arX

iv:1

607.

0400

7v2

[nuc

l-th]

18

Oct

201

6

∆ (1232) effects in density-dependent relativistic Hartree-Fock theory and neutron stars

Zhen-Yu Zhu1, Ang Li1∗, Jin-Niu Hu2†, Hiroyuki Sagawa3,41Department of Astronomy, Xiamen University, Xiamen 361005, China

2School of Physics, Nankai University, Tianjin 300071, China3RIKEN Nishina Center, RIKEN, Wako 351-0198, Japan

4Center for Mathematics and Physics, University of Aizu,Aizu-Wakamatsu, Fukushima 965-8560, Japan

(Dated: October 10, 2018)

The density-dependent relativistic Hartree-Fock (DDRHF)theory is extended to include∆-isobars for thestudy of dense nuclear matter and neutron stars. To this end,we solve the Rarita-Schwinger equation for spin-3/2 particle. Both the direct and exchange terms of the∆-isobars’ self-energies are evaluated in details. Incomparison with the relativistic mean field theory (Hartreeapproximation), a weaker parameter dependence isfound for DDRHF. An early appearance of∆-isobars is recognized atρB ∼ 0.28fm−3, comparable with thatof hyperons. Also, we find that the∆-isobars’ softening of the equation of state is mainly due tothe reducedFock contributions from the coupling of the isoscalar mesons, while the pion contributions are negligibly small.We finally conclude that with typical parameter sets, neutron stars with∆-isobars in their interiors could be asheavy as the two massive pulsars whose masses are precisely measured, with slightly smaller radii than normalneutron stars.

PACS numbers: 21.60.Jz, 21.30.Fe, 26.60.-c, 26.60.Kp

I. INTRODUCTION

Recently the early appearance of∆-isobars in dense nu-clear matter has intrigued many illuminating studies for theinterest of both neutron stars (NSs) [1–6] and heavy ioncollision [7]. In particular, very compact stellar configura-tions [1, 2] are reached due to the introduction of∆-isobars asrequired by some current radius measurements [8–10]. How-ever, there is the∆ puzzle [1] similar with the hyperon puz-zle [11–13] that the corresponding maximum NSs’ massescould not fulfill the constraint from the recent two precisely-measured 2-solar-mass pulsars PSR J1614-2230 [14, 15] andPSR J0348+0432 [16].

Previous studies have suggested the two-family scenario ofcompact stars to resolve this problem [3], we here go beyondthe previously-employed relativistic mean field (RMF) theoryto include the contributions due to the exchange (Fock) termsand the pseudo-vectorπ-meson couplings which are effec-tive only through exchange terms. Therefore, we employ thedensity-dependent relativistic Hartree-Fock (DDRHF) the-ory [17] to study this problem.

As one of the most advanced nuclear many-body modelbased on the covariant density functional theory, DDRHFpresents a quantitative description of nuclear phenomena [18–24] with a similar accuracy as RMF. In DDRHF, the Lorentzcovariant structure is kept in full rigor, which guaranteesall well-conserved relativistic symmetries. Previously,it hasbeen demonstrated that the isoscalar Fock terms are essentialfor the prediction of NS properties [18, 19]. Also, we expectthat the density-dependence [20, 25] introduced in the presentstudy for all meson couplings would make difference on the

∗liang@xmu.edu.cn†hujinniu@nankai.edu.cn

∆-matter study.

Besides the∆-puzzle problem in NSs, we are also inter-ested in how the DDRHF results depend on the uncertain∆-meson coupling constants, since there are hardly previousstudies except with RMF. Presently, there is no constraint on∆-ρ coupling. Several constraints exist forσ ,ω couplingsfrom the quark model [26–28], finite-density QCD sum-rulemethods [29], quantum hadrodynamics [30, 31], and labora-tory experiments [32–34]. Based on the quark counting argu-ment [26, 27], there are the universal couplings between nu-cleons,∆-isobars with mesons, namely,xλ = gλ ∆/gλ N = 1 (λlabels meson). A theoretical analysis [28] of M1 giant reso-nance and Gamow-Teller transitions in nuclei found 25−40%reductions of the transition strength due to the couplings to∆ isobars, while recent experiments [32–34] bring down thequenching value to be at most 10−15% due to the couplingto ∆-isobars. This means that the∆ couplings of isoscalarmesons (σ ,ω) should be weaker than those of quark modelprediction, namelyxσ ,xω ≤ 1 are suggested by these experi-ments. Also, at the saturation densityρ0, a possible smaller∆-isobars’ vector self-energy was indicated [29] than the cor-responding value for the nucleon [i.e.,xω(ρ0) <∼ 1], whilethe∆-isobars’ scalar self-energy was difficult to predict. In theanalysis of Ref. [31] in Hartree approximation, the differencebetweenxσ andxω was found to bexω = xσ −0.2. A recentstudy, however, concluded that the cross sections involvedarenot sensitive toxω andxσ either [30]. In the present study,for the purpose of comparing with previous RMF studies [4],we scale the density-dependence of∆-meson coupling in areasonable range according to previous constraints forσ ,ω ,ρmesons. Theπ meson coupling is fixed to bexπ = 1.0 here-after.

We provide the DDRHF formula extended to include∆-isobars in Sect. II. Our results and discussions will be givenin Sect. III, before drawing conclusions in Sect. IV.

2

II. DDRHF MODEL WITH ∆-ISOBARS

Here we extend the DDRHF calculation to∆-matter.The model system consists of nucleonsψN , ∆-isobars(∆−,∆0,∆+,∆++) which are treated as a Rarita-Schwingerparticleψα

∆ , and the effective fields of two isoscalar mesons(σ andω) as well as two isovector mesons (π andρ). Theeffective Lagrangian density can be written as three parts:

L = L0+LIN +LI∆, (1)

where theL0 denotes the free Lagrangian density of baryonsand mesons,LIN denotes the interaction Lagrangian densitybetween mesons and nucleons, andLI∆ denotes the interac-tion Lagrangian between mesons and∆-isobars. More physicsdetails can be found in Ref. [17], and we write explicitly hereonly theLI∆ term which is newly introduced in the presentwork:

LI∆ = −gσ∆ψ∆α σψα∆

−gω∆ψ∆αγµ ω µψα∆ +

fω∆

2M∆ψ∆α σµν∂ ν ω µψα

∆

−gρ∆ψ∆α γµρµ ·τ∆ψα

∆ +fρ∆

2M∆ψ∆ασµν ∂ ν

ρµ ·τ∆ψα

∆

− fπ∆

mπψ∆αγ5γµ∂ µ

π ·τ∆ψα∆ , (2)

where the∆ isospin operatorτ∆ has the form of a 4×4 ma-trix. M∆ denotes the bare∆-isobar mass. We follow Ref. [17]and neglect the small tensor coupling ofω meson for bothnucleons and∆-isobars in our calculations.

From the Lagrangian in Eq. (1), following the standard pro-cedure in Ref. [17], one can derive the Hamiltonian as:

H =

∫

t=0ψN(−iγ ·∇+MN)ψNd3x

+

∫

t=0ψ∆α(−iγ ·∇+M∆)ψα

∆ d3x

+12 ∑

λ

∫

t=0d3xd4x′ ψa(x)ψb (x′)Γab

λ Dλ (x− x′)ψb (x′)ψa(x),

(3)

where the indexλ represents each meson (λ = σ ,ω ,π ,ρ),anda,b represent nucleon or∆-isobar.Γab

λ are the vertices be-tween mesons and baryons as listed in details in Eqs. (43-48)of Appendix A.Dλ (x− x′) are the propagator of each meson,respectively.

In the mean field approximation, the field operator of nu-cleon should satisfy the Dirac equation:

(−iγµ∂µ +MN +ΣN)ψ(x) = 0, (4)

while the field operator of∆-isobars must be described by theRarita-Schwinger equations as spin 3/2 particle,

(−iγµ∂µ +M∆ +Σ∆)ψ(x)α = 0, γα ψα(x) = 0, (5)

with the corresponding self-energiesΣN ,Σ∆ to be determinedself-consistently.MN is the bare nucleon mass.

The field operators of nucleons and∆-isobars could be ex-panded in the second quantization forms:

ψ(x, t) = ∑κ

[

fκ (x)e−iEκ tbN

κ + gκ(x)eiE ′

κ tdN†κ

]

, (6)

ψ†(x, t) = ∑κ

[

f †κ (x)e

iEκ tbN†κ + g†

κ(x)e−iE ′

κ tdNκ

]

, (7)

ψα(x, t) = ∑κ

[

f ακ (x)e−iEκ tb∆

κ + gακ (x)e

iE ′κ td∆†

κ

]

, (8)

ψ†α(x, t) = ∑

κ

[

f †κα (x)e

iEα tb∆†κ + g†

κα(x)e−iE ′

κ td∆κ

]

. (9)

Hereb(N/∆)κ andb(N/∆)†

κ are the annihilation and creation oper-

ators for nucleons/∆-isobars, whiled(N/∆)κ andd(N/∆)†

κ are theannihilation and creation operators for the correspondinganti-particles, respectively. The no-sea approximation is adoptedin this work. Then the contributions of antiparticles are ne-glected, and only the particle terms of the baryon field opera-tors are reserved.

For nuclear matter, the Dirac spinorfκ(x) or Rarita-Schwinger spinorf α

κ (x) could be written as:

fκ(x) = u(p,κ)eip·x, (10)

f ακ (x) = uα(p,κ)eip·x, (11)

where theu(p,κ) anduα(p,κ) are the solutions of the follow-ing equations, respectively:

(γ ·p+MN +ΣN)u(p,κ) = γ0Eu(p,κ), (12)

(γ ·p+M∆+Σ∆)uα(p,κ) = γ0Euα(p,κ). (13)

HereE is the single particle energy in nuclear matter.κ con-tains both spins and isospinτ components. From now on, weomit for simplicity the indexτ.

In consistent with the rotational invariance of the in-finite nuclear matter, the baryon self-energyΣi (i =n, p, ∆−, ∆0, ∆+, ∆++), produced by the meson exchanges,can be written quite generally as

Σi(p) = ΣiS(p)+ γ0Σi

0(p)+γ · pΣiV (p) (14)

whereΣS, Σ0, ΣV are respectively the scalar, timelike andspacelike-vector components of the self-energy.

Then Eq. (12) is simplified as

(γ ·p∗+M∗)u(p,s) = γ0E∗u(p,s), (15)

where the starred quantities are defined by

p∗(p) = p+ pΣV (p), (16)

M∗(p) = M+ΣS(p), (17)

E∗(p) = E(p)−Σ0(p). (18)

M∗ is the usual scalar effective mass of the baryon, andE∗2 =p∗2+M∗2.Eq. (15) has the similar form with the Dirac equation of a

free particle. The positive energy solution is

u(p,s) =

(

E∗+M∗

2E∗

)1/2

1σ ·p∗

E∗+M∗

χs, (19)

3

whereχs is the wave function of spin. The normalization con-dition of Dirac spinor is chosen as

u†(p,s)u(p,s) = 1. (20)

While for the∆-isobars,uα(p,s∆) can be obtained from theangular momentum coupling of spin 1 and spin 1/2 as fol-lows:

uα(p,s∆) = ∑s1,s2

(1s112

s2|32

s∆)εα(p,s1)u(p,s2), (21)

where (1s112s2|3

2s∆) is the Clebsch-Gordan coefficient andεα(p,s1) is the unit vector of spin 1 particle. Introducingthe detailed values of Clebsch-Gordan coefficients in Eq. (21),one can write explicitly the wave function of∆-isobars for allspin components:

uα(p,±32) = εα (p,±1)u(p,±1

2), (22)

uα(p,±12) =

√

23

εα(p,0)u(p,±12)

+

√

13

εα (p,±1)u(p,∓12).

(23)

The unit vectorεα(p,s1) are determined by

γα uα(p,s∆) = 0. (24)

The normalization condition of the Rarita-Schwinger spinorsatisfies

u†α(p,s∆)u

α(p,s∆) =−1. (25)

Next we define some useful quantities as

P ≡ |p∗|E∗ , M ≡ M∗

E∗ . (26)

They can also be expressed byu(p,s) anduα(p,s∆):

MN(p) =12 ∑

su(p,s)u(p,s), (27)

PN(p) =12 ∑

su(p,s)γ · pu(p,s); (28)

M∆(p) = −14 ∑

s∆

uα(p,s∆)uα(p,s∆), (29)

P∆(p) = −14 ∑

s∆

uα(p,s∆)γ · puα(p,s∆), (30)

where the coefficient 1/4 in Eqs. (29-30) is generated by spin3/2.

The energy density of nuclear matter will be obtainedthrough calculating the Hamiltonian density with the Hartree-Fock wave function of baryons,

|HF〉= ∏b†(p,κ)|0〉, (31)

0.5 1.0 1.5 2.0 2.5 3.01.0

1.5

2.0

2.5

3.0

crit /

0

x

x =0.8 x =1.0 x =1.0 x =1.0 x =1.0 x =1.2 x =1.0 x =1.0(RMF)

FIG. 1: (Color online)∆ critical densityρcrit∆ as a function ofxρ ,

for (xσ ,xω ) = (0.8,1.0),(1.0,1.0),(1.0,1.2). Previous RMF calcu-lations [4] with(xσ ,xω ) = (1.0,1.0) are also shown for comparison.

where|0〉 is the physical vacuum. With this wave function,the energy density of nuclear matter could be written as:

ε =1Ω〈HF |H|HF〉= 〈T 〉+ 〈V〉, (32)

whereΩ is the volume of the system.〈T 〉 and〈V 〉 are the ki-netic part and potential part of the energy density, respectively.〈V 〉 can be further decomposed into direct (Hartree) term〈VD〉and exchange (Fock) term〈VE〉 (See details in Appendix B forthe calculations of the energy density).

The self-energies for nucleons/∆-isobars are obtained bydifferentiating〈V 〉 with respect tou(p,s)/uα(p,s∆). The re-sults for all self-energy components are presented in Ap-pendix D.

For neutron star matter, there are not only baryons but alsoleptons (electrons and muons). They are in equilibrium underweak processes, which leads to the following relations amongthe involved chemical potentials:

µn = µp + µe, µµ = µe, (33)

µn = µ∆− − µe, µn = µ∆0, (34)

µn = µ∆+ + µe, µn = µ∆++ +2µe. (35)

The chemical potentials are determined by the relativisticenergy-momentum relation at Fermi momentum (p = kF ),

µi = Σi0(kF,i)+E∗

i (kF,i), (36)

µl =√

k2F,l +m2

l , (37)

wherei = n, p, ∆−, ∆0, ∆+, ∆++ andl = e,µ . Furthermore,the baryon number conservation and charge neutrality are im-posed in neutron star matter as,

ρB = ρBn +ρBp +ρB∆− +ρB∆0 +ρB∆+ +ρB∆++, (38)

ρe +ρµ = 2ρB∆++ +ρB∆+ +ρBp. (39)

4

0.0 0.2 0.4 0.6 0.8 1.00

200

400

600

800

1000

1200

M*(MeV

)

B(fm-3)

Mn* Mp*

(a) Ne

M *

0.0 0.2 0.4 0.6 0.8 1.0

B(fm-3)

Mn* Mp*

(b) N e0

200

400

600

800

1000

1200 M -* M 0* M +* M ++*

FIG. 2: (Color online) Effective baryonic massM∗ as a function ofthe baryon densityρB for both Neµ matter andN∆eµ matter. Thecalculations are done for one representative parameter set(PKO1) atfixed meson coupling of(xσ ,xω ,xρ ) = (0.8,1.0,0.5).

The pressure is calculated following the thermodynamics re-lation as

P(ρB) = ρ2B

∂∂ρB

ερB

= ∑j=i,l

ρB jµ j − ε +ρ2B ∑

j=i,l

µ jdYj

dρB. (40)

HereYj =ρ j/ρB is the relative fraction of each particle speciesj, respectively.

Once the equation of state (EoS),P(ε), is specified, the sta-ble configurations of a NS then can be obtained from the wellknown hydrostatic equilibrium TOV equations as in our pre-vious work [12, 35–40]:

dP(r)dr

= −Gm(r)ε(r)r2

[

1+ P(r)ε(r)

][

1+ 4πr3P(r)m(r)

]

1− 2Gm(r)r

, (41)

dm(r)dr

= 4πr2ε(r), (42)

whereP(r) is the pressure of the star at radiusr, andm(r) isthe total star mass inside a sphere of radiusr (G is the gravi-tational constant).

III. RESULTS AND DISCUSSION

We begin this section with the sensitivity study of theuncertain∆-meson couplings with one representative modelparameter set (PKO1 [20]). As mentioned in the intro-duction, the calculations are done by varying the density-dependence features of∆ couplings withσ ,ω ,ρ mesons inreasonable ranges. The resulting∆ critical densitiesρcrit

∆ /ρ0are shown in Fig. 1 as functions of the presently-unknownxρ in the range of 0.5− 3.0, for three cases of(xσ ,xω ) of

0.0 0.2 0.4 0.6 0.8 1.010-3

10-2

10-1

100

Yi

B(fm-3)

n

p

e

-

FIG. 3: (Color online) Relative fractionYi = ρi/ρB of N∆eµ mat-ter as a function of the baryon densityρB with the representativeparameter set (PKO1) and fixed meson coupling of(xσ ,xω ,xρ ) =(0.8,1.0,0.5).

(0.8,1.0),(1.0,1.0),(1.0,1.2). Previous calculations usingRMF [4] with (xσ ,xω ) = (1.0,1.0) are also shown for com-parison.

We first notice a similarly linearxρ dependence for DDRHFand RMF, namely, the largerxρ , the largerρcrit

∆ /ρ0. However,due to remarkable adjustment of the coupling strengthes in-troduced by the density dependence in the present study, twomain differences are present: One is an even earlier appear-ance of∆-isobars, atρcrit

∆ /ρ0 = 1.45, to be compared withρcrit

∆ /ρ0 = 2.1 with RMF [4], for the same choice of me-son coupling of(xσ ,xω ,xρ) = (1.0,1.0,1.0); The other is aweakerxρ dependence, which is not subject to the modifica-tions of xσ ,xω as one can notice from all three solid curvesin Fig. 1. The increase ofxσ (xω) leads to an earlier/later∆appearance, which can be attributed to the attraction/repulsivenature of theσ(ω) coupling with∆-isobars in the dominateHartree contributions.

In Fig. 1, very similar∆ critical density is found for RMFat (xσ ,xω ,xρ) = (1.0,1.0,0.5), and DDRHF at(xσ ,xω ,xρ) =(0.8,1.0,0.5). Therefore, we do the following calculationsmainly using one fixed meson coupling of(xσ ,xω ,xρ) =(0.8,1.0,0.5) with the same representative PKO1 param-eter set. We mention here that the choice ofxσ =gσ∆(ρB)/gσN(ρB) = 0.8 corresponds to a 20% relaxation [25]of the QCD result atρB = ρ0 [29].

Fig. 2 shows the baryonic effective masses of Eq. (34). Forthe study of∆-isobar effects, the results forNeµ matter in theleft panel are compared to those ofN∆eµ matter in the rightpanel. All baryonic effective massM∗ decrease with densitybecause of the attractiveσ field for nucleons and∆-isobars.A slightly lower value forM∗

n thanM∗p in both panels is due

to its larger fraction in the matter. Similarly, we can tell themerging sequence for∆-isobars as∆−,∆0,∆+,∆++, follow-

5

0.0 0.2 0.4 0.6 0.8 1.00

200

400

600

800Pressure(MeV

/fm3 )

B(fm-3)

PKO1 PKO2 PKO3 PKA1 PKO1 w.o.

FIG. 4: (Color online) Total pressureP for N∆eµ matter as a func-tion of the baryon densityρB with all four available DDRHF param-eter sets (PKO1, PKO2, PKO3, PKA1), at fixed meson-∆ couplingof (xσ ,xω ,xρ ) = (0.8,1.0,0.5). The results without∆-isobars withPKO1 are also shown for comparison.

ing the requirements of chemical equilibrium of Eqs. (34-35).With ∆− first appearing atρB ∼ 0.28 fm−3, the proton popu-lation increases under the condition of charge neutrality,thenM∗

p decreases to be close toM∗n in the right panel forN∆eµ

matter.The detailed composition ofN∆eµ matter is given in Fig. 3

as a function of the baryonic density. It is clear that the∆-isobars appear in the same sequence expected in theM∗ studyin Fig. 2. After its early presence,∆− quickly replaces neu-tron and achieves a comparable (even higher) population withproton at high densities. It is a nature consequence of the inter-play between the baryon number conservation and the chargeneutrality.

The resulting EoSs are then presented in Fig. 4. Hereall four available DDRHF parameter sets (PKO1, PKO2,PKO3 [20], and PKA1 [21]) are employed and compared. Wesee that at the∆− threshold density around 0.28 fm−3, thepressure curves ofN∆eµ matter start to differ from (actuallybe softer than) those ofNeµ matter. The modification of the∆− threshold density due to different model parameter em-ployed is shown to be quite limited (only≤ 5.99%). For ex-ample it is 0.2805 fm−3, 0.2775 fm−3, 0.2715 fm−3, 0.2640fm−3 for PKO1, PKO2, PKO3, PKA1, respectively. Thisweak-dependence feature was not observed in previous RMFcalculations [5]. In particular, as shown in Fig. 2 of Ref. [5],to substitute the parameter set from NL1 to TM1, the mod-ification in the∆ threshold in the nonlinear Walecka modelis ∼ 37.6%; Similarly, a change from T1 to T3 for a chiralhadronic model resulted in a corresponding modification of∼ 28.9%. In addition, we notice that in the calculations of thenonlinear Walecka model [5], the∆-excited matter is found tobe the ground state of nuclear matter instead of normalNeµ

0.0 0.2 0.4 0.6 0.8 1.0-200

0

200

400

600

Pressure(MeV

/fm-3)

B(fm-3)

Total with Total w.o. Hartree with Hartree w.o. Fock with Fock w.o.

FIG. 5: (Color online) Total pressureP as a function of the baryondensityρB for both Neµ matter (in dashed lines) andN∆eµ matter(in solid lines). The corresponding contributions from theHartreechannels (in blue) and Fock channels (in red) also shown. Thecalcu-lations are done for one representative parameter set (PKO1) and atfixed meson coupling of(xσ ,xω ,xρ ) = (0.8,1.0,0.5).

matter, which might not be physically valid. We also mentionhere that a much delayed appearance of∆-isobars (the earliestof 3.22ρ0 for T1 parameter set) was found in the calculationof the chiral hadronic model [5], even with an enhancedσ -meson coupling ofxσ = 1.39. The last point demonstrated inFig. 4 for DDRHF is that, with increasing density, the PKA1(PKO3) parameter set provides as the softest (stiffest) EoS,with the PKO1 and PKO2 cases in the middle. Moreover, theweak-dependence feature continues at high densities, withthepressure only lifting∼ 12.8% from softest PKA1 to the stiffestPKO3 at the highest density plotted here (1.0 fm−3).

In the following two figures, we continue to use PKO1 asthe representative parameter set to discuss in details the sep-arate contributions from various meson terms for the EoSs.The meson couplings are also fixed to be(xσ ,xω ,xρ) =(0.8,1.0,0.5).

In Fig. 5, the total pressures of bothNeµ matter andN∆eµmatter are plotted together with the corresponding contri-butions from the Hartree terms and Fock terms. One canimmediately notice that there are discontinuous in separateHartree/Fock contributions at threshold densities of∆-isobars,as a result of the third term in the right hand side of Eq. (40).Both the Hartree channel and Fock channel become less re-pulsive once a new type of∆-isobars appears. Also, for bothtwo cases of with and without∆-isobars, the total pressuresare dominated by the Hartree channels, which stay repulsiveat high densities. However, the Fock contribution could be de-creased to be attractive at high densities (around 0.86 fm−3)after the presence of∆− and∆0. One may then conclude thatthe ∆ softening on the EoS is largely due to a reduced Fockcontribution introduced by∆-isobars.

6

0.0 0.2 0.4 0.6 0.8 1.0

0

100

200

300

400Pressure(MeV

/fm-3)

B(fm-3)

with with with with w.o. w.o. w.o. w.o.

(a) Hartree

0.0 0.2 0.4 0.6 0.8 1.0

B(fm-3)

with with with with w.o. w.o. w.o. w.o.

(b) Fock

-30

0

30

60

90

120

FIG. 6: (Color online) Pressure contributions from different mesonsfor both the Hartree channels (left panel) and Fock channels(rightpanel), with (in solid lines) or without (in dashed lines)∆-isobars.The calculations are done for one representative parameterset(PKO1) and at fixed meson coupling of(xσ ,xω ,xρ ) = (0.8,1.0,0.5).

To further understand this point, we separate the contribu-tions from different mesons in Fig. 6, where the Hartree/Fockpart is presented in the left/right panel. We see that the con-tributions from the isovector mesons (π , ρ) are much smallerthan those of the isoscalar mesons (σ , ω). Also, although atρB = ρ0 theω contribution is comparable with theσ contribu-tion, its increasing with density is more pronounced than thatof the σ contribution. As a result, the pressures in both theHartree part and the Fock part are dominated by the repulsiveω couplings to both nucleons and∆-isobars. In the right panelfor the Fock part, we can easily observe strong decreasing ofthe repulsiveσ ,ω couplings to the∆-isobars. This is the mainreason why the Fock term contributes a negative pressure athigh densities in Fig. 5. In particular, the reducing∆-ω Fockcoupling is enhanced once a new type of∆-isobars appears.A similar conclusion has been previously found when Longet al. [18] studied theΛ-matter for theΛ-ω coupling withinDDRHF.

Finally the NS structures are calculated based on the EoSsobtained in the present study. The results are shown in Fig. 7,where the star’s gravitational mass is shown as a functionof the central density (the stellar radius) in the left (right)panel. All four available DDRHF parameter sets are em-ployed and the calculations are done for three cases of∆-meson coupling of(xσ ,xω ) = (0.8,1.0),(1.0,1.0),(1.0,1.2).We see that the introducing of∆-isobars leads to a decreaseof the predicted maximum mass for NSs, as expected fromadding a new degree of freedom to the system. For example at(xσ ,xω) = (0.8,1.0), a maximum mass of 2.50M⊙ (2.43M⊙)without ∆-isobars is lowered to 2.24M⊙ (2.14M⊙) for PKO3(PKA1). The mass (radius) difference between PKO3 andPKA1 are 1.45% (0.10%) without∆-isobars, 2.46% (1.34%)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5(a)

M/M

C(fm-3)

PKO1 w.o. PKA1 w.o. PKO2 (0.8, 1.0) PKO3 (0.8, 1.0) PKA1 (0.8, 1.0) PKO1 (0.8, 1.0) PKO1 (0.8, 1.2) PKO1 (1.0, 1.2)

12 14 16

(b)

PSR J1614-2230

PSR J0348+0432

R(km)

0.0

0.5

1.0

1.5

2.0

2.5

FIG. 7: (Color online) Gravitational massM as a function of thecentral densityρc (the stellar radiusR) in the left (right) panel. Thecalculations with PKO1 parameter set are done for three cases of ∆-meson coupling of(xσ ,xω ) = (0.8,1.0),(1.0,1.0),(1.0,1.2) at fixedxρ = 0.5. The calculations with PKA1, PKO2, PKO3 parameter setsare done for one case of∆-meson coupling of(xσ ,xω ) = (0.8,1.0).The results without∆-isobars for PKO1 and PKA1 are also shownfor comparisons, together with two recent massive stars: PSR J1614-2230 [14, 15] and PSR J0348+0432 [16]. The cross in the(xσ ,xω ) =(1.0,1.2) curve is the point beyond which no physical solution isfound.

with ∆-isobars, although the symmetry energy parameters dif-fer considerably, i.e.,(Esym,L) = (32.99 MeV,83.00 MeV)for PKO3, and(Esym,L) = (36.02 MeV,103.50 MeV) forPKA1. These weak sensitivities of star predictions on symme-try energy parameters (especiallyL) were demonstrated be-fore [18] in the case without∆-isobars. Model studies withRMF using constant meson couplings, however, show muchlarger sensitivities [18]. We then conclude here that the in-troduction of density-dependence for meson couplings couldlead to concentrated results on predictions of star properties.

Moreover, the maximum mass increases/decreases with in-creasingxω/xσ , as expected from the stiffened/softened EoSobtained above. To take PKO1 as an example, a maximummass of 2.45M⊙ without ∆-isobars is lowered to 2.37M⊙,2.19M⊙ corresponding to(xσ ,xω ) = (0.8,1.2),(0.8,1.0). Theradius also smaller, from 12.38 km down to 12.28 km, 11.62km, respectively. It is important to point out here that lowerL values are suggested [41–43] by both nuclear experiments,chiral effective theory, and some neutron star radii. NewDDRHF parameter sets should be proposed in line with them.The cross in the(xσ ,xω ) = (1.0,1.2) curve is the point wherethe∆ population is close to that of nucleons, and no physicalsolution is found after that point. In that case, the HF potentialfor spin-3/2 ∆-isobars can not be self-consistently obtained,even the in-medium∆ effective mass is still far from zero.

7

IV. CONCLUSIONS

Summarizing, we have extended the DDRHF theory to in-clude the∆-isobars for the interest of dense matter and NSs.For this purpose, we solve the Rarita-Schwinger equationfor spin-3/2 particle with the∆ self-energy determined self-consistently. Four mesons (the isoscalarσ andω as well asisovectorρ andπ) are included for producing interaction be-tween the baryons. All four available parameter sets (PKO1,PKO2, PKO3, PKA1) are employed to explore the model uti-lization. Also, the calculations are done with various choicesof the uncertain meson-∆ couplings following the constrainsreported in the various analysis of experimental data.

We found that∆-isobars appear early in the nuclear matter,soften the EoS of NS matter, and reduce the NSs’ maximummass. We observed a controlled behaviour of the results, withrespect to the change of the model parameter set and differ-ent meson-∆ couplings. In particular, the∆ softening of the

EoS is due to the large decrease in Fock channel, mainly fromthe isoscalar mesons. On the other hand, the pion contribu-tions are found to be negligibly small. Finally, we concludethat within DDRHF, a NS with∆-isobars in its core can beas heavy as the two recent massive stars whose masses areaccurately measured, with a slightly smaller radius than thecorresponding normal NSs.

V. ACKNOWLEDGMENTS

This work was supported by the National Natural ScienceFoundation of China (Grants Nos. 11405090 and U1431107).

VI. APPENDIX A

The verticesΓabλ (1,2) between mesons and baryons are:

Γabσ (1,2) =−gσagσb, (43)

Γabω (1,2) = gωagωbγµ(1)γµ(2), (44)

Γabπ (1,2) =− fπa fπb

m2π

(γ5/qτb)1(γ5/qτa)2, (45)

ΓabρV(1,2) = gρagρbγµ(1)τb(1)γµ(2)τa(2), (46)

ΓabρT(1,2) =

fρa fρb

4MaMb(qλ σ µλ

τb)1(qνσµντa)2, (47)

ΓabρVT

(1,2) =i fρagρb

2Maγµ(1)τb(1)q

νσµν(2)τa(2)−i fρbgρa

2Mbqνσ µν(1)τb(1)γµ(2)τa(2), (48)

wherea,b denote nucleon or∆-isobars.

VII. APPENDIX B

Here we calculate the energy density of the system. we first write the Hamiltonian densityH (Eq. (3)) in terms ofu(p,κ) anduα(p,κ), with the corresponding kinetic energy termT and the potential termV as

T = ∑p,κ1,κ2

uN(p,κ1)(γ ·p+MN)uN(p,κ2)bN†p,κ1

bNp,κ2

+ ∑p,κ1,κ2

u∆α(p,κ1)(γ ·p+M∆)uα∆ (p,κ2)b

∆†p,κ1

b∆p,κ2

, (49)

V =12 ∑

λ∑

q,p1,p2

∑κ1,κ2,κ3,κ4

uN(p1,κ1)uN(p2,κ2)ΓNN

λm2

λ +q2uN(p2+q,κ3)uN(p1−q,κ4)b

N†p1,κ1

bN†p2,κ2

bNp2+q,κ3

bNp1−q,κ4

+12 ∑

λ∑

q,p1,p2

∑κ1,κ2,κ3,κ4

uN(p1,κ1)u∆β (p2,κ2)ΓN∆

λm2

λ +q2uβ

∆(p2+q,κ3)uN(p1−q,κ4)bN†p1,κ1

b∆†p2,κ2

b∆p2+q,κ3

bNp1−q,κ4

+12 ∑

λ∑

q,p1,p2

∑κ1,κ2,κ3,κ4

u∆α(p1,κ1)uN(p2,κ2)Γ∆N

λm2

λ +q2uN(p2+q,κ3)u

α∆ (p1−q,κ4)b

∆†p1,κ1

bN†p2,κ2

bNp2+q,κ3

b∆p1−q,κ4

+12 ∑

λ∑

q,p1,p2

∑κ1,κ2,κ3,κ4

u∆α(p1,κ1)u∆β (p2,κ2)Γ∆∆

λm2

λ +q2uβ

∆(p2+q,κ3)uα∆(p1−q,κ4)b

∆†p1,κ1

b∆†p2,κ2

b∆p2+q,κ3

b∆p1−q,κ4

.

(50)

8

TABLE I: FunctionsAλ , Bλ , Cλ of λ = σ ,ω,π,ρT ,ρV couplings for calculating〈VEN 〉.

λ Aλ Bλ Cλσ −g2

σ θσ −g2σ θσ g2

σ φσω −2g2

ω θω 4g2ω θω 2g2

ω φωρV −2g2

ρ θρ 4g2ρ θρ 2g2

ρ φρ

π(

fπmπ

)2

m2πθπ

(

fπmπ

)2

m2π θπ

(

fπmπ

)2

[−(p2+ p′2)φπ +2pp′ θπ ]

ρT

(

fρ2M

)2

m2ρ θρ 3

(

fρ2M

)2

m2ρ θρ

(

fρ2M

)2

[−2(p2+ p′2−m2ρ/2)φπ +4pp′ θρ ]

ρV T D = 12fρ gρ

2M(− p′ θρ + pφρ )

With Hartree-Fock wave function in Eq. (31), the kinetic term is calculated as:

〈T 〉= ∑k=n,p

1π2

∫ pF,k

0p2d p(MNMk + pPk)+ ∑

i=∆−,∆0,∆+,∆++

2π2

∫ pF,i

0p2d p(M∆Mi + pPi). (51)

The potential term can be decomposed into the direct term〈VD〉 and the exchange term〈VD〉, respectively. For the direct potentialterm:

〈VD〉 = −12

g2σN

m2σ

ρ2SN

+12

g2ωN

m2ω

ρ2BN

+12

g2ρN

m2ρ

ρ2BN3

+gσN gσ∆

m2σ

ρSN ρS∆ −gωN gω∆

m2ω

ρBN ρB∆ −gρN gρ∆

m2ρ

ρBN3ρB∆3 (52)

−12

g2σ∆

m2σ

ρ2S∆+

12

g2ω∆

m2ω

ρ2B∆

+12

g2ρ∆

m2ρ

ρ2B∆3

,

where, the densities of nucleon and∆-isobars are defined as

ρBN = ρBn +ρBp, ρB∆ = ρB++ +ρB+ +ρB0 +ρB− , (53)

ρBNk=

1π2

∫ pF,k

0p2d p, ρB∆i

=2

π2

∫ pF,i

0p2d p, (54)

ρSN = ρSn +ρSp, ρS∆ = ρS++ +ρS+ +ρS0 +ρS− , (55)

ρSNk=

1π2

∫ pF,k

0p2d pM(p), ρS∆i

=2

π2

∫ pF,i

0p2d pM(p), (56)

ρBN3 = ρBp −ρBn , ρB∆3 = 3ρB++ +ρB+ −ρB0 −3ρB− (57)

The calculation of exchange contribution from nucleons and∆-isobars can be changed into trace calculations as

〈VEN 〉=12 ∑

λ∑

τ1,τ2

∑p1,p2

1

m2λ +q2

Tr[DN(p1,τ1)ΓNNλ (2)DN(p2,τ2)ΓNN

λ (1)], (58)

〈VE∆〉=12 ∑

λ∑

τ1,τ2

∑p1,p2

1

m2λ +q2

Tr[Sαβ∆ (p1,τ1)Γ∆∆

λ (2)S∆,αβ (p2,τ2)Γ∆∆λ (1)]. (59)

HereDN(p,τ) is the propagator of nucleons:

DN(p,τ) = ∑s

u(p,s,τ)u(p,s,τ) = /p ∗+M∗

2E∗ , (60)

andSαβ∆ (p,τ) is the propagator of∆-isobars:

Sαβ∆ (p,τ) = ∑

suα

∆(p,s,τ)uβ∆(p,s,τ) = /p∗+M∗

2E∗

[

−gαβ +13

γα γβ +2

3M∗2 pα pβ − 13M∗ (pα∗γβ − pβ∗γα)

]

(61)

9

The nucleonic exchange terms of Eq. (58) for symmetric nuclear matter were calculated in Ref. [17]. We write the formula herefor asymmetric nuclear matter (k, l denote neutron or proton):

〈VEN 〉=1

(2π)4 ∑k,l

∫ pF,k

0pd p

∫ pF,l

0p′d p′

12

[

∑λ

Ωλkl

(

Aλ +Bλ MM′+Cλ PP′)

+12

Ωρkl

(

PM′ D(p, p′)+ P′M D(p′, p)

)

]

, (62)

whereP(p)≡ P, P(p′) ≡ P′ are used in order to shorten the notation. The matrix elementΩλkl is related to the isospin part and

will be evaluated in Appendix C. FunctionsAλ ,Bλ ,Cλ ,D for λ = σ ,ω ,π ,ρV ,ρT ,ρTV are shown in Table I. The functions inTable I are defined as

θλ (p, p′) = ln

[

m2λ +(p− p′)2

m2λ +(p+ p′)2

]

, (63)

φλ (p, p′) =p2+ p′2+m2

λ2pp′

θλ (p, p′)+2. (64)

Now we turn to the∆ parts in Eq. (59). We first do trace calculation for each meson, defining

Rλ ≡ Tr[Sαβ∆ (p1,τ1)Γ∆∆

λ (2)S∆,αβ (p2,τ2)Γ∆∆λ (1)]. (65)

We have

Rσ =19

g2σ∆(−10b−14a−8aa−4a2a), (66)

Rω =19

g2ω∆(32b−4a+16aa−8a2a), (67)

Rπ =19

(

fπ∆

mπ

)2

(−20c1c2+6ad+6bd+4c21

M2

M1+4c2

2M1

M2−8a2c1c2+4a2ad+4aad), (68)

RρV =19

g2ρ∆(32b−4a+16aa−8a2a), (69)

RρT =19

(

fρ∆

2M∆

)2

(−24c1c2+6ad+22bd−16a2c1c2+4a2ad+14aad), (70)

RρTV =19

fρ∆ gρ∆

2M∆(44c1−44c2−8ac2+8ac1−24a2c2+24a2c1)M1M2, (71)

whereM(p1)≡ M1, M(p2)≡ M2 have been used in order to shorten the notation.The coefficients ˆa, a, b, c1, c2, c1, c2, d are defined as:

a =p∗1 · p∗2M∗

1M∗2=

1

M(p1)M(p2)− P(p1)P(p2)

M(p1)M(p2)cosθ , (72)

a = 1− P(p1)P(p2)cosθ , (73)

b = M(p1)M(p2), (74)

c1 =p∗1 ·qE∗

1= P(p1)p2cosθ − P(p1)p1, (75)

c2 =p∗2 ·qE∗

1= P(p2)p2− P(p2)p1cosθ , (76)

c1 =P(p1)

M(p1)p2cosθ − P(p1)

M(p1)p1, (77)

c2 =P(p2)

M(p2)p2−

P(p2)

M(p2)p1cosθ , (78)

d = q2. (79)

Then the exchange term of energy for∆-isobars could be given as (herei, j = ∆−,∆0,∆+,∆++):

〈VE∆〉=1

(2π)4 ∑λ

∑i, j

∫ pF,i

0pd p

∫ pF, j

0p′d p′∑

m

12

(

∑λ 6=ρVT

Ωλi jΛλ HmAm

λ +ΛρVT Ωρi j J

mBm

)

, (80)

10

TABLE II: FunctionsHm andAmλ (m = 1,2...12) of λ = σ ,ω,π,ρT ,ρV couplings for calculating〈VE∆〉.

HmAm

λ Amσ Am

ω Amπ Am

ρVAm

ρT

H1 MM′ −10θ 32θ 6dθ 32θ 22dθH2 cons. −14θ −4θ 6dθ −4θ 6dθH3 1/MM′ −8θ 16θ 4dθ 16θ 8dθH4 1/M2M′2 −4θ −8θ 4dθ −8θ 4dθH5 PP′ 14φ 4φ 20pp′ θ −20(p2+ p′2)φ −6dφ +20pp′ η 4φ 24pp′ θ −24(p2+ p′2)φ −6dφ +24pp′ ηH6 PP′/MM′ 16φ −32φ −8dφ −32φ −16dφH7 PP′/M2M′2 12φ 24φ 8pp′ θ −8(p2+ p′2)φ −12dφ +8pp′ η 24φ 16pp′ θ −16(p2+ p′2)φ −12dφ +16pp′ ηH8 P2P′2/MM′ −8η 16η 4dη 16η 8dηH9 P2P′2/M2M′2 −12η −24η −16pp′ φ +16(p2+ p′2)η +12dη −16pp′ ϕ −24η −32pp′ φ +32(p2+ p′2)η +12dη −32pp′ ϕH10 P3P′3/M2M′2 4ϕ 8ϕ 8pp′ η −8(p2+ p′2)ϕ −4dϕ +8pp′ ϑ 8ϕ 16pp′ η −16(p2+ p′2)ϕ −4dϕ +16pp′ ϑH11 P2M′/M 0 0 4p2θ −8pp′ φ +4p′2 η 0 0

H12 P′2M/M′ 0 0 4p′2 θ −8pp′ φ +4p2η 0 0

TABLE III: Functions Jm andBmλ (m = 1,2...12) of λ = ρV T couplings for calculating〈VE∆〉.

J1 PM′ J7 P2P′/M B1 −44pθ +44p′ φ B7 8pφ −8p′ ηJ2 P′M J8 PP′2/M′ B2 −44p′ θ +44pφ B8 8p′ φ −8pηJ3 P/M J9 P2P′/M2M′ B3 −8pθ +8p′ φ B9 48pφ −48p′ ηJ4 P′/M′ J10 PP′2/MM′2 B4 −8p′ θ +8pφ B10 48p′ φ −48pηJ5 P/M2M′ J11 P3P′2/M2M′ B5 −24pθ +24p′ φ B11 −24pη +24p′ ϕJ6 P′/MM′2 J12 P2P′3/MM′2 B6 −24p′ θ +24pφ B12 −24p′ η +24pϕ

where the matrix elementΩλi j is related to the isospin part and will be evaluated in Appendix C. Λλ are the coupling constants

of different meson with∆-isobars:

Λσ =19

g2σ∆

, Λω =19

g2ω∆, (81)

Λπ =19

(

fπ∆

mπ

)2

, ΛρT =19

(

fρ∆

2M∆

)2

, (82)

ΛρV =19

g2ρ∆

, ΛρVT =19

(

fρ∆ gρ∆

2M∆

)

. (83)

Hm,Amλ , J

m andBmλ are the functions related to momenta and meson masses, as listed in the Tables II and III. The functionsη ,

ϕ , ϑ in those tables are defined as:

ηλ (p, p′) =p2+ p′2+m2

λ2pp′

φλ (p, p′), (84)

ϕλ (p, p′) =p2+ p′2+m2

λ2pp′

ηλ (p, p′)+23, (85)

ϑλ (p, p′) =p2+ p′2+m2

λ2pp′

ϕλ (p, p′). (86)

VIII. APPENDIX C

For the isoscalar mesons (σ , ω), the isospin part of the exchange potential requires that the initial and final isospin wavefunctions to be identity, the matrix elementΩkl then corresponds that of a unit matrix.

11

TABLE IV: Matrix elementsΩkl of isovector mesons for the isospin part in〈VEN 〉.

lk

n p

n 1 2p 2 1

While for isovector mesons (ρ ,π), the isospin part in the exchange contributions of Eqs. (58-59) could be written as:

χ†2τ∆χ1χ†

2τ∆χ1, (87)

whereχi is the isospin wave functions of baryon.The isospin matrices for nucleons are the usual Pauli matrix. Then the matrix elementΩkl for nucleons can be evaluated

(shown in Table IV).The isospin matrices for∆-isobar are given as:

τ∆1 =

0√

3 0 0√3 0 2 0

0 2 0√

30 0

√3 0

, τ∆2 =

0 −√

3i 0 0√3i 0 −2i 00 2i 0 −

√3i

0 0√

3i 0

, τ∆3 =

3 0 0 00 1 0 00 0 −1 00 0 0 −3

. (88)

Then the matrix elementΩi j for ∆-isobars can be evaluated (shown in Table V).

IX. APPENDIX D

The components of self-energy for nucleons are given as (herek, l denote neutron or proton):

ΣkS(p) = −

(

gσN

mσ

)2

ρsN −gσNgσ∆

m2σ

ρS∆

+1

(4π)2

1p

∫ pkF

0p′d p′

∑λ=σ ,ω,

ρ ,π

Bλ (p, p′)M(p′)+12

P(p′)D(p, p′)

+2

(4π)2

1p

∫ pl(6=k)F

0p′d p′

[

∑λ ′=ρ ,π

Bλ ′(p, p′)M(p′)+12

P(p′)D(p, p′)

]

, (89)

Σk0(p) =

(

gωN

mω

)2

ρBN +gωNgω∆

m2ω

ρB∆ −(

gρN

mρ

)2

ρBN3τk −gρNgρ∆

m2ρ

ρB∆3τk

+1

(4π)2

1p

∫ pkF

0p′d p′ ∑

λ=σ ,ω,ρ ,π

Aλ (p, p′)

+2

(4π)2

1p

∫ pl(6=k)F

0p′d p′ ∑

λ ′=ρ ,πA′

λ (p, p′), (90)

ΣkV (p) =

1(4π)2

1p

∫ pkF

0p′d p′

P(p′) ∑

λ=σ ,ω,ρ ,π

Cλ (p, p′)+12

M(p′)D(p, p′)

+2

(4π)2

1p

∫ pl(6=k)F

0p′d p′

[

P(p′) ∑λ ′=ρ ,π

Cλ ′(p, p′)+12

M(p′)D(p, p′)

]

. (91)

12

TABLE V: Matrix elementsΩi j of isovector mesons for the isospin part in〈VE∆〉.

ji ∆++ ∆+ ∆0 ∆−

∆++ 9 6 0 0∆+ 6 1 8 0∆0 0 8 1 6∆− 0 0 6 9

For ∆-isosbars, the three components are given as (herei, j = ∆−,∆0,∆+,∆++ andλ = σ ,ω ,ρ ,π):

ΣiV (p) =

132π2 ∑

jΩi j

1p

∫ p jF

0p′d p′

[

∑λ

Λλ

(

P′A5λ +

P′

MM′ A6λ +

P′

M2M′2A7

λ +2PP′2

MM′ A8λ +

2PP′2

M2M′2A9

λ +3P2P′3

M2M′2A10

λ +2PM′

MA11

λ

)

+19

fρ∆ gρ∆

2M∆

(

M′B1+1

MB3+

1

M2M′ B5+

2PP′

MB7+

P′2

M′ B8+2PP′

M2M′ B9+

P′2

MM′2B10+

3P2P′2

M2M′ B11+2PP′3

MM′2B12)]

,

(92)

ΣiS(p) =−gσN gσ∆

m2σ

ρSN −g2

σ∆

m2σ

ρS∆ +1

32π2 ∑j

Ωi j1p

∫ p jF

0p′d p′

[

∑λ

Λλ

(

M′A1λ − 1

M2M′ A3λ − 2

M3M′2A4

λ − PP′

M2M′ A6λ − 2PP′

M3M′2A7

λ − P2P′2

M2M′ A8λ − 2P2P′2

M3M′2A9

λ − 2P3P′3

M3M′2A10

λ − P2M′

M2A11

λ +P′2

M′ A12λ

)

+19

fρ∆ gρ∆

2M∆

(

P′B2− P

M2B3− 2P

M3M′ B5− P′

M2M′2B6− P2P′

M2B7− 2P2P′

M3M′ B9− PP′2

M2M′2B10− P3P′2

M3M′ B11− P2P′3

M2M′2B12)]

,

(93)

Σi0(p) =

gωN gω∆

m2ω

ρBN +gρN gρ∆

m2ρ

ρBN3τi +g2

ω∆

m2ω

ρB∆ +g2

ρ∆

m2ρ

ρB∆3τi +1

32π2 ∑j

Ωi j1p

∫ p jF

0p′d p′

[

∑λ

Λλ (A2λ + H3A3

λ +3H4A4λ + H6A6

λ +2H7A7λ + H9A9

λ )+ΛρTV (J3B3+ J4B4+2J5B5+2J6B6+ J9B9+ J10B10)

]

.

(94)

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