Chiral Symmetry Restoration and Deconfinement to Quark Matter in

Neutron Stars

V.A. Dexheimera∗, S. Schrammb†

aPhysics Department, Gettysburg College , Gettysburg, United States

bCSC, FIAS, ITP, Johann Wolfgang Goethe University, Frankfurt am Main, Germany

We describe an extension of the hadronic SU(3) non-linear sigma model to include quarks. As a result, weobtain an effective model which interpolates between hadronic and quark degrees of freedom. The new parametersand the potential for the Polyakov loop (used as the order parameter for deconfinement) are calibrated in orderto fit lattice QCD data and reproduce the QCD phase diagram. Finally, the equation of state provided by themodel, combined with gravity through the inclusion of general relativity, is used to make predictions for neutronstars.

1. Introduction

Dense matter hadronic models, such as the onesused to describe neutron stars, work in a deter-mined range of energies [1–3]. They can, in ad-dition, fulfill some symmetries of the underlyingtheory (QCD) such as chiral symmetry [4–6] andscale invariance [7,8] but they do not include de-confinement to quark matter. On the other hand,the broadly used bag-model [9] includes quark de-grees of freedom but does not include chiral sym-metry. Models such as the quark-NJL and quarksigma-models [10–12] include that symmetry, butdo not directly incorporate hadronic degrees offreedom or quark confinement.

Our goal is to construct an effective model thatcontains hadronic and quark degrees of freedompresent at different densities/temperatures butthat can also coexist in a mixed phase. This al-lows the deconfinement phase transition to be asteep first order as well as a smooth crossover andcases in between. The last two possibilities, de-spite being predicted by lattice QCD, cannot beachieved by the usual procedure that connects, byhand, two different models with separate equa-tions of state for hadronic and quark matter atthe chemical potential at which the pressure of

∗vantoche@gettysburg.edu†schramm@th.physik.uni-frankfurt.de

the quark EOS exceeds the hadronic one [13].In order to achieve this goal, we employ a single

model for the hadronic and quark phases. Ourextension of the hadronic SU(3) non-linear sigmamodel also includes quark degrees of freedom in aspirit similar to the PNJL model [14], in the sensethat it is a non-linear sigma model that uses thePolyakov loop Φ as the order parameter for thedeconfinement. This is a quite natural idea, sincethe Polyakov loop is related to the Z(3) symmetry,that is spontaneously broken by the presence ofquarks. In QCD the Polyakov loop Φ is definedvia Φ = 1

3Tr[exp (i∫

dτA4)], where A4 = iA0 isthe temporal component of the SU(3) gauge field.

2. The Model

The Lagrangian density of our non-linear sigmamodel becomes:

L = LKin + LInt + LSelf + LSB − U, (1)

where besides the kinetic energy term forhadrons, quarks, and leptons (included to insurecharge neutrality) the terms:

LInt = −∑

i ψ̄i[γ0(giωω + giφφ + giρτ3ρ)

+m∗i ]ψi, (2)

Nuclear Physics B (Proc. Suppl.) 199 (2010) 319–324

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doi:10.1016/j.nuclphysbps.2010.02.051

LSelf = − 12 (m2

ωω2 + m2ρρ

2 + m2φφ2)

−g4

(ω4 + φ4

4 + 3ω2φ2 + 4ω3φ√2

+ 2ωφ3

√2

)

+ 12k0(σ

2 + ζ2 + δ2) − k1(σ2 + ζ2 + δ2)2

−k2

(σ4

2 + δ4

2 + 3σ2δ2 + ζ4)

−k3(σ2 − δ2)ζ − k4 ln (σ2−δ2)ζ

σ2

0ζ0

, (3)

LSB = m2πfπσ +

(√2m2

kfk − 1√2m2

πfπ

)ζ, (4)

represent (in mean field approximation) the in-teractions between baryons (and quarks) and vec-tor and scalar mesons, the self-interactions of thescalar and vector mesons and an explicit chiralsymmetry breaking term, responsible for produc-ing the masses of the pseudo-scalar mesons. ThePolyakov-loop potential U will be discussed indetail below. The underlying flavor symmetryof the model is SU(3) and the index i denotesthe baryon octet and the three light quarks. Themesons included are the vector-isoscalars ω andφ, the vector-isovector ρ, the scalar-isoscalars σand ζ (strange quark-antiquark state) and thescalar-isovector δ. The isovector mesons affectisospin-asymmetric matter and are consequentlyimportant for neutron star physics. The mesonsare treated as classical fields within the mean-fieldapproximation [15]. A detailed discussion of thepurely hadronic part of the Lagrangian, that de-scribes very well nuclear saturation properties aswell as nuclei properties can be found in [16,4,17].

Finite-temperature calculations include a heatbath of hadronic and quark quasiparticles withinthe grand canonical potential of the system. Fi-nite temperature calculations also include a gas offree pions and kaons. As they have very low mass,they dominate the low density/ high temperatureregime. All calculations were performed consid-ering zero net strangeness except the zero tem-perature star matter case. The reason for suchexception is that the time scale in neutron stars islarge enough for strangeness not to be conserved.

The effective masses of the baryons and quarksare generated by the scalar mesons except for asmall explicit mass term M0 (equal to 150 MeV

for nucleons, 354 MeV for hyperons, 5 MeV forup and down quarks and 150 MeV for strangequarks) and the term containing the Polyakovfield Φ:

M∗B = gBσσ + gBδτ3δ + gBζζ

+M0B+ gBΦΦ2, (5)

M∗q = gqσσ + gqδτ3δ + gqζζ

+M0q+ gqΦ(1 − Φ). (6)

With the increase of temperature/density, the σfield (non-strange chiral condensate) decreases itsvalue, causing the effective masses of the hadronsto decrease towards chiral symmetry restoration.The Polyakov loop assumes non-zero values withthe increase of temperature/density and, due toits presence in the baryons effective mass (Eq. 5),suppresses their propagation. On the other hand,the presence of the Polyakov field in the effectivemass of the quarks, included with a negative sign(Eq. 6), insures that they will not be present atlow temperatures/densities.

The behavior of the order parameters of themodel is shown in Fig. 1 for neutron star matterat zero temperature. The calculations for neutronstars assume charge neutrality (which is essentialfor their stability) and beta equilibrium. In thiscase, the chiral symmetry restoration, which is acrossover for purely hadronic matter, turns into afirst order phase transition by the influence of thestrong first-order transition to deconfined matter.The fact that the value of the chiral condensateincreases at a certain chemical potential is due tothe fact that it is connected to the baryon den-sity value that decreases at the phase transition(the baryonic number of quarks is one third ofthe hadronic one). The model is consistent in thesense that both order parameters are related.

The coupling constants related to the Polyakovloop in the effective mass formulas (Eq. 5 andEq. 6) are high but still finite. This causes the ef-fective masses of the degrees of freedom not effec-tively present in each phases to be high but alsofinite (Fig. 2). The effective normalized massesof the particles are directly related to the onsetthese particles appearance in the system.

V.A. Dexheimer, S. Schramm / Nuclear Physics B (Proc. Suppl.) 199 (2010) 319–324320

1000 1200 1400 1600μ

B (MeV)

0

0.2

0.4

0.6

0.8

1 σ/σ0Φ

Figure 1. Order parameters for chiral restorationand deconfinement to quark matter for star mat-ter at zero temperature.

The potential U for the Polyakov loop reads:

U = (a0T4 + a1μ

4 + a2T2μ2)Φ2

+a3T40 ln (1 − 6Φ2 + 8Φ3 − 3Φ4). (7)

It is based on [18,19] and adapted to also includeterms that depend on the chemical potential. Thetwo extra terms (that depend on the chemical po-tential) are not unique, but the most simple nat-ural choice to reproduce the main features of thephase diagram at finite densities. The couplingconstants for the baryons (already shown in [17])are chosen to reproduce the vacuum masses of thebaryons and mesons, nuclear saturation proper-ties, and asymmetry energy as well as the hyperonpotentials. The vacuum expectation values of thescalar mesons are constrained by reproducing thepion and kaon decay constants. The coupling con-stants for the quarks (gqω = 0, gqφ = 0, gqρ = 0,gqσ = −3.00, gqδ = 0, gqζ = −3.00, T0 = 270/200MeV, a0 = −1.85, a1 = −1.44x10−3, a2 = −0.08,a3 = −0.40, gBΦ = 1500 MeV, gqΦ = 500 MeV)are chosen to reproduce lattice data as well asknown information about the phase diagram.

More specifically, the parameters a0 and a3 ofthe Polyakov potential were fit to reproduce thesame pressure functional as the one in the modelpresented in Refs. [18,19] for the quenched case

1000 1200 1400 1600μ

B (MeV)

0

500

1000

1500

2000

2500

3000

M*/

QB

i (M

eV)

p, nΛu, ds

Figure 2. Effective normalized mass of baryonsand quarks for star matter at zero temperature.

for the range of temperature of interest (Fig. 3).In this way we ensure the agreement with latticeat the zero chemical potential limit. The param-eter T0 was chosen to be 270 MeV for quenchedcalculations reproducing a very strong first orderphase order transition at 270 MeV temperature(as in Ref. [18]). The parameter T0 was re-scaledfor the calculations including quarks and hadronsalso as in Ref. [18]. In our case it changed fromT0 = 270 to 200 MeV reproducing, at zero chem-ical potential, a peak on the susceptibility for thePolyakov loop at T = 190 MeV (Fig. 4).

The functional form of the potential in ourmodel is very similar to the one in Refs. [18,19]with a leading term proportional to temperatureto the fourth power. At the moment, the otherpowers of temperature (third and second) werenot considered to avoid a large number of pa-rameters. In our case, the potential also con-tains a term proportional to chemical potentialto the fourth power, with parameter a2 (chosento reproduce a phase transition from hadronic toquark matter at a value of four times saturationdensity for star matter at zero temperature), anda mixed term between temperature and chemi-cal potential, with parameter a1 (chosen to re-produce a critical endpoint at chemical potentialμc = 354 MeV and temperature Tc = 167 MeV

V.A. Dexheimer, S. Schramm / Nuclear Physics B (Proc. Suppl.) 199 (2010) 319–324 321

150 200 250 300 350 400T (MeV)

1000

2000

3000

4000

P (M

eV/f

m3 )

Refs. [18, 19] potentialour potential

Figure 3. Pressure for the quenched case.

for symmetric matter and zero net strangeness inaccordance with lattice data from Fodor and KatzRef. [20]). Together, these parameters lead themodel to reproduce the QCD phase diagram atlarge densities. More complicated structures forthe Polyakov potential will be analyzed in futurework.

As can be seen in Fig. 5 the transition fromhadronic to quark matter obtained is a crossoverfor small and vanishing chemical potentials. Be-yond the critical end point, a first order transitionline begins. The critical temperatures for chiralsymmetry restoration coincide with the ones fordeconfinement. Since the model is able to repro-duce nuclear matter saturation at realistic valuesfor the saturation density, nuclear binding energy,as well as the compressibility and asymmetry en-ergy, we also reproduce results at low densities forthe nuclear matter liquid-gas phase transition.

3. Application to Neutron Stars

In the same way we used lattice QCD data tocalibrate and test the model for high tempera-tures and low chemical potentials, we can cal-culate neutron star properties to test the modelin the high-density/low-temperature regime. Itis important to notice that up to this point,the charge neutrality was considered to be local,

0 100 200 300T (MeV)

0

0.2

0.4

0.6

0.8

1 σ/σ0Φ

Figure 4. Order parameters for chiral symmetryrestoration and deconfinement to quark matter atzero chemical potential.

meaning that each phase was separately chargeneutral. At finite temperature the two phasescontain mixtures of hadrons and quarks, whichare dominated by hadrons or quarks, dependingon the respective phase. At vanishing tempera-ture there is no mixture, i.e. the system exhibitsa purely hadronic or purely quark phase.

It is important to notice that the quarks aretotally suppressed in the hadronic phase but thehadrons are suppressed in the quark phase untila certain chemical potential (above 1700 MeV forT = 0). This threshold, which is higher than thedensity in the center of neutron stars, establishesa limit for the applicability of the model.

For a more realistic approach we allow thetwo phases to be charge neutral when combined(global charge neutrality) following [21]. In thiscase, the particle densities change in the coexis-tence region causing quarks and baryons to ap-pear and vanish in a smoother way (Fig. 6).

The density of electrons and muons is signifi-cant in the hadronic phase but not in the quarkphase. The reason for this behavior is that be-cause the down and strange quarks are also neg-atively charged, electrons are not necessary forcharge neutrality, and only a small amount ofleptons remains to ensure beta equilibrium. The

V.A. Dexheimer, S. Schramm / Nuclear Physics B (Proc. Suppl.) 199 (2010) 319–324322

0 500 1000 1500 2000μ

B (MeV)

0

50

100

150

200

T (

MeV

)

symmetric matterneutron star matter

Figure 5. Phase diagram. The lines represent firstorder transitions and the circles mark the criticalend-points.

hyperons, in spite of being included in the calcu-lation, are suppressed by the appearance of thequark phase. Only a very small amount of Λ’sappear right before the phase transition. Thestrange quarks appear after the other quarks andalso do not make substantial changes in the sys-tem.

The possible neutron star masses and radii arecalculated by solving the Tolmann-Oppenheimer-Volkof equations [22,23]. The solutions forhadronic (same model but without quarks) andhybrid stars are shown in Fig. 7, where besidesour equation of state for the core, a separate equa-tion of state was used for the crust [24]. Themaximum mass supported against gravity in ourmodel is 2.1M� in the first case and approxi-mately 2.0M� in the second. The predicted radiiare in the observed range, being practically thesame for hadronic or hybrid stars.

Although the stars that contain a phase of purequark matter in our model are not stable, starsthat contain a core of mixed phase (surroundedby a hadronic shell) are. In this case, the mixedphase could extend up to approximately 2 km ofradius but with only a small quark fraction.

1000 1200 1400 1600μ

B (MeV)

0.001

0.01

0.1

1

QB

iρ i (fm

-3)

pnΛeμuds

Figure 6. Population (baryonic densities) for starmatter at zero temperature using global chargeneutrality.

4. Conclusions

In spite of the fact that our model is relativelysimple, it is the only one able to take into accountdifferent degrees of freedom and consequently al-low steep as well as smooth transitions betweendifferent phases. The model is in accordance withlattice QCD data and the phase diagram it re-produces is able to describe a broad variety ofregimes, from compact stars to heavy ion colli-sions. Calculations along this line are in progress[25].

We conclude that our model is suitable forthe description of neutron stars, since it predictsmasses and radii in agreement with observations.Although it does not predict stable stars contain-ing pure quark matter, the model allows starsthat contain a core of mixed phase that can ex-tend up to approximately 2 km of radius. Even inthis case, the reduced maximum star mass is stillhigher than the most massive pulsars observed.

A major advantage of our work compared toother studies of hybrid stars is that we can studyin detail the way in which chiral symmetry isrestored and the way deconfinement occurs athigh temperature/density. Since the propertiesof the physical system, e.g. the density of parti-

V.A. Dexheimer, S. Schramm / Nuclear Physics B (Proc. Suppl.) 199 (2010) 319–324 323

8 10 12 14 16 18R (Km)

0

0.5

1

1.5

2

2.5

M/M

o

hadronic starhybrid star

12.8 13 13.2 13.4

1.8

1.9

2

Figure 7. Mass-radius diagram

cles in each phase, are directly connected to thePolyakov loop it is not surprising that we obtaindifferent results in a combined description of thedegrees of freedom compared to a simple match-ing of two separate equations of state.

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