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Modification of nucleon spectral function in the nuclear medium from QCD sum rules
Collaborators: Philipp Gubler(ECT*), Makoto Oka
Tokyo Institute of Technology Keisuke Ohtani
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Outline
•Nucleon QCD sum rule in vacuum
•Summary
•QCD sum rules
•Nucleon QCD sum rule in nuclear matter
•Introduction
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Introduction
・ Hadron properties in the nuclear medium
N
N
N
N
N N
N
N
NN
N
Probe hadron
Nuclear matter
The properties of the hadron are modified in the medium.
Partial restoration of the chiral symmetry
Interaction with the nucleons in the nuclear matter
We focus on the nucleon ground state and its negative parity excited state.
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IntroductionMass spectrum of the nucleons
p, n
N(1440)
N(1535)
N(1650)
Positive parity Negative parity
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IntroductionMass spectrum of the nucleons
p, n
N(1440)
N(1535)
N(1650)
Positive parity
• It is predicted that Chiral symmetry breaking cause these difference.
Negative parity
• The mass difference between nucleon ground state and N(1535) is about 600 MeV.
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IntroductionMass spectrum of the nucleons
p, n
N(1440)
N(1535)
N(1650)
Positive parity
• It is predicted that Chiral symmetry breaking cause these difference.
Negative parity
• The mass difference between nucleon ground state and N(1535) is about 600 MeV.
To investigate these properties from QCD, non perturbative method is needed.
When chiral symmetry is restored, the mass spectrum will change.
Analysis of QCD sum rule in nuclear matter
• The mass difference between nucleon ground state and N(1535) is about 600 MeV.
?In the nuclear matter
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is calculated by the operator product expansion (OPE)
Non perturbative contributions are expressed by some Condensates.
QCD sum rulesHadronic spectral function
, ・・・
An order parameter of chiral symmetry
Application of the analyses in the nuclear matter.
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QCD sum rulesVacuum Nuclear matter
N
N
N
N
N N
N
N
NN
N
Probe hadron
In nuclear matterIn vacuum
Probe hadron: Ground state of nuclear matter
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Nuclear matter
N
N
N
N
N N
N
N
NN
N
In nuclear matterIn vacuum
Probe hadron
Modification:
Chiral condensate :
New condensate:
Probe hadron
: Ground state of nuclear matter
QCD sum rulesVacuum
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QCD sum rules
τ 、 s: parameterOPE side
Gaussian sum rule
Vacuum Nuclear matter
is calculated by OPETransformation
We extract the information on the spectral function with maximum entropy method (MEM).
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contains the term
Nucleon QCD sum rule in vacuumThe behavior of the OPE data in the vacuum
The difference between positive parity and negative parity is mainly caused by chiral condensate term.
Positive parity OPENegative parity OPE
term
Parity: + Parity: -
Mass spectrum of the nucleons
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930MeV
Positive parity Negative parity
1550 MeV
In both positive and negative parity, the peaks are found.In the negative parity state, the peak correspond to the N(1535) or (and) N(1650).
Nucleon QCD sum rule in vacuum
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The behavior of the OPE data in the nuclear matter
Vacuum Nuclear matter
N
N
N
N
N N
N
N
NN
N
Probe nucleon
In nuclear matterIn vacuum
Probe hadron
: Ground state of nuclear matter
Nucleon QCD sum rule in nuclear matter
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Positive parity OPE Negative parity OPE
term
term
ρN: nuclear matter density
The behavior of the OPE data in the nuclear matter
Vacuum Nuclear matter
Parity: + Parity: -
Mass spectrum of the nucleonsVacuum
Nucleon QCD sum rule in nuclear matter
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contain the terms
Positive parity: OPE data decreases.
Negative parity: OPE data slightly increases.
ρ = 0.25ρNPositive parity OPE Negative parity OPE
term
term
ρN: nuclear matter density
The behavior of the OPE data in the nuclear matter
Parity: + Parity: -
?Mass spectrum of the nucleons
contains the term
Nucleon QCD sum rule in nuclear matter
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Positive parity OPE Negative parity OPE
term
term
ρN: nuclear matter density
The behavior of the OPE data in the nuclear matter
Parity: + Parity: -
?Mass spectrum of the nucleonsρ = 0.5ρN
contain the terms
contains the term
Positive parity: OPE data decreases.
Negative parity: OPE data slightly increases.
Nucleon QCD sum rule in nuclear matter
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Positive parity OPE Negative parity OPE
term
term
ρN: nuclear matter density
The behavior of the OPE data in the nuclear matter
Parity: + Parity: -
?Mass spectrum of the nucleonsρ = 0.75ρN
contain the terms
contains the term
Positive parity: OPE data decreases.
Negative parity: OPE data slightly increases.
Nucleon QCD sum rule in nuclear matter
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Positive parity OPE Negative parity OPE
term
term
ρN: nuclear matter density
The behavior of the OPE data in the nuclear matter
Parity: + Parity: -
?Mass spectrum of the nucleonsρ = 1.0ρN
contain the terms
contains the term
Positive parity: OPE data decreases.
Negative parity: OPE data slightly increases.
Nucleon QCD sum rule in nuclear matter
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Positive parity OPE dataNegative parity OPE data
term
term
: n=n0
: ρ=0.75ρ0
: ρ=0.5ρ0
: ρ=0.25ρ0
: Vacuum
Positive parity Negative parity
ρN : nuclear matter density
Vacuum
Nucleon QCD sum rule in nuclear matter
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ρ = 0.25ρN Positive parity OPE dataNegative parity OPE data
term
term
: n=n0
: ρ=0.75ρ0
: ρ=0.5ρ0
: ρ=0.25ρN
: Vacuum
ρN : nuclear matter density
Positive parity Negative parity
Nucleon QCD sum rule in nuclear matter
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ρ = 0.5ρN Positive parity OPE dataNegative parity OPE data
term
term
: n=n0
: ρ=0.75ρ0
: ρ=0.5ρN
: ρ=0.25ρN
: Vacuum
ρN : nuclear matter density
Positive parity Negative parity
Nucleon QCD sum rule in nuclear matter
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ρ = 0.75ρN Positive parity OPE dataNegative parity OPE data
term
term
: n=n0
: ρ=0.75ρN
: ρ=0.5ρN
: ρ=0.25ρN
: VacuumPositive parity Negative parity
ρN : nuclear matter density
: ρ=1.0ρ0
Positive parity Negative parity
Nucleon QCD sum rule in nuclear matter
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ρ = 1.0ρN Positive parity OPE dataNegative parity OPE data
term
term
: n=n0
: ρ=0.75ρN
: ρ=0.5ρN
: ρ=0.25ρN
: VacuumPositive parity Negative parity
ρN : nuclear matter density
: ρ=1.0ρN
Positive parity Negative parity
The values of the energies are obtained.
Nucleon QCD sum rule in nuclear matter
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Vacuum Nuclear matter
N N
N
N
N
N
N N
N
Probe nucleon
Propagator: Propagator:
N
NN
N
Pole of positive energy state:
Effective mass:
Pole of negative energy state:
Nucleon QCD sum rule in nuclear matter
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Investigation of effective masses and vector self energies
210 q
210 q
u
By fitting the phenomenological side and OPE side, we can investigate the self energies.
+ (contribution of negative parity states)
are calculated by OPE are expressed by the propagators in physical energy region
Nucleon QCD sum rule in nuclear matter
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ρN: nuclear matter density
Nucleon QCD sum rule in nuclear matter
Vacuum ρ=0.25 ρN ρ=0.5 ρN ρ=0.75 ρN ρ=1.0 ρN
810
120
151040
660
270
1460
80
490
430
1430
110
290
640
1410
130
1550
930
0
0
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•We analyze the nucleon spectral function by using QCD sum rules with MEM
Summary
• The information of not only the ground state but also the negative parity excited state is extracted
• We find that the difference between the positive and negative parity spectral function is mainly caused by the chiral condensate.
•We apply this method to the analyses in nuclear medium and investigate the effective masses and the vector self-energies.
•As the density increases, the effective masses decrease and the vector self-energies increase.