Effects of Brown-Rho scaling in nuclear matter, neutron stars and finite nuclei T.T.S. Kuo ★ ★...
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Transcript of Effects of Brown-Rho scaling in nuclear matter, neutron stars and finite nuclei T.T.S. Kuo ★ ★...
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
€
VNN
€
VNN
€
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
€
m *
m≈1− C
ρ
ρ 0
,
€
C ≈ 0.2
€
VNN
€
E0
A≈ −16
€
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
€
E0
A≈ −16
€
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).
€
E0 = E0free + ΔE0
€
ΔE0 = dλ Ym (ij,λ )Ym* (kl,λ )
ijkl<Λ
∑m
∑0
1
∫ ij |Vlow−k | kl
€
Ym*(kl,λ ) = Ψm (λ ,A − 2) alak Ψ0(λ , A)
€
AX + BY = ωX
€
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
€
Vlow−k
€
Vlow−k
€
VNN
€
VNN
€
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
€
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 )
€
m *
m≈1− C
ρ
ρ 0
€
m *
m≈1− C(
ρ
ρ 0
)γ ,
€
γ=0.3
1
2
Skyrme 3b-forces (TBF)
€
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 )
€
Vlow−k
€
Vlow−k
€
VSkyrme = V2b (i, j) + V3b (i, j,k)i< j<k
∑i< j
∑
€
V3b
with BR
1
2
We have 3 calculations for EOS:
with BR
unscaled - plus TBF
€
Vlow−k
€
Vlow−k
€
Vlow−k
€
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
€
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.
€
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)
€
Vlow−k
€
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
€
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
€
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
€
Vlow−k
Ring-diagram EOSs, CD-Bonn with
and without TBF
€
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)
€
Vlow−k
€
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:
€
ρ =ρn + ρ p + ρ e + ρ μ
€
ρp = ρ e + ρ μ
€
μe = μμ
€
μn = μ p + μ e
Proton fraction ( ) of β-stable neutron stars
€
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
€
(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
€
VρT (r) =
fNρ2
4πmρτ 1 ⋅τ 2 −S12
1
(mρ r)3+
1
(mρ r)2+
1
3mρ r
⎡
⎣ ⎢
⎤
⎦ ⎥e−mρ r
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
€
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)
€
Vlow−k
€
MGT = ψ f σ (i)τ +(i) ψ i
i=1,2
∑ = − 6(xa − yb / 3)
from BonnB with BR-scaled (m , m , m )
€
VNN ρ ω σ
€
ψi = x 1S0 + y 3P0
€
ψ f = a 3S1 + b 1P1 + c 3D1
€
14C :
€
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
€
MGT for
€
14C→14N decay
GT0
€
ρ ≈0.75ρ 0
€
MGT ≈ 0at
Ericson (1993) scaling:
Leads to non-linear BRS
Calculations with this scaling for m , m , m in progress
Recall BR scaling is
€
q q(ρ)
q q(0)=
1
1+ ΣN ρ / fπ2mπ
2
€
m *
m≈
1
1+ Dρ /ρ 0
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 3
€
D ≡ ΣN ρ 0 / fπ2mπ
2
€
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
€
m *
m≈1− C
ρ
ρ 0
€
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