对称能与非核子自由度及其它
蒋维洲
东南大学物理系
合作者：Bao-An Li，陈列文、任中洲
学生：张广华、张东睿、杨荣瑶、向
仟飞
May 9, 2013 @ USTC
Outline
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Introduction to quantum many-body
approaches and the unsettled
Renormalization, vacuum, dark
energy?
Real puzzle of symmetry energy or
mal-definition?
Symmetry energy and quarks &
hyperons
In neutron stars, effect of dark matter?
Concluding remarks
May 9, 2013 @ USTC
从宇宙大爆炸到黑洞形成
10-10--100 s
质子
Proton
neutron
After 1 Billion years
中子
中子星(R~10km)
黑洞
May 9, 2013 @ USTC
核结构——量子多体问题
•
•
•
•
•
•
宏观模型、宏观-微观模型
准粒子、费米独立气体模型
Hartree Approximation
Hartree-Fock (反对称化）
Brueckner-Hartree-Fock（since1960’s)
Relativistic models (since 1970’s)
Dirac旋量（大分量、小分量），自旋，膺
自旋对称性，极化散射
May 9, 2013 @ USTC
Field-theoretic Hamiltonian approach


i ( x, t )
H (t )  d x 
i
 L , L is Lagrangian,
t
i 

L
i 
 (i / t )


3
Omitting the retardation effect, the Hamiltonian is

1

d


2 
H  d 3 x ( x)( iγ    M ) ( x)
3
xd 3 x ( x) ( x)i (1,2) ( x) ( x)
i
May 9, 2013 @ USTC
Based on the plane wave expansion, it can be written in the
momentumspace
1
HT V
2
with
T

u ( p1 , a1 )( γ  p  M )u ( p2 , a2 )b p1a1b p 2 a 2
p1 , a1 , p2 , a2 ,
 
1
V
u ( p1  q, a1 )u ( p1  q, a2 )i (1,2) *2
2
m

q
p1 p2 , q a1 , a2 , 1
i
a1 , a2 , 2
 u ( p2 , a2 )u ( p1 , a1 ) b p q ,a b p1  q ,a2 b p2 ,a2 b p1 ,a1
1
May 9, 2013 @ USTC
1
Hartree-Fock approximation
Energydensity (withoutretardation effect):
1
2
P erformoperatorcommutations in potentialenergy :
 Oˆ   | b  b   b b
| 
  0 | H | 0  T    V 
0
p1  q , a1
p2  q , a 2
p2 , a 2
p1 ,a1
0
   0 | bp2  q ,a2 ( p2 , p1  q a1 ,a2  bp2 ,a2 b p1  q ,a1 )bp1 ,a1 | 0 
   0 | bp1 ,a2 bp1 ,a1 a1 ,a2  p2 , p1  q | 0    0 | bp2  q ,a2 bp2 ,a2 b p1  q ,a1 bp1 ,a1 | 0 
Using theprojectionoperator | i  i |  1, it becomes
i
 Oˆ   a2 ,a1 a1 ,a2  p2 , p1  q    0 | bp2 q ,a2 bp2 ,a2 | i  i | b p  q ,a bp1 ,a1 | 0 
1
i
1
HartreeFock approximation : takingi  0 & omittingall intermediate states
 Oˆ      
   | b  b
|    | b  b
| 
a2 ,a1 a1 ,a2
p2 , p1  q
0
p2  q , a 2
p2 , a 2
0
0
p1  q , a1
p1 ,a1
  a2 ,a1 a1 ,a2  p2 , p1  q   q ,0 a2a2  a1a1  Fock term Hartreeterm
May 9, 2013 @ USTC
0
Relativistic Hartree-Fock
|
Relativistic Hartree-Fock-Brueckner
May 9, 2013 @ USTC
No quantum effect
• Radiation correction requires the quantization
of Coulomb field
• In strong fields, the renormalization required
• Relativistic Hartree Approximation: vacuum
polarization of mesons (Chin, AP108, 301 (1977).)
No ring energy calculation done in Relativistic Hartree-Fock-Brueckner framework!
Two-body correlation is not enough to depict pairing correlation: BCS, Bogoliubov,…
May 9, 2013 @ USTC
Jiang & Li, Modern Physics Letters A
Vol. 23, No. 40May
(2008)
3393
9, 2013 @ USTC
2 纳米管中真空能量的计算
•真空是无限的，而实际的世
界在一定的边界内是有限的。
例如，无限的真空能量在平
行的不带电的导体板间是有
限的，它带来了可观的吸引
力。
•我们的工作在计算同心柱环
（多层碳纳米管）的真空能
量，给出环层间的吸引。
May 9, 2013 @ USTC
May 9, 2013 @ USTC
Dark Energy
What Is Vacuum?
The Vacuum Energy?
Relevant
symmetry breaking
Are to
WeDynamical
In Vacuum?
Or Explicit symmetry breaking?

3


8 G
2HNf
| mq  0 | : q( 0)q( 0) : | 0 | ,
m '
F. R. Urban and A. R. Zhitnitsky, Phys. Lett.,
c  cQCDcgr av .
B688, 9 (2010; Phys. Rev, D 80, 063001
May 9, 2013 @ (2009);
USTC Nucl. Phys. B835, 135 (2010); also
from Prof. Ma Wei-Xing
  c
“Normal Matter”
4%
Dark Energy
73%
Dark Matter
23%
May 9, 2013 @ USTC
To Recognize the SYMMETRY ENERGY
Liquid-drop model for nuclei
E=
W. D. Myers, W.J. Swiatecki, P. Danielewicz,……
In isospin asymmetric matter:
E(,  )  E(,0)  Esym ( ) 2  O( 4 ),   (n   p ) / 
May 9, 2013 @ USTC
Status quo of Equation of state
Early stage: Nuclear Potentials by Fitting low energy
NN phase shifts and properties of deuteron
Most important:
Nuclear EoS
(1) Saturation properties:
ρ0  0.16 0.03fm3
E/A  M  16  1MeV
κ  230  5MeV, Yo99,PRL82,691
May 9, 2013 @ USTC
Brockmann,Machleidt，PRC42,1965(1990)
P.Danielewicz, R.Lacey,W.G.Lynch,
Science 298(2002)1592
Collective flow data from high energy heavy-ion reactions
May 9, 2013 @ USTC
L.W.Chen,et.al., PRC72, 064309 (05)
May 9, 2013 @ USTC
Fuchs, et.al., arXiv:nucl-th/0511070
Diverse trends from data
1. Xiao, et.al PRL102, 062502 (2009)
2. Feng&Jin, PLB683 (2010) 140
3. P. Russotto,W. Trauntmann, Q.F. Li et al., PLB697, 471(2011)
31.6(ρ/ρ)γ,with γ=0.9+/-0.4：almost linear
May 9, 2013 @ USTC
May 9, 2013 @ USTC
May 9, 2013 @ USTC
例子:
Horowitz,& Piekarewicz,
PRL86,5647
Horowitz,& Piekarewicz, PRC64
062802
May 9, 2013 @ USTC
ApJ2010
May 9, 2013 @ USTC
Nuclear matter symmetry energy aroundρ0
Current constraints on Esym (ρ0) and L from nuclear reactions and structures
(1) TF+Nucl. Mass (1996)
Myers/Swiatecki, NPA 601, 141 (1996)
(2) Iso. Diff. (IBUU04, 2005)
L.W. Chen et al., PRL94, 032701 (2005);
B.A. Li/L.W. Chen, PRC72, 064611(2005)
(3) Isoscaling (2007)
D. Shetty et al., PRC76, 024606 (2007)
(4) PDR in 130,132Sn (2007) (LAND/GSI)
A. Klimkiewicz et al., PRC76, 051603(R)(2007)
(5) Iso. Diff. & double n/p (ImQMD, 2009)
M.B. Tsang et al., PRL102, 122701 (2009);
(6) IAS+LDM (2009)
Danielewicz/J. Lee, NPA818, 36 (2009)
(7) DM+N-Skin (2009)
M. Centelles et al., PRL102, 122502 (2009);
M. Warda et al., PRC80, 024316 (2009)
(8) PDR in 68Ni and 132Sn (2010)
(10) Opt. Pot. (2010)
A. Carbon et al., PRC81, 041301(R)(2010)
C. Xu et al., PRC82, 054607 (2010)
(9) SHF+N-Skin (2010)
(11) Nucl. Mass (2010)
L.W. Chen et al., PRC82, 024321 (2010)
M. Liu et al., PRC82, 064306 (2010) May 9, 2013 @ USTC
Slide from Lie-Wen Chen
Why not I(I+1)?
RMF study on finite nuclei suggests E ~T(T+1), Ban,
et.al. Phys.Lett.B633:231-236,2006.
In nuclear matter, the parabolic approximation
suggests E~T*T
E(,  )  E(,0)  Esym ( ) 2  O( 4 ),   (n   p ) / 
Why so uncertain at high
densities?
May 9, 2013 @ USTC
Tensor force
Factor to soften greatly the Symmetry Energy:
C. Xu and B. A. Li, PRC81, 064612:
I. Vidana, A. Polls, and C. Providencia,
PRC 84, 062801(R)
May 9, 2013 @ USTC
Revisit the tensor force
• Mainly from exchange terms.
• Pion & rho: the most important ingredients of the chiral
perturbative theory.
• Rho meson becomes important in the tensor force due to the
in-medium mass dropping associated with the partial
restoration of the chiral symmetry.
• Vacuum effect on the symmetry energy: vacuum condensate
• Applications in superheavy nuclei, nuclei near drip lines, highspin states…
May 9, 2013 @ USTC
Non-nucleonic degrees of freedom
• Threshold for hyperons reaches at high
densities; 2ρ0-4ρ0
• Hadron-Quark transition may occur for nucleon
overlaps >2ρ0
• In astrophysical extraction of the symmetry
energy, there may be the effect of unknown
dark matter.
• What is the effect of these components in
dense matter?
May 9, 2013 @ USTC
EoS with inclusion of hyperons
 /   E / A  e0 (  )  E sym (  ) 2 ,
  (  n -  p )/( -   ).
Definition of symmetry energy:
E sym
 2 ( /  )
1 2  N2 1 k F2  N

 C

2
*

2

6
E
F 
 0
at given hyperon fraction.
1

E sym (  ) 
( n   p )(1 
)
4

for charge neutral and chemical equilibrated matter
May 9, 2013 @ USTC
RMF models: SLC and SCLd
Demorest, et al., Nature 467, 1081(2010)：PSR J1614-2230
Jiang，Li，and Chen, PLB653，184 （07）；
PRC76, 054314 (07); ApJ756，56（12）
RMF model NL3: Lalazissis, et. al. PRC 55, 540(97).
May 9, 2013 @ USTC
Esym
May 9, 2013 @ USTC
RMF model NL3
May 9, 2013 @ USTC
Symmetry energy with quarks
  (1  Y ) H  Y Q
  (1  Y ) H  Y Q
H
Q
 /    /  ( , Y )  e0 (  , Y )  E sym
(  , Y ) H2  E sym
(  , Y ) Q2 ,
 H  (  n -  p )/(  n   p ).
a function of α
Y is the quark phase proportion, determined by Gibbs conditions.
We just consider the symmetric matterα=0 1  2 ( /  )
E sym 
2
 2
 0
p H  pQ , u  n / 3  2e / 3, d  s  n / 3  e / 3
May 9, 2013 @ USTC
Critical density
Calculation detail: Charge chemical potential neglected here!
May 9, 2013 @ USTC
H
Esym
May 9, 2013 @ USTC
Symmetry energy of quark flavor
May 9, 2013 @ USTC
With Λ hyperons
May 9, 2013 @ USTC
Tolman-Oppenheimer-Volkoff (TOV)方程
质量~1.4M¤，半径~11km
中子星
May 9, 2013 @ USTC
含暗物质及其相互作用的中子星
暗物质与正常物质
只有引力相互作用：
May 9, 2013 @ USTC
Symmetry energy extraction with
Dark matter
Steiner:PRL 108, 081102 (2012)
Effect of Dark Matter?
May 9, 2013 @ USTC
Free mass of dark matter 10GeV
May 9, 2013 @ USTC
Remarks not to summary:
•
•
Quantum many-body problems are still problems, some of
which are far off unsettled.
Tensor force? three-body force…
•
Symmetry Energy is a mirror to reflect all these problems.
•
Vacuum effect on symmetry energy? non-nucleonic
degrees of freedom disturb Symmetry Energy.
•
We should not have to regard Symmetry Energy as a
beauty in mind, but a field beyond which we can grasp
something, different, new, or even fundamental…
•
For us, Dark Energy & Dark Matter are still a dream, not a
story.
May 9, 2013 @ USTC
Thank you for your attention!
Acknowledgment
• 兰州重离子加速器国家实验室理论物理中心
• National Natural Sciences Foundation of China
under grant Nos.10405031, 10235030 ,
11275048
• China Jiangsu Provincial Natural Science
Foundation under GrantNo.BK2009261
May 9, 2013 @ USTC