Neutron star equations of state with optical potential constraint

Nuclear matter and neutron stars are studied in the framework of an extended relativistic mean-field (RMF) model with higher-order derivative and density dependent couplings of nucleons to the meson fields. The derivative couplings lead to an energy dependence of the scalar and vector self-energies of the nucleons. It can be adjusted to be consistent with experimental results for the optical potential in nuclear matter. Several parametrisations, which give identical predictions for the saturation properties of nuclear matter, are presented for different forms of the derivative coupling functions. The stellar structure of spherical, non-rotating stars is calculated for these new equations of state (EoS). A substantial softening of the EoS and a reduction of the maximum mass of neutron stars is found if the optical potential constraint is satisfied.


Introduction
The recent observation of two pulsars with approximately two solar masses [1,2] presents a severe challenge to the theoretical description of cold highdensity matter in β-equilibrium. The equation of state (EoS) has to be sufficiently stiff in order to support such high masses of compact stars. Many models 5 that are solely based on nucleonic (neutrons and protons) and leptonic (electrons and muons) degrees of freedom are able to reproduce maximum neutron star masses above two solar masses if the effective interaction between the nucleons becomes strongly repulsive at high baryon densities. However, additional hadronic particle species can appear at densities above two or three times the 10 nuclear saturation density n sat ≈ 0.16 fm −3 . In most cases, these additional degrees of freedom lead to a substantial softening of the EoS resulting in a reduced maximum mass of the compact star below the observed values. This feature is well-known for models with hyperons -the so-called "hyperon puzzle", see, e.g., [3,4] and references therein -but was also observed in approaches that take 15 excited states of the nucleons such as ∆(1232) resonances into account, see, e.g., [5,6] and references therein. Usually, only specifically designed interactions can avoid the problem of too low maximum masses.
Successful models of the baryonic contribution to the stellar EoS should be scrutinized whether they comply with other experimental contraints, e.g. with 20 respect to the employed interactions. In the center of compact stars very high baryon densities are reached exceeding several times n sat and the corresponding Fermi momenta of the particles are much larger than those at saturation. This is particularly significant for models with only nucleonic degrees of freedom. Hence, not only the density dependence of the effective in-medium interaction 25 between nucleons but also their momentum dependence becomes relevant. For densities near n sat this information is contained in the optical potential of nucleons that can be extracted from the systematics of elastic proton scattering on nuclei, see, e.g., [7,8]. A saturation of the real part of the optical potential is observed at high kinetic energies approaching 1 GeV. The momentum depen-30 dence of the in-medium interaction is also crucial in simulations of heavy-ion collisions [9].
Typical approaches for the baryonic contribution to the stellar EoS are energy density functionals that originate from nonrelativistic or relativistic meanfield models of nuclear matter. The most prominent cases among the former 35 class are Skyrme energy density functionals, see reference [10] for an overview of different parametrizations. They are derived originally from the zero-range Skyrme interaction with a two-body contribution, which is an expansion up to second order in the particle momenta, and a density dependent three-body contribution, which is included in order to reproduce saturation properties of 40 nuclear matter. Obviously, an extrapolation of the model to high momenta is questionable given the limited form of the momentum dependence. Examples of the latter class, frequently denominated covariant density functionals, can be inferred from relativistic Lagrangian densities. In conventional models, see reference [11] for a wide collection of different parametrizations, a nucleon optical 45 potential in the medium can be derived from the relativistic scalar and vector self-energies, see section 5. It exhibits a linear increase with energy, which is in contradiction with the expectation from experiment. In general, one would expect that the nucleon self-energies itself depend explicitly on the particle momentum or energy as, e.g., in Dirac-Brueckner calculations of nuclear matter 50 [12]. However, this is not realized in standard RMF models.
There are particular extensions of relativistic mean-field (RMF) models that contain nucleon self-energies with an explicit energy or momentum dependence. This dependence cannot be introduced in a relativistic model in a simple parametric form because it affects, e.g., the definition of the conserved currents. In extended systematic approaches new derivative couplings between the nucleon and meson fields are introduced that allow to reproduce the energy dependence of the optical potential as extracted from experiments. One of the earliest RMF models with scalar derivative couplings was presented in reference [13]. A rescaling of the nucleon fields removed the explicit momentum dependence of the self-energies but lead to a considerable softening of the EoS. More general couplings of the mesons to linear derivatives of the nucleon fields were considered in reference [14] with an application to uniform nuclear matter. With appropriately chosen coupling constants a reduction of the optical potential was found as compared to the strong linear energy dependence in convential RMF models. The model was further extended in reference [15] assuming a density dependence of the couplings. It was successfully applied to the description of finite nuclei. Notable new features were the increase of the effective nucleon masses (usually rather small in order to explain the strong spin-orbit interaction in nuclei) and correspondingly higher level densities close to the Fermi energy 70 in nuclei in better concordance with expectations from experiments. Though, using couplings only linear in the derivatives leads to a quadratic dependence of the optical potential on the kinetic energy with a decrease for energies exceeding 1 GeV. Couplings to all orders in the derivative of the nucleons were introduced in the so-called nonlinear derivative (NLD) model [16,17] assuming 75 a particular exponential dependence on the derivatives but no density dependence of the couplings. The general formalism was developed and applied to infinite isospin symmetric and asymmetric nuclear matter. In reference [18] the approach was slightly modified with a derivative momentum dependence of Lorentzian form and additional nonlinear self-couplings of the σ meson field in 80 order to improve the description of characteristic nuclear matter parameters at saturation. The application of this version of the NLD model to stellar matter yielded a maximum neutron star mass of 2.03 M sol barely satisfying the observational constraints but the dependence of the result on the model parameters was not explored in detail.

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In this work we introduce a more flexible extension of the nonlinear derivative model assuming density dependent meson-nucleon couplings in addition. Instead of using derivative operators that generate an explicit momentum dependence of the self-energies, we will use a functional form that leads to an energy dependence. This approach will also be more suitable for a future ap-90 plications of the DD-NLD approach to nuclei since the relevant equations and their numerical implementation are simplified. Here, the equations of state of symmetric and asymmetric nuclear matter will be calculated for different choices of the derivative coupling operators that lead to a saturation of the optical potential at high energies as derived from experiments. They are compared to 95 the results of a standard RMF model with density dependent couplings that is consistent with essentially all modern constraints for the characteristic nuclear matter parameters at saturation. The parameters of the DD-NLD models are chosen such that these saturation properties are reproduced. The effect of the optical potential constraint on the mass-radius relations of neutron stars will be 100 studied.
The paper is organized as follows: In section 2 the Lagrangian density of the DD-NLD approach is presented. The field equations in mean-field approximation and the energy-momentum tensor will be derived. The relevant equations for the case of infinite nuclear matter will be considered in more detail in section 105 3. The parametrization of the density dependent couplings and the functional form of the derivative coupling functions is discussed in section 4. Results for the energy dependence of the optical potential, the EoS of nuclear matter and the mass-radius relation are presented in section 5 for various versions of the model. Conclusions are given in section 6. Detailed expression for various densities are 110 collected in Appendix A.

Lagrangian density and field equations of the DD-NLD model
In most RMF models the effective interaction between nucleons is described by an exchange of mesons. Usually, σ and ω mesons are introduced to consider the attractive and repulsive contributions to the nucleon-nucleon potential, respectively. They are represented by isoscalar Lorentz scalar and Lorentz vector fields σ and ω µ . In order to model the isospin dependence of the interaction, the exchange of ρ mesons is included. It is denoted by the isovector Lorentz vector field ρ µ in the following. The Lagrangian density in the DD-NLD approach 1 in a symmetrized form and of free mesons of the isoscalar ω meson and the isovector ρ meson, respectively. The arrows in equation (2) denote the direction of differentiation.

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Standard RMF models assume a minimal coupling of the nucleons to the meson fields leading to for the interaction contribution to the total Lagrangian density L with mesonnucleon couplings Γ i (i = σ, ω, ρ). We assume that they depend on the vector 1 Natural units withh = c = 1 are used in the following.
density n v , see equation (26) for the explicit definition. In the derivative coupling model the nucleon field Ψ (Ψ) is replaced in L int by D m Ψ (D m Ψ) with operator functions D m , which can be different for the various mesons m = σ, ω, ρ. They can be expanded in a series n . The argument x contains derivatives i∂ β that act on the nucleon field. More specifically we write as a hermitian Lorentz scalar operator with an auxiliary Lorentz vector v β = (v 0 , v) and a scalar factor s. Hence the interaction contribution in the NLD model is written as with coefficients Obviously, no derivatives appear for the choice D m = 1 (corresponding to d (m) n = 120 δ n0 ) and the standard form (5) is recovered. When spatially inhomogeneous systems with Coulomb interaction are considered, L mes and L int can be complemented with the appropriate contributions. Since only uniform matter is considered in the following, we do not give them here explicitly.

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The field equations of nucleons and mesons are derived from the generalized Euler-Lagrange equation for all fields φ r = Ψ, Ψ, σ, ω µ , ρ µ of the model. Details can be found in references [16,18]. The Dirac equation for the nucleons looks formally the same as in standard RMF approaches but the scalar (Σ) and vector (Σ µ ) self-energie operators now contain the derivative operators D m . They are given by and with the 'rearrangement' contribution of the coupling functions. In the case of inhomogeneous systems and a nonvanishing three-vector component v of the auxiliary vector v β , additional contributions in (15) and (16) will appear. In the present application of the DD-NLD model, however, we do not consider this case. The field equations of the mesons 130 are found as with source terms containing derivative operators. The conserved baryon current in the DD-NLD model is given by with the norm operator where ∂ µ p D m is the derivative of D m operator with respect to the momentum and . . . denotes the summation over all occupied states. The current (22) is not identical to the vector current which is used to define the vector density appearing as the argument of the coupling functions Γ i . The energy-momentum tensor assumes the form Then the energy density ε and pressure p are found from ε = T 00 and p = 3 i=1 T ii /3, respectively.

DD-NLD model for nuclear matter
In the case of stationary nuclear matter, the equations simplify considerably since the system is homogeneous and the meson fields, which are treated as classical fields, are constant in space and time. Positive-energy solutions of the Dirac equation (14) are plane waves Ψ i = u i exp (−ip µ i x µ ) for protons and neutrons with Dirac spinors u i , which are normalized according to with the time component of the norm operator (23). They depend on the effective mass and effective momentum related by the dispersion relation The derivative i∂ β in the D m operators can be replaced by the corresponding four-momentum p β i = (E i , p i ) resulting in a simple function D m depending on the energy E i and the momentum p i of the nucleon.
Using the identity the conserved current and the energy-momentum tensor can be written as and respectively, with the four-momentum and spin degeneracy factors κ i = 2. The integration runs over all momenta p with modulus lower than the Fermi momenta p F i in the no-sea approximation. They are defined through the individual nucleon densities Without the preference for a particular direction in infinite nuclear matter, the spatial components of the Lorentz vector meson fields vanish and the aux-140 iliary vector in equation (7) is set to v β = δ β0 such that the D m functions only depend on the nucleon energy E i . Without isospin changing processes, only the third component of the isovector ρ field has to be considered in the field equations for the mesons. Using the abbreviations ω = ω 0 and ρ = ρ 0 3 the meson fields are immediately obtained from with source densities n σ , n ω , and n ρ . The self-energies simplify to with τ 3,i = 1 (−1) for protons (neutrons) and the 'rearrangement' contribution independent of the nucleon energy. The dispersion relation reads if we introduce the energy-dependent scalar potentials S i (E) = Σ i and vector potentials V i (E) = Σ 0 i . Explicit expressions for the various densities and thermodynamic quantities of the DD-NLD model are given in Appendix A.

Parametrization of the DD-NLD model
In order to reduce the number of independent parameters we demand that the

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Explicit values of the model parameters are given in table 1 with two choices of the cut-off parameter Λ for the cases of Lorentzian and exponential functions D. Note that the coefficients of the function f ω are identical for all five parametrizations due to the constraints. The reference density n ref is not necessarily identical to the saturation density n sat because in the case of explicit 180 derivative couplings the vector density n v is different from the conserved baryon density n B = J 0 = n p + n n .

Results
The nonlinear derivative couplings are introduced in the RMF model in order to improve the energy dependence of the optical potential U opt . The elastic proton scattering on nuclei of different mass number A can be well described in Dirac phenomenology with scalar (S) and vector (V ) potentials, which smoothly vary with A and the energy of the projectile [7,8]. From these global fits the optical potential in symmetric nuclear matter at saturation density is obtained as a function of the kinetic energy E kin = E − m nuc in the limit A → ∞. There are different definitions of the nonrelativistic optical potential when it is derived from relativistic scalar and vector self-energies. Here we use the form with S = Σ p and V = Σ 0 p as in references [14,15,16,17,18]. In conventional RMF models without derivative couplings, the scalar and vector potentials are 185 constant in energy and the optical potential (48) is just a linear function in energy. This is clearly seen in figures 1 and 2 as a full black line for the calculation with the parametrization D1. In contrast, the optical potentials derived from the scalar and vector potentials in Dirac phenomenology from two different fits [7] are much smaller at high energies and exhibit a saturation for 190 E kin approaching 1 GeV. At low energies, the optical potentials from expriment behave more similar as that of the theoretical model concerning the absolute strength and the energy dependence. In figure 1 (2) the result for the DD-NLD parametrization D2 (D3) is depicted for two values of the cut-off parameter Λ. Here, a reasonable description of the experimental optical potential is achieved 195 due to the energy dependence of the nucleon self-energies. The difference between parametrizations D2 and D3 is not very significant. The dependence of U opt on Λ is stronger for D2. The deflection of the DD-NLD curve for that of the standard RMF model D1 appears at lower kinetic energies for the D3 parametrization as compared to the D2 case. In the DD-NLD model, the op- neutron-proton asymmetries. Because there are no experimental data available for these general cases, we refrain from presenting the results here. But see reference [14] for the systematics with density in the linear derivative coupling model.

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The reduction of the optical potential at high kinetic energies, which originates from the energy dependence of the self-energies, is also reflected in the equation of state. In figure 3 the energy per nucleon E/A (without the rest mass contribution) is depicted as a function of the baryon density n B in symmetric nuclear matter (left panel) and neutron matter (right panel). In both cases, 210 a substantial softening of the EoS is found as compared to the standard RMF calculation with parametrisation D1. The effect is stronger for an exponential energy dependence of the self-energies (D3) than for the case of a Lorentzian dependence (D2). By construction, all EoS are identical at the saturation density n sat .

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The DD-NLD model can be used to predict the properties of neutron stars. Here, the EoS of stellar matter is required. It is obtained by adding the contribution of electrons to the energy density and pressure of the baryons. The conditions of charge neutrality and β equilibrium fix the lepton density and proton-neutron asymmetry uniquely. Since the present model calculations treat 220 only homogeneous matter, a suitable EoS for the crust of neutron stars has to be added at low densities. We use the standard Baym-Pethick-Sutherland (BPS) crust EoS [21]. The mass-radius relation of neutron stars is found finally by solving the Tolman-Oppenheimer-Volkoff equations [22,23]. It is shown for the five models of this work in figure 4 together with the masses of the two most mas-

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There are several aspects that have to be taken into account in constraining models of dense matter for the application to neutron stars. In phenomenological models, the characteristic saturation properties of nuclear matter and the density dependence of the effective interaction are usually addressed. However, less attention is paid to its energy or momentum dependence.

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Introducing non-linear derivative couplings into RMF models, it is possible to generate an energy dependence of the nucleon self-energies such that the optical potential in nuclear matter, which is extracted in Dirac phenomenology from elastic proton-nucleus scattering experiments, can be well described up to energies of 1 GeV. Considering density-dependent nucleon-meson couplings at 250 the same time, a very flexible model is obtained. Its parameters can be fitted to the usual nuclear matter constraints even for different functional forms of the energy dependent couplings. The energy dependence of the self-energies causes a softening of the EoS at high densities, both for symmetric nuclear matter and pure neutron matter. As 255 a result, it becomes more difficult to obtain very massive neutron stars consistent with the observational constraints. The results of the our study indicate that the optical potential constraint has to be taken seriously into account in the development of realistic phenomenological models for dense matter.
In the present work, an explicit energy dependence of the nucleon-meson 260 couplings was favored. It allows to apply the DD-NLD approach to the description of nuclei without major difficulties. Such an independent investigation of the model will permit a better control on the parameters. Work in this direction is in progress. and vector source densities the total source densities