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Nuclear Symmetry Energy from QCD Sum Rule

Phys.Rev. C87 (2013) 015204 

Recent progress in hadron physics -From hadrons to quark and gluon- Feb. 21, 2013

Kie Sang JEONGSu Houng LEE

(Theoretical Nuclear and Hadron Physics Group)

Yonsei University

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Motivation 1 – KoRIA plan

• Rare Isotope Accelerator Plan

(Quoted from Physics Today November 2008)

Nuclear Symmetry Energy plays a key role in Rare Isotope and Neutron Star study

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Motivation 2 – RMFT QCD SR• Dirac phenomenology of nucleon-nucleus scatter-

ing

• This tendency naturally comes from RMFT

Dirac optical potentialStrong scalar attraction Re S<0

Strong vector repulsion Re V>0

QHD model, DBHF approximation etc.

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• For symmetric nuclear mat-ter, this large cancelation mechanism between the self energies can be understood in QCD degree of freedom

• Could asymmetric nuclear matter be understood in QCD degree of freedom?

• We applied QCD Sum Rule to asymmetric nuclear matter

Physical Review C 49, 464 (1993)(Thomas Cohen et al. (1992))

Motivation 2 – RMFT QCD SR

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Early attempt for finite nuclei• Liquid drop model • Total shifted energy

Total shifted states number

Nuclear Symmetry En-ergy

This simple concept can be generalized to infinite matter case

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For infinite nuclear matter• Energy per a nucleon

• Density expansion of single nucleon energy

• Averaged single nucleon energy

Nuclear Symmetry Energy

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Mean field approximation• RMFT type quasi-nu-

cleon propagator

• QCD Sum Rule is a well established method for investigating the quasi-hadronic state in nuclear medium

• Quasi-particle on the Fermi sea

(Up to linear density order)

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• Correlation function

• Energy dispersion relation

• Phenomenological ansatz in hadronic d.o.f.

Contains quantum number of pro-ton

QCD Sum Rule

Contains all possible hadronic resonance states in QCD degree of freedom

Equating both sides, nucleon self ener-gies can be expressed in QCD degree of freedom

is medium four-veloc-ity

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• Self energies near quasi-nucleon pole

• At short distance, Wilson coefficient can be ob-tained by perturbative calculation

Contain all non-perturbative con-tributionFor example,

(These figures are quoted from Ph.D. thesis of Thomas Hilger)

Kinetic part is ex-cluded

QCD Sum Rule

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• Linear gas approximation

• Iso-scalar/vector operators in light quark flavor

is well known from many previous sum rule studies

In-medium condensate in asymmetric matter

Iso-scalar opera-tor

Iso-vector opera-tor

can be related with by appropriate ratio factor which comes from chiral perturbation, DIS structure functions… etc.

Nucleon expectation value X Nucleon density

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Borel transformation• To emphasize quasi-nucleon pole contribution

• Borel transformed invariants

For OPE side

For phenomenological side, equivalent weight function has been used

With continuum correc-tion

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Self energies with OPEs• Self energies can be obtained by taking ratio

• New symbols for self energies

To treat self energies in terms of density order and asymmetric factor, self energies need to be re-arranged

Power of in the first index represents the density power orderPower of I in the second index represents the iso-spin order

Numerator of total self energy

Denominator of total self energy

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QCD sum rule Formula• General expression

• Expression up to linear density order

• We need both sum rule

We also need general expression which contains up to 2nd order density to check density behavior of Nuclear Symmetry Energy

Linear gas approxima-tion

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Sum Rule analysis up to dimension 5• Main ingredient?• Nuclear Symmetry En-

ergy

Nuclear Symmetry Energy is ~40 MeV and do not strongly depend on Consistency with previous studies in order of magnitudeMainly consists of `potential’ like part -> Phenomenological ansatz works well

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4-quark Operator Product Expansion• By using Fierz rearrangement we have found the exact

four-quark operators in the nucleon sum rule

With constraint from `zero identity’ and as-sumed P, T symmetry of medium ground state

Among these, except for , all proton expectation value of twist-4 operators can be estimated from experiment (DIS)

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Twist-4 operator from DIS data

(SHL et al., Physics Letter B 312 (1993) 351-357)

Where, Ku, Kd and Kud are

From these result…

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Twist-4 operator from DIS data• Table for Twist-4 matrix elements for the nucleon

SR

Only first set gives physically meaningful results

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• Iso-vector scalar / vector decomposition of Nu-clear Symmetry Energy

• RMFT result

• In the result without twist-4 ops., both self energies give weak contribution

• Which part gives re-ducing contribution in our QCD sum rule?

Vector meson exchange -> RepulsiveScalar meson exchange -> Attrac-tive(V.Baran, et al. Physics Reports 410 (2005)

335–466)

Iso-vector meson exchange

Nuclear Symmetry Energy with Twist-4 ops.

Contribution of Twist-4 Ops. mimics rho-delta meson exchange chan-nel in RMFT

Esy

m (G

eV)

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DIS data will be improved• Table for Twist-4 matrix elements has some un-

certainty• Jefferson Lab has a plan for accelerator up-

grade

• This plan will lead to more precise structure func-tions for Twist-4 matrix elements

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At extremely high density?• The origin of nuclear reaction

in asymmetric matter may comes from Multi-quark oper-ators like Twist-4 ops.

• At high density, phase of nu-clear matter may become quark phase, and the domi-nating dynamics will be QCD

• We may make naïve guess that this kind of prediction for dense condition might be justified by QCD via study-ing multi-quark operators at high density

• Model predictions for Nu-clear Symmetry Energy at extreme condition

(V.Baran, et al. Physics Reports 410 (2005) 335–466)

C. Xu, Bao-an Li, Lie-wen Chen, Che Ming Ko. / Nuclear Physics A 865 (2011) 1–16

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Conclusion• We have successfully reproduced numerical value of the

Nuclear Symmetry Energy of previous studies and found exact four-quark operators in the nucleon sum rule.

• Twist-4 matrix elements give important contribution to Nu-clear Symmetry Energy. Its contribution mimics the iso-vector meson exchange channel in recent RMFT models.

• Our study shows that Nuclear Symmetry Energy can be un-derstood via QCD, its fundamental origin may comes from quark and gluon degrees of freedom.

• Extremely high density behavior remains unclear, but this also might be understood via QCD. Related study is now in progress.