aa r X i v : . [ nu c l - t h ] J a n Momentum dependent mean-fields of (anti)hyperons
T. Gaitanos, A. Chorozidou
Department of Theoretical Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greeceemail: [email protected]
Abstract
We investigate the in-medium properties of hyperons and anti-hyperons in the framework of theNon-Linear Derivative (NLD) model. We focus on the momentum dependence of in-mediumstrangeness optical potentials. The NLD model is based on the simplicity of the well-establishedRelativistic Mean-Field (RMF) approximation, but it incorporates an explicit momentum depen-dence on a field-theoretical level. The extension of the NLD model to the (anti)baryon-octet isformulated in the spirit of SU(6) and G-parity arguments. It is shown that with an appropriatechoice of momentum cut-offs the Λ , Σ and Ξ optical potentials are consistent with recent studiesof the chiral effective field theory and Lattice-QCD calculations over a wide momentum region.In addition, we present NLD predictions for the in-medium momentum dependence of Λ -, Σ -and Ξ -hyperons. This work is important for future experimental studies such as CBM, PANDAat the Facility for Antiproton and Ion Research (FAIR). It is relevant for nuclear astrophysics too. Keywords:
Equations of state of hadronic matter, optical potential, in-medium hyperonpotentials.
1. Introduction
Astrophysical observations on particularly massive neutron stars [1, 2, 3] have driven thenuclear physics and astrophysics communities to detailed investigations of the nuclear equationof state (EoS) under conditions far beyond the ordinary matter [4]. On one hand, theoretical andexperimental studies on heavy-ion collisions over the last few decades concluded a softening ofthe high-density EoS in agreement with phenomenological and microscopic models [5, 6, 7].On the other hand, the observations of two-solar mass pulsars [1, 2, 3] together with additionalconstraints on the high-density limit of the speed of sound [8] gave some controversial insightson the EoS of compressed baryonic matter. They provide an upper limit for the neutron star massby excluding soft-type hadronic EoS’s at high baryon densities.Compressed baryonic matter may consist not only of nucleons. It can include fractions ofheavier baryons, when their production is energetically allowed. These are the hyperons Λ , Σ and Ξ as a part of the irreducible representations of SU(3). While the nucleon-nucleon (NN) in-teraction is very well known, the hyperon interactions are still not fully understood. Indeed, thereare many experimental data for NN-scattering in free space and inside hadronic media (finitenuclei, heavy-ion collisions, hadron-induced reactions) allowing a precise determination of the Preprint submitted to Elsevier January 22, 2021
N-interaction. Concerning the strangeness sector (hyperon-nucleon (YN) or hyperon-hyperon(YY) interactions), there exist phenomenological and microscopic models with predictions forthe in-medium hyperon properties at matter densities close to saturation and at higher densities.However, the experimental access to the strangeness sector is still scarce. A common predictionof theoretical models is a considerable softening of the hadronic EoS at high densities by addingto a system more degrees of freedom such as strangeness particles. The inclusion of hyperonsinto nuclear approaches made many of them, which were successfully applied to nuclear systems(nuclear matter, finite nuclei, nuclear reactions), incompatible with the astrophysical observationsof two-solar mass pulsars [1, 2]. This is the so-called hyperon-puzzle [9, 10]. This puzzle hasreceived recently theoretical attraction by a new observation of a quite massive neutron star [3].A comprehensive theoretical view concerning the microscopic descriptions of in-medium prop-erties of the baryon-octet is given in Ref. [11]. There exist also theoretical reviews based on theRMF approximation, see for instance Refs. [12, 13, 14].It is thus of great interest to address the in-medium behaviour of hyperons in nuclear matter,as we do in this work. We use an alternative RMF approach based on the fact, that compressedmatter consists of particles with high relative momenta. Therefore, not only the density de-pendence, but the momentum dependence of the in-medium interactions is important too. Thereason for doing so is that conventional RMF-models do not explain the empirical saturation ofthe in-medium interactions of high-momenta (anti)nucleons. In terms of SU(6) this issue appearsfor high-momenta (anti)hyperons too. This is the Non-Linear Derivative (NLD) model [15]. Itretains the basic RMF Lagrangian formulation, but it includes higher-order derivatives in the NN-interaction Lagrangians. It has been demonstrated that this Ansatz corrects the high-momentumbehaviour of the interaction, makes the EoS softer at densities just above saturation, but at thesame time it reproduces the two-solar mass pulsars at densities far beyond saturation [15]. Herewe extend the NLD approach by including strangeness into the nuclear matter and discuss themomentum dependence of the in-medium hyperon potentials.
2. The NLD Model for the baryon octet
In this section we briefly introduce the non-linear derivative (NLD) model and extend it to thebaryon octet. A detailed description of the NLD model for nucleons can be found in Ref. [15].The NLD-Lagrangian is based on the conventional Relativistic Hadro-Dynamics (RHD) [16] andit reads as L = 12 X B h Ψ B γ µ i −→ ∂ µ Ψ B − Ψ B i ←− ∂ µ γ µ Ψ B i − X B m B Ψ B Ψ B − m σ σ + 12 ∂ µ σ∂ µ σ − U ( σ )+ 12 m ω ω µ ω µ − F µν F µν + 12 m ρ ~ρ µ ~ρ µ − ~G µν ~G µν − m δ ~δ + 12 ∂ µ ~δ ∂ µ ~δ + L σint + L ωint + L ρint + L δint . (1)2he sum over B runs over the baryonic octet Ψ B =(Ψ N , Ψ Λ , Ψ Σ , Ψ Ξ ) T (2)with Ψ N =( ψ p , ψ n ) T , Ψ Λ = ψ Λ (3) Ψ Σ =( ψ Σ + , ψ Σ , ψ Σ − ) T , Ψ Ξ = ( ψ Ξ , ψ Ξ − ) T (4)for the isospin-doublets Ψ N and Ψ Ξ , isospin-triplet Ψ Σ and the neutral Ψ Λ . The interactionsbetween the nucleon fields are described by the exchange of meson fields. These are the scalar σ and vector ω µ mesons in the isoscalar channel, as well as the scalar ~δ and vector ~ρ µ mesonsin the isovector channel. Their corresponding Lagrangian densities are of the Klein-Gordon andProca types, respectively. The term U ( σ ) = bσ + cσ contains the usual selfinteractions of the σ meson. The notations for the masses of fields in Eq. (1) are obvious. The field strength tensorsare defined as F µν = ∂ µ ω ν − ∂ ν ω µ , ~G µν = ∂ µ ~ρ ν − ∂ ν ~ρ µ for the isoscalar and isovector fields,respectively. In the following we restrict to a minimal set of interaction degrees of freedom. Inthe iso-scalar sector, the σ - and ω -fields are obviously considered. In the iso-vector channel, wekeep the vector, iso-vector ρ -meson field and neglect the δ -field.The NLD interaction Lagrangians contain the conventional RHD combinations between thebilinear baryon- and linear meson-fields, however, they are extended by the inclusion of non-linear derivative operators −→D , ←−D for each baryon species B : L σint = X B g σB h Ψ B ←−D B Ψ B σ + σ Ψ B −→D B Ψ B i , (5) L ωint = − X B g ωB h Ψ B ←−D B γ µ Ψ B ω µ + ω µ Ψ B γ µ −→D B Ψ B i , (6) L ρint = − X B g ωρ h Ψ B ←−D B γ µ ~τ Ψ B ~ρ µ + ~ρ µ Ψ B ~τ γ µ −→D B Ψ B i , (7)for the isoscalar-scalar, isoscalar-vector and isovector-vector vertices, respectively. The ar-rows on the non-linear operator D B indicate the direction of their action. The only differencewith respect to the conventional RHD Lagrangian is the presence of additional operator functions −→D B , ←−D B . As we will see, they will regulate the high momentum component of hyperons. Forthis reason we will call them as regulators too. The operator functions (or regulators) −→D B , ←−D B are hermitian and generic functions of partial derivative operator. That is, −→D B := D (cid:16) −→ ξ B (cid:17) and ←−D B := D (cid:16) ←− ξ B (cid:17) with the operator arguments −→ ξ B = − ζ αB i −→ ∂ α , ←− ξ B = i ←− ∂ α ζ αB . The fourvector ζ µB = v µ / Λ B contains the cut-off Λ B and v µ is an auxiliary vector. These regulators areassumed to act on the baryon spinors Ψ B and Ψ B by a formal Taylor expansion with respect tothe operator argument. The functional form of the regulators is constructed such that in the limit Λ B → ∞ the original RHD Lagrangians are recovered, that is, −→D B = ←−D † B → .3he presence of higher-order partial derivatives in the Lagrangian mediate a modification ofthe field-theoretical prescriptions. As discussed in detail in the original work of Ref. [15], thegeneralized Euler-Lagrange equations as well as the Noether-currents contain additional infiniteterms of higher-order partial derivative contributions. However, the main advantage of the NLDapproach relies on the fact that these terms can be resummed to compact expressions.From the generalized Euler-Lagrange formalism we obtain the equations of motion for thedegrees of freedom in the NLD model. The meson field equations of motion read ∂ α ∂ α σ + m σ σ + ∂U∂σ = 12 X B g σB h Ψ B ←−D B Ψ B + Ψ B −→D B Ψ B i , (8) ∂ µ F µν + m ω ω ν = 12 X B g ωB h Ψ B ←−D B γ ν Ψ B + Ψ B γ ν −→D B Ψ B i , (9) ∂ µ G µν + m ρ ~ρ ν = 12 X B g ρB h Ψ B ←−D B γ ν ~τ Ψ B + Ψ B ~τ γ ν −→D B Ψ B i , (10)for the isoscalar-scalar, isoscalar-vector and isovector-vector exchange mesons, respectively.Each baryon-field obeys a Dirac-equation of the following type [ γ µ ( i∂ µ − Σ µB ) − ( m B − Σ sB )] ψ B = 0 , (11)with the selfenergies Σ µB and Σ sB defined as Σ µB = g ω B ω µ −→D B + g ρB ~τ B · ~ρ µ −→D B , (12) Σ sB = g σB σ −→D B . (13)Both Lorentz-components of the selfenergy, Σ µ and Σ s , show an explicit linear behaviour withrespect to the meson fields σ , ω µ and ~ρ µ as in the standard RHD. However, they contain an ad-ditional dependence on the regulators. General expressions for the Noether-current and energy-momentum tensor can also be derived. We give them below in the RMF approximation.The RMF application of the NLD formalism to static hadronic matter follows the same proce-dure as in the conventional RHD. The spatial components of the meson fields in Minkowski- andisospin-spaces vanish, ω µ → ( ω , ~ and ~ρ µ → ( ρ , ~ ) . For simplicity, we denote in the fol-lowing the remaining isospin component of the isovector fields as ρ µ . The solutions of the RMFequations start with the usual plane wave ansatz ψ B ( s, ~p ) = u B ( s, ~p ) e − ip µ x µ where B standsfor the various isospin states of the baryons and p µ = ( E, ~p ) is the single baryon 4-momentum.The application of the non-linear derivative operator D B to the plane wave Ansatz of the spinorfields results in regulators D B which are now functions of the scalar argument ξ B = − v α p α Λ B .That is, they depend explicitly on the single baryon momentum p (with an appropriate choice ofthe auxiliary vector v α ) and on the cut-off Λ B , which may differ for each baryon type B . Eachbaryon fulfils a Dirac equation with the same form as in Eq. (11) and with corresponding ex-plicitly momentum dependent scalar and vector selfenergies. Their vector components are givenby Σ µp = g ωN ω µ D N + g ρN ρ µ D N , (14) Σ µn = g ωN ω µ D N − g ρN ρ µ D N , (15)4 µ Λ = g ω Λ ω µ D Λ , (16) Σ µ Σ + = g ω Σ ω µ D Σ + g ρ Σ ρ µ D Σ , (17) Σ µ Σ − = g ω Σ ω µ D Σ − g ρ Σ ρ µ D Σ , (18) Σ µ Σ = g ω Σ ω µ D Σ , (19) Σ µ Ξ − = g ω Ξ ω µ D Ξ − g ρ Ξ ρ µ D Ξ , (20) Σ µ Ξ = g ω Ξ ω µ D Ξ + g ρ Ξ ρ µ D Ξ . (21)Similar expressions result for the scalar selfenergies. In the following the scalar and time-likecomponent of the baryon selfenergy will be denoted as S B and V B , respectively. Note that theselfenergies are explicitly momentum dependent due to the regulators D B = D B ( p ) as spec-ified below. The solutions of the Dirac equation are the standard Dirac-spinors with a propernormalization N B u B ( s, ~p ) = N B ϕ s ~σ · ~pE ∗ B + m ∗ B ϕ s , (22)but now for quasi-free baryons B with an in-medium energy E ∗ B := E B − V B ( p ) , (23)and a Dirac mass m ∗ B := m B − S B ( p ) . (24)At a given momentum the single particle energy E is obtained from the in-medium on-shellrelation (23). These expressions are needed for evaluation of expectation values, for instance,the source terms of the meson-field equations. For the definition of the nuclear matter we needa conserved nucleon density. It is obtained from the time-like component of the Noether-current J µ defined as J µ = κ (2 π ) X B = p,n Z | ~p |≤ p FB d p Π µB Π B (25)with the generalized -momentum Π µB = p ∗ µB + m ∗ B (cid:16) ∂ µp S B (cid:17) − (cid:16) ∂ µp Σ βB (cid:17) p ∗ Bβ (26)and the usual effective -momentum p ∗ µB = p µ − Σ µB . (27)5he EoS (Equation of State) is obtained from the time-like components of the energy-momentumtensor. In nuclear matter the resummation procedure of the NLD model results in the followingexpression T µν = X B κ (2 π ) Z | ~p |≤ p FB d p Π µB p ν Π B − g µν hLi , (28)from which the energy density ε ≡ T and the pressure P can be calculated, see for detailsRef. [15]. Finally, the NLD meson-field equations in the RMF approach to nuclear matter can beresummed to the following forms m σ σ + ∂U∂σ = X B g σB D ψ B D B ψ B E = X B g σB ρ sB , (29) m ω ω = X B g ωB D Ψ B γ D B Ψ B E = X B g ωB ρ B , (30)with the scalar and vector density sources ρ sB = κ (2 π ) Z | ~p |≤ p FB d p m ∗ B Π B D B ( p ) , (31) ρ B = κ (2 π ) Z | ~p |≤ p FB d p E ∗ B Π B D B ( p ) . (32)The isovector densities are calculated through the standard isospin relations. For a hyperon witha given momentum relative to nuclear matter at rest (at a given nucleon density and isospinasymmetry) the mesonic sources contain only nucleons, that is B = p, n .The meson-field equations of motion show a similar structure as those of the standard RMFapproximation. However, the substantial difference between NLD and other conventional RMFmodels appears in the source terms which now contain in addition the momentum-dependentregulators D B . This is an important feature of the NLD model. The cut-off leads naturally to aparticular suppression of the vector field at high densities or high Fermi-momenta in agreementwith phenomenology, as discussed in detail in the previous work [15]. This feature is absent inconventional RHD approaches, except if one introduces by hand additional scalar/vector self-interactions.The key observable for general discussions related to momentum or energy dependenciesof in-medium hadronic potentials is the Schroedinger-equivalent optical potential U opt , which isa complex quantity. The imaginary part describes the scattering processes of a given particle,e.g., a hyperon, with a nucleon of the nuclear matter. The real part of the optical potential isrelated to the mean-field that a particle, e.g., a hyperon with a given momentum, experiencesin the nuclear medium at a given density and isospin-asymmetry. The imaginary part of U opt cannot be calculated within a conventional RMF prescription. In RMF models one is usually6LD parameters Λ sN Λ vN g σN g ωN g ρN b c [ GeV] [ GeV] [ fm ]0 .
95 1 .
125 10 .
08 10 .
13 3 .
50 15 . − . Bulk saturation properties ρ sat E b K a sym [ fm ] [ MeVA ] [ MeV] [ MeV]0 . − .
30 251 30
Table 1: (Top) NLD parameters: meson-nucleon couplings g mN , ( m = σ, ω, ρ ) , σ self-interaction constants b, c ,and NLD cut-off for scalar ( Λ sN ) and vector ( Λ vN ) meson-nucleon isoscalar vertices. The isovector meson-nucleoncut-off is the same as the isoscalar-vector one. (Bottom) Bulk saturation properties of nuclear matter: saturationdensity ρ sat , binding energy per nucleon E b , compression modulus K and asymmetry parameter a sym in the NLDmodel. See Ref. [15] for more details. interested in the real part of an optical potential that can be then examined in more realisticsystems, for instance, in heavy-ion collisions or hadron-induced reactions within a relativistictransport theory. The missing imaginary part is then modelled within a collision term in termsof cross sections for elastic, quasi-elastic and inelastic channels with a proper counting of Pauli-Blocking effects.In the NLD model one cannot calculate precisely the imaginary part of U opt . However, theNLD approach contains an explicit momentum dependence of the mean-fields, and thus, of theoptical potential. This particular feature allow us to give, at least, estimations for the imaginarypart of an optical potential too. This will be discussed in the case of the anti-hyperons, and wewill mainly focus the study here on the real part of the optical potentials.The real part of the Schroedinger-equivalent optical potential for hyperons is obtained froma non-relativistic reduction of the Dirac-equation and reads U Bopt = − S B + E B m B V B + 12 m B (cid:0) S B − V B (cid:1) . (33)It describes the in-medium interaction of a baryon species B , e.g., a hyperon, with a momentum p (or single-particle energy E B = E B ( p ) , see Eq. (23)) relative to nuclear matter at rest at agiven density and isospin asymmetry. We will use Eq. (33) to compare the NLD results with themicroscopic calculations from χ -EFT and Lattice-QCD for the hyperon in-medium potentials.
3. Results and discussion
We briefly give the status of the NLD model for the in-medium nucleons, before starting thediscussion on the in-medium hyperon potentials. As in detail discussed in [15], a momentumdependent monopole form D ( p ) = Λ Λ + ~p (34)7or the regulators turned out to be very effective for a simultaneous description of the low andhigh density nuclear matter properties. An example is shown in table 1 for the extracted satura-tion properties together with the model parameters. It is seen that the NLD model leads to a verygood description of the empirical values at saturation. The NLD EoS is rather soft and similarto the density dependence of Dirac-Brueckner-Hartree-Fock microscopic calculations. At highdensities, however, the NLD EoS becomes stiff. This feature makes a prediction of the maximummass of neutron stars of M ⊙ possible even with a soft compression modulus. Note that the NLDmodel gives a correct description of the Schroedinger-equivalent optical potential for in-mediumprotons and antiprotons simultaneously by imposing G-parity only [15]. For the strangeness sector we consider again nuclear matter at rest, at a given density, isospin-asymmetry and at zero temperature, in which hyperons ( Λ , Σ , Ξ ) are situated at a given momen-tum relative to the nuclear matter at rest. The quantity of interest will be the optical potential U opt of the in-medium hyperons , see Eq. (33). Since there is no experimental information onthe momentum dependence of the in-medium hyperonic potentials, we use for our comparisonsthe recent microscopic calculations from Refs. [17] (see also Ref. [18]) and [19] as a guidance.They are based on the χ -EFT approach in Next-To-Leading (NLO) order and to Lattice-QCD.In the NLD calculations we assume for the in-medium hyperon interactions no additionalparameters except of the strangeness cut-off of the hyperons. That is, the various hyperon-nucleon couplings are fixed from the corresponding nucleon-nucleon ones by means of SU(6).The hyperon cut-offs retain their monopole form as in Eq. (34). In particular, they take the form D Y ( p ) = Λ γ Λ γ + ~p , (35)with γ = σ, ω, ρ indicating the cut-off values for the hyperon-nucleon σ, ω - and ρ -vertices,respectively, and Y = Λ , Σ , Ξ denotes the hyperon type. In principle, one could use a singlecut-off Λ γ = Λ γ = Λ γ for each meson-hyperon vertex. However, in order to describe the non-trivial momentum dependence of the microscopic calculations as precise as possible we allowfor different cut-off values for the vector-isoscalar ω - and vector-isovector ρ -hyperon vertices, asshown in Eq. (35). For the isoscalar meson-hyperon interactions a single cut-off Λ σ = Λ σ = Λ σ for each hyperon type is used. This prescription was found to be the most appropriate onewhen comparing to the microscopic calculations. In fact, the scalar-like interactions are in anycase better controlled with increasing density (respectively momentum) by m ⋆ /E ⋆ -suppressionfactors while the vector-like vertices do not include them, besides the NLD-regulators in thesource terms of the meson-field equations (31,32). Note that Π = E ⋆ for momentum-dependentregulators Π = E ⋆ and for each baryon type B. Similar studies concerning the peculiar role ofthe vector ω -meson exist in the literature. For instance, in Refs. [20, 21, 22] non-linear quadratic ω -field contributions were considered as an alternative approach for the vector-like interactionLagrangian leading to more complex density dependencies of their mean-fields. In the NLDmodel all higher-order non-linear terms are summed up into regulators. The novel feature ofNLD is that these regulators mediate a non-linear density and, at the same time, a non-linear8 momentum p [fm -1 ] -60-40-2002040 U op t [ M e V ] χ EFT (LO) χ EFT (NLO)
Juelich ’04NLD
Figure 1: Optical potential of Λ -hyperons as function of their momentum p in symmetric nuclear matter at saturationdensity. The NLD-results (thick-solid curve) are compared with χ -EFT microscopic calculations (taken from [17])at different orders LO (band with closed dashed borders) and NLO (band with closed solid borders) [17]. Furthermicroscopic calculations from the J¨ulich group (dot-dashed curve) are shown too [23]. momentum dependence of in-medium potentials not only for nucleons, but for hyperons too.This will become clear in the following discussions.At first, the cut-offs of the hyperons have to be determined. The strangeness- S = 1 cut-offsare adjusted to the corresponding hyperonic optical potentials at saturation density of symmetricand cold nuclear matter from χ -EFT calculations. This is shown in Fig. 1 for the optical potentialof Λ -hyperons. The gray bands correspond to the microscopic calculations at different orders in χ -EFT, while the solid curve represents the NLD result. At low momenta the Λ in-mediuminteraction is attractive, but it becomes repulsive at high momenta. The non-trivial momentumdependence in NLD arises from the explicitly momentum dependent regulators which show uptwice: in the scalar and vector selfenergies and in the source terms of the meson fields. As aconsequence, the cut-off regulates the Λ -potential not only at zero momentum, but particularlyover a wide momentum region. The in-medium Λ -potential does not diverge with increasing p -values (not shown here), but it saturates. Furthermore, the in-medium Λ -potential at zero kineticenergy leads to a value of U Λ opt ≃ −
28 MeV , which is consistent with the NLO-calculationsand also consistent with phenomenology. Therefore it exists an appropriate choice of cut-offregulators that do reproduce the microscopic calculations over a wide momentum range up to9 cut-off Σ cut-off Ξ cut-off Λ σ Λ ω Λ ω Λ ρ Λ ρ Λ σ Λ ω Λ ω Λ ρ Λ ρ Λ σ Λ ω Λ ω Λ ρ Λ ρ Table 2: Λ , Σ and Ξ cut-offs for σ - ( Λ σ ), ω - ( Λ ω , ) and ρ -hyperon-nucleon ( Λ ρ , ) vertices in units of GeV . In thecases for Σ and Ξ the isospin cut-offs ( Λ r , ) are relevant for the charged particles only. For the Σ -hyperon differentcut-off values Λ ρ , are used for Σ − (upper line) and for Σ + (bottom line). p ≃ very well. A similar picture occurs for the in-medium potential of Σ -hyperons, asshown in Fig. 2. The NLD cut-off for the Σ -particles can be regulated in such way to reproducea repulsive potential at vanishing momentum with a weak momentum dependence at finite Σ -momentum. Again, the NLD calculations are able to describe the microscopic χ -EFT results inNLO very well. The corresponding values for the strangeness cut-offs are tabulated in 2. Even ifthe origin of the cut-offs is different between the NLD model and the microscopic calculations,it may be interesting to note that these NLD cut-off values are close to the region between 500and 650 GeV used in the χ -EFT calculations.We emphasize again the non-trivial momentum dependence of the in-medium hyperon-potentials,as manifested in the χ -EFT calculations at different orders, see for instance Ref. [17]. Thisprescription modifies the momentum dependencies in such a complex way, which cannot be re-produced in standard RMF models by imposing SU(6) arguments. Furthermore, any standardRMF model leads to a divergent behaviour of optical potentials at high momenta. Note that aweak repulsive character of the Σ -potential, as proposed by the microscopic calculations, cannotbe achieved in conventional RMF. The momentum-dependent NLD model resolves these issueseffectively through momentum cut-offs of natural hadronic scale. Since we are dealing withhadronic matter, values of hadronic scale in the GeV -regime for the NLD regulators seem to bean adequate choice.So far we have discussed the momentum dependence of the Λ and Σ hyperons at saturationdensity (Figs. 1 and 2). These comparisons served also as a guideline for the NLD cut-offs for the Λ and Σ baryons. Now we discuss the predictive power of the NLD approach by comparing inmore detail the density and momentum dependence of the NLD formalism with the microscopic χ -EFT calculations. This is shown in Figs. 3 and 4, where the momentum dependence of the Λ (Fig. 3) and Σ (Fig. 4) particles is displayed again, but now at various densities of symmetricnuclear matter. At first, the Λ and Σ optical potentials become more repulsive with increasingnuclear matter density in NLD. However, the non-trivial momentum and density dependence,as manifested in the NLD selfenergies and the meson-field sources, weakens the in-mediumpotentials with increasing momentum. In particular, the NLD model predicts astonishingly wellthe complex microscopic behaviours in momentum and at various densities of symmetric nuclearmatter.In asymmetric matter besides the standard iso-scalar and iso-vector vertices ( σ and ω mesonfields, respectively) the iso-vector and Lorentz-vector ρ -meson must be taken into account. In10 -1 ]-30-20-10010203040 U op t [ M e V ] χ EFT (LO) χ EFT (NLO)
NLDJuelich ’04
Figure 2: Same as in Fig. 1, but for Σ -hyperon. NLD we assume again a monopole form for the ρ -meson coupling to the hyperons too by usingthe coupling constant of table 1 and the cut-off values of table 2 for the isospin sector. Relevantare the cut-off values Λ ρ , for the charged Σ ± -hyperons. They have been fixed from the corre-sponding χ -EFT calculations for Σ − and Σ + at saturation density. The NLD calculations for theneutral Λ - and Σ -hyperons are free of parameters here.The results for pure neutron matter at three different baryon densities are summarized inFig. 5. The NLD model does predict the general microscopic trends. In particular, in the caseof the neutral hyperons ( Λ and Σ ), where within the RMF approximation the ρ -meson doesnot appear at all, one would expect identical results between symmetric and pure neutron matter(at same total baryon density and momentum). This is in fact not the case. There is an inherentisospin dependence in the source terms of the meson-field equations, see Eqs. (30) even for the σ -and ω -fields. The upper limits in those integrals (31, 32) are different for protons and neutronsbetween symmetric and asymmetric nuclear matter at the same total density. This leads to adifferent cut value in the regulators D p,n and thus to a different result between symmetric andasymmetric matter. This NLD feature induces a hidden isospin dependence which is qualitativelyconsistent with the microscopic calculations at the three total densities as indicated in Fig. 5 forthe ”isospin-blind” hyperons. Concerning the charged Σ ± -hyperons, the comparison betweenNLD and χ -EFT calculations is obviously at best for densities close to saturation. In general,the NLD predictions follow satisfactorily the details of the microscopic in-medium potentials as11unction of momentum and matter density.Finally we discuss the in-medium properties of the cascade-hyperons as shown in Figs. 6and 7 for symmetric nuclear matter (SNM) and pure neutron matter (PNM). Here we apply forcomparison the latest microscopic calculations from Lattice-QCD. The same NLD scheme withappropriate monopole-type regulators leads to the results in Fig. 6 for symmetric nuclear matterat saturation density. It is seen that a simple monopole-like regulator with hadronic cut-off valuescan explain the microscopic Lattice calculations. Indeed, a soft attractive potential for in-medium Ξ -hyperons is obtained in the NLD model over a wide momentum range. The prediction of NLDis then displayed in Fig. 7 for pure neutron matter but at the same total density at saturation as inthe previous figure. The hidden isospin-dependence modifies slightly the momentum dependenceof the neutral Ξ -hyperon. In this case the Lattice calculations are reproduced only qualitativelyby the NLD model, while for the charged cascade partner ( Ξ − ) the comparison between NLD andLattice is very well for pure neutron matter at saturation and over a broad region in single-particlecascade-momentum.In the future experiments such as those at FAIR the in-medium properties of anti-hadronswill be investigated too. We thus give predictions for anti-hyperon in-medium potentials too.We recall the novel feature of the NLD formalism [15], that is, a parameter free predictions foranti-baryon optical potentials in the spirit of G-parity. In fact, once the cut-off parameters arefixed from saturation properties, the application of NLD to anti-matter gave very successful re-sults by imposing G-Parity only. Note that in conventional RMF models one has to introduce byhand additional scaling factors in order to reproduce the weak attractiveness of the anti-protonoptical potential at vanishing momenta [24]. We therefore use the same NLD formalism forthe description of anti-hyperons too and performed additional calculations for the Λ , Σ and Ξ optical potentials as function of momentum and density. These results are shown in Fig. 8 foranti- Λ (left), anti- Σ (middle) and anti- Ξ optical potentials versus their momentum at three den-sities of symmetric nuclear matter. Due to the negative sign in the Lorentz-vector component ofthe hyperon self-energy these potentials are in general attractive over a wide momentum range.Compared to the anti-proton potential at saturation these potentials are less attractive with a sim-ilar dependence on single-particle momentum.Since for anti-hyperons we make predictions and for anti-particles in general one may ex-pect significant contributions to the imaginary part of U opt too, we briefly discuss the imaginarypart of the anti-hyperon optical potentials too. An exact treatment of the imaginary part of theoptical potential is not possible within an RMF model. However, within the NLD approach onecan estimate the strength of Im U opt from dispersion relations [15]. This prescription was suc-cessfully applied to the antiproton case in a previous work (see Ref. [15]), thus we apply it herefor the anti-hyperons too. The results for
Im U opt are shown in the same figure 8 by the thincurves. One generally observes a strong contribution to the in-medium anti-hyperon interactionsfrom the imaginary parts of the optical potentials too. These contributions are quite similar tothe imaginary potential of antiprotons with a value around -150
MeV at very low kinetic ener-gies (see for instance in [15] the second citation of 2015). However, in the antiproton-case theimaginary potential is rather strong relative to its real part, while for anti-hyperons both parts ofthe potential are sizeable. Even if the NLD results for the
Im U opt are only estimations, we cangive a physical interpretation. In antinucleon-nucleon scattering annihilation can occur through12 -1 ]-30-20-10010203040 U op t [ M e V ] -1 -1 -1 Figure 3: Optical potential of Λ -hyperons versus their momentum at various densities of symmetric nuclear matter,as indicated by the Fermi-momenta in units of 1/fm − . The NLD calculations at these three Fermi-momenta (thick-solid, thick-dashed and thick-dot-dashed curves) are compared to the χ -EFT calculations at NLO [17]. the production of light pions. On the other hand, the interaction of anti-hyperons with nucleonscan happen via the production of the heavier kaons due to strangeness conservation, which mayinfluence the imaginary potential at low energies. This might be one reason why the imaginarypart of the anti-hyperon optical potential is comparable with its corresponding real part particu-larly at very low energies. These calculations can be applied to anti-hadron induced reactions inthe spirit of relativistic transport theory and can be tested in the future experiments at FAIR.
4. Summary
We have investigated the properties of strangeness particles inside nuclear matter in theframework of the NLD approach. The NLD model is based on the simplicity of the relativisticmean-field theory, but it includes the missing momentum dependence in a manifestly covariantfashion. This is realized by the introduction of non-linear derivative series in the interaction La-grangian. In momentum space this prescription leads to momentum dependent regulators, whichare determined by a cut-off. The NLD approach does not only resolve the optical potential issuesof protons and antiprotons at high momenta, but it affects the density dependence. That is, thecut-off regulators make the EoS softer at densities close to saturation and stiffer at very highdensities relevant for neutron stars.Because of the successful application of the NLD model to infinite nuclear matter (and tofinite nuclei [25]), it is a natural desire to extend this approach to hadronic matter by takingstrangeness degrees of freedom into account. This is realized in the spirit of SU(6) symmetry.We applied the NLD model to the description of in-medium hyperon interactions for ordinary13 -1 ]-10010203040506070 U op t [ M e V ] -1 -1 -1 Figure 4: Same as in Fig. 3, but for the Σ -hyperons. nuclear matter. It was found that the strangeness cut-off regulates the momentum dependence ofthe optical potentials of hyperons in multiple ways. At first, the optical potentials do not divergewith increasing hyperon momentum. Furthermore, the NLD model predicts an attractive Λ -optical potential at low momenta, which becomes repulsive at high energies and finally saturates.In particular, it is possible to predict a weak and repulsive in-medium interaction for Σ -hyperonsinside nuclear matter at saturation density. These results are in consistent agreement with cal-culations based on the chiral effective field theory. Regarding Ξ -hyperons, the NLD predictionsturned out to be in agreement with recent Lattice-QCD calculations. In symmetric nuclear matterthe cascade optical potential is attractive and it follows the Lattice-QCD results. In pure neutronmatter the isospin-separation as predicted by the NLD model agrees with the Lattice-QCD be-haviours qualitatively. While the potential of the neutral cascade particle remains attractive, the Ξ − -hyperon shows a weak repulsion in neutron matter. The weak repulsion of those hyperonsmay likely effect to a stiffer EoS for neutron star matter.We briefly discussed the imaginary part of U opt of anti-hyperons too. These estimationsindicate a significant contribution of the imaginary part to the anti-hyperon dynamics that couldbe explored in anti-hadron induced reactions. For instance, the present calculations can be testedin anti-proton induced reactions and in reactions with secondary Ξ -beams, as they are planned atFAIR in the future PANDA experiment.Obviously this study is relevant not only for hadron physics, but also for nuclear astrophysics.The application of the NLD approach to β -equilibrated compressed matter is under progress, inorder to investigate the hyperon-puzzle in neutron stars. Another interesting application concernsthe dynamics of neutron star binaries. To do so, an extension to hot and compressed hadronicmatter is necessary and under progress too. Note that the NLD formalism is fully thermodynam-14 U op t [ M e V ] -1 ]0 1 2 3 0 1 2 3 4 Λ Σ + Σ Σ - Figure 5: Optical potentials for hyperons (as indicated) versus their momentum for pure neutron matter. Solidcurves with symbols indicate the NLD calculations while pure curves without symbols are the microscopic χ -EFTresults at NLO from Ref. [17]. Green pairs (circles-solid for NLD and solid for χ -EFT) refer to low density of p F = 1 fm − , red pairs (diamonds-dashed for NLD and dashed for χ -EFT) refer to saturation density of p F = 1 . fm − and blue pairs (triangles-dot-dashed for NLD and dot-dashed for χ -EFT) refer to a density of p F = 1 . fm − . ically consistent, which is an important requirement before applying it to hot and dense systems.In summary, we conclude the relevance of our studies for future experiments at FAIR and fornuclear astrophysics. Acknowledgments
This work is partially supported by COST (THOR, CA 15213) and by the European Union’sHorizon 2020 research and innovation programme under grant agreement No. 824093. We alsoacknowledge H. Lenske and J. Haidenbauer for fruitful discussions and for providing us the χ -EFT calculations. ReferencesReferences [1] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels,
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