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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.