高密度原子核物質の対称エネルギー
熱場の量子論とその応用 研究会
磯部忠昭
理研仁科センター

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原子核: 陽子と中性子からなる多体系• コアを持たない自己だけからなる量子多体系• 微視的な描像と巨視的な描像を併せ持つ• 一粒子の重ね合わせbulkな系• 1990年くらいまではZ~Nの安定な原子核を使った実験が盛んだった• 1990年以降は不安定核の生成技術があがり、Z≠N原子核を使った実験が盛んに• 陽子数と中性子数を独立にコントロールする
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Contents• 原子核状態方程式(EOS)と対称エネルギー• これまでの原子核実験による対称エネルギーの研究• 宇宙物理学における対称エネルギーの重要性• 重イオン衝突実験によるEOSと対称エネルギーの研究• 重イオン衝突におけるパイオン測定を通した対称エネルギーの研究

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Important term in nuclear EOS :Symmetry energy• The term depends on δ𝐸𝐸 𝑇𝑇, 𝜌𝜌, 𝛿𝛿 = 𝐸𝐸 𝑇𝑇, 𝜌𝜌, 𝛿𝛿 = 0 + 𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 (𝑇𝑇, 𝜌𝜌)𝛿𝛿 2 + 𝑂𝑂(𝛿𝛿 4 )𝛿𝛿 = (𝜌𝜌𝑛𝑛 − 𝜌𝜌𝑝𝑝 )/𝜌𝜌
• T~0の近似で議論する事が多い• 原子核構造測定による制限が強い為• 中性子星の様な極低温マターを議論する上ではT~0の極限で問題ない為• 後に出てくるHICでは考慮しないといけないが、考慮されない事が多い

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Important term in nuclear EOS :Symmetry energy
𝐸𝐸 𝑇𝑇, 𝜌𝜌, 𝛿𝛿 = 𝐸𝐸 𝑇𝑇, 𝜌𝜌, 𝛿𝛿 = 0 + 𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 (𝑇𝑇, 𝜌𝜌)𝛿𝛿 2 + 𝑂𝑂(𝛿𝛿 4 )𝛿𝛿 = (𝜌𝜌𝑛𝑛 − 𝜌𝜌𝑝𝑝 )/𝜌𝜌

arXiv:1303.0064

𝛿𝛿 = 1, N=A
𝛿𝛿 = 0, N=Z7

Parameters to discuss the EOS quantitatively
Only the numerical values of K, L, and S• Assume low T𝐸𝐸 𝜌𝜌, 𝛿𝛿 = 𝐸𝐸 𝜌𝜌, 𝛿𝛿 = 0 + 𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 (𝜌𝜌)𝛿𝛿 2 + 𝑂𝑂(𝛿𝛿 4 )𝐾𝐾0 𝜌𝜌 − 𝜌𝜌0𝐸𝐸 𝜌𝜌, 𝛿𝛿 = 0 = 𝐸𝐸 𝜌𝜌 = 𝜌𝜌0 +23𝜌𝜌02𝛿𝛿 = (𝜌𝜌𝑛𝑛 − 𝜌𝜌𝑝𝑝 )/𝜌𝜌+ 𝑂𝑂

𝐾𝐾𝑠𝑠𝑠𝑠𝑠𝑠 𝜌𝜌 − 𝜌𝜌0𝜌𝜌 − 𝜌𝜌0𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 𝜌𝜌 ~𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 𝜌𝜌 = 𝜌𝜌0 + 𝐿𝐿+3𝜌𝜌023𝜌𝜌023𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 𝜌𝜌 ~𝐸𝐸𝑘𝑘 𝜌𝜌 + 𝐸𝐸𝑝𝑝 𝜌𝜌𝛾𝛾 : Fermi Gas model

2+ 𝑂𝑂

8Parameters to discuss the EOS quantitatively
Only the numerical values of K, L, and S
arXiv:1303.0064

stiff
Slope :L𝛿𝛿 = 1, N=A

Incompressibility :K𝛿𝛿 = 0, N=Zsoft• Qualitatively theEOS is expressed
as soft or stiff.

Symmetry Energy at ρ=ρ0 :S• Soft matter is easy
to be compressed.
9Experimental constraint for nuclear EOS isgiven only around ρ~ρ0, N~ZLattimer

Esym(ρ)=12.5(ρ/ρo)2/3 +C (ρ/ρo)neutrinosphere10Constraint on nuclear symmetry energy
based on nuclear structure• Nuclear skin thickness: Slope parameter (L) and NeutronSkin• Large L ⇔ Small Esym in low ρ ⇔ Thick neutron skin
• Small L ⇔ Large Esym in low ρ ⇔ Thin neutron skinDensity distribution of protons
and neutrons in a nucleus

X. Roca-Maza et al., PRL106, 252501 (2011)11

Motivation to study symmetry term ofnuclear EOS:
important for nuclear/astro-physics• Nuclear structure• Sustain neutron rich matter of heavy ions/super heavy ions.
• Information for beyond the super heavy elements.

• Supernovae process• Symmetry term affects supernovae explosion dynamics.

• Neutron star structure• Supreme asymmetry nuclear matter• Gravitational wave

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Courtesy of K. Kotake

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Two representative (non-polytrope) EOSs
in recent supernova simulations• Only one EOS was available for simulation
till ~2000. （Lattimer & Swesty’91)

incompressibility

K = 180,220,315 MeV• Made in Japan EOS.
EOS needs to cover: 0 ≤ 𝑇𝑇 ≤ 100 MeV
104 ≤ 𝜌𝜌 ≤ 1015 g/cm30 ≤ 𝑌𝑌𝑒𝑒 ≤ 0.6

K = 281 MeV （stiff）(Shen et al ’98)

 5ρ014Current issue on supernovae
simulation related to EOS• 2000年代後半~ Succeeded to reproduce the explosion
by employing 2D, 3D models.
• Available with HPC.

• Still the released energy from explosion is not large
enough to explain the observation.
• 1~2 order smaller• Prefer soft EOS to reproduce the explosion.

• Softer EOS leads to a more compact PNS, stronger instabilities,earlier explosions.

• EOS at ρ=2~3ρ0,F(proton)~0.3-0.35 isimportant.

Couch 201215

Neutron Star: balance of gravity (pulls in) andnuclear force (pushes out): Masses vs. Radii
Strange quark matter

Lattimar & PrakashNuclear matter

Average mass of NS

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Strong constraint on EOS byobservation of heavy neutron star
• Mass of J1614-2230 pulser is 2 solar mass
Demorest et al., Nature 467 (2010) 1081

Shapiro delay

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Neutron Star: balance of gravity (pulls in) andnuclear force (pushes out): Masses vs. Radii
Strange quark matter

Lattimar & PrakashNuclear mattertoo soft18

Observation of 2 solar mass NS forced
nuclear physicist to consider 3NF
NN
NNN (T=3/2??)
NY
NNY?
NYY?
YYY?

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Nuclear matter – Y, q matter x-over234• The crossover from hadronic matter to quark matter at ~3ρ0.
• The quark matter is strongly interacting in the crossover
region and has a stiff equation of state.
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Masuda PTEP 2013, 073D01

Recent observations of Neutron Stars (Radii)

S. Guillot, et al Astrophys. J.
772, 7 (2013), 1302.0023

Lattimar & Prakash

too soft
Steiner
Suleimanov

Very small Neutron Star radius rules out nearly all EOS21

Density dependence of symmetry energy
at supra-saturation density
Wiringa, Fiks, & Fabrocini 1988

Skyrme interactions

Neutron Star obs.
or HIC• Above saturation density, the symmetry energy
density dependence may have a different
energy dependence than Skyrme interactions. 22

Constraining the EOS at high densities byheavy ion collisions
Au+Au collisions E = 1 AGeV

pressurecontours

densitycontours• Constraints for N~Z system

23Constraining the EOS at high densities byheavy ion collisions

Au+Au collisions, PRC87 067601• Constraints for N~Z system

BES@RHIC
8ρ0 @CBM
5ρ0 @JPARC-HI

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Constraining the EOS at high densities byheavy ion collisions
Au+Au collisions E = 1 AGeV

pressurecontours

densitycontours• Two main observable consequences of the high pressuresthat are formed:– Nucleons deflected sideways in the reaction plane. (v1)
– Nucleons are “squeezed out” above and below the25
reaction plane. (v2)

Flow studies of the symmetric matter EOS
Flow inHIC

Transport
EOS
theoryMean field U• Theoretical tool: transport theory:• QMD: Quantum Molecular Dynamics• BUU: Boltzmann-Uehling-Uhlenbeck eq. (Bertsch Phys. Rep. 160, 189(1988).).

• Each nucleon is represented by ~1000 test particles that propagate
classically under the influence of the self-consistent mean field U
and subject to collisions due to the residual interaction.
• They can describe nucleon flows, the nucleation of weakly boundlight particles and the production of nucleon resonances.
• The production of heavier fragments is a difficult problem.
• Anti-Symmetrized Molecular Dynamics (AMD) or other.
• Sensitive to fluctuations in the mean field that give rise to spinodal
decomposition.

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Theoretical problem: constraining themomentum dependence of mean field
Danielewicz, Nucl. Phys. A 673, 375.

• Momentum dependence ofN-N interaction.
• Well established forsymmetric matter.
• At low energies, it can bedescribed by effective mass,m*:

• 𝑝𝑝2

1
1−2𝑚𝑚∗
2𝑚𝑚

𝜕𝜕𝑈𝑈
2= 𝑝𝑝
�2
𝜕𝜕𝑝𝑝 𝑝𝑝2 =027

Theoretical problem: constraining σNN
PRC57(1998)R1032• With free cross-sections (red/pink) transport
overestimates stopping data.

• Different reduced cross section can explain data.

• Size of elementary cross section should not exceed theaverage interparticle separation?

𝜎𝜎 = 𝜎𝜎0 tanh 𝜎𝜎𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 /𝜎𝜎02−𝜎𝜎0 = 𝑦𝑦𝜌𝜌 3• Cross section modified by Pauli principle and density of
nucleonic states, from many-body calculations
28PRC55(1997)3006

Directed transverse flow
targetpxyprojectile• Event has “elliptical” shape in
momentum space, with thelong axis in the reaction plane.
• The data display the “s” shapecharacteristic of directedtransverse flow.
• Should be symmetric but the
TPC has inefficiencies at
y/ybeam< -0.2.
• Slope 𝑑𝑑 𝑃𝑃𝑥𝑥 /𝐴𝐴 /𝑑𝑑𝑑𝑑 isdetermined at-0.2<y/ybeam<0.3

Partlan, PRL 75, 2100 (1995).

Au+Au collisions
EOS TPC data

Ebeam/A(GeV)y/ybeam (in C.M)

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Determination of symmetric matter EOS
from nucleus-nucleus collisions
Danielewicz et al., Science 298,1592 (2002).

𝜕𝜕 𝑝𝑝𝑥𝑥 /𝐴𝐴𝜕𝜕𝑦𝑦• The curves labeled by Knm
represent calculations withparameterized Skyrme mean fields.
• Adjusted to find the pressure that
replicates the observed transverseflow.

• The boundaries represent therange of pressures obtained forthe mean fields that reproduce thedata.
• They also reflect the uncertainties
from the effective masses and inmedium cross sections.
30Determination of symmetric matter EOS
from nucleus-nucleus collisions
Danielewicz et al., Science 298,1592 (2002).

In plane

Out ofplane• The curves labeled by Knm
represent calculations withparameterized Skyrme mean fields.
• Adjusted to find the pressure that
replicates the observed transverseflow.

• The boundaries represent therange of pressures obtained forthe mean fields that reproduce thedata.
• They also reflect the uncertainties
from the effective masses and inmedium cross sections.
31Constraints from collective flow on EOS at
ρ>2ρ0.
E/A (ρ, δ) = E/A (ρ,0) + δ2⋅S(ρ)δ = (ρn- ρp)/ (ρn+ ρp) = (N-Z)/A≈1• Note: analysis required additional
constraints on m* and σNN .

• The symmetry energy dominates theuncertainty in the n-matter EOS.

• Flow confirms the softening of the EOS
at high density.

• Both laboratory and astronomical
constraints on the density dependenceof the symmetry energy are needed.

• Constraints from kaon production areconsistent with the flow constraints
and bridge gap to GMR constraints.

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How about the symmetry energy at ρ>ρ0stiffsoft33

One of the high density probes for
Esym: pion production• Larger values for ρn/ρp at high density: softer Esym
causes stronger emission of negative pions.
• In delta resonance model, Y(π-)/Y(π+)≈(ρn/ρp)2• In equilibrium, μ(π+)-μ(π-)=2(μp-μn)

10%34Beams for the experimental study with pion
ratiosZhang et al., PRC 80 (2009) 034616

Xiao et al., PRL 102 (2009) 062502• Sensitivity to symmetry energy is larger for neutron-richbeams.
• Sensitivity increases with decreasing incident energy.
35Symmetry energy studies at ρ~2ρ0• Densities of ρ≈2ρ0 can be achieved at E/A ≈400 MeV HIC.
• Esym(ρ) influences diffusion of neutrons from dense overlap
region at b=0.

• Diffusion is greater in neutron-rich dense region is formed for stiffer
Esym(ρ).
Yong et al., Phys. Rev. C 73, 034603 (2006)

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Present status of Esym(ρ)
Adapted Horowitz, et al., J. Phys.
G: Nucl. Part. Phys. 41 (2014).

• Need to deal with momentum dependence of mean field.

• Important for non-zero temperatures and non-equilibrium systems• Difficulty of the study with HIC  discrepancy at ρ~2ρ0• Systematic study is necessary.

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Pion production: quantitative comparison
is necessary
Yong et al., PRC 73 (2006) 034603• Pion spectra, ratios of pion spectra and pion flows
ratios can provide more quantitative comparisons,and tests of theory.
• However the Coulomb interaction strongly
influences such observables.
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Pion double ratio
Yong et al., PRC 73 (2006) 034603

𝑌𝑌 𝜋𝜋 − 132+124 /𝑌𝑌 𝜋𝜋 + 132+124
𝑌𝑌 𝜋𝜋 − 112+112 /𝑌𝑌 𝜋𝜋 + 112+112• Yong et al., have proposed measuring pion double ratios
involving two systems with the same charge but different
neutron number.

• Can remove the sensitivity to Coulomb and retain sensitivity tosymmetry energy.

• Largely removes sensitivity to difference between π- and π+
acceptances.
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Pion production is under ctrl.?
JAM+AMD
Ikeno et al., PRC 93 (2016) 044612

Reaction rate of Delta

Production

Absorption~60% of pions absorbed.
Sensitivity on Esym decreases after
taking into account the clustering effect40

Heavy RI Collision program @RIKEN-RIBF• Experimental project to give a constrain on the density
dependent symmetry energy main for higher dense region.
• Systematic measurement in same Z but different N system
realized with heavy RI beam.

• Interesting comparison because the Coulomb interaction is the same for bothto first order. Coulomb also strongly influences the pion ratios.
• ρ~2ρ0 nuclear matter at RIBF energy (E/A=300MeV).
Primary Beam Target Ebeam/A(N-Z/A)sys

238U

132Sn

124Sn

300

124Sn

112Sn

300

0.15

124Xe

108Sn

112Sn

300

0.09

112Sn

124Sn

300

0.15

• Successfully finished.
• 2016 Apr. – Jun.

0.2241Heavy RI collision experimental setup
Chamber in SAMURAI magnet B=0.5T

132Sn

NeuLAND

Beam Tracker

STQ

MWPC typeTime Proj.
Chamber
12k channel

Measure differential flow
and yield ratios for (π+ &π-), (p & n), (3H & 3He) inHeavy RI Collisions
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Collaboration for New Device at RIBF:
SPiRIT

SAMURAI Pion Reconstruction and Ion Tracker
International Collaboration aiming to study
density dependent symmetry energy
through Heavy RI Collision experiments.

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Charged particle measurement with TPC
132Sn + 124Sn @E/A=300MeV

dE/dX (A.U.)

TPC Top view

PID by TPC (132Sn + Nat. Sn @E/A=300MeV)

pπ− π+dt

preliminaryp/Q (MeV/c)

Clearly seen pion signals.
S/N of pion will be improved after applying cuts for clean tracks.

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Magnetic effect on pion production

PRC84.064605• Magnetic effect caused by HIC seems not tobe minor effect.
• 5 x 1015 G (~1017@RHIC energy)

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Neutron and proton observables• Sensitive to the “symmetryenergy” and the “n and peffective mass difference”.

• Mean field for symmetry
matter has large uncertainty.
• Significant influence on theexcitation energy.
• 𝑇𝑇 2 ∝ 𝐸𝐸 ∗ /𝐴𝐴 � 𝑚𝑚𝑛𝑛 /𝑚𝑚𝑛𝑛∗

Ratio of neutron over proton spectraout of the reaction plane.
Li et al., Phys. Rep. 464 (2008) 113• Most models predict thedifferences between:• neutron and proton flow
• t and 3He flows

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Neutron and proton transverse flow

Li et al., arXiv:nucl-th/0504069• Transverse directed flow isusually obtained byplotting the meantransverse momentum<px> vs. the rapidity y.
• The neutron-proton
differential flow is defined
here to be:w i = 1 (-1); neutron (proton)47

Symmetry energy effects on the mixed
hadron-quark phase at high ρ?
MIT-Bag
Energy density:ε(free gas) + B

hadronQuark:
Fermi onlysymmetric
neutron

PhysRevC.83.014911
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Symmetry energy effects on phase
diagram.
Symmetric

Asymmetric• Large isospin effects at high densities and finite
temperatures.
G.Y. Shao et al., NICA White Paper
PhysRevD.84.034028 (2011) 49

Phase Diagram of Strongly Interacting Matter

NICA White Paper

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Summary• Nuclear EOS and its symmetry energy is studied
through astrophysical observables and nuclearexperiments.
• Supernovae explosion  prefer soft EOS• Heavy NS  support stiff EOS• Strong constraint• Small radii NS  soft EOS• Pion production in HIC  soft EOS?
• Experimental study on the symmetry energy forhigh dense region started by using heavy RI
collision.

• Systematic study with various observables, energies
and systems are necessary to understand one by one.
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