A new consistent Neutron Star Equation of State from a Generalized Skyrme model

We propose a new equation of state for nuclear matter based on a generalized Skyrme model which is consistent with all current constraints on the observed properties of neutron stars. This generalized model depends only on two free parameters related to the ranges of pressure values at which different submodels are dominant, and which can be adjusted so that mass-radius and deformability constraints from astrophysical and gravitational wave measurements can be met. Our results support the Skyrme model and its generalizations as good candidates for a low energy effective field-theoretic description of nuclear matter even at extreme conditions such as those inside neutron stars.

Introduction.-The modern understanding of strong interactions in the Standard Model of particle physics is based on the theory of Quantum Chromodynamics (QCD), a non-abelian gauge theory where the fundamental degrees of freedom are carried by the quark and gluon fields. Despite its great success at very high energies, we are unable to achieve the same precision in the low-energy regime using full QCD, since the theory becomes nonperturbative. In particular, theoretical computations of the properties of baryons and nuclei from QCD are extremely difficult even for the smallest nuclei, and phenomenological models are usually employed, instead. The Skyrme model [1] offers an alternative approach to this problem, by considering baryons (and nuclei) as topological solitons of a nonlinear field theory of mesons, which corresponds to an effective field theory for lowenergy QCD in the large N c expansion. This field of research has experienced significant progress in recent years, as different generalizations of this model, like the addition of higher derivative terms [2] or additional degrees of freedom (DoF)-e.g., vector mesons [3][4][5][6]-, or more general potential terms [7], have been proposed to better reproduce the observed nuclear properties [8][9][10][11][12].
On the other hand, the first observations of gravitational waves by LIGO opened a new window for the exploration of matter at ultra high densities, like at the cores of Neutron Stars (NS), which are thought to be the most dense objects allowed by General Relativity (GR) before collapsing to a black hole. Indeed, recent [13] (and prospect) observations of mergers of NS binaries will allow us to constrain the equation of state (EoS) of nuclear matter at such densities. In particular, since the Skyrme model (and its generalizations) allow to find star-like solutions when coupled to GR, these observations may serve us to determine whether the (generalized) Skyrme model is a consistent way to describe the properties of nuclei and nuclear matter at a large range of scales in a unified manner. Different models for NS as Skyrme solitons have been previously proposed, for example, in [14,15]. These models are interesting from a theoretical point of view, because they allow to obtain the EoS of NS cores from a relatively simple field theoretic description. However, none of the Skyrmion star models present in the literature have achieved a good agreement with current observational data of NS [16]. In this paper, we present an EoS for NS based on a generalized Skyrme model which satisfies all recent observational constraints of NS, such as the maximum mass limit or the deformability as measured in coalescent binary systems.
In this article we will use units in which c = 1. Skyrme crystals.-The Skyrme model is an effective field theory of strong interactions at low energies which emerges in the large N c limit of QCD. It is defined via the Lagrangian (1) with f π the pion decay constant and e the Skyrme coupling constant. Also, L µ = U † ∂ µ U is the left invariant Maurer-Cartan form associated to the SU(2)-valued Skyrme field U (x), and U = U(U ) is a potential. For the pion mass potential U π = (1/2) tr (1 − U ), the parameter µ is related to the pion mass m π via µ = (1/2)f π m π .
In order to obtain finite energy configurations, one imposes constant boundary values of U at |x| → ∞, so that the physically relevant Skyrme field configurations define maps U : S 3 → SU (2) S 3 , and thus the Skyrme model presents topological solitons (Skyrmions), whose topological charge equals the topological degree of these maps, the baryon density current. The Skyrme model (1) describes an interacting theory for the Goldstone bosons arXiv:2006.07983v2 [hep-th] 18 Jun 2020 associated to the (broken) chiral symmetry, but baryons emerge as topological solitons, whose topological charge corresponds to the baryon number [17]. Furthermore, the Skyrme model has been applied to the study of matter at extremely high densities, required to describe the EoS of NS. To do so, one needs to find the lowest energy solutions of the Skyrme model for the very large baryon number of NS, typically N ∼ N ∼ 10 57 .
It is well known [16] that the lowest energy solutions of the standard Skyrme model (described by the Lagrangian density (1)) for very large baryon number consist of crystalline cubic lattices of B = 4 Skyrmions-which can be thought of as α particles. The energy per baryon of such solutions as a function of the lattice parameter of the unit cell, l, has been numerically shown to be [18]: We fit the values of energy (per baryon) and lattice length corresponding to the minimum energy configuration, E 0 = 923.32 MeV and l −3 0 = n 0 = 0.16 fm −3 , to reproduce the properties of infinite nuclear matter [19]. Note that our values slightly differ from those originally proposed in [18] due to the different fit [20] [21]. From this expression one may obtain the energy per baryon as a function of the pressure [14], i.e., the EoS of the Skyrme crystal (at zero temperature).
The BPS model.-Since it is an effective theory, the Skyrme model can be extended by adding higher order terms to the Lagrangian. The only possible Lorentzinvariant extra term with at most second order time derivatives of the Skyrme field is [8] with λ a coupling parameter. Thus, the generalized Skyrme model Lagrangian reads L gen SK = L SK + L 6 . Unfortunately, neither large B solutions for the generalized model L gen SK nor the corresponding EoS have been found, to our knowledge. However, at sufficiently high densities -for instance, those which occur at the core of a neutron star, which can reach several times the nuclear saturation density n 0 -, the sextic term (4) provides the most important contribution to the EoS, related to the omega meson repulsion of nuclear matter [22]. The sextic term alone defines a barotropic perfect fluid with energy density ρ 6 = λ 2 π 4 n 2 = p (see below), where p is the pressure and n the baryon number density. The EoS ρ 6 = p is maximally stiff with a speed of sound equal to 1, which explains its dominance at high density.
L gen SK has another interesting submodel which will be relevant for us, the so-called BPS Skyrme model L BP S = L 6 − µ 2 U(U ). This model supports topological soliton configurations saturating a BPS energy bound [8], hence the name of the model. Minimally coupling this submodel to gravity, we obtain its stressenergy tensor which still is of the perfect fluid form, T µν BP S = (p + ρ)u µ u ν − pg µν , with the following definitions (here g := |det{g ρσ }|), (5) and ρ = p + 2µ 2 U. Further, the proper baryon number Note that this perfect fluid is, in general, non-barotropic, since the potential term U introduces a dependence on the Skyrme field in p and ρ, such that no simple algebraic relation can be found between them. Nevertheless, one may still perform a mean-field approximation and obtain an effective, barotropic EoS for the BPS Skyrme fluid, which offers the interesting possibility to compare the results obtained within the exact and the mean-field approaches [23]. In the case of interest here, however, we will introduce a constant effective potential µ 2 U = ρ 0 = const., which implies the barotropic EoS ρ = ρ 6 + ρ 0 = λ 2 π 4 n 2 + ρ 0 = p + 2 ρ 0 already at the full field-theory level.
The generalized model.-Both the standard Skyrme model and the BPS submodel have been previously used to describe nuclear matter inside NS [16]. However, it is clear from these attempts that the true equation of state for Skyrme matter should take into account both models in a unified fashion, because the results from approximating the full model with either of the two submodels deviate from the most recent observational data of NS, and do so in opposite directions. For example, the maximum mass of NS are either too small (for pure skyrmion crystals) or too large (for BPS Skyrmion stars) as compared with the current constraints [16]. As explained, the generalized Skyrme model has not been solved yet for large baryon number. Nevertheless, we may still obtain some information of these high baryon number solutions by scaling arguments of the energy terms for the different submodels of the complete Lagrangian. The equivalence between pressure and scaling allows us to write the energy per baryon of the Skyrmion crystal at any pressure (i.e. σ = 1) as a simple function of σ = l/l 0 , Obviously, the contributions from the term proportional to σ becomes negligible for large pressure, whereas the term proportional to σ −1 dominates in this regime (σ 1). Next, consider the sextic term contribution to the energy (and energy per baryon) of a fluid element Ω, , which transforms as E 6 → σ −3 E 6 under a scaling of spacetime coordinates. This implies that the sextic contribution will dominate the energy per baryon at sufficiently high pressure. Therefore, we may assume that a solution of the complete model will tend to a solution for the BPS submodel at high pressure, with an asymptotic energy per baryon of E 6 /B = ρ 6 /n = λπ 2 √ p. This is, therefore, the asymptotic behavior of the energy per baryon at high pressure also for the full model. On the other hand, as the pressure decreases to a certain value (which depends on λ), E 6 /B becomes of the order of the energy per baryon of the Skyrme crystal, and the BPS approximation to the complete solution will start to fail. For even lower p, the contribution of E 6 /B will be subleading in comparison to the Skyrme crystal.
This supports the idea that a phase transition of some kind must take place within this generalized model, between the crystalline phase of the standard Skyrme model and the perfect fluid phase of the BPS model. A quantitative prediction of the pressure value p P T where this phase transition occurs would require the knowledge of the full solution or, at least, the value of the parameter λ, because the contribution to the energy per baryon of the sextic term strongly depends on λ.
In [15], the BPS submodel was used to model the full neutron star core and, therefore, the model parameters λ and µ were fitted to match with the infinite nuclear matter approximation at zero pressure. In the present case, however, the Skyrme crystal describes the low-pressure region and, therefore, should be fitted to nuclear matter. The value of λ will be determined, instead, by the condition of a continuous transition between the crystal and the fluid phases, see below.
From the previous considerations, we can construct a generalized EoS which takes into account both the standard Skyrme and BPS submodels at different regimes, based on simple assumptions on the behavior of the full solutions in the low and high pressure regimes, without knowing these solutions explicitly. Indeed, we will assume that the low pressure solutions of the complete model are still Skyrme crystals whose energy is approximately described by (3). Therefore, for sufficiently low values of p, we may approximate the energy density of the solutions for the complete model by where ρ SK (p) = E(l(p)) · l(p) −3 , being E(l) the energy per baryon of the Skyrme crystal (eq. (3)) and l(p) the lattice length of the crystal. We approximate ρ 6 (p) p also for small p although, strictly speaking, we know this to be the correct expression only for large p. The reasons are that i) this term is small for small p and vanishes for p → 0, so that we recover the Skyrme crystal in this limit; ii) using ρ 6 (p) p is certainly a better approximation for the generalized model than not including a ρ 6 term at all; and iii) ρ 6 (p) p is the correct behavior for the fluid at high pressure, such that (7) guarantees a continuous transition from the crystal to the fluid phase.
In this fluid high-pressure phase, the sextic term will provide the most important contribution, and the complete solutions can be well described by a BPS Skyrme model. Thus, we can model this behavior by introducing a certain value of the pressure, p P T , above which the solutions are described by a BPS fluid, and define the following energy density for the generalized model, Hence, the energy density contribution from the crystal freezes at its value at p P T and becomes constant, playing the role of an effective potential for the BPS Skyrme model. The p dependence for p > p P T is taken into account by ρ 6 , which is known to provide the leading contribution for large p. In the following section, we will see that the value of p P T determines the maximum mass of a NS, so we may adjust the value of p P T to agree with the current maximum mass limit for NS. To obtain the baryon density n in the generalized model, we use the well-known Euler relation ρ = −p + ∂ρ ∂n n, which yields a differential equation for n, that we integrate using n(p = 0) ≡ n 0 = 0.16 fm −3 as initial condition to obtain the curve n(p). The result, and the corresponding EoS ρ(p), are shown in Fig. 1, where other EoS have been included for comparison.
The generalized Skyrme EoS (8), by construction, only describes nuclear matter above nuclear saturation [27]. Below saturation density, nuclear matter in a NS is known to be in a rather inhomogeneous state, resulting from a competition between nuclear and electromagnetic forces (e.g., "nuclear pasta" phases, [28]). In principle, the (generalized) Skyrme model can be coupled to the electromagnetic interaction, so these low-density phases are fundamentally within its scope. Full fieldtheoretical calculations for this coupled system and for large B are, however, not feasible, and a macroscopic (hydrodynamical) treatment is currently unknown. On the other hand, the standard methods of nuclear physics, such as many-body techniques, can be used to describe these low-density NS crust regions and are completely reliable there. This motivates us to consider a hybrid version of (8) in which, at a sufficiently low density n * (or, equivalently, p * ), a neutron star crust EoS ρ BCPM (p) is glued, Concretely, we choose the BCPM EoS of [19], based on the Brueckner-Hartree-Fock (BHF) approach (plus the BCPM density functional for the crust). For the crust and the outer core n n 0 , nuclear matter is well understood, and standard nuclear physics EoS like [19] should provide a precise description of NS matter (the BCPS EoS turns out to be numerically very similar to SLy4). Observational constraints.-In the hybrid EoS proposed above there are only two free parameters, namely the values of p * and p P T corresponding to the low and high density parts of the hybrid EoS. Here we show that recent astrophysical and gravitational wave observations actually tightly constrain the value ranges for both parameters. For example, from the mass-radius curves for different values of these parameters, we find that only the value of p P T affects the maximum NS mass in the model. Thus, the maximum mass limit for nonrotating NS of M/M = 2.16 +0. 17 −0.15 proposed in [29] allows us to constrain the value of p P T such that the maximum NS mass in the hybrid model is consistent with this limit.
In Fig. 2 we show different mass-radius curves of the hybrid model corresponding to different values of p P T , together with the maximum mass limit of [29]. We can see a good agreement, for any pair (p * , p PT ) within the ranges p * ∈ [2, 6] MeV/fm −3 and p P T ∈ [50, 80] MeV/fm −3 , with the most likely mass-radius relation for the NS corresponding to the GW170817 event [13]. The observed gravitational waveform can also be used to place direct constraints on the tidal deformability of NS. Indeed, the waveform produced by the coalescence of two NS at the early phase of the inspiral depends on the underlying EoS mostly through the tidal Love number [33]. However, the individual Love numbers for the two stars cannot be disentangled in the observed gravitational waveform. Instead, what is measured is the so-called effective tidal deformabilityΛ, a mass weighted average of the deformabilities of the individual stars in the merger [34]. Similarly, the two component masses are not measured directly, but the chirp mass, M c = m 1 q 3/5 /(1 + q) 1/5 where q = m 1 /m 2 is the mass ratio, can actually be tightly constrained. In the case of the GW170817 event, 10 11 12 13 14 R (km)  1). We represent the most probable M-R region from combined observations of GW and Heavy pulsars (NICER) [30], the GW170817 event [13] and the maximum mass constraint of [29] obtained from its EM counterpart GRB170817A. Also, other constraints from NICER, chiral EFT and multimessenger observations are represented, adapted from [31] and [32].
the chirp mass was constrained to 1.188 +0.004 −0.002 at the 90% confidence level, and the mass ratio was constrained to be in the range 0.7 − 1 within the same confidence level, whereas the effective tidal deformability was inferred to be smaller than 800 [35]. Such measurements allow to reduce the set of Skyrme models able to reproduce the NS properties. Following [36], we have solved the Einstein equations for slowly rotating Skyrmion stars in the hybrid model using the Hartle-Thorne formalism [37,38] and obtained the dimensionless tidal deformability of stars described by this model as a function of their TOV mass. On the other hand, since the chirp mass of the binary progenitor of GW170817 is well measured, for any given EoS the effective deformability reduces to a simple EoS-dependent function of the mass ratio. These curves, together with the constraints commented above, are represented in Fig.  3, from where it follows that our new EoS is compatible with the data from [35] for the ranges of p * and p PT considered. Future measurements of the tidal deformability of NS will allow us to further constrain these ranges, since we find that the curvesΛ(q) depend on the particular values of both parameters.
Conclusions.-In this letter, we propose a completion of standard nuclear physics EoS at low densitiesknown to be reliable there-by an EoS based on the generalized Skyrme model in the uncharted territory above nuclear saturation density n 0 . In the simplest version of Skyrme models, where electromagnetic effects, quantum corrections or the proton-neutron mass difference are not taken into account, they can describe nuclear matter only for n ≥ n 0 , by construction. The use of the generalized Skyrme model at high densities is based on the assumptions that i) strong-interaction effects (nuclear repulsion) are more important than degeneracy pressures in that region, ii) the extended character of nucleons-which is automatic in the Skyrme model-is relevant at high pressure and iii) nucleons are the only relevant DoF inside NS cores (no exotic contributions). This last assumption is shared by many NS models.
We find that the resulting EoS provides an excellent description of NS properties, compatible with all constraints, among them the latest ones from LIGO. Our EoS contains two parameters which have a clear physical interpretation as transitions between standard nuclear matter and the Skyrme crystal (p * ) and between this crystal and a Skyrme fluid (p P T ). In particular, we predict a transition between a crystalline and a fluid regime for 50 ≤ p P T · fm 3 /MeV ≤ 80, whose precise position and nature (phase transition or crossover) can be determined by more precise NS binary observations.