1. 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

2. Nucleon QCD sum rule in vacuum Summary QCD sum rules Nucleon QCD sum rule in nuclear matter Introduction

3. ・ Hadron properties in the nuclear medium N N N N N N N N N N 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.

4. Mass spectrum of the nucleons p, n N(1440) N(1535) N(1650) Positive parity Negative parity

5. Mass 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.

6. Mass 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

7. is calculated by the operator product expansion (OPE) Non perturbative contributions are expressed by some Condensates. Hadronic spectral function, ・・・ An order parameter of chiral symmetry Application of the analyses in the nuclear matter.

8. Vacuum Nuclear matter N N N N N N N N N N N Probe hadron In nuclear matter In vacuum Probe hadron ： Ground state of nuclear matter

9. Nuclear matter N N N N N N N N N N N In nuclear matter In vacuum Probe hadron Modification: Chiral condensate ： New condensate: Probe hadron ： Ground state of nuclear matter Vacuum

10. τ 、 s: parameter OPE side Gaussian sum rule Vacuum Nuclear matter is calculated by OPE Transformation We extract the information on the spectral function with maximum entropy method (MEM).

11. contains the term The 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 OPE Negative parity OPE term Parity: +Parity: - Mass spectrum of the nucleons

12. 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).

13. The behavior of the OPE data in the nuclear matter Vacuum Nuclear matter N N N N N N N N N N N Probe nucleon In nuclear matterIn vacuum Probe hadron ： Ground state of nuclear matter

14. Positive parity OPE Negative parity OPE term ρ N : nuclear matter density The behavior of the OPE data in the nuclear matter Vacuum Nuclear matter Parity: +Parity: - Mass spectrum of the nucleons Vacuum

15. contain the terms Positive parity: OPE data decreases. Negative parity: OPE data slightly increases. ρ ＝ 0.25ρ N Positive parity OPE Negative parity OPE term ρ N : nuclear matter density The behavior of the OPE data in the nuclear matter Parity: +Parity: - ？ Mass spectrum of the nucleons contains the term

16. Positive parity OPE Negative parity OPE term ρ N : nuclear matter density The behavior of the OPE data in the nuclear matter Parity: +Parity: - ？ Mass spectrum of the nucleons ρ ＝ 0.5ρ N contain the terms contains the term Positive parity: OPE data decreases. Negative parity: OPE data slightly increases.

17. Positive parity OPE Negative parity OPE term ρ N : nuclear matter density The behavior of the OPE data in the nuclear matter Parity: +Parity: - ？ Mass spectrum of the nucleons ρ ＝ 0.75ρ N contain the terms contains the term Positive parity: OPE data decreases. Negative parity: OPE data slightly increases.

18. Positive parity OPE Negative parity OPE term ρ N : nuclear matter density The behavior of the OPE data in the nuclear matter Parity: +Parity: - ？ Mass spectrum of the nucleons ρ ＝ 1.0ρ N contain the terms contains the term Positive parity: OPE data decreases. Negative parity: OPE data slightly increases.

19. Positive parity OPE data Negative parity OPE data term : n=n 0 : ρ=0.75ρ 0 : ρ=0.5ρ 0 : ρ=0.25ρ 0 : Vacuum Positive parity Negative parity ρ N : nuclear matter density Vacuum

20. ρ ＝ 0.25ρ N Positive parity OPE data Negative parity OPE data term : n=n 0 : ρ=0.75ρ 0 : ρ=0.5ρ 0 : ρ=0.25ρ N : Vacuum ρ N : nuclear matter density Positive parity Negative parity

21. ρ ＝ 0.5ρ N Positive parity OPE data Negative parity OPE data term : n=n 0 : ρ=0.75ρ 0 : ρ=0.5ρ N : ρ=0.25ρ N : Vacuum ρ N : nuclear matter density Positive parity Negative parity

22. ρ ＝ 0.75ρ N Positive parity OPE data Negative parity OPE data term : n=n 0 : ρ=0.75ρ N : ρ=0.5ρ N : ρ=0.25ρ N : Vacuum Positive parity Negative parity ρ N : nuclear matter density : ρ=1.0ρ 0 Positive parity Negative parity

23. ρ ＝ 1.0ρ N Positive parity OPE data Negative parity OPE data term : n=n 0 : ρ=0.75ρ N : ρ=0.5ρ N : ρ=0.25ρ N : Vacuum Positive parity Negative parity ρ N : nuclear matter density : ρ=1.0ρ N Positive parity Negative parity The values of the energies are obtained.

24. Vacuum Nuclear matter N N N N N N N N N Probe nucleon Propagator: N N N N Pole of positive energy state: Effective mass: Pole of negative energy state:

25. Investigation of effective masses and vector self energies 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

26. ρ N : nuclear matter density Vacuum ρ=0.25 ρ N ρ=0.5 ρ N ρ=0.75 ρ N ρ=1.0 ρ N 810 120 1510 4040 660 270 1460 80 490 430 1430 110 290 640 1410 130 1550 930 0 0

27. We analyze the nucleon spectral function by using QCD sum rules with MEM 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.