work is the presentation and tabu-lated publication of a variety of EoS models, which can be y c n B [fm -3 ] FIG. 4. Phase diagram (Charge fraction y c vs baryon density n ) of asymmetric matter at zero temperature. The boundaryof the phase transition is shown as black lines, while the lightblue lines connect corresponding quark and hadron pointsthat fulfil the Gibbs conditions. used in astrophysical applications to study the influenceof a first-order phase transition at high densities. Publi-cation of tabulated data will be done on the specialisedwebsite http://eos.bastian.science , which intends tocollect all future tabulations for both astrophysical appli-cations and HIC as well as the CompOSE [30] database,which is a widely used repository in astrophysics, afterthis work is published. A. Nomenclature
Further publications with equation of states areplanned, both for astrophysical applications as well asfor HIC, which are based on different hadronic and quarkmodels and can vary in the type of phase-transition. Toform a unified nomenclature of the current and followingEoS the naming scheme
RDF a.b.c is introduced. Here a is the index of the set of parametrizations, which mostlyhave the same formalism and are presented in the samepublication, here is always a = 1 . The index b is the run-ning index inside a published set and c is occurring onlyin the published file names, in case of technical updates,which do not affect the physics description. B. Parameters
In table I one can find all considered sets of parame-ters. The sets
RDF 1 . and RDF 1 . are the initialones, which were developed for supernova simulations byFischer et al. [8]. For the later study of binary neutronstar mergers [7] the amount of sets was supplementedby RDF 1 . . . . RDF 1 . , which represent a systematicvariation of onset density and latent heat (density jump).Finally here two more sets of parameters ( RDF 1 . and RDF 1 . ) are added to address the demand of a very Name Hadron √ D α ω ω ω (cid:48) ρ n ∆ n M onset M max [ MeV ] [ fm ] [ MeV fm ] [ MeV fm ] [ fm ] [ MeV fm ] [ fm − ] [ fm − ] [ M (cid:12) ] [ M (cid:12) ] RDF 1 . DD2F
265 0 . − . . .
025 80 0 .
530 0 .
109 1 .
57 2 . . DD2Fvex
250 0 . . . . .
466 0 .
057 1 .
37 2 . . DD2F
240 0 .
36 1 . . .
015 80 0 .
536 0 .
125 1 .
58 2 . . DD2F
240 0 .
34 1 . . .
015 80 0 .
579 0 .
083 1 .
68 2 . . DD2F
240 0 .
38 1 . . .
015 80 0 .
498 0 .
106 1 .
48 2 . . DD2F
240 0 . − . . .
015 80 0 .
545 0 .
120 1 .
60 2 . . DD2F
240 0 .
47 7 . . .
015 80 0 .
562 0 .
030 1 .
62 2 . . DD2
240 0 .
45 1 . . .
015 55 0 .
285 0 .
255 0 .
94 2 . . DD2
240 0 .
63 10 . . . .
265 0 .
189 0 .
81 2 . TABLE I. List of parameters for each of the sets presented in the current work. The first two columns show the name of theset and the hadronic model, see details in the main text. Columns 3-8 present the parameters of the quark model, which areexplained in section II C. The last four columns show some representative observables of the parameter sets. Here n and ∆ n are the onset density and the density jump of the quark-hadron phase transition at neutron star conditions. For resulting coldneutron star configurations, the maximal mass M is presented, as well as the lowest mass M of neutron stars with a quarkcore. early onset densities, e.g., to study mergers of hybridcompact stars. RDF 1 . . . . RDF 1 . use the softer DD2F as hadronicmodel, while RDF 1 . has an excluded volume ver-sion of DD2F with (its) parameters α dd2fev = 2 . and n dd2fev0 /n sat = 2 . , but which has only minimal ef-fect on the onset density (visible in fig. 5). RDF 1 . and RDF 1 . use the stiffer DD2 to obtain a signif-icant lower onset density of the phase transition. Theparameters of the quark model are already thoroughlydiscussed in [6]. Here α varies the onset of the phase tran-sition, ω is linear and ω is higher order vector coupling,while parameter ω (cid:48) ensures causality. The highest orderof vector interaction adjusts the maximal neutron starmass, while the lower order modifies the mixed phase be-haviour. The new parameter ρ is the coupling strenghtto the isovector-vector meanfield and therefore varies thebehaviour of the symmetry energy, as discussed later.All sets of parameters are designed to fulfil commonconstrains. By the choice of hadronic EoS the behaviouruntil saturation is determined, which fulfils the boundaryof chiral effective field theory ( χ EFT) [31], constraintson the symmetry energy and its slope [32, 33] and otherproperties of saturation density, which are well within themargin of experimental margins. Relevant for densitiesabove saturation density are the maximal mass of neu-tron stars [23, 34] and the flow constraint [13]. Causal-ity is always preserved, meaning the speed of sound inmedium c = (cid:16) ∂p∂ε (cid:17) { s/n j } is always smaller then the speedof light in vacuum.An overview of the phase diagrams of all parametersets can be found in fig. 9. The intended variation oflow temperature phase transition can be clearly seen inthe different panels. Furthermore one can see the typ-ical temperature dependant features of two-phase ap-proaches, which are in detail explained in section IV A.Note here, that by construction the transition is alwaysof first-order. This is contradicting to results of Lat- p [ M e V f m - ] n B [fm -3 ]DD2FDD2RDF 1.1RDF 1.2RDF 1.3RDF 1.4RDF 1.5RDF 1.6RDF 1.7RDF 1.8RDF 1.9 FIG. 5. Equation of state shown as pressure p versus baryondensity n B for each parameter set. The grey shaded areadenotes the flow constraint [13]. M [ M ⊙ ] R [km]DD2FDD2RDF 1.1RDF 1.2RDF 1.3RDF 1.4RDF 1.5RDF 1.6RDF 1.7RDF 1.8RDF 1.9
FIG. 6. Mass-radius relations M ( R ) for cold neutron star con-figurations for each parameter set including the two hadronicreference EoS. Also shown are the experimental constraintsof precise high mass measurements and the recent NICERresults. tice QCD calculations [35, 36], which predict a smoothcrossover transition at vanishing densities. Therefore thepresented EoS is not applicable for high energy HIC, butrather specialised in the investigation of astrophysicalphenomena.In fig. 5 is shown how the EoS behaves at zero tem-perature for the iso-spin symmetric case. Until the onsetdensity, the EoS follows exactly the purely hadronic refer-ence curve. All parametrizations which are based on theDD2F EoS are fulfilling well the flow constraint [13]. Thehadronic EoS DD2 is rather stiff at super saturated den-sities and violates the flow constraint. Anyway the twoparametrizations ( RDF 1 . and RDF 1 . ), which arebased on it, have a phase transition before the applicabil-ity of the constraint, resulting in a significant softening.Because the phase transition is rather big, it is eventuallyto soft to perfectly lie within the constraint, which is anacceptable discrepancy.The fig. 6 compares the mass-radius relations for coldneutron stars, together with the highest neutron starmasses, which could be measured precisely [22, 23, 34].Results from radius measurements, done by the NICERmission [37], were not taken into account, while creatingthe parameter sets, but they do not contradict our resultswith given accuracy. IV. DISCUSSION
Aim of the discussion section is to address particularfeatures which occur in the presented EoSs. The dis-cussed parameter set is
RDF 1 . , because particulareffects are most pronounced. A. Temperature dependence of the phasetransition
In fig. 7 is shown how the phase transition is con-structed. For a given set of temperatures the pres-sure over baryon chemical potential is shown for both(hadronic and quark) EoSs. Here only low and intermedi-ate temperatures are discussed, because at high temper-atures the applicability of the two-phase approach ends.For simplicity only pure symmetric matter is shown, be-cause the charge chemical potential can then be assumedto be zero for both phases. This simplification is onlyused in this picture for presentation and not used in theactual data or any other pictures, because the chargechemical potential of the hadronic model is not zero( ≈ − ), due to the mass difference of neutrons andprotons.The hadronic model features at low temperatures( T c ≈ . ) the liquid gas phase transition in formof a van-der-Waals wiggle. For temperatures beyond thecritical temperature, the lines are monotonously risingwith an increasing gradient. The quark model showsalways a rather steep gradient, starting from negative p [ M e V f m - ] µ B [MeV]T = 0T = 10T = 20T = 30T = 40T = 50T = 60T = 70T = 80 FIG. 7. Phase construction for different temperatures in the
RDF 1 . set of the model, in the case of symmetric matter.The solid lines show the hadronic EoS, while the correspond-ing dashed lines are from the quark EoS. The grey dash-dottedline represent the phase transitions, here following the cross-ings of hadronic and quark EoS for each value of temperature. pressures. This bag-model like behaviour origins fromthe confinement mechanism and is in details explainedin Kaltenborn et al. [6]. Also can be seen that thequark model is much more affected by temperature thenthe hadron model, which can be explained by the lowermasses of quarks.Applying Gibbs conditions and assuming that temper-ature and charge chemical potential is already equal inboth phases, the phase transition is at the crossing pointof hadronic and quark line in fig. 7. With rising tem-perature, one can clearly see, that the model features amonotonous decrease of transition pressure and (baryon)chemical potential.The baryon density in fig. 7 can be derived from itsgradient n B = (cid:0) ∂p (cid:14) ∂µ B (cid:1) T,µ C . Given the gradient of thequark model, especially around the crossing points, isnot much temperature dependent, the transition densityfrom mixed phase to pure quark matter is rather con-stant wrt. temperatures, as it can be seen in the phasediagram in fig. 9. The hadron model on the other hand,has its crossing point at different areas of the EoS, whichfeature different gradients. Therefore the onset densityof the phase transition is highly temperature dependent.Note here, that this is a typical feature of two-phase con-structions, as discussed in section IV D. B. Leptonic contributions and charge neutrality
In this work, the QCD phase transition is considereda phenomenon of strong interacting particles. This as-sumption has several reasons and implications. One ofthem is the fact that in dynamical applications, suchas supernovae, even electrons can not considered to bein chemical or even thermal equilibrium with the sys-tem. In such case, the leptons need to be treated sepa-0 y c n B [fm -3 ]RDF 1.9DD2 FIG. 8. Baryonic charge fraction y c for neutron star condi-tions (see eq. (38)) for the hadronic reference EoS DD2 andthe hybrid EoS RDF 1 . . rately, e.g. via Boltzmann equation, and the discussionof non-equilibrium effects goes beyond the scope of thismanuscript.Problematic here is appearance of an additional charge– the lepton number – which needs to be taken into ac-count while fulfilling Gibbs conditions, even though inastrophysical applications, the lepton chemical potentialis dictated by the assumption of electric charge neutralityof the system. A most appropriate construction would bedone with three chemical potentials, resulting in a four-dimensional (additionally temperature) phase diagram.For astrophysical applications then a three dimensionalcut of electric charge neutrality can then be extracted.If taken into account, at constant chemical potential,the quark EoS has a higher particle density. This leadsto a stronger contribution (like pressure) by leptons anda resulting lowering of the onset of the phase transi-tion. Not including leptons during the construction ofthe phase transition and adding them afterwards resultsalways in a smearing effect, hence all resulting effects areweaker than in a more appropriate treatment and pre-sented results can be seen as conservative. Eventuallythe position or even existence of the phase transition isnot known and its effects can be studied by systematicvariation of also unknown parameters of quark couplingsat high density. The inclusion of leptons would have theneed of readjusting these parameters. C. Description of nuclear clusters
The current work focuses on the high density part ofthe equation of state and implications of a possible phasetransition to deconfined quark matter. In order to haveclear signatures of these high density features, a well es-tablished treatment of the nuclear cluster formation atlow densities was chosen.Direct improvements can be achieved by using updatedexperimental and theoretical data for nuclear binding energies [38, 39]. More sophisticated models are directquantum-statistical descriptions of light clusters [40–42]or generalised density functionals with a depleting bind-ing energy which change the composition significantly[43].
D. The two-phase approach
As was discussed in section II F, the current work isbased on the so-called two-phase approach, where thequark model and the hadron model are completely inde-pendent. Even though it is a commonly used approach,particularly in astrophysics, it has some well-known dis-advantages, which significantly limit its usabilityThe most obvious problem is that it always gives afirst-order transition by construction, which is a contra-diction to the Lattice QCD results, predicting a smoothcross over at low densities. In this publication the con-struction of a first-order transition is the goal and there-fore is focused on the high density regime of the QCDtransition, which is relevant for astrophysics. Makingthe model applicable for HIC would demand a cross-overtransition at low densities and hence imply a critical end-point, in case of the existence of a first-order phase tran-sition at high densities.Another problem, as already discussed by Bastian andBlaschke [44], is the fact, that both hadron and quarkEoS are modelled completely separately and their physicsdoes not need to have the same footing. Effects of onemodel, might occur in the other phase and is by construc-tion suppressed. An example would be chiral restoration,which is usually modelled for quarks, but might occuralready in the hadronic phase [45]. Furthermore, sub-structure effects in the hadronic phase like Pauli-blockingare not taken into account as well as possible remainingbound states or resonances in the quark phase. The effectof Pauli blocking has been explored in a non-relativisticapproximation [18, 46] and show similar effects as ex-cluded volume models like Typel [47].A solution to most of those problems can be achievedby applying a unified approach, where quarks andhadrons are introduced on the same level, preferablywhere hadrons are described as bound states of quarksas in the cluster-expansion of Bastian et al. [48]. Thisapproach obtains a CEP, depending on the parametrisa-tion of the interactions [49]. Substructure effects can beincluded by the appropriate exchange diagrams. Finally,a sophisticated formulation of cluster mean field wouldallow to formulate quark and hadron interactions consis-tently, which makes it possible to adjust quark interac-tion parameters to hadronic constraints and have effectslike chiral restoration already in the hadronic phase.A more pragmatic approach is presented by Typel andBlaschke [50], where a density dependent excluded vol-ume formalism is used to achieve a phase transition ona thermodynamic level. Due to the temperature depen-dence of this model, it features a critical endpoint at finite1temperature. Since it does not actually describe quarks,this model is limited to applications, which are only sen-sitive to thermodynamic quantities and to microscopicquantities.
V. SUMMARY
Presented is a microscopical quark-hadron model withan effective interaction potential on the level of the mean-field approximation. The descriptions of quark matterand hadron matter is done separately and the resultingEoS are merged via a thermodynamically consistent two-phase construction. Dependencies on temperature andbaryonic charge fraction are included naturally by use offermion distribution functions.Alternative attempts to study the effect of first-orderquark-hadron phase transitions in hot astrophysics aredone by Roark and Dexheimer [51], using a chiral-meanfield model to describe quark matter and applying a two-phase approach to obtain a first-order phase tran-sition with different incorporation of leptonic degrees offreedom. The work of Sagert et al. [52] also applies a two-phase approach for the phase transition, but uses a ther-modynamic bag model to describe quark matter, whichcan not describe neutrons stars of two solar masses andshould therefore be considered outdated. The alternativescenario of a cross-over at low temperatures is exploredfor hyperonic models by Marques et al. [53] or for quark-hadron EoS in, e.g., Baym et al. [54].
ACKNOWLEDGMENTS
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RDF 1 . and RDF 1 .9