• Quarkmatterandhybridstars • Neutronstarproperties • Hypernuclei • BHFapproachofhypernuclearmatter TheoryofNuclearMatterandNeutronStars

Hans-Josef Schulze Catania

Theory of Nuclear Matter and Neutron Stars

M. Baldo & G.F. Burgio & U. Lombardo & I. Vidaña & H.-J. S. : Catania H. Chen & A. Li & Z.H. Li & H.Q. Song & X.R. Zhou & W. Zuo : China T. Maruyama & S. Chiba & T. Tatsumi & N. Yasutake : Japan

T. Rijken : Nijmegen

• BHF approach of hypernuclear matter

• Hypernuclei

• Neutron star properties

• Quark matter and hybrid stars

PRC 61, 055801 (2000) PRC 69, 018801 (2004) PRD 70, 043010 (2004) A&A 451, 213 (2006) PRC 73, 058801 (2006) PRC 74, 047304 (2006) PRD 74, 123001 (2006) PRD 76, 123015 (2007) PRC 78, 028801 (2008) PRC 83, 025804 (2011) PRC 84, 035801 (2011) PRD 84, 105023 (2011) A&A 551, A13 (2013) PRD 91, 105002 (2015) EPJA 52, 21 (2016) PRC 94, 024322 (2016) PRC 96, 044309 (2017)

http://chandra.harvard.edu

8...30 M

_⊙







http://chandra.harvard.edu

8...30 M

_⊙







∼ 2800 known neutron stars 2500 pulsars

10% in binary systems

∼10⁸ in our galaxy ?

http://chandra.harvard.edu

8...30 M

_⊙







∼ 2800 known neutron stars 2500 pulsars

10% in binary systems

∼10⁸ in our galaxy ?

Neutron Star or

Black Hole formation ?

A Theorist’s View of a Neutron Star:

A huge nucleus: ∼ 10

⁵⁷

nucleons :

n pe μ n pe μ ⁻ npe μ −

mesons?Λ

quarks?

M ∼ 1 . . . 2 M_⊙

T≪1MeV ≈10¹⁰K

∼ 10 km

∼ ρ₀

∼ 10ρ₀

ROSAT image of Puppis A

The only “laboratory” for ρ

_B

∼ 10ρ

₀

in the Universe !

Need EOS of nuclear matter including hyperons and quarks

Hypernuclear Matter in the Neutron Star:

ρ = ρ

_N

+ ρ

_Y

bb b b b

b b

b b

b

V

_YY

V

_NN

V

_NY

N = qqq: n

p (939 MeV) Y = qqs:

qss:

Λ

⁰

(1116 MeV)



⁻⁰⁺

(1193 MeV)



⁻⁰

(1318 MeV)

V

_NN

: Argonne, Bonn, Paris, ... potential V

_NY

: Nijmegen (NSC89,NSC97,ESC08...) V

_YY

: ? (no scattering data)

In free space weak decay: Y → N + π etc. (cτ ≈ 8 cm)

In dense nucleonic medium the decay is Pauli-blocked !

We need to compute the energy density of this system ...

Brueckner Theory of (Hyper)Nuclear Matter:

• Effective in-medium interaction G from potential V:

G = V + V G

= + G

e_k = m + k²

2m + U(k)

self-consistent parameter-free !

Results: binding energy ε(ρ

_n

,ρ

_p

,ρ

_Λ

,ρ

_

) =

^P



P

k<k_F^()

h

e

^()_k

−

^U₂^(k)i

s.p. properties, cross sections, ...

K.A. Brueckner and J.L. Gammel; PR 109, 1023 (1958) for nuclear matter Extension to hypernuclear matter ...

• Framework: Brueckner-Bethe-Goldstone hole-line expansion

+ G

+ + · · · +

^O

(κ

⁴

) E

A = 3 5

k

²

F

2m

≈

^h

22 − 40 + 2 + ?

ⁱ

MeV

ρ = ρ₀, symmetric matter, V₁₈ potential

• Expansion parameter κ ∼ ρV

_core

≈ 0.2 :

bb b

b b

b b

b b

b bb b

b b

c d

κ ≡

P

k

n(k > k

_F

)

P

k<k_F

= ρ

Z

d

³

r

^D

|η(r )|

^2E

S,T

= N V

_core

V =

c d

3

 − ϕ : defect function

- Hierarchy of n-body correlations/clusters within hard-core range c, avg. distance d:

- Justified for hard-core potentials

• Correlation parameter for different NN potentials:

κ ≈ ρV

_core

(ρ)

Small up to large density Hard vs. soft potentials

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36

N3LO500

N3LO450 Paris

AV18

Nij93

BonnB

CDBonn

(fm -3

)

• Binding energy up to three hole lines:

B/ A = T + E

₂

+ E

₃

Hole-line expansion appears well converged, but misses slightly for AV18 the empir- ical saturation point of nu- clear matter

0.0 0.1 0.2 0.3 0.4

-40 -35 -30 -25 -20 -15 -10 -5 0

cont. choice

gap choice

B/A(MeV)

2HL

0.0 0.1 0.2 0.3 0.4 0.5

AV18

CDBONN

N3LO500

N3LO450

2HL+3HL

-3

• Diagrams up to 3HL:

B.D. Day, PRC 24, 1203 (1981)

G

( a ) ( b )

( c ) ( d )

( e )

T⁽³⁾

( f )

T⁽³⁾ = + + +

+ + + · · ·

2HL

3HL

Fadeev calculation:

Three-Nucleon Forces:

1

3 N^∗ = Δ, R, ...

2

μ= π,ρ, σ, ω

+ N + . . .

• Only small effect required [δ(B/ A) ≈ 1 MeV at ρ

₀

]

• Model dependent, no final theory yet

• Use and compare microscopic and phenomenological TBF...

• Microscopic TBF of P. Grangé et al., PRC 40, 1040 (1989):

Exchange of π, ρ, σ, ω via Δ(1232), R(1440), NN¯

Parameters compatible with two-nucleon potential (Paris,V₁₈,...)

• Urbana IX phenomenological TBF:

Only 2π-TBF + phenomenological repulsion Fit saturation point

• BHF binding energy and saturation point of nuclear matter:

0.1 0.2 0.3 0.4 0.5 0.6

-30 -20 -10 0 10 20 30 40 50 60 70

PAR: Paris

V14: Argonne V14

V18: Argonne V18

A: Bonn A

B: Bonn B

C: Bonn C

CD: CD-Bonn

R93: Reid93

N93: Nijmegen93

NI: Nijmegen I

NII: Nijmegen II

N3: N3LO

IS

PAR+TNF

V14+TNF

V18+TNF

B+TNF

N93+TNF

V18+UIX

V18+v+UIX*(AP R)

A(DBHF)

B(DBHF)

C(DBHF)

B/A(MeV)

(fm -3

)

0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44

-28 -24 -20 -16 -12

B B

NI R93

N93 NII V14 PAR V18

C V18+UIX

V14 PAR A C

BHF

BHF+TB F

VCS

DBHF

B/A(MeV)

(f m -3

) V18+v UIX*

V18

N93;

B CD

N3

IS

A

Coester band:

Nuclear mass formula

• Dependence on NN potential

• TBF needed to improve saturation properties

Include Hyperons:

• Technical difficulty: coupled channels:

n n

Λ Λ

G =

n n

Λ Λ

+

n n

p

{n n}

Λ {Λ 

⁻



⁰

} Λ

+ . . .

Include Hyperons:

• Technical difficulty: coupled channels:

n n

Λ Λ

G =

n n

Λ Λ

+

n n

p

{n n}

Λ {Λ 

⁻



⁰

} Λ

+ . . .

Λ Λ

Λ Λ

G =

Λ Λ

Λ Λ

+

Λ { Λ 

⁰



⁰



⁻

} Λ Λ { Λ Λ 

⁰



⁺

} Λ

+

Λ { n p } Λ Λ { 

⁰



⁻

} Λ

+ . . .

Example BHF Results: Input:

• Hyperon-nucleon potentials (NSC89) vs. Paris NN:

-800 -600 -400 -200 0 200 400 600 800

0 1

nΛ → nΛ nΛ → nΛ nΛ → nΛ nΛ → nΛ

V (MeV)

1S₀

3S₁

3SD₁

0 1

nΛ → nΣ⁰ nΛ → nΣ⁰ nΛ → nΣ⁰ nΛ → nΣ⁰

0 1

nΛ → pΣ⁻ nΛ → pΣ⁻ nΛ → pΣ⁻ nΛ → pΣ⁻

r (fm)

0 1

nΣ⁻ → nΣ⁻ nΣ⁻ → nΣ⁻ nΣ⁻ → nΣ⁻ nΣ⁻ → nΣ⁻

0 1

NN NN NN NN

“Soft” cores, Strong coupling NΛ ↔ N

Example BHF Results: Output:

s.p. potentials Λ eff. mass & mean field

-100 -80 -60 -40 -20 0 20 40 60

0 1 2 3 4

ρ_N = 0.17 fm^-3 , ρ_Λ/ρ_N = 0.0

k [fm^-1]

Re U [MeV]

A18+TBF NN & ESC08 NY+YY Potentials

U ^(N)_B U_B N

Λ Σ Ξ

0 1 2 3 4 5

ρ_N = 0.17 fm^-3 , ρ_Λ/ρ_N = 0.5

k [fm^-1]

0.7 0.8 0.9 1

(m*/m)Λ

ESC08 NSC89

-50 -40 -30 -20 -10 0

0 0.1 0.2 0.3

ρ_N [fm^-3] VΛ , UΛ0 [MeV]

V_Λ U_Λ⁰

Hyperons are weaker bound than nucleons

Vela pulsar

Neutr on Stars

«Recipe» for Neutron Star Structure Calculation:

Brueckner results: ε ({ρ

_

}) ;  = n, p, e, μ, Λ, , , d, s, ...

Chemical potentials: μ

_

= ∂ε

∂ρ

_

Beta-equilibrium: μ

_

= b

_

μ

_n

− q

_

μ

_e

Charge neutrality:

^P_



_

q

_

= 0

Composition: 

_

(ρ)

Equation of state: p (ρ) = ρ

²

d(ε/ρ)

dρ (ρ, 

_

(ρ)) TOV equations: dp

dr = − Gmε r

²

(1 + p/ε)(1 + 4πr

³

p/m) 1 − 2Gm/ r

dm

dr = 4πr

²

ε

Structure of the star: ρ(r ), M(R) etc.

«Recipe» for Neutron Star Structure Calculation:

Brueckner results: ε ({ρ

_

}) ;  = n, p, e, μ, Λ, , , d, s, ...

Chemical potentials: μ

_

= ∂ε

∂ρ

_

Beta-equilibrium: μ

_

= b

_

μ

_n

− q

_

μ

_e

μ_e = μ_μ = μ_n−μ_p μ_⁻ = 2μ_n − μ_p μ_⁰ = μ_Λ = μ_n μ⁺ = μp

Charge neutrality:

^P_



_

q

_

= 0 Composition: 

_

(ρ)

Equation of state: p (ρ) = ρ

²

d(ε/ρ)

dρ (ρ, 

_

(ρ)) TOV equations: dp

dr = − Gmε r

²

(1 + p/ε)(1 + 4πr

³

p/m) 1 − 2Gm/ r

dm

dr = 4πr

²

ε

Structure of the star: ρ(r ), M(R) etc.

• Generic implications for EOS and stellar structure:

0 0.1 0.2

0 1 2 3 4 5

ρ / ρ

₀

x

i

p

e

Σ

⁻

Λ µ

0 1 2

8 10 12 14

rotation

R (km)

M/M

O

•

nucleons + hyperons

+ quarks

• Hyperon onset occurs at ρ ∼ 2...3 ρ

₀

• Softer EOS

• NS structure including hyperons

. . . and including quark matter

Vela pulsar

Data ?

Observational Data: Masses

Courtesy of J. Lattimer

The heaviest neutron stars:

Recent: ∼ 1.97M_⊙ (Nature 09466)

∼ 2.01M_⊙ (Science 340)

No combined (M, R) measurements

!

(Would practically fix the EOS)

Observational Data: Radii

Neutron Star Radius Results

Ozel et al. 2015

Six Burst Sources Six qLMXBs

F. Özel et al., APJ820, 28 (2016)

The measurement is difficult: currently no accurate results

J0617 in IC 443

BHF Results ...

• Composition of neutron star matter:

10⁻² 10⁻¹ 10⁰

Paris + TBF V₁₈ + TBF n

p e−

µ⁻

0 0.2 0.4 0.6 0.8 1 1.2

Baryon density ρ (fm⁻³)

10⁻² 10⁻¹ 10⁰

Relative fractions

p e⁻

µ⁻

Σ⁻ Λ 10⁻²

10⁻¹ 10⁰

e⁻ µ⁻

Σ⁻

Λ (a)

(b) n

p

(c) n

No hyperons

Free hyperons

Interacting hyperons

(⁻ repulsive, Λ attractive) NY interaction determines Y onset

• EOS of neutron star matter:

0 0.2 0.4 0.6 0.8 1 1.2

n,p,e⁻,µ⁻ n,p,e⁻,µ⁻, free Y n,p,e⁻,µ⁻, interacting Y

V₁₈ + TBF

0 0.2 0.4 0.6 0.8 1 1.2

Baryon density ρ (fm⁻³)

0 50 100 150 200 250 300 350 400 450 500

Pressure p (MeV fm−3 )

n,p,e⁻,µ⁻ n,p,⁻,µ⁻,free Y

n,p,e⁻,µ⁻, interacting Y

Paris + TBF

Strong softening due to hyperons !

(More Fermi seas available)

• Mass-radius relations with different nucleonic TBF:

8 10 12 14 16

0.0 0.5 1.0 1.5 2.0 2.5

0.0 0.4 0.8 1.2 1.6

BOB V18 N93 UIX

M/M ⊙

R (km)

with hyperons

ρ_c(fm^-3)

NSC89 NY potential No YY

No hyperon TBF

Large variation of M

_max

with nucleonic TBF

Self-regulating softening due to hyperon appearance

(stiffer nucleonic EOS → earlier hyperon onset)

• Mass-radius relations with different nucleonic TBF:

8 10 12 14 16

0.0 0.5 1.0 1.5 2.0 2.5

0.0 0.4 0.8 1.2 1.6

BOB V18 N93 UIX

M/M ⊙

R (km)

with hyperons

ρ_c(fm^-3) 11.5-13.0 km

NSC89 NY potential No YY

No hyperon TBF

Large variation of M

_max

with nucleonic TBF

Self-regulating softening due to hyperon appearance

(stiffer nucleonic EOS → earlier hyperon onset)

• Mass-radius relations using different NY ,YY potentials:

0 1 2

8 10 12 14 16

R (km)

M/M S

V18+UIX

V18+UIX+NSC89 V18+UIX+NSC97 V18+UIX+ESC08

Maximum mass independent of potentials !

Maximum mass too low (< 1.4 M

_⊙

) !

Proof for “quark” matter inside neutron stars ?

• Mass-radius relations using different NY ,YY potentials:

0 1 2

8 10 12 14 16

R (km)

M/M S

V18+UIX

V18+UIX+NSC89 V18+UIX+NSC97 V18+UIX+ESC08

0 50 100 150 200

ε,p [MeV fm-3 ]

ε/10 p

BHF(AV₁₈+UIX+NSC89) , M/M_S = 1.20 BHF(AV₁₈+UIX+NSC89) , M/M_S = 1.20 BHF(AV₁₈+UIX+NSC89) , M/M_S = 1.20 BHF(AV₁₈+UIX+NSC89) , M/M_S = 1.20 BHF(AV₁₈+UIX+NSC89) , M/M_S = 1.20

0 0.5 1

0 2 4 6 8 10 12

ρ[fm-3 ]

r [km]

n p Σ Λ

Maximum mass independent of potentials !

Maximum mass too low (< 1.4 M

_⊙

) !

Proof for “quark” matter inside neutron stars ?

Quark Matter EOS of Dense Matter:

• Problem: No “exact” results from QCD:

Large theoretical uncertainties, limited predictive power

• Current strategy:

Use available eff. quark models (MIT, NJL, CDM, DSM, ...) in combination with the hadronic EOS

• An important constraint (from heavy ion collisions):

In symmetric matter phase transition not below ≈ 3ρ

₀

E.g., the simplest (MIT) quark model requires

a density-dependent bag “constant”:

ε

_Q

= B + ε

_kin

+ α

_s

× ...

B(ρ) = B

_∞

+ (B

₀

− B

_∞

) exp

−β

ρ/ ρ

₀²

• A more sophisticated approach: Dyson-Schwinger model:

S(q) Dρσ(k)

Γ^a_σ(q, p)

λ^a 2γ_ρ

H. Chen et al.,

PRD 84, 105023 (2011) PRD 86, 045006 (2012) PRD 91, 105002 (2015) EPJA 52, 291 (2016) PRD 96, 043008 (2017)

(p) =

Z

d

⁴

p

(2π )

⁴

S(q) λ

^

2 γ

_ρ

D

_ρσ

(k )

^

σ

(q, p)

Compute the quark propagator S(q) from QCD

0.0 0.3 0.6 0.9 1.2 1.5

0 100 200 300 400 500 600

B[MeVfm

-3 ]

[fm -3

] RB-1

1BC-0.8

BC-0.25

RB-2

1BC-1.7

BC-0.93

MIT

Allows to calculate

the bag constant:

• Different quark EOS’s: bag models, color dielectric model:

0 0.5 1 1.5 2

8 10 12 14 16

R (km) M/M O

•

0 0.5 1 1.5 2

ρ_c (fm^-3)

V18

V18 & NSC89 V18 & B=90 V18 & B(ρ) V18 & CDM V18 & O(α_s²) V18 & FCM

NJL, FCM, Dyson-Schwinger models: hyperons prevent phase transition

Maximum masses: 1.5...1.9 M

_⊙

, Radii are different !

• Details of the phase transition: neutron star profiles:

Bulk Gibbs Screened Gibbs Maxwell

ε/10 p

0 1 2 3 4 5 6 7 8 9 10

r [km]

n p u

d s

BHF[V18+UIX+NSC89] & MIT[B=100,α=0,σ=40] , M/M_S=1.40

ε/10 p

0 1 2 3 4 5 6 7 8 9

r [km]

p n u

d s

0 100 200 300

ε,p [MeV fm-3 ]

ε/10 p

0 0.5 1

0 1 2 3 4 5 6 7 8 9

ρB[fm-3 ]

r [km]

n p u d s

• Hyperons replaced by strange quark matter

• Very different possible internal structures

• Surface tension + screening enforce ‘quasi’ Maxwell

construction (exact for σ

¦

70 MeV/fm

²

)

• Mass-radius relations with different hadron-quark phase transition constructions:

H Q

⁻⁻⁻

−−

−−

−−−−−+ ++ ++ + +

++ + ++

∼50 fm

e.m. interaction vs. surface tension :

8 10 12 14 16

R [km]

0.0 0.5 1.0 1.5 2.0

M/M

nucleon hyperon

hyp+quark B=100 hyp+quark B=B_eff (ρ_B)

Maxwell

Bulk Gibbs

σ=40

• Screened Gibbs constr.

very close to Maxwell construction

• Maximum mass indepen-

dent of phase transition

Hans-Josef Schulze Catania

Summary:

•

Neutron star physics probes the 4 fundamental interactions:

- Gravitation: Densest object in the Universe - Strong: Nuclear EOS

- Weak: Beta-equilibrium of matter, Neutrino physics - EM: Charge-neutrality, Mixed-phase structures, Crust

Conclusions:

•

Hyperons cannot be ignored !

•

BHF EOS with hyperons predicts M_mx not above ∼ 1.7M_⊙

•

Need “quark matter” to reach higher masses

•

Currently M_mx ≈ 1.9M_⊙ for hybrid stars in this approach

However:

However:

We do not know dark matter.

However:

We do not know dark matter.

We do not know dark energy.

However:

We do not know dark matter.

We do not know dark energy.

Do we know GR at 10 ρ ₀ ?