Solution of the Hyperon Puzzle within a Relativistic Mean-Field Model

The equation of state of cold baryonic matter is studied within a relativistic mean-field model with hadron masses and coupling constants depending on the scalar field. All hadron masses undergo a universal scaling, whereas the coupling constants are scaled differently. The appearance of hyperons in dense neutron star interiors is accounted for, however the equation of state remains sufficiently stiff if a reduction of the $\phi$ meson mass is included. Our equation of state matches well the constraints known from analyses of the astrophysical data and the particle production in heavy-ion collisions.


Introduction
A nuclear equation of state (EoS) is one of the key ingredients in the description of neutron star (NS) properties [1], supernova explosions [2] and heavy-ion collisions [3,4]. A comparison of various EoSs in how well they satisfy various empirical constraints was undertaken in Ref. [5] for the EoSs obtained within relativistic mean-field models (RMF) and some more microscopic calculations and in Ref. [6] for the Skyrme models. It turns out difficult to reconcile the constraint on the maximum NS mass, which must be larger than 1.97 M ⊙ after the recent measurements reported in [7,8], and the upper constraints on the stiffness of the EoS extracted from the analyses of heavy-ion collisions (HICs) [3,4]. Another relevant constraint on the EoS of the NS matter is imposed by the direct Urca (DU) processes, like n → p + e +ν, which occurs as soon as the nucleon density exceeds some critical value n n DU . The occurrence of these very efficient processes is hardly compatible with NS cooling data, if the value of the NS mass, at which the central density becomes larger than n n DU , is M n DU < 1.5M ⊙ (the so-called "strong" DU constraint) [9,5]. There should be M n DU < 1.35M ⊙ (the "weak" DU constraint) [10,5], since 1.35 M ⊙ is the mean value of the NS mass distribution, as it follows from the analysis of the observational data on NSs in binary systems. The DU problem appears in the EoSs with the linear density dependence of the symmetry energy, except, maybe, most stiff ones. All the standard RMF EoSs and the microscopic Dirac-Brueckner-Hartree-Fock (DBHF) EoS, suffer of this linear dependence. On the contrary, variational calculations of the Urbana-Argonne group with A18 + δv + UIX* forces [11], as well as the RMF models with density dependent hadron coupling constants [12], generate a weaker growth of the symmetry energy with the density, and the problem with the DU reactions is avoided. The later models are also able to describe NSs, as heavy as those in Refs. [7,8].
The problems worsen if strangeness is taken into account, because the population of new Fermi seas of hyperons leads to softening of the EoS and reduction of the maximum NS mass. By employing a recently constructed hyperonnucleon potential, the maximum masses of NSs with hyperons are computed to be well below 1.4M ⊙ [13]. Also, within RMF models one is able to explain observed massive NSs only if one artificially prevents the appearance of hyperons, cf. [13,14] and references therein. This is called in the literature, the "hyperon puzzle". So, the difference between NS masses with and without hyperons proves to be so large for reasonable hyperon fractions in the standard RMF approach, that in order to solve the puzzle one has to start with very stiff purely nuclear EoS, that hardly agrees with the results of the microscopically-based variational EoS [11] and the EoS calculated with the help of the auxiliary field diffusion Monte Carlo method [15]. Such an EoS would also be incompatible with the restrictions on the EoS stiffness extracted from the analysis of nucleon and kaon flows in heavy-ion collisions [3,4]. All suggested explanations require additional assumptions, see discussion in [16]. For example, the inclusion of an interaction with a φ-meson mean field, and the usage of smaller hyperon-nucleon coupling constant ratios following the SU(3) symmetry relations [17], as well as other modifications performed within the standard RMF approach, all help to increase the NS mass.
There is also another part of the "hyperon puzzle", which attracted less attention so far. With the hyperon coupling constants introduced with the help of the SU(6) symmetry relations the critical densities for the appearance of first hyperons prove to be rather low, n H DU ∼ 3n 0 , cf. [18,19]. However with the appearance of the hyperons the efficient DU reaction on hyperons, e.g. Λ → p + e +ν, occurs that may cause very rapid cooling of the NSs with M > M H DU , M H DU is the NS mass, at which the central density reaches the value n H DU . In Ref. [10] two of us formulated a RMF model, in which hadron masses and meson-baryon coupling constants are dependent of the σ mean field. A working model MW(n.u., z=0.65) labeled in [5] as KVOR model has been constructed. This model was shown in [5] to satisfy appropriately the majority of experimental constraints known to that time. In Ref. [20] the particle thermal excitations were incorporated and the model was successfully applied to description of heavy-ion collisions. However, even without hyperons the KVOR EoS with the added (BPS) crust EoS from [21] yields M KVOR max = 2.01M ⊙ that fits the new constraint [7,8] only marginally.
In the present Letter we will show that within the RMF models with hadron masses and coupling constants dependent of the σ mean field one is able to overcome the mentioned above problems and to construct the appropriate EoS with hyperons satisfying presently known experimental constraints.
Following the approach of Ref. [10], the scalar field enters as a dimensionless variable f = g σN χ σN (σ)σ/m N and the meson-baryon coupling constants g mB , B = (N, H), are made σ-dependent with the help of scaled coupling constants, where x σH = g σH /g σN and ξ σH (f ) = χ σH (f )/χ σN (f ). Taking into account the equations of motion for vector fields, the energy density of the cold infinite matter with an arbitrary isospin composition is recovered from the Lagrangian of the model in the standard way, see Ref. [10]: where x ω(ρ)B = g ω(ρ)B /g ω(ρ)N , x φB = g φB /g ωN , and The values of the isospin projections for baryons t 3i follow from the Gell-Mann-Nishijima relation t 3B = Q B − (1 + S B )/2, where Q B and S B are the baryon electric charge and strangeness, respectively. The baryon densities are related to the baryon Fermi momentum as n B = p 3 F,B /3π 2 and the fermionic kinetic energy density is The dimensionless coupling constants are C M = gMN mN mM . Since the ratios x φH are determined through g ωN , the φfield contribution enters the energy density with the constant C φ = C ω m ω /m φ . Here we take m ω = 783 MeV, m φ = 1020 MeV. Bare meson masses of other mesons and their couplings enter the energy only via the combinations C M , and the scaling functions Φ m and χ m enter only through the scaling factors Therefore we actually do not need to determine Φ m (f ) and χ m (f ) separately, but only η m (f ) combinations. The selfinteraction of the scalar field introduced usually in RMF models through a potential U (f ) is hidden now in the definition of η σ (f ). The equation of motion for the remaining field variable f follows from the minimization of the energy density ∂E[f ]/∂f = 0 . If we suppress φ and H terms, put and put all other scaling functions to unity, we recover the energy density functional of the standard non-linear Walecka σ-ω-ρ model.
For the NS matter sustained in the β-equilibrium the Fermi momenta of a baryon can be expressed through the baryon chemical potential, µ B , as p 2 is related to the nucleon and electron chemical potentials as µ B = µ n − Q B µ e . Solving the system of equations for p F,B and making use of the electroneutrality condition B Q B n B = n e + n µ , where the lepton densities are n e = µ 3 e /3π 2 , n µ = (µ 2 e − m 2 µ ) 3/2 /3π 2 , we can express the hadron densities and the total energy density through the total baryon density n = B n B . The total pressure is calculated as P = i=B,l µ i n i − E.
The parameters of the nucleon sector are tuned to reproduce the properties of nuclear matter at saturation: the saturation density n 0 , the binding energy per nucleon E 0 , the effective nucleon mass m * N , the compressibility modulus K and the symmetry energy J. The coupling constants of hyperons with vector mesons are interrelated by SU(6) symmetry relations [22]: Table 1 Characteristics of KVOR and MKVOR models at saturation The coupling constants of hyperons with the scalar mean field are constrained with the help of the hyperon binding energies per nucleon E H bind in the isospin symmetric matter (ISM) at n = n 0 given by [18]: The repulsive Σ potential prevents the appearance of Σ hyperons in all models considered below. The NS configuration follows from the solution of the Tolman-Oppenheimer-Volkoff equation. We construct a resulting pressure as a function of the density by using the crust BPS EoS [21] for n ≤ 0.45n 0 , a cubic spline interpolation for 0.45n 0 < n ≤ 0.6n 0 , and the pressure for beta-equilibrium matter (BEM) given by our hadronlepton model for n > 0.6n 0 .

Models
We discuss now two models constructed according to the principles described above. One is the formal extension of the KVOR model from Ref. [10] to hyperons, which we call the KVORH model now. On this example we demonstrate the problems, which appear when one includes hyperons. For the second model, labeled as MKVOR (or MKVORH when hyperons are included), we propose new set of scaling functions. In Table 1 we present the saturation parameters for both models and the coefficients of the expansion of the nucleon binding energy per nucleon near the nuclear saturation density n 0 , in terms of small ǫ = (n − n 0 )/3n 0 and β = (n n − n p )/n parameters.

KVORH model
Reference [10] proposed a set of input parameters for the RMF model, which matches the APR EoS (in the relativistic HHJ parameterization of [23]) up to n < ∼ 4n 0 , see Eq. (61) in [10]. To fulfill the DU constraint Ref. [10] introduced the scaling function η ρ (f ) = 1, see Eq. (63) in [10]. Thus the model labeled as KVR in Ref. [5] was constructed. The idea behind the KVOR modification of the KVR model was to demonstrate that introducing the additional scaling function η ω = 1 one can increase the maximum value of the NS mass without a sizable change of the EoS for densities n < ∼ 4n 0 . The coupling constants for the KVOR model are given in Eq. (58) in Ref. [10]. The KVOR model produces the maximum NS mass M max = 2.01M ⊙ , and the critical proton density for the DU reaction threshold on neutrons n n DU = 3.92n 0 corresponding to M n DU = 1.76M ⊙ . In the KVORH model the parameters x σH deduced from hyperon binding energies in Eq. (5) are x σΛ = 0.599 , x σΣ = 0.282 , x σΞ = 0.305 .
The baryon concentrations for the KVORH model are depicted in Fig. 1 as functions of the baryon density n. For n < 0.6n 0 the curves presented in Fig. 1 should be replaced by those computed within the EoS of the crust. The KVORH model produces the maximum NS mass M max = 1.66M ⊙ . The critical density for the appearance of first hyperons, which is simultaneously in this case the critical density for the onset of DU reactions on Λs, is n Λ DU = 2.82n 0 , and the corresponding NS mass at which first Λs appear in the NS center is M Λ DU = 1.38M ⊙ . The total strangeness concentration (the ratio of the number of strange quarks to the total number of quarks) in the NS with the maximum mass is f S = 0.034 .

MKVORH model
Reference [10] showed that the EoS is more sensitive to the value of m * N (n 0 ) than to the compressibility K. The smaller m * N (n 0 ) is in a certain RMF model, the larger is the value of the maximum NS mass. The input parameters for the new MKVOR model are listed in Table 1 together with   Table 2 Parameters of the MKVOR model the corresponding parameters of the nuclear binding energy per nucleon at saturation. We took in the MKVOR model a smaller value of m * N (n 0 ) than in the KVOR model and a smaller value of the compressibility, K = 240 MeV, that agrees with canonical value K = 240 ± 20 MeV extracted from the analysis of giant monopole resonances (GMR) [24].
The scaling functions of the MKVOR model are as follows: The parameters of the model and of the scaling functions are collected in Table 2. In the model with hyperons (MKVORH) we use the σH coupling constant ratios deduced from hyperon binding energies in ISM following Eq. (5) for ξ σH = 1: x σΛ = 0.607 , x σΣ = 0.378 , x σΞ = 0.307 .
Note that the value x σΛ ≃ 0.61 is close to the best value derived from hypernuclei x σΛ ≃ 0.62 in Ref. [22]. To have an opportunity for an increase of the maximum NS mass in the model with hyperons, we incorporate the φ-meson mean field with a scaled φ meson mass (version labeled MKVORHφ) and, additionally, allow for a scaling of the σH coupling constants, ξ σH (n) = 1 (version MKVORHφσ).
In the model MKVORHφ we exploit the very same scaling of the φ-meson mass as for other hadrons Φ φ = 1 − f and take χ φH = 1. This implies the scaling function We use the minimal model, assuming parameterization Eq. (4) for the vector-meson-hyperon coupling constants, and the σH coupling constants from Eq. 5M ⊙ ), we should notice that for reactions on hyperons this constraint might be a bit soften since the baryon part of the squared matrix element of the DU reaction on Λs is 25 times smaller than that for the DU reaction on neutrons [25]. The effect of the ξ σH (n) scaling we demonstrate at hand of the MKVORHφσ model, where we take ξ σH (n) such that ξ σH (n ≤ n 0 ) = 1, and assume that ξ σH (n) decreases with an increase of n and vanishes for densities n > min{n H DU }. This means that effectively we will exploit vacuum hyperon masses for n > min{n H DU }. Note that the KVOR model extended to high temperatures in Ref. [20] (called there as the SHMC model) matches well the lattice data up to T ∼ 250 MeV provided σB coupling constants for all baryons except the nucleons, are artificially suppressed, that partially motivates our choice of suppressed values ξ σH .
The baryon concentrations from the MKVORHφ and MKVORHφσ models are shown in Fig. 1. The proton fractions of the MKVORHφ and MKVORHφσ models are smaller than that for the KVORH model. We also see that inclusion of the φ scaling (11) reduced the hyperon population. The reduction of the σH coupling constants prevents the appearance of Λ and Ξ 0 hyperons and shifts the threshold density of the Ξ − appearance to higher values. Without Λs the reaction Ξ − → Λ + e − +ν e does not occur and the DU threshold is determined by the DU reactions on nucleons. Replacing the values of f (n) depicted in Fig. 1 for the BEM in Eq. (1) one can recover the density dependence of the effective hadron masses and from Eqs. (9) and (11) that of the effective coupling constants. Within the MKVORHφσ model we get M max = 2.29 M ⊙ , n n DU = 3.69 n 0 (M n DU = 2.09 M ⊙ ), and the total strangeness concentration in the heaviest NS is reduced to f S = 6.2 · 10 −3 .
Applying the φ-mass scaling and the ξ σH scaling to the KVORH model we obtain for the KVORHφ model M max = 1.88 M ⊙ and that the first hyperons, Λs, appear at the density n Λ DU = 2.81 n 0 (M Λ DU = 1.37 M ⊙ ). The strangeness fraction is f S = 0.035. For the KVORHφσ model we find M max = 1.96 M ⊙ , f S = 9.2·10 −3 . The first among hyperons appear Ξ − s, therefore the DU threshold is shifted to n n DU = 3.96 n 0 (M n DU = 1.77 M ⊙ ). Below we compare how well the EoSs obtained within the MKVORHφσ and KVORH models satisfy various phenomenological constraints.

Symmetry energy and nucleon optical potential
Constraints on the density dependence of the symmetry energy [parameterJ(n) in Eq. (7)] are extracted in [26] from the study of the analog isobar states and in [27] from the electric dipole polarizability of 208 Pb nuclei. They are shown in Fig. 2 (left panel) by the shaded and hatched regions, respectively, together with the symmetry energies calculated in the KVOR and MKVOR models. We see that Shaded area shows the constraint from the study of analog isobar states (AIS) in [26]. Hatched area is the constraint from the electric dipole polarizability (α D ) in 208 Pb [27]. (Right) The nucleon optical potential as a function of the nucleon kinetic energy for ISM at n = n 0 for the KVOR, cf. [20], and MKVOR models. Shaded area shows the extrapolation from finite nuclei to the nuclear matter from [29].
the both models follow the lower boundary of the region. The dependence of the nucleon optical potential on the nucleon kinetic energy in the ISM at n = n 0 is shown in Fig. 2 (right panel). The shaded region is extracted from the atomic nucleus data [28] and recalculated to the case of the infinite nuclear ISM in [29]. The KVOR model describes the nucleon optical potential for low and high particle energies but does not describe it for intermediate energies.
The MKVOR model describes the nucleon optical potential rather well for nucleon energies E N − m N < ∼ 400 MeV. To fit appropriately the data at higher particle energies, the momentum dependence of the N N interaction would be required that is not present in the mean-field approach. The iso-vector part of the optical potential U (n) opt − U (p) opt is less constrained by the data, therefore we do not show it.

Particle flow in heavy-ion collisions
The analysis of the transverse and elliptical flow data in HICs allowed [3] to extract a constraint on the pressure of the ISM as a function of the nucleon density and to reconstruct the pressure for the purely neutron matter (PNM) with some assumptions about the density dependence of the symmetry energyJ(n) (soft or stiff one). The analysis of the kaon production in HICs in Ref. [4] might provide some restrictions on the pressure at lower densities. These constraints are shown in Fig. 3 together with the results of the GMR data analysis taken from [30] and the pressure calculated in the KVOR and MKVOR models. The constraints rule out a very stiff EoS. We see that the KVOR model satisfies the requirements. The MKVOR model fulfills the constraints, for n < 4n 0 in ISM. In the PNM the MKVOR curve fits the hatched region with stif J(n), whereas KVOR is softer.  Fig. 3. Pressure as a function of the nucleon density in ISM and PNM for the KVOR and MKVOR models. Hatched areas show the empirical constraints from the analyses of a particle flow in HICs in [3], kaon production in HICs [4], and the GMR data [30].

DU constraint
In the BEM the DU process on neutrons, n → p + e +ν e , can occur only if the proton fraction is high enough so that the Fermi momenta of neutrons, protons and electrons (p F,i=n,p,e ) satisfy the inequality p F,n ≤ p F,p + p F,e . Usually, RMF models yield uncomfortably low values of the threshold densities for these reactions and correspondingly low values of the NS mass, M n DU , at which the process begins to occur in the NS center. Every star with a mass only slightly above M n DU cools down fast due to the DU process, even in the presence of nucleon pairing, and becomes invisible for the thermal detection within few years [9]. Most of single NSs have likely masses below 1.5M ⊙ in accordance with the type-II supernova explosion scenario [2] and the population synthesis analysis [31]. Therefore, it is natural to believe that majority of the pulsars, which surface temperatures have been measured, have masses M < ∼ 1.5M ⊙ . The analysis of these data in the existing cooling scenarios supports the constraint M n DU > ∼ 1.5M ⊙ [32,33]. The adequate description of the novel precision data on the cooling of the Cassiopea A also requires the absence of the DU reactions [32].
In the presence of Λ hyperons, reactions Λ → p + e +ν may occur. As seen in Fig. 1, proton concentrations for the KVORH, MKVORHφ models do not exceed the neutron-DU threshold for n < n Λ DU , and for higher densities the DU reactions on Λs start occurring. For the MKVORHφσ model Λs do not appear at all and the DU processes occur for n > n n DU .

Gravitational mass versus baryon mass constraint
The unique double-neutron-star system J0737-3039 with two millisecond pulsars provided an important constraint on the nuclear EoS. The gravitational mass of one of the companions (B) is very low M G = 1.249 ± 0.001M ⊙ [34] which implies a very peculiar mechanism of its creation -a type-I supernova of an O-Ne-Mg white dwarf driven hydrostatically unstable by electron captures onto Mg pulsar derived in Refs. [35] and [36] labeled by 1 and 2, respectively. The empty rectangle demonstrates the change of the constraint [35] by 1% M ⊙ . (Right) The NS mass-radius relations for the KVORH, MKVORHφ MKVORHφσ models compared with constraints from the isolated NS RX J1856.53754 [37], QPOs in the LMXBs 4U 0614+09 [39], the millisecond pulsar PSR J0437-4715 [40], and the Byesian probability analyses (BPA) [1]. The horizontal lines border the uncertainty range for the mass of PSR J0348+0432 [8].
and Ne. Knowing this mechanism Refs. [35,36] calculated the number of baryons in the pulsar and the corresponding baryon mass: M B = 1.366-1.375M ⊙ [35] and M B = 1.358-1.362 [36]. The constraint of Ref. [35] can be released by 1% M ⊙ because of a possible baryon loss and a critical mass variation due to carbon flashes during the collapse. Therefore one can speak about "strong" (without the mass loss) and "weak" (with the mass loss) constraints on the EoS, respectively. Microscopically motivated EoSs, like the relativistic DBHF EoS [38], the APR EoS [11], the diffusion Monte-Carlo one [15], and many RMF-based models do not fulfill the strong constraint of Ref. [35]. Many EoSs do not satisfy the constraint of [36] and even the weak constraint of Ref. [35], cf. [5]. In Fig. 4 (left panel) we plot the gravitational NS mass M G versus the baryon mass M B . The KVOR model matches marginally the weak constraint of Ref. [35], whereas the MKVOR model matches marginally both the result from Ref. [36] and the strong constraint of [35], the latter was not reproduced by the EoSs considered in Ref. [5]. Note that hyperons do not appear in the NS with the mass of 1.25M ⊙ , therefore the curves for the models KVORH and KVOR coincide, as well as the curves for the MKVOR, MKVORHφ and MKVORHφσ models. Comparing particle concentration for different models shown in Fig. 1 and the baryon mass of the NS, we observe a correlation: the smaller is the proton fraction within the density interval n 0 < n < ∼ 2.5n 0 , the better the given EoS satisfies the baryon-mass constraint. For n ∼ n 0 the value of the proton fraction is correlated with the values of J and L in Eq. (7). The value of J may vary only a little (from 28 MeV to 34 MeV or even in a narrower interval) but L is badly known, presently. With a decrease of L curves in Fig. 4 (left panel) are shifted to the right. For L < 40 MeV (at fixed other parameters) the MKVOR curves would pass through the shaded area 1. The smaller L is for the given EoS, the less the proton fraction is in the relevant density interval n 0 < n < ∼ 2.5n 0 , and the better the constraint is satisfied.

Mass and radius constraints
In Fig. 4 (right panel) we show mass-radius relations of NSs for the KVORH, MKVORHφ and MKVORHφσ models. Largest precisely-measured masses of NSs are 1.97 ± 0.04M ⊙ for PSR J1614-2230 [7] and 2.01±0.04M ⊙ for PSR J0348+0432 [8]. The MKVORHφ and MKVORHφσ models can describe these high-mass NSs, whereas the KVORH model fails badly. Experimental information about heavier NSs is plagued with large experimental errors and with additional theoretical uncertainties, see Ref. [1] for a review. In Fig. 4 (right panel) we confront our models with other constraints derived from the quasi-periodic oscillations in the low-mass X-ray binary 4U 0614+091 [39]) and thermal spectra of the nearby isolated NS RX J1856.5-3754 [37]. More details on these constraints can be found in Ref. [5]. In contrast to the mass determination, there are no accurate estimates of NS radii. Some constraints were derived recently from the X-ray spectroscopy of PSR J04374715 with the proper account for the system geometry [40] and from the Bayesian probability analysis of several X-ray burst sources in Ref. [1]. These constraints are also shown in Fig. 4. We see that the MKVORHφ and MKVORHφσ satisfy the mass-radius constraints and produce the radii of NSs in a narrow interval 11.7 ± 0.5 km for star masses M > 0.5M ⊙ .

Concluding remarks
We constructed relativistic mean-field models with scaled hadron masses and coupling constants including a φ meson mean field and hyperons. The hyperon-vectormeson (ω, ρ, φ) coupling constants obey the SU(6) symmetry relations. The treatment of the φ meson so, that its mass scales in the same way as the masses of nucleon and hadrons but the φ coupling constants do not scale, allows to fulfill the empirical constraints on the maximum neutron star mass; see curve for the MKVORHφ model in Fig. 4 (right panel). Thus the "hyperon puzzle" is naturally resolved within such model. Other constraints are also satisfied except that the model MKVORHφ produces a rather low threshold value of the neutron star mass M Λ DU ≃ 1.44M ⊙ for the occurrence of direct Urca reactions on Λs. The model MKVORHφσ, where the hyperon masses do not change in medium, eliminates this deficiency and satisfies appropriately all other constraints discussed in this Letter, which are known from the analyses of atomic nuclei, heavy-ion collisions and neutron star data. Moreover, the maximum neutron star mass increased. As an interesting finding, we indicate that the smaller the proton fraction is in the density interval n 0 < n < 2.5n 0 and the smaller the value of L is at n 0 , the better the baryon-gravitational mass constraint is fulfilled.