Effects of Brown-Rho scaling in nuclear matter, neutron stars

and finite nuclei

T.T.S. Kuo

★

★

Collaborators:

H. Dong (StonyBrook), G.E. Brown (StonyBrook)R. Machleidt (Idaho), J.W. Holt (TU Munchen), J.D. Holt (Oak Ridge)

Brown-Rho (BR) scaling of in-medium mesons (in medium) ≠ (in free space) ? ? Using one-boson exchange (OBE) models, we have studied effects of BR scaling in nuclear matter neutron stars finite nuclei

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VNN

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VNN

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VNN

Brown-Rho scaling: in-medium meson mass m is ‘dropped’ relative to m in vacuum

ρis nuclear matter density, ρ is that at

saturation.

How to determine the Cs ? ?

We adopt: fixing Cs by requiring BR-scaled OBE

(BonnA and Nijmegen) giving symmetric nuclear matter

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m *

m≈1− C

ρ

ρ 0

,

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C ≈ 0.2

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VNN

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E0

A≈ −16

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ρ0 ≈ 0.16 -3

*

0

MeV and fm

Symmetric (N=Z) nuclear matter equation of state (EOS):

Most theories can not ‘simultaneously’

reproduce its binding energy and saturation density

This difficulty is well known (the ‘Coester’ band).

-3

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E0

A≈ −16

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ρ0 ≈ 0.16

MeV

fm

Coester band

B.-A. Li el at., Phys. Rep. 464, 113 (2008)

We calculate nuclear matter EOS using a ring-diagram method:

The pphh ring diagrams are included to all orders.

Each vertex isIn BHF and DBHF, only first-order G-matrix

diagram included. €

Vlow−k

EOS with all-order pphh ring diagrams:Ground state energy

The transition amplitudes Y are given by RPA equations

and it is equivalent to treating nuclear matter as a system of“quasi bosons” (quasi-boson approximation).

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E0 = E0free + ΔE0

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ΔE0 = dλ Ym (ij,λ )Ym* (kl,λ )

ijkl<Λ

∑m

∑0

1

∫ ij |Vlow−k | kl

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Ym*(kl,λ ) = Ψm (λ ,A − 2) alak Ψ0(λ , A)

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AX + BY = ωX

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B*X + A*Y = −ωY

is used in our nuclear matter calculation.

It is obtained by ‘integrating’ out the k >Λ components

of , namely

is a smooth (no hard core) potential, and reproduces phase shifts of up to

0

),,(),(),(),,(

22

2'

0

2'2'

iqk

kkqTqkVdqqkkVkkkT NN

NN

0

),,(),(),(),,(

22

2'

0

2'2'

iqp

ppqTqpVdqqppVpppT klowklow

klowklow

);,,(),,( 2'2' pppTpppT klow ),( ' pp

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Vlow−k

€

Vlow−k

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VNN

€

VNN

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2 -1(We use Λ~ 3 fm )

Ring-diagram EOSs for N=Z nuclear matter

with from CDBonn and BonnA ,

and Λ= 3 and 3.5 fm €

Vlow−k

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VNN

Empirical values: €

E0

A≈ −16

€

ρ0 ≈ 0.16

MeV

fm-3

-1

Linear BR scaling (BR ), not suitable for large ρ

Non-linear BR scaling (BR )

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m *

m≈1− C

ρ

ρ 0

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m *

m≈1− C(

ρ

ρ 0

)γ ,

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γ=0.3

1

2

Skyrme 3b-forces (TBF)

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V3b = t3δ(r r i −

r r j )δ(

r r j −

r r k ) →

1

6(1+ x3Pσ )t3δ(

r r 1 −

r r 2)ρ(

r r av )

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Vlow−k

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Vlow−k

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VSkyrme = V2b (i, j) + V3b (i, j,k)i< j<k

∑i< j

∑

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V3b

with BR

1

2

We have 3 calculations for EOS:

with BR

unscaled - plus TBF

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Vlow−k

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Vlow−k

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Vlow−k

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Vlow−k

Ring-diagram EOSs for N=Z nuclear matter (Λ=3.5 fm )

-1

Effects of BR scaling ≈ that of Skyrme TBF

alone, too soft

with BR , too stiff

with BR and plus TBF satisfactory

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Vlow−k

1

2

Ring-diagram EOS for N=Z nuclear matter using

( plus TBF ) with CDBonn, BonnA, Λ =3 and 3.5 fm

A common t =2000 MeVfm used for all cases.

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Vlow−k

-1

63

Can we test EOSs and BRs at high densities ( ρ ≈ 5ρ ) ?

Heavy-ion scattering experiments (e.g. Sn

+ Sn )

Neutron stars where ρ≈ 8ρ

0

132132

0

Experiment constraint for N=Z nuclear matter

Danielewicz el at., Science 298, 1592(2002)

Comparison with the Friedman-Pandharipande (FP) neutron matter EOS

solid lines: FP

various symbols: + TBF

dotted line: only (CDBonn)

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Vlow−k

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Vlow−k

Tolman-Oppenheimer-Volkov (TOV) equations for neutron stars:

To solve TOV, need EOS for energy density vs pressure.

Neutron star outer crust ( ρ<~3×10 fm ), Nuclei EOS of Baym, Pethick and Sutherland (BPS)Neutron star core ( >~4×10 M c/km ), Extrapolated polytrope EOS Ring-diagram EOS used for intermediate region

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dp

dr= −

GM(r)ε(r)

c 2r2

[1+p(r)

ε(r)][1+

4πr3 p(r)

M(r)c 2]

1−2GM(r)

c 2r

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dM(r)

dr= 4πr2ε(r)

-4 2 3p

-4 -3

Mass-radius trajectories of pure neutron starsRing-diagram EOSs, CD-Bonn with and

without TBFCausality limit: the straight line in the upper

left core

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Vlow−k

Ring-diagram EOSs, CD-Bonn with

and without TBF

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Vlow−k

Density profile for Maximum mass Pure Neutron stars

Pure neutron stars’ moment of inertiaCD-Bonn with and without TBFMiddle solid points are the empirical constraint

(Lattimmer-Schutz)

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Vlow−k

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I ≈ (0.237 ± 0.008)MR2 ×[1+ 4.2M

M

km

R+ 90(

M

M

km

R)4 ]

Neutron stars with β-stable ring diagram EOS:

Consider medium including p, n, e, μEquilibrium conditions:

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ρ =ρn + ρ p + ρ e + ρ μ

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ρp = ρ e + ρ μ

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μe = μμ

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μn = μ p + μ e

Proton fraction ( ) of β-stable neutron stars

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χ

Ring and nuclei-crust EOSTop four rows with TBF, bottom without TBF

POTENTIALS

M [M ]

R [km] I [M km ]

CDBonn 1.80 8.94 60.51

Nijmegen 1.76 8.92 57.84

BonnA 1.81 8.86 61.09

Argonne V18

1.82 9.10 62.10

CDBonn (V =0)

1.24 7.26 24.30

Mass, radius and moment of inertia of β-stable neutron stars

3b

2

Carbon-14 decay

This β-decay has a long half-life T ≈ 5170 yrs

Tensor force is important for this long life time

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(MGT ≈ 0)

1/2

Tensor forces from π- and ρ-mesons are of opposite signs:

m decrease substantially at nuclear matter density

m remains relatively constant (Goldstone boson)

BR scaling is to decrease the tensor force at finite density

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VρT (r) =

fNρ2

4πmρτ 1 ⋅τ 2 −S12

1

(mρ r)3+

1

(mρ r)2+

1

3mρ r

⎡

⎣ ⎢

⎤

⎦ ⎥e−mρ r

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

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VπT (r) =

fNπ2

4πmπ τ 1 ⋅τ 2 S12

1

(mπ r)3+

1

(mπ r)2+

1

3mπ r

⎡

⎣ ⎢

⎤

⎦ ⎥e

−mπ r ⎛

⎝ ⎜

⎞

⎠ ⎟

ρ

π

Shell model calculations (2 holes in p-p shell) using

LS-coupled wave functions:

Gamow-Teller transition matrix

element (Talmi 1954)

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Vlow−k

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MGT = ψ f σ (i)τ +(i) ψ i

i=1,2

∑ = − 6(xa − yb / 3)

from BonnB with BR-scaled (m , m , m )

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VNN ρ ω σ

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ψi = x 1S0 + y 3P0

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ψ f = a 3S1 + b 1P1 + c 3D1

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14C :

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14N :

ρ/ρ x y a b c M 0

0.844 0.537

0.359

0.168

0.918

-0.615

0.25

0.825

0.564

0.286

0.196

0.938

-0.422

0.50

0.801

0.599

0.215

0.224

0.951

-0.233

0.75

0.771

0.637

0.154

0.250

0.956

-0.065

1.00

0.737

0.675

0.103

0.273

0.956

0.074

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MGT for

€

14C→14N decay

GT0

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ρ ≈0.75ρ 0

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MGT ≈ 0at

Ericson (1993) scaling:

Leads to non-linear BRS

Calculations with this scaling for m , m , m in progress

Recall BR scaling is

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q q(ρ)

q q(0)=

1

1+ ΣN ρ / fπ2mπ

2

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m *

m≈

1

1+ Dρ /ρ 0

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 3

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D ≡ ΣN ρ 0 / fπ2mπ

2

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m *

m≈1− C

ρ

ρ 0

with

ρ ω σ

Summary and outlook:Effects from BR scaling is important and

desirable for

nuclear matter saturation, neutron stars and C β-decay.

At densities (<~ρ ), BR scaling is likely linear,

but at high densities it is an OPEN question.

BR scaling is similar to Skyrme

0

14

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m *

m≈1− C

ρ

ρ 0

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V3b = t3δ(r r i −

r r j )δ(

r r j −

r r k )

Thanks to organizers

A. Covello, A. Gargano, L.Coraggio and N. Itako