Momentum dependent mean-field dynamics of compressed nuclear matter and neutron stars

# Momentum dependent mean-field dynamics of compressed nuclear matter and neutron stars

Theodoros Gaitanos    Murat M. Kaskulov Institut für Theoretische Physik, Universität Giessen, D-35392 Giessen, Germany
August 5, 2019
###### Abstract

Nuclear matter and compact neutron stars are studied in the framework of the non-linear derivative (NLD) model which accounts for the momentum dependence of relativistic mean-fields. The generalized form of the energy-momentum tensor is derived which allows to consider different forms of the regulator functions in the NLD Lagrangian. The thermodynamic consistency of the NLD model is demonstrated for arbitrary choice of the regulator functions. The NLD approach describes the bulk properties of the nuclear matter and compares well with microscopic calculations and Dirac phenomenology. We further study the high density domain of the nuclear equation of state (EoS) relevant for the matter in -equilibrium inside neutron stars. It is shown that the low density constraints imposed on the nuclear EoS and by the momentum dependence of the Schrödinger-equivalent optical potential lead to a maximum mass of the neutron stars around which accommodates the observed mass of the J1614-2230 millisecond radio pulsar.

###### pacs:
21.65.-f, 21.65.Mn, 25.40.Cm
preprint: 02-02

## I Introduction

Relativistic hadrodynamics (RHD) of interacting nucleons and mesons provide a simple and successful tool for the theoretical description of different nuclear systems such as nuclear matter, finite nuclei, heavy-ion collisions and compact neutron stars Serot:1984ey (). Starting from the pioneering work of Duerr Duerr:1956zz (), simple RHD Lagrangians have been introduced Walecka:1974qa (); Serot:1997xg () and since then many different extensions of RHD approach, which rely on relativistic mean-field (RMF) approximation, have been developed. They describe the saturation mechanism in nuclear matter and generate a natural mechanism for the strong spin-orbit force in nuclei. An energy dependence of the Schrödinger-equivalent optical potential Cooper:1993nx (); Hama:1990vr () is thereby included as a consequence of a relativistic description. However, when using the standard RHD Lagrangian in RMF approximation, the nucleon selfenergies become simple functions of density only, and do not depend explicitly on momentum of the nucleon. As a consequence a linear energy dependence of the Schrödinger-equivalent optical potential with a divergent behavior at high energies arises Weber:1992qc (). This well-known feature contradicts Dirac phenomenology Cooper:1993nx (); Hama:1990vr (); Typel2002299 ().

To solve this issue one may go beyond the mean-field approximation in a quantum field theoretical framework by a systematic diagrammatic expansion of nucleon selfenergies. For instance, in Dirac-Brueckner-Hartree-Fock (DBHF) Haar:1986ii (); Brockmann:1996xy (); Muther2000243 () calculations the nucleon selfenergies indeed depend on both the density and single particle momentum. They reproduce the empirical saturation point of nuclear matter as well as the energy dependence of the optical potential at low energies. However, the DBHF approach has its apparent limitations at high energies and densities relevant, for instance, in heavy-ion collisions where its application within transport theory turns out to be intricate Botermans1990115 (); Buss:2011mx (). Also the thermodynamic consistency of the DBHF calculations is not obvious Hugenholtz:1958 ().

As an alternative approach to ab-initio DBHF calculations for the nuclear many-body systems a phenomenological treatment of the problem in the spirit of the RMF approximation is still considered as a powerful tool. However, the simple Lagrangian of RHD Walecka:1974qa (); Serot:1997xg () has to be further modified for a quantitative description of static nuclear systems such as nuclear matter and/or finite nuclei. Therefore, it is mandatory to introduce new terms, e.g., including non-linear self interactions of the scalar Boguta:1977xi () and/or vector Sugahara1994557 () meson fields, or to modify existing contributions in the Lagrangian, e.g., introducing density dependent meson-nucleon couplings PhysRevLett.68.3408 (); Fuchs:1995as (); Typel:1999yq (). The model parameters have to be then fitted to properties of nuclear matter and/or atomic nuclei, since, they cannot be derived in a simple manner from a microscopic description.

The momentum dependence of in-medium interactions becomes particularly important in description of nuclear collision dynamics such as heavy-ion collisions. Indeed, analyses of proton-nucleus scattering data Cooper:1993nx (); Hama:1990vr () show that the proton-nucleus optical potential starts to level off already at incident energies of about MeV. Thus, other RMF approaches have been developed by including additional non-local contributions, i.e., by introducing Fock-terms, on the level of the RMF selfenergies leading to a density and momentum dependent interactions Weber:1992qc (). However, such a treatment is not covariant and also its numerical realization in actual transport calculations is rather difficult Weber:1992qc (). Another approach has been proposed in Zimanyi:1990np () and more recently in Typel:2002ck (); Typel:2005ba () by introducing higher order derivative couplings in the Lagrangian of RHD. In Ref. Zimanyi:1990np () such gradient terms have been studied with the conclusion of a softening of the nuclear EoS. In another study of Ref. Typel:2005ba () both the density dependence of the nuclear EoS and the energy dependence of the optical potential have been investigated. While the modified interactions of meson fields with nucleons explain the empirical energy dependence of the optical potential, a stiff EoS at high densities results from an introduction of an explicit density dependence of the nucleon-meson couplings with additional parameters. The impact of momentum dependent RMF models on nuclear matter bulk properties and particularly on the high density domain of EoS relevant for neutron stars is presently less understood.

The purpose of the present work is to develop a relativistic and thermodynamically consistent RMF model, which provides the correct momentum dependence of the nucleon selfenergies and agrees well with available empirical information on nuclear matter ground state, in a self consistent Lagrangian framework. Some steps in this direction have been already done in Refs. Gaitanos:2011yb (); Gaitanos:2011ej (); Gaitanos:2009nt () where the concept of non-linear derivative meson-nucleon Lagrangian has been introduced. However, the calculations of Refs. Gaitanos:2011yb (); Gaitanos:2011ej (); Gaitanos:2009nt () were based on a particular exponential form of the regulators in the RHD Lagrangian and a detailed study of nuclear matter ground state properties has not been done. In the present work the generalized form of the energy-momentum tensor in the NLD model is derived and allows to consider different regulator functions in the Lagrangian. The thermodynamic consistency of the NLD model is demonstrated for arbitrary choice of the regulators. A thorough study of the properties of nuclear matter around saturation density is further performed. The model describes the bulk properties of the nuclear matter and compares well with microscopic calculations and Dirac phenomenology. We also investigate the high density region of the NLD EoS relevant for the neutron stars. It is found that the low density constraints imposed on the nuclear matter EoS and by the momentum dependence of the Schrödinger-equivalent optical potential lead to a maximum mass of the neutron stars around . It is demonstrated that the high density pressure-density diagram as extracted from astrophysical measurements Ozel:2010fw (); Steiner:2010fz () can be well described with nucleonic degrees of freedom only.

## Ii Field theory with higher derivatives

The non-linear derivative (NLD) model is based on a field-theoretical formalism which accounts for the higher-order derivative interactions in the RHD Lagrangian. As a consequence, the conventional RHD mean-field theory based on minimal interaction Lagrangians has to be extended to the case of higher-order non-linear derivative functionals.

For that purpose we consider the most general structure of a Lagrangian density with higher-order field derivatives, i.e.

 L(φr(x),∂α1φr(x),∂α1α2φr(x),⋯,∂α1⋯αnφr(x)), (1)

where it is supposed that has continuous derivatives up to order with respect to all its arguments, that is

 ∂α1⋯αnφr(x)≡∂∂xα1⋯∂∂xαnφr(x)≡∂α1⋯∂αnφr(x),

where is a four index and denotes the coordinates in Minkowski space. The order can be a finite number or . The subscript denotes different fields, for instance, in the case of the spinor fields one would have and .

The derivation of the generalized Euler-Lagrange equations of motion follows from the variation principle for the action with the Lagrangian of Eq. (1), where one considers , , , , as independent generalized coordinates. The Euler-Lagrange equations are obtained from principle of least action

 δS=0, (2)

where is given by

 δS=∫d4xδL(φr,∂α1φr,∂α1α2φr,⋯,∂α1⋯αnφr) (3)

and is obtained by the variation of the generalized coordinates

 φr ⟶φr+δφr, ∂α1φr ⟶∂α1φr+δ∂α1φr, ∂α1α2φr ⟶∂α1α2φr+δ∂α1α2φr, ,⋯, ∂α1⋯αnφr ⟶∂α1⋯αnφr+δ∂α1⋯αnφr, (4)

with vanishing contributions on the surface of the integration volume as the boundary condition. The variation of the Lagrangian density with respect to all degrees of freedom reads

 δL= [∂L∂φrδφr+∂L∂(∂α1φr)∂α1δφr+∂L∂(∂α1α2φr)∂α1α2δφr +⋯+∂L∂(∂α1⋯αnφr)∂α1⋯αnδφr]. (5)

As a next step one inserts Eq. (5) into Eq. (3) and then performs successively partial integrations, e.g., one partial integration for the second term in Eq. (5), two partial integrations for the third term in Eq. (5), and partial integrations for the last term. This procedure results in to the following integrand in Eq. (3)

 δL= [∂L∂φr−∂α1∂L∂(∂α1φr)+∂α1α2∂L∂(∂α1α2φr) +⋯+(−)n∂α1⋯αn∂L∂(∂α1⋯αnφr)]δφr (6)

up to -divergence terms, which by Gauss law do not contribute to the action in Eq. (3). Thus, one arrives to the following generalized Euler-Lagrange equation

 ∂L∂φr+n∑i=1(−)i∂α1⋯αi∂L∂(∂α1⋯αiφr)=0. (7)

The Noether theorem follows from invariance principles of the Lagrangian density, Eq. (1), with respect to infinitesimal variations of the generalized coordinates and their argument (see for notations Appendix A). As further shown in Appendix B, the requirement of invariance of the Lagrangian density, Eq. (1), with respect to global phase transformations

 φr(x)⟶φ′r(x)=e−iϵφr(x) (8)

leads to a continuity equation for a conserved Noether current . The latter is given by the following expression

 (9)

In fact, for the Noether current consists of an infinite sequence of tensors with increasing rank order. Furthermore, each of the different tensors in Eq. (9) contains again infinite series terms of higher-order derivatives with respect to the Lagrangian density. They are given by the following expressions

 Kμr =n∑i=1(−)i+1i−1∏j=1∂αj∂L∂(∂μαjφr), (10) Kμσ1r =n∑i=1(−)i+1i−1∏j=1∂αj∂L∂(∂μαjσ1φr), Kμσ1σ2r =n∑i=1(−)i+1i−1∏j=1∂αj∂L∂(∂μαjσ1σ2φr), ⋮ Kμσ1⋯σnr =n∑i=1(−)i+1i−1∏j=1∂αj∂L∂(∂μαjσ1⋯σnφr).

The derivation of the energy-momentum tensor proceeds in a similar way, see Appendix B. Now the field arguments are transformed, but not the fields them self. In particular, invariance of the Lagrangian density (1) with respect to a constant displacement of the coordinates

 xμ⟶x′μ=xμ+δμ, (11)

implies a continuity equation for the energy-momentum tensor which takes the following form

 Tμν=Kμr∂νφr+Kμσ1r∂νσ1φr+Kμσ1σ2r∂νσ1σ2φr+⋯+Kμσ1⋯σnr∂νσ1⋯σnφr−gμνL. (12)

The -component of the energy-momentum tensor describes the energy density and the spatial diagonal components are related to the pressure density. These equations form a background for the construction and application of the NLD formalism presented in the proceeding sections. They will further provide a thermodynamically consistent framework for the calculation of the EoS in mean field approximation in terms of energy and pressure densities.

## Iii The non-linear derivative model

In this section we introduce the non-linear derivative (NLD) model and derive the equations of motion for the relevant degrees of freedom. The NLD approach is essentially based on the Lagrangian density of RHD Duerr:1956zz (); Walecka:1974qa (); Serot:1997xg (), which is given by

 L= 12[¯¯¯¯Ψγμi→∂μΨ−¯¯¯¯Ψi←∂μγμΨ]−m¯¯¯¯ΨΨ−12m2σσ2+12∂μσ∂μσ−U(σ) + 12m2ωωμωμ−14FμνFμν+12m2ρ→ρμ→ρμ−14→Gμν→Gμν−12m2δ→δ2+12∂μ→δ∂μ→δ+Lint (13)

where denotes the nucleon spinor field in the Lagrangian density of a Dirac-type. In a spirit of RHD, the interactions between the nucleon fields are described by the exchange of meson fields. These are the scalar and vector mesons in the isoscalar channel, as well as the scalar and vector mesons in the isovector channel. Their corresponding Lagrangian densities are of the Klein-Gordon and Proca types, respectively. The term contains the usual selfinteractions of the meson. The notations for the masses of fields in Eq. (13) are obvious. The field strength tensors are defined as , for the isoscalar and isovector fields, respectively.

In conventional RHD approaches the interaction Lagrangian is given by Walecka:1974qa (); Serot:1997xg ()

 Lint=Lσint+Lωint+Lρint+Lδint, (14)

where

 Lσint=gσ¯¯¯¯ΨΨσ, (15)
 Lωint=−gω¯¯¯¯ΨγμΨωμ, (16)
 Lρint=−gρ¯¯¯¯Ψγμ→τΨ→ρμ, (17)
 Lδint=gδ¯¯¯¯Ψ→τΨ→δ, (18)

and contains the meson-nucleon interactions with coupling strengths and denotes the isospin Pauli operator.

In the NLD model the momentum dependence of fields is realized by the introduction of non-linear derivative operators in the interaction Lagrangian of conventional RHD. These additional operators regulate the high momentum components of the RMF fields in the interaction vertices and can be interpreted as cut-off form factors. This is in spirit of boson-exchange models where the phenomenological cut-off is an indispensable part of any microscopic description of meson-nucleon interaction Machleidt:1987hj (); Erkelenz:1974uj (). In the RMF (Hartree) approximation to RHD only bare Lorentz structures corresponding to the point-like meson-nucleon interactions are taken into account and the high momentum components of fields are not suppressed due to the missing nucleon finite size effect. The NLD model attempts to account for the suppression of the high momentum part of the nucleon field in the meson-nucleon interaction on a field-theoretical level.

The NLD interaction Lagrangians contain the conventional meson-nucleon RHD structures, however, they are extended by the inclusion of non-linear derivative operators into the meson-nucleon vertices. The NLD interaction Lagrangians followed here read

 Lσint=gσ2[¯¯¯¯Ψ←DΨσ+σ¯¯¯¯Ψ→DΨ], (19)
 Lωint=−gω2[¯¯¯¯Ψ←DγμΨωμ+ωμ¯¯¯¯Ψγμ→DΨ], (20)
 Lρint=−gρ2[¯¯¯¯Ψ←Dγμ→τΨ→ρμ+→ρμ¯¯¯¯Ψ→τγμ→DΨ], (21)
 Lδint=gδ2[¯¯¯¯Ψ←D→τΨ→δ+→δ¯¯¯¯Ψ→τ→DΨ]. (22)

As one can see, the only difference with respect to the conventional RHD interaction Lagrangian is the presence of additional operators which serve to regulate the high momentum component of the nucleon field. The hermiticity of the Lagrangian demands . The operator functions (regulators) are assumed to be generic functions of partial derivative operator and supposed to act on the nucleon spinors and , respectively. Furthermore, these regulators are assumed to be smooth functions. Therefore, the formal Taylor expansion of the operator functions in terms of partial derivatives generates an infinite series of higher-order derivative terms

 →D:=D(→ξ)= n→∞∑j=0∂j∂→ξjD|→ξ→0→ξjj!, (23) ←D:=D(←ξ)= n→∞∑j=0←ξjj!∂j∂→ξjD|←ξ→0. (24)

The expansion coefficients are given by the partial derivatives of with respect to the operator arguments and around the origin. The operators are defined as where the four vector contains the cut-off and is an auxiliary vector. The functional form of the regulators is constructed such that in the limit the following limit holds . Therefore, in the limit the original RHD Lagrangians are recovered.

In the most general case the NLD formalism can be extended to the case of multiple variable regulators. In particular, we can assume the non-linear operator to be a multi-variable non-linear function of higher-order partial derivatives, which are given by the following Taylor expansion

 →D:=D(→ξ1,→ξ2,→ξ3,→ξ4)= n→∞∑i1=0n→∞∑i2=0n→∞∑i3=0n→∞∑i4=0∂i1+i2+i3+i4∂→ξi11∂→ξi22∂→ξi33∂→ξi44D|{→ξ1,→ξ2,→ξ3,→ξ4}→0→ξi11→ξi22→ξi33→ξi44i1!i2!i3!i4!, (25) ←D:=D(←ξ1,←ξ2,←ξ3,←ξ4)= n→∞∑i1=0n→∞∑i2=0n→∞∑i3=0n→∞∑i4=0←ξi11←ξi22←ξi33←ξi44i1!i2!i3!i4!∂i1+i2+i3+i4∂←ξi11∂←ξi22∂←ξi33∂←ξi44D|{←ξ1,←ξ2,←ξ3,←ξ4}→0. (26)

Then Eqs. (25) and (26) can be rearranged into the terms with increasing order with respect to the partial derivatives, see for details Appendix C. The operators are defined in a similar way as before

 →ξi=−ζαii→∂α , ←ξi=i←∂αζαi, (27)

with () in this case. As we will show latter on, this representation allows to generate any desired form of the regulator function, i.e., momentum and/or energy dependent monopole, dipole etc. functions.

The derivation of the equation of motion for the Dirac field follows the generalized Euler-Lagrange equations, Eq. (7), to the NLD-Lagrangian density using the Taylor form of the regulators. This obviously will generate an infinite number of partial derivative terms in the equations of motions. However, as shown in detail in Appendix D these infinite series can be resummed (up to terms containing the derivatives of the meson fields) to the following Dirac equation

 [γμ(i∂μ−Σμ)−(m−Σs)]Ψ=0, (28)

where the selfenergies and are given by

 Σμ = gωωμ→D+gρ→τ⋅→ρμ→D+⋯ , (29) Σs = gσσ→D+gδ→τ⋅→δ→D+⋯. (30)

Here both Lorentz-components of the selfenergy, and , show an explicit linear behavior with respect to the meson fields , , and as in the standard RMF. However, they contain an additional dependence on regulator functions.

The additional terms in Eqs. (29) and (30) containing the meson field derivatives are denoted by multiple dots. All these contributions can be also resummed. However, in the mean-field approximation to infinite nuclear matter, which will be discussed in the next section, these terms vanish. On the other hand, they will be needed in the description of finite systems, such as finite nuclei and heavy-ion collisions. Therefore, for simplicity we do not consider these terms here, and postpone the effect of these terms for future studies.

The derivation of the meson field equations of motion is straightforward, since here one has to use the standard Euler-Lagrange equations

 ∂L∂φr−∂α∂L∂(∂αφr)=0, (31)

where now and . The following Proca and Klein-Gordon equations are obtained

 ∂α∂ασ+m2σσ+∂U∂σ=12gσ[¯¯¯¯Ψ←DΨ+¯¯¯¯Ψ→DΨ], (32) ∂μFμν+m2ωων=12gω[¯¯¯¯Ψ←DγνΨ+¯¯¯¯Ψγν→DΨ], (33) ∂μ→Gμν+m2ρ→ρν=12gρ[¯¯¯¯Ψ←Dγν→τΨ+¯¯¯¯Ψ→τγν→DΨ], (34) ∂α∂α→δ+m2σ→δ=12gδ[¯¯¯¯Ψ←D→τΨ+¯¯¯¯Ψ→τ→DΨ]. (35)

Finally, we provide the general expressions for the Noether theorems within the NLD formalism. The evaluation of the conserved baryon current results from the application of the generalized expression for , Eq. (9), to the Lagrangian density of the NLD model. As shown in detail in Appendix E, a systematic evaluation of the higher-order field derivatives of the NLD Lagrangian and the resummation procedure result in

 Jμ=¯¯¯¯ΨγμΨ− 12gσ[¯¯¯¯Ψ←ΩμΨ−¯¯¯¯Ψ→ΩμΨ]σ+12gω[¯¯¯¯Ψ←ΩμγαΨ−¯¯¯¯Ψγα→ΩμΨ]ωα + 12gρ[¯¯¯¯Ψ←Ωμγα→τΨ−¯¯¯¯Ψγα→Ωμ→τΨ]→ρα−12gδ[¯¯¯¯Ψ←Ωμ→τΨ−¯¯¯¯Ψ→Ωμ→τΨ]→δ+⋯. (36)

The new non-linear derivative operators in Eq. (III), and , denote the derivatives of and with respect to their operator argument and (see Appendix E). The first term in Eq. (III) corresponds to the standard expression of the RHD models and the additional contributions arise due to the additional higher-order field derivatives in the Noether theorem, Eq. (9).

The energy-momentum tensor, , is determined according to Eq. (12). The evaluation procedure, which is similar to that one for the Noether current, results in the following NLD expression for

 Tμν= 12¯¯¯¯Ψγμi→∂νΨ−12¯¯¯¯Ψi←∂νγμΨ + 12gσ[¯¯¯¯Ψ→Ωμi→∂νΨ+¯¯¯¯Ψi←∂ν←ΩμΨ]σ−12gω[¯¯¯¯Ψγα→Ωμi→∂νΨ+¯¯¯¯Ψi←∂ν←ΩμγαΨ]ωα − 12gρ[¯¯¯¯Ψ→τγα→Ωμi→∂νΨ+¯¯¯¯Ψi←∂ν←Ωμγα→τΨ]→ρα+12gδ[¯¯¯¯Ψ→τ→Ωμi→∂νΨ+¯¯¯¯Ψi←∂ν←Ωμ→τΨ]→δ − gμνL+⋯. (37)

The first line in Eq. (III) is just the usual kinetic RHD contribution to , while the additional kinetic terms originate from the evaluation of the higher-order derivatives in Eq. (12). These terms will be important for the the thermodynamic consistency of the model and the validation of the Hugenholtz-Van Hove theorem Weisskopf:1957 (); Hugenholtz:1958 (). Again the terms not shown in Eqs. (III) and (III) describe the contribution of terms containing the derivatives of the meson fields.

## Iv RMF approach to infinite nuclear matter

In the mean-field approximation the mesons are treated as classical fields. Infinite nuclear matter is described by a static homogeneous, isotropic, spin and isospin-saturated system of protons and neutrons. In this case, the spatial components of the Lorentz-vector meson fields vanish with , and in isospin space only the neutral component of the isovector fields survive, i.e., and . For simplicity, we denote in the following the third isospin components of the isovector fields as and .

The derivation of the RMF equations starts with the usual plane wave ansatz

 Ψi(s,→p)=ui(s,→p)e−ipμxμ, (38)

where stands for protons () or neutrons () and is a single nucleon 4-momentum. The application of the non-linear derivative operator to the plane wave ansatz of the spinor fields results in

 D(→ξ)Ψi=D(ξ)ui(s,→p)e−ipμxμ, (39)
 ¯¯¯¯ΨiD(←ξ)=D(ξ)¯¯¯ui(s,→p)e+ipμxμ, (40)

where the regulators in the r.h.s. of above equation are now functions of the scalar argument .

With the help of Eqs. (38) and (39) one gets the Dirac equation similar to Eq. (28) with selfenergies given by

 Σμvi=gωωμD+gρτiρμD , (41)
 Σsi=gσσD+gδτiδD , (42)

where now for protons () and for neutrons (). We note again that in the RMF approximation to infinite matter the additional terms including the meson field derivatives vanish. This largely simplifies the formalism, since these terms which show up in the original Dirac equation, see Eq. (28), do not appear any more.

The solutions of the Dirac equation takes the form

 ui(s,→p)=Ni⎛⎜ ⎜ ⎜ ⎜⎝φs→σ⋅→pE∗i+m∗iφs⎞⎟ ⎟ ⎟ ⎟⎠, (43)

with spin eigenfunctions , the in-medium energy

 E∗i:=E−Σ0vi , (44)

and the Dirac mass

 m∗i:=m−Σsi . (45)

For a given momentum the single particle energy is obtained from the in-medium on-shell relation

 E∗2i−→p2=m∗2i . (46)

The factor is determined from the normalization of the probability distribution, that is . In the conventional RMF models the baryon density is given by the familiar expression and the normalization condition would result in . In the NLD model one has to use Eq. (9) for the Noether current by keeping in mind that the infinite series of meson field derivatives vanish in the RMF approach to nuclear matter. In this case, see again Appendix E for details, the conserved baryon current is resummed up to infinity and the result reads

 Jμ= ∑i=p,n[⟨¯¯¯¯ΨiγμΨi⟩ (47) +gσ⟨¯¯¯¯Ψi[∂μpD]Ψi⟩σ−gω⟨¯¯¯¯Ψi[∂μpD]γαΨi⟩ωα −gρτi⟨¯¯¯¯Ψi[∂μpD]γαΨi⟩ρα+gδτi⟨¯¯¯¯Ψi[∂μpD]Ψi⟩δ].

The -component of the Noether current describes the conserved nucleon density , from which also the relation between the Fermi momentum and is uniquely determined. In particular, using the Gordon identity and Eqs. (41) and (42) for the RMF selfenergies, one obtains

 Jμ =κ(2π)3∑i=p,n∫|→p|≤pFid3pN2i ×[p∗μiE∗i+(∂μpΣsi)m∗iE∗i−(∂μpΣβvi)p∗iβE∗i],

where is a spin degeneracy factor, stands for the proton or neutron Fermi-momentum and the effective momentum is given by

 p∗μi=pμ−Σμvi. (48)

One defines now a new in-medium -momentum as

 Πμi=p∗μi+m∗i(∂μpΣsi)−(∂μpΣβvi)p∗iβ, (49)

and arrives to the following expression

 Jμ=κ(2π)3∑i=p,n∫|→p|≤pFid3pN2iΠμiE∗i. (50)

On the other hand, the general definition of the baryon current results from the covariant superposition of all the occupied in-medium on-shell nucleon positive energy states up to the proton or neutron Fermi momentum Weber:1992qc ()

 Jμ= κ(2π)3∑i=p,n∫|→p|≤pFid4p (51) × Πμiδ(p∗μip∗iμ−m∗2i)2Θ(p0).

In the NLD approach the mean-field selfenergies depend explicitly on the single-particle momentum . Therefore, using the properties of the -function the time-like component can be integrated out explicitly. The result reads

 Jμ=κ(2π)3∑i=p,n∫|→p|≤pFid3pΠμiΠ0i. (52)

Comparing Eq. (52) with the equation for the NLD current, Eq. (50), one gets the following result for the normalization

 Ni=√E∗i+m∗i2E∗i ⎷E∗iΠ0i, (53)

and the bilinear products between the in-medium spinors of protons and neutrons are given by

 ¯¯¯ui(p)ui(p)= m∗iΠ0i, (54) ¯¯¯ui(p)γ0ui(p)= 1. (55)

Eq. (55) ensures also the proper normalization of the probability distribution, i.e., .

In our first work Gaitanos:2009nt (), where the non-linear derivative model has been proposed, the correction terms proportional to the partial derivatives of the selfenergies with respect to the single-particle momentum in Eq. (49) were not taken into account. Even if their contributions are small at low densities, these terms will be included in the present calculations which attempt to consider also the high density domain of the EoS in neutron stars. On the other hand, the inclusion of these terms is crucial for a fully thermodynamically consistent formalism and is independent of the particular form of the cut-off functions. Note that, the additional cut-off dependent terms in the baryon and energy densities of Ref. Gaitanos:2009nt () are now canceled by the proper normalization conditions.

The energy-momentum tensor in NLD is obtained by applying the Noether theorem for translational invariance. In nuclear matter the resummation procedure results in the following expression

 Tμν= ∑i=p,n[⟨¯¯¯¯ΨiγμpνΨi⟩ +gσ⟨¯¯¯¯Ψi[∂μpD]pνΨi⟩σ−gω⟨¯¯¯¯Ψi[∂μpD]γαpνΨi⟩ωα−gρτi⟨¯¯¯¯Ψi[∂μpD]γαpνΨi⟩ρα+gδτi⟨¯¯¯¯Ψi[∂μpD]pνΨi⟩δα] −gμν⟨L⟩. (56)

The evaluation of the expectation values in Eq. (IV) can be done in a similar way as for the current with the result

 Tμν=∑i=p,nκ(2π)3∫|→p|≤pFid3ppνΠ0i[p∗μi+m∗i(∂μpΣsi)−p∗αi(∂μpΣαi)]−gμν⟨L⟩. (57)

Using Eq. (49) one arrives to the final expression for the energy-momentum tensor in the NLD formalism, which can be written in the following form

 Tμν=∑i=p,nκ(2π)3∫|→p|≤pFid3pΠμipνΠ0i−gμν⟨L⟩, (58)

from which the energy density and the pressure can be calculated, i.e.,

 ε= ∑i=p,nκ(2π)3∫|→p|≤pFid3pE(→p)−⟨L⟩ , (59) P= 13∑i=p,nκ(2π)3∫|→p|≤pFid3p→Πi⋅→pΠ0i+⟨L⟩. (60)

Eqs. (59) and (