High-density skyrmion matter and neutron stars Prashanth Jaikumar1, Manjari Bagchi2, and Rachid Ouyed3 ABSTRACT 8 0 We examine neutron star properties based on a model of dense matter com- 0 2 posed of B=1 skyrmions immersed in a mesonic mean field background. The n model realizes spontaneous chiral symmetry breaking non-linearly and incor- a J porates scale-breaking of QCD through a dilaton VEV that also affects the 5 mean fields. Quartic self-interactions among the vector mesons are introduced 2 on grounds of naturalness in the corresponding effective field theory. Within a ] plausible range of the quartic couplings, the model generates neutron star masses h t and radii that are consistent with a preponderance of observational constraints, - l c including recent ones that point to the existence of relatively massive neutron u stars M ∼ 1.7M and radii R ∼12-14 km. If the existence of neutron stars n ⊙ [ with such dimensions is confirmed, matter at supra-nuclear density is stiffer than 2 extrapolations of most microscopic models suggest. v 6 3 Subject headings: neutron stars, equation of state, skyrme model 4 3 . 8 0 7 1. Introduction 0 : v Neutron star astronomy, initiated by the serendipitous discovery of the first radio i X pulsar (Hewish et al. 1968), has since found over 1700 similar “rotation-powered” neutron r a stars (ATNF Pulsar Catalogue). Pulsar timing measurements inradio binaries yield a simple (unweighted) mean neutron star mass hMi ∼1.4M . In contrast, measurement of general- ⊙ relativistic parameters in neutron star-white dwarf binaries suggest hMi ∼1.6M , owing to ⊙ a few exceptionally large inferred neutron star masses in the latter case. Examples include the binary component PSR J0621+1002 (Nice et al. 2007) with 1.7+0.10M at the 1σ level −0.16 ⊙ 1Institute of Mathematical Sciences, C.I.T Campus, Taramani, Chennai 600113India; E-mail: [email protected] 2Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005 India 3Department of Physics and Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4 Canada – 2 – and a more stringent value of M ≥1.68M at the 2σ level set by a neutron star binary in ⊙ the Terzan 5 cluster (Ransom et al. 2005). Constraining neutron star radii is harder, due to uncertainties in atmospheric modelling and distance estimates, but bounds from thermally emitting neutron stars eg. RXJ 1856.5-3754(Walter & Lattimer 2002; Ho et al. 2007) imply that the canonical range of 10-12km is exceeded. These observations have oriented attention towards ’atypical’ neutron stars with large mass and possibly large radius. An independent determination of mass (2.10±0.28M ) and radius (13.8±1.8km) of the ⊙ bursting neutron starinthelow-mass X-raybinary(LMXB) EXO 0748-676hasbeen claimed (O¨zel 2006), based on an accurate determination of gravitationally red-shifted Fe and O ab- sorption lines (Cottam et al. 2002). However, the evolving nature of the source has compli- cated further observational tests of the same. Furthermore, Doppler tomography of emission lines in the mass-transfer stream between the neutron star and its less massive companion suggests a more canonical value of 1.35M for the former (Pearson et al. 2006). NASA’s ⊙ upcoming Constellation-X mission will improve on such spectral measurements, shrinking systematic errors on mass and radius even further. With due caution on the observational front, the confirmation of a mass ∼2.0M for a neutron star strongly constrains the equa- ⊙ tion of state of dense matter, ruling out the possibility of extreme softening at high den- sities. It also implies an upper bound on the energy density of observable cold and dense matter (Lattimer & Prakash 2005). A large radius R ∼13-14 km for a 1.4M neutron star ⊙ implies a stiff symmetry energy at densities of [1-2]n , where n is the saturation density of 0 0 nuclear matter. These connections between the nuclear physics of dense matter and neutron star observations have been the focus of recent reviews (Steiner et al. 2005; Sedrakian 2006; Page & Reddy 2006; Lattimer & Prakash 2007). Uncertainties in theoretical aspects of many-body interactions at n&[1-2]n lead to pre- 0 dictions for the mass versus radius curve that vary widely depending on the equation of state (EoS), with the maximum mass M ranging from 1.4-2.7M and radius at maximum mass max ⊙ R from 9-14km (Lattimer & Prakash 2001). Constraints from astrophysical observations max and terrestrial laboratory data have whittled this range down (Li & Steiner 2006) so that most microscopic models of longstanding for nuclei and nuclear matter would struggle to explain the existence of relatively heavy neutron stars (2M ) which also have a large ra- ⊙ dius (R∼13km). In fact, only few stiff equations of state eg., MPA1 (Mu¨ther et al. 1987), MS0 (Mu¨ller & Serot 1996) and PAL1 (Prakash et al. 1988a) are consistent with the EXO 0748-676constraint at the1σ level, andeven they failif thematter accreted onto theneutron star is helium-rich 1. However, while satisfying astrophysical constraints, it is also important 1The largest source of systematic error in extracting the mass and radius of EXO 0748-676 comes from the accreted mass fraction of Hydrogen 0.3<X <0.7 (O¨zel 2006). – 3 – to keep in mind constraints from laboratory data on strongly interacting matter around nu- clear saturation density. This point was nicely brought out in recent papers by Li & Steiner (2006), and Kl¨ahn et al. (2006). In this context, our goal in this paper is to explore further a recently proposed model of a skyrmion fluid (Ouyed & Butler 1999; Jaikumar & Ouyed 2006), henceforth referred to as OBJ,thatwasshowntoleadtoaverystiffequationofstateandconsequentlygeneratealarge maximum mass as well as radius for a neutron star. However, it was pointed out (Lattimer, private communication) that the rapid rise of the compressibility and symmetry energy just above saturation density in this model puts it at odds with experimental constraints from collective flow data (Danielewicz et al. 2002) and isospin diffusion studies in medium-energy heavy-ion collisions (Tsang et al. 2004). In this work, we determine the extent to which we can satisfy these constraints by extending the skyrmion fluid model to include higher- order interactions among the vector mesons that are theoretically motivated by arguments of naturalness in the corresponding effective field theory. The mass and radius predictions of the extended model, henceforth referred to as JOM, are also confronted with constraints set by the observation of X-ray burst oscillations, kiloHertz quasi-periodic oscillations in LMXBs and thermal emission from neutron stars. We include observational uncertainties wherever they may impact our conclusions. Our phenomenological model is able to satisfy a preponderance ofthese constraints. Weemphasize at theoutset thatourmodel, initscurrent form, has not been investigated for its applicability to nuclei or more complex phases of matter at sub-saturation densities. Our EoS presently applies only to infinite nuclear matter and neutron-rich matter in the range [1-5]n . 0 This paper is presented as follows. In § 2, we discuss some conventional EsoS for neutron stars; in § 3 we revisit the Ouyed-Butler-Jaikumar (OBJ) model for skyrmion stars and motivate higher-order interactions that serve to tune the stiffness of the equation of state such that laboratory constraints are met. The main features of the mass versus radius curves are explained in § 4. We compare the results obtained in the skyrmion star model to predictions of other neutron star models in light of observational bounds in § 5. Our conclusions are in § 6. 2. Equations of State for Neutron Stars An equation of state for dense matter is a relation between pressure and energy (or baryon) density, usually derived from an underlying microscopic model or effective the- ory for strong interactions. To apply to neutron stars, it should be able to generate at least a 1.4M static neutron star with a radius in the 10-14km range. The connection ⊙ – 4 – to the underlying microscopic theory can be formulated in several ways: examples include relativistic mean field theory, non-relativistic potential models, relativistic Dirac-Brueckner- Hartree-Fock theory etc (Arnett & Bowers 1977; Lattimer & Prakash 2001). As we are con- cerned with recent findings of relatively heavy neutron stars, we consider three stiff eos: MS0 (Mu¨ller & Serot 1996), APR (Akmal et al 1998) and UU (Wiringa et al. 1988). (i) Mu¨ller & Serot (1996) used a relativistic theory of point-like nucleons interacting via mesonic degrees of freedom. These are the neutral scalar (σ) and vector (ω) fields, plus the isovector ρ meson. In this model, like or unlike meson-meson interactions are encoded by terms that are polynomials in the fields. By demanding a match to the properties of nuclear matter at saturation, they obtained a sequence of EsoS that depend on the coupling constants of the polynomial interactions. The stiffest eos (MS0) corresponds to vanishing couplings and yields a maximum mass of 2.7M . This model is consistent with a large ⊙ neutron star mass and radius ∼14km for static configurations. Presently, the 1σ limits on theradiusoftheburstingneutronstarsourceEXO0748-676are13.8±1.8km. However, there is no fundamental symmetry principle that requires the higher-order couplings to vanish. Introducing natural values for these couplings drastically reduces the maximum mass to ≤ 2.0M . ⊙ (ii)Akmaletal.(1998)obtainedtheAPREoSbasedontheArgonneυ nucleon-nucleon 18 interaction (Wiringa et al. 1995),UrbanaIXthree-nucleoninteraction(Pudliner et al. 1995) and a relativistic boost term (Forest et al. 1995) as microscopic input. This EoS gives a maximum neutron star mass of 2.2M and does not lie within the 1σ limits on the mass- ⊙ radius estimate of the neutron star in EXO 0748-676, even assuming the accreted matter is mostly hydrogen. (iii) EoS UU is obtained via similar variational methods applied to an older two-nucleon and three-nucleon interaction (Urbana υ +UVII). This model yields a maximum mass of 14 2.2 M at radius 10 km (Wiringa et al. 1988). However, a radius larger than about 11km ⊙ is not supported by this equation of state for a static neutron star with mass greater than 1.4M . ⊙ Our purpose in selecting and highlighting these EsoS is two-fold. Firstly, these models are representative of complementary philosophies behind constructing an equation of state for dense matter: as in (i) forgo the connection to laboratory data on vacuum two-nucleon interactions and focus instead on the empirical properties of large nuclei and infinite nuclear matter within a relativistic theory; or as in (ii) and (iii) insist on a satisfactory description of available data on the structure and interaction of few nucleon systems (free or bound) in a non-covariant approach. Secondly, these are among the stiffest equations of state that arise from hadronic degrees of freedom alone. It is possible that hybrid equations of state – 5 – that allow for quark matter at high density can be almost just as stiff (Alford et al. 2007), but we will not consider quark matter EsoS in this work. These threeEsoSareusedinourneutronstarmass-radius plots(Figs.4,5)andcompared with the EsoS for dense skyrmion matter which we now describe. 3. The Skyrmion fluid 3.1. Nuclear Matter Phenomenology Before the advent of Quantum Chromodynamics (QCD), T. H. R. Skyrme proposed a description of baryons as topological solitons in a mesonic field theory that realizes spon- taneous chiral symmetry breaking in non-linear fashion (Skyrme 1961). This model is now qualitatively supported by studies of large N QCD which suggest that mesonic degrees of c freedoms are fundamental and baryons arise as solitons. When augmented by the inclusion of low-lying vector mesons (m ≤1GeV) and flavor symmetry breaking effects, the Skyrme V model can provide a reasonable description of static baryon properties such as mass split- tings, charge radii and magnetic moments (Schecter & Weigel 2000). In the 2-nucleon sector, the problem of finding a sufficiently attractive isoscalar central and spin-orbit force within the Skyrme model at distances 1fm≤ r ≤2fm has held up progress. One solution is to in- clude a dilaton field that mocks up scale-breaking in QCD and provides attraction in these channels. To make progress towards an equation of state for a skyrmion fluid, K¨albermann (1997) introduced a self-consistent model that incorporates medium effects through the response of the dilaton2 σ and the isoscalar ω-field to a smooth density distribution obtained by integratingoverthecollectiveco-ordinatesoftheSkyrmion. Thisisequivalenttoanensemble of non-interacting B=1 Skyrmions, a valid picture upto a separation of 0.8fm (n .5n ). A 0 subsequent work (Jaikumar & Ouyed 2006) extended this model to asymmetric matter by incorporating the ρ meson in the spirit of standard mean field approaches. The Lagrangian for the Skyrme model, augmented by the σ,ω fields, and including isospin-breaking effects from the ρ meson as well as explicit scale-breaking effects from the dilaton and quark masses is given in Jaikumar & Ouyed (2006). A fit to nuclear matter phenomenology is achieved throughadditionalparametersinthedilatonpotential, which is givenby (K¨albermann 1997) 2Since chiral symmetry is broken non-linearly,the usual σ field as the chiral partner of the pion does not appear in the theory. – 6 – V(σ) = B[1+e4σ(4σ −1)+a (e−σ −1) (1) 1 +a (eσ −1)+a (e2σ −1)+a (e3σ −1)], 2 3 4 where B ≈ (240 MeV)4 is related to the Bag constant (the non-perturbative glue that breaks scale-invariance in QCD). The six unknown parameters of the model are a -a , g the ω- 1 4 w N coupling and g , the ρ-N coupling. To determine the a ’s, the following constraints are ρ i imposed: the scale anomaly condition dVσ| = 0 which implies that a = a +2a +3a , dσ σ=0 1 2 3 4 the stationarity w.r.t σ , viz., ∂E/∂σ = 0, a binding energy/nucleon of -16 MeV for infinite 0 0 nuclear matter at saturation density (n = 0.16fm−3), and a choice of the compressibility 0 K. E is the energy density of the fluid and σ is the non-vanishing mean field value of 0 the time component of the σ field. At saturation, σ is determined by the choice of the 0 effective mass M, which then also fixes g . The choice of symmetry energy and effective w mass at saturation fixes g . Once the a ’s are determined, σ is generally obtained from ρ i 0 its equation of motion for an arbitrary density. The a ’s show very weak dependence on i the choice of K in the range 200-300 MeV, while displaying more sensitivity to the choice of effective mass. Without further modifications, the model displays a sharp rise in the compressibility just above saturation, rising from K=240 MeV (our choice) to K ∼ 2000 MeVfora10%increaseinbaryondensity. Similarly, thesymmetry energyrisestoosteeply in thisrangetobeconsistent withexperimental constraints(see§3.4). These inconsistencies are a consequence of the specific form of the dilaton potential, which is essential to preserve the trace anomaly relation (scale-breaking). Therefore, a modification of V(σ) is not desirable. It is the exponential sensitivity of the curvature of the potential V(σ) to the dilaton VEV that drives the compressibility to large values. One way to address this issue is to view the Skyrme model in the mean field approximation as an effective field theory of hadrons, so that the Lagrangian can be extended to include higher-order terms (meson self-interactions) that parameterize unknown physics at a more microscopic level. This rationale is also employed in Mu¨ller & Serot (1996) although their model has an explicit σ meson, point-like nucleons and no scalar-vector mixing, while our model has a dilaton as the only scalar, and includes scalar-vector mixing. If the meson fields are viewed as relativistic functionals, the higher- order interactions can be thought of as parts of an effective potential that determines their mean field values at a particular density through the stationarity of the effective action associated to the Skyrme lagrangian (Furnstahl et al. 1996). They therefore modify the density dependence of the meson fields as well as the properties of the background skyrmion fluid that couples to these fields. We restrict ourselves to quartic self-interactions in the ρ and ω fields with coupling constants whose value can be surmised by naturalness. Then, higher-order terms such as six or eight-meson self-interactions do not substantially change the results obtained in the quartic case. In the next section, we implement this procedure – 7 – to obtain an acceptable behaviour of the compressibility and symmetry energy in dense Skyrmion matter. 3.2. Mean-field equations The additional quartic interactions take the form ξ L = L0 +Lint; Lint = + g4ω4, (2) ω ω ω ω 4! ω 1 1 L0 = − Fω Fω,µν + e2σm2ω2−g nω (3) ω 4 µν 2 ω ω for the ω field and χ L = L0 +Lint; Lint = + g4ρ4, (4) ρ ρ ρ ρ 4! ρ 1 1 g L0 = − Fρ Fρ,µν + e2σm2ρ2 − ρn ρ (5) ρ 4 µν 2 ρ 2 I for the ρ field. Here, the F’s are the standard field-strength tensors for the abelian (ω) and non-abelian cases (ρ). Working at T = 0, we have k3 +k3 k3 −k3 Fp Fn Fp Fn n = ,n = , (6) (cid:16) 3π2 (cid:17) I (cid:16) 3π2 (cid:17) where k is the Fermi momentum of species i = neutron,proton. Then, the Skyrme La- Fi grangian is compactly expressed as L = L +L +L +L −V(σ), (7) 2 4 ω ρ whereL ,L involve gradientsoftheSkyrmionprofile. Forthedensities ofinterest (n ≤ n ≤ 2 4 0 5n )wheretheapproximationofnon-overlapping skyrmions isvalid, thisprofiledropsofffast 0 enough that the mean-field averaging simply counts the number of individual Skyrmions in a given volume, equivalent to a non-interacting Fermi gas model. At n ≥ 5n , we expect cor- 0 rections to our mean field model from Skyrmion overlap. We also do not expect such a mean field treatment to apply at densities much lower than saturation density, since Skyrmions do not form a uniform fluid there. Thus, our model is restricted to (n ≤ n ≤ 5n ). For the ω 0 0 and ρ fields, the equations of motion read as follows: a ω3 +b ω +c = 0; (8) ω 0 ω 0 ω – 8 – ξg4 a = ω,b = e2σ0m2, c = −g n. (9) ω 6 ω ω ω ω a ρ3 +b ρ +c = 0; (10) ρ 0 ρ 0 ρ χg4 g nδ a = ρ, b = e2σ0m2, c = ρ , (11) ρ 6 ρ ρ ρ 2 where ω ,ρ denote mean field values and δ=(1-2x), with x=(n -n)/2n being the proton 0 0 I fraction of neutron-rich matter (x=1/2 for symmetric matter). The magnitudes 3 of the quartic couplings ξ,χ are estimated from the naturalness argument for an effective field the- ory, viz., that the co-efficients of the various terms in the Lagrangian, through a given order of truncation, should be of the same size when expressed in an appropriate dimensionless form. Thus, we find eσ0m 2 eσ0m 2 ξg2 ∼ 12 ω ; χg2 ∼ 192 ρ . (12) ω M ρ M (cid:18) 0 (cid:19) (cid:18) 0 (cid:19) Wechoose the effective mass, given by M = eσ0M , tobe 600MeVat saturationdensity, 0 so thateσ0=2/3forabarenucleon mass M =900MeV (neglecting the∼40 MeVcontribution 0 from explicit symmetry breaking). Since the model is fit to saturation properties, g itself ω depends on ξ, whose value must be chosen so as to satisfy the naturalness condition above. Furthermore, real solutions to the equation of motion for the dilaton Eqn.(16) cease to exist beyond a small range of couplings. This restricts us to 0.1 ≤ ξ ≤ 0.3 and 1.0 ≤ χ ≤ 2.0. These values differ from those in Mu¨ller & Serot (1996) due to additional factors of e2σ0 from the dilaton (metric) that appear in the fitting expressions for g and g and the qualitatively ω ρ different form of the dilaton potential (it contains all powers in σ). The energy densities corresponding to the vector mean fields are a ω4 b ω2 a ρ4 b ρ2 E = − ω 0 − ω 0 −c ω , E = − ρ 0 − ρ 0 −c ρ , (13) ω ω 0 ρ ρ 0 4 2 4 2 so the total energy density is then E = E +E +E +V(σ), (14) kin ω ρ k E (E2 +k2) M4 k +E E = F F F F − ln F F , kin 8π2 8π2 M n,p (cid:20) (cid:18) (cid:19)(cid:21) X where the kinetic energy E (k ) comes from a Lorentz boost of the static Skyrmion to kin F momentum k (K¨albermann 1997). In Eqn.(14), E = k2 +M2 since the Dirac effective F Fi Fi q 3We choose the sign of the couplings to be positive since this guarantees zero mean fields at vanishing source density (baryon/isospin). – 9 – mass is the same for both neutrons and protons. The binding energy is E/n−M and the 0 pressure is given by g ρ δ ρ 0 P = n 0.5 E +g ω − −E. (15) F ω 0 2 " ! # n,p X The effective mass M is determined at any density from the equation of motion for σ 0 k3E dV F F +4E + −b ω2 −b ρ2 = 0. (16) 4π2 kin dσ ω 0 ρ 0 " # 0 n,p X 3.3. Compressibility For symmetric matter, δ = 0 and ρ vanishes. The compressibility is defined as 0 dP ∂2(E/n) ∂(E/n) K = 9 = 9 n2 +2n . (17) dn ∂n2 ∂n (cid:20) (cid:21) In the present case, this is equivalent to ∂2E ∂2E dσ K = 9n − ; (18) ∂n2 ∂n∂σ dn (cid:20) (cid:18) (cid:19) (cid:21) dσ ∂2E ∂2E = / , dn ∂n∂σ ∂σ2 (cid:18) (cid:19) (cid:18) (cid:19) ∂2E k2 g2 = F + ω , (19) ∂n2 3nE b +3a ω2 F ω ω 0 ∂2E M2 2g b ω ω ω 0 = − , (20) ∂n∂σ E b +3a ω2 F ω ω 0 ∂2E 3nM2 dV d2V (b +a ω2) = −4 + +6b ω2 ω ω 0 . ∂σ2 E dσ dσ2 ω 0(b +3a ω2) (cid:20) F 0 0 ω ω 0 (cid:21) For the choice K=240 MeV, binding energy=-16 MeV and M=600 MeV at saturation density, the best fit a values for various ξ are listed in Table 1, correcting an unfortunate i error in their values quoted for ξ=0 in Jaikumar & Ouyed (2006). – 10 – For ξ=0, at saturation, freedom in choosing the fit parameters a allows for a deli- i cate cancellation between various contributions to the compressibility such that K is small (∼200-300 MeV). This is achieved largely as a result of fine-tuning d2V/dσ2 away from its natural scale ∼ B. As we move to slightly higher density, this fine tuning cannot be re- covered since V(σ ) is exponentially sensitive to changes in the σ VEV. Consequently, the 0 contribution from the ω-meson appearing in the ∂2E/∂n2 term dominates, pushing the com- pressibility to unnaturally large values, as in the OBJ model. For ξ 6= 0, a 6= 0 and ω ω 0 being large, it is clear from Eqn.(19) that ∂2E/∂n2 and hence K is substantially reduced, as in the JOM model. This effect is reflected in the EoS of symmetric matter, shown in the upper panel of Fig.1. By definition, the slope of the pressure-density curve is pro- portional to the compressibility. The hatched region represents constraints on the EoS of symmetric matter from the analysis of collective flow data in nucleus-nucleus collisions at (1-2)GeV/nucleon (Danielewicz et al. 2002). While the ξ = 0 curve (OBJ) is clearly too stiff, choosing natural values of the coupling (0.1 ≤ ξ ≤ 0.3) provides a considerable im- provement, and the JOM model is able to satisfy the flow constraint for ξ ∼ 0.25. The APR EoS, which has a phase transition at n ∼2n , is also shown for comparison. 0