Delta ma(er in a parity doublet model

Transcription

1 Delta ma(er in a parity doublet model Masayasu Harada (Nagoya (May 22, 2017) Based on Yusuke Takeda, Youngman Kim, M.Harada, arxiv: See also M. Harada, Y.L. Ma, D. Suenaga, Y. Takeda, arxiv: Y. Motohiro, Y.Kim, M.Harada, Phys. Rev. C 92, (2015); Erratum: Phys. Rev. C 95, (2017). 17/05/23 JAEA 1

2 Introduc)on 17/05/23 JAEA 2

3 of Hadrons Origin of Mass? of Us = One of the Interes)ng problems of QCD 17/05/23 JAEA 3

4 Mass generated by the spontaneous chiral symmetry breaking chiral symmetry broken phase at vacuum chiral symmetric phase at high T and/or density qq 0 (chiral condensate) qq = 0 The spontaneous chiral symmetry breaking is expected to generate a part of hadron masses. It causes mass difference between chiral partners. 17/05/23 JAEA 4

5 Phase diagram of Quark-Gluon system high temperature 1 trillion kelvin Early Universe Chiral Symmetry Restra)on Deconfinement of Quarks Spontaneous Chiral Symmetry Breaking Confinement of Quarks Neutron Star High density 100 million ton/cm 3 17/05/23 JAEA 5

6 Dense nuclear ma(er by a Skyrme-crystal model Y. L. Ma, M. Harada, H.K. Lee, Y. Oh, B. Y. Park, M. Rho, PRD88 (2013) Y. L. Ma, M. Harada, H.K. Lee, Y. Oh, B. Y. Park, M. Rho, PRD90 (2014) There exists the half-skyrmion phase where the space-average of the chiral condensate vanishes. Nucleon mass decreases with density in the normal phase, while it is stable In the half-skyrmion phase. σ d 3 x qq z Nucleon mass ~ 2 ρ 0 density normal hadron half-skyrmion phase Chiral invariant mass? phase 17/05/23 JAEA 6

7 Chiral Invariant Mass of Hadrons? Parity doublet model for light baryons In [C.DeTar, T.Kunihiro, PRD39, 2805 (1989)], N*(1535) is regarded as the chiral partner to the N(939) having the chiral invariant mass. m B = m 0B + m qq chiral invariant mass spontaneous chiral symmetry breaking How much mass of nucleon is from the spontaneous chiral symmetry breaking? What is the value of the chiral invariant mass? 17/05/23 JAEA 7

8 Phase diagram of Quark-Gluon system high temperature 1 trillion kelvin Early Universe Chiral Symmetry Restra)on Deconfinement of Quarks Spontaneous Chiral Symmetry Breaking Confinement of Quarks Neutron Star High density 100 million ton/cm 3 17/05/23 JAEA 8

9 In [Y. Motohiro, Y.Kim, M.Harada, Phys. Rev. C 92, (2015)], we studied nuclear ma(er using a parity doublet model, and showed some relakons between the chiral invariant mass of nucleon and the phase structure. We also presented a density dependence of the nucleon mass, which changes refleckng the parkal chiral symmetry restorakon. What happens to the masses of other hadrons in nuclear ma(er? In this talk, I focus on the mass of Delta baryon in ma(er. 17/05/23 JAEA 9

10 π - /π + rako in low-energy Heavy-Ion Collision T.Song, C.M.Ko, Phys. Rev. C91, (2015) Solid and dashed curves correspond to two different models which predict different values for the symmetry energy. Red curves include the mass shin of Delta baryon. The effect of symmetry energy is reversed by the effect of mass shin of Δ. E/A 17/05/23 JAEA 10

11 Delta ma(er in Neutron Star T.Schurhoff, S. Schramm, V.Dexheimer, Astrophys. J. 724, L74 (2010). B. J. Cai, F. J. Fa(oyev, B. A.Li, W.G.Newton, Phys. Rev. C 92, (2015). There may exist Delta ma(er in ρ B > 2 ρ 0. 17/05/23 JAEA 11

12 Delta Ma(er in low-e HIC? In [Y.Takeda, Y.Kim, M.Harada, arxiv: ], we study the possibility of Delta ma(er using a hadronic model based on the parity doublet structure for Delta baryons and nucleons. There are some differences of HIC with Neutron star: e.g. Weak interackon does not contribute in HIC. Hadrons including strange quark can be neglected in low-energy HIC. 17/05/23 JAEA 12

13 Outline 1. Introduckon 2. Nuclear ma(er from a parity doublet model 3. Effeckve mass of Delta in nuclear ma(er 4. Delta ma(er 5. Predickons of Delta ma(er 6. Summary 17/05/23 JAEA 13

14 2. Nuclear ma(er from a parity doublet model Y. Motohiro, Y.Kim, M.Harada, Phys. Rev. C 92, (2015) 17/05/23 JAEA 14

15 Parity Doublet model C.DeTar, T.Kunihiro, PRD39, 2805 (1989) D.Jido, M.Oka, A.Hosaka, PTP106, 873 (2001) An excited nucleon with negative parity such as N*(1535) is regarded as the chiral partner to the N(939) which has the positive parity. These nucleons have a chiral invariant mass in addition to the mass generated by the spontaneous chiral symmetry breaking. In this model, the origin of our mass is not only the chiral symmetry breaking. 17/05/23 JAEA 15

16 Determinakon of the parameters at vacuum Masses of parity eigenstates Determinakon of parameters at vacuum (D.Jido et al., PTP106, 873 (2001)) Inputs : m + = 939 MeV, m - = 1535 MeV, σ 0 = f π = 93 MeV, and g πn+n- = 0.7 obtained from Γ N* πn = 75 MeV. Outputs : m 0 = 270 MeV, g 1 = 9.8, g 2 = 16. Global fit in an extended model (S.Gallas et al., PRD82, (2010) ) shows m 0 = MeV. 17/05/23 JAEA 16

17 Nuclear ma(er in parity doublet models A parity doublet model including omega meson with 4-point interackon is used in a Walecka-type mean field analysis. Large value of m 0 is needed to reproduce the incompressibility. Rho meson is further included with 4- point interackon. m 0 > 800 MeV is needed to have 100 < K < 400 MeV In our analysis [Y.Motohiro, Y.Kim, M.Harada, Phys. Rev. C 92, (2015)], we construct a model with a 6- point interackon of sigma, but without 4-point interackon for vector mesons. Our results show that K = 240 MeV is reproduced for m 0 = MeV. D.Zschiesche et al., PRC75, (2007) V.Dexheimer et al., PRC77, (2008) 17/05/23 JAEA 17

18 Model parameters : Inputs from vacuum phenomenology We include sigma and omega mesons as well as the pion and rho meson in our model We have 11 parameters. 3 parameters in the scalar potential µ,, 6 : V = 1 2 µ masses for,,! : m, m, m! 5 parameters in the baryon sector m 0 : chiral invariant mass ; g 1, g 2 : Yukawa couplings g NN, g!nn : NN and!nn couplings We determine 10 parameters for a given value of the chiral invariant mass m 0. We use the following 6 inputs at vacuum (MeV) 18 17/05/23 JAEA

19 Inputs from medium property We calculate the thermodynamic potenkal in the nuclear medium in our model, using the mean field approximakon. Then, we determine the 4 remaining parameters from the following physical inputs for a given value of the chiral invariant mass m 0. Nuclear saturakon density Binding energy at normal nuclear density Incompressibility = 240 MeV Symmetry energy : E sym (ρ 0 ) = 31 MeV 17/05/23 JAEA 19

20 Binding Energy, Pressure Y. Motohiro, Y.Kim, M.Harada, Phys. Rev. C 92, (2015); Erratum: Phys. Rev. C 95, (2017). Binding Energy Pressure E/A-m + [MeV] m 0 =500 MeV ρ n /ρ B =0.5 ρ n /ρ B =1.0 (a) ρ B [fm -3 ] P [MeV fm -3 ] m 0 =500 MeV ρ n /ρ B =0.5 ρ n /ρ B =0.6 ρ n /ρ B =0.7 ρ n /ρ B =0.8 ρ n /ρ B =0.9 ρ n /ρ B =1.0 (b) ρ B [fm -3 ] ρ B (fm 3 ) ρ B (fm 3 ) m N0 = 500 MeV 17/05/23 JAEA 20

21 Density Dependence of Mean fields σ - σ (MeV) m N0 = 500 MeV m N0 =500 m N0 =700 m N0 =900 m N0 = 700 MeV h!i = g!nn m 2! B (MeV) m N0 = 500, 700, 900 MeV m N0 = 900 MeV ρ 0 B /ρ An experiment shows f π* /f π ~ 0.8 at ρ B = ρ 0. [K. Suzuki et al., Phys. Rev. Le(. 92, (2004)] We use m N0 = 500, 700 MeV as typical examples 17/05/23 JAEA 21

22 Effeckve masses of nucleons We define effeckve masses of nucleons by including effects of exchanging the sigma and omega mesons in the mean field approximakon in symmetric nuclear ma(er, following our recent work [M.Harada, Y.L.Ma, D.Suenaga, Y.Takeda, arxiv: ]. N N σ m (e ) N = m N + g!nn! m N = 1 appleq (g 1 + g 2 ) m 2 0 (g 2 g 1 ) 940 m N0 =500 m N0 =700 N N ω m N (eff) /05/23 JAEA 22

23 3. Effeckve mass of Delta in nuclear ma(er 17/05/23 JAEA 23

24 Chiral Partner Structure of Delta Δ(1232) and Δ(1700) are regarded as chiral partners. m ± = D.Jido, T.Hatsuda, T.Kunihiro, PRL84, 3252 (2000) D.Jido, M.Oka, A.Hosaka, PTP106, 873 (2001) q (g 1 + g 2 ) m 2 0 (g 1 g 2 ) We regard m Δ0 as a parameter. We use masses of D(1232) and D(1700) as inputs to determine the values of g Δ1 and g Δ2 for a given value of m Δ0. We study dependence of our results on the value of m Δ0. 17/05/23 JAEA 24

25 Effeckve masses vs. Chemical potenkal m (e ) = q (g 1 + g 2 ) m 2 0 (g 1 g 2 ) + g!! m N0 = m D0 = 500 MeV m N0 =m Δ0 =500 symmetric matter g ωδδ =g ωnn g ωδδ =g ωnn /2 µ B g ωδδ = g ωnn m N0 = m D0 = 700 MeV m N0 =m Δ0 =700 symmetric matter g ωδδ =g ωnn g ωδδ =g ωnn /2 µ B m Δ (eff) and µ B g ωδδ = 0.5 g ωnn µ 950 B 950 µ B m Δ (eff) and µ B 3 3 Mass of Delta baryon becomes smaller than the chemical potenkal µ B. This indicate that the Delta baryon is populated in the ground state, forming Delta Fermi sea. Delta Ma(er? 17/05/23 JAEA 25

26 4. Delta Ma(er 17/05/23 JAEA 26

27 Thermodynamic Potenkal We constructed the thermodynamic potenkal at mean field level including the effect of Delta baryon. = 1 2 m m2!! µ m X Z dk 2 k2 (µ E ) (µ E ) =p,n contribukon of the Delta baryon X Z dk 2 2 k2 (µ a E a) (µ a E a) a=++,+,0, µ ++ = µ B µ I, µ + = µ B µ I µ 0 = µ B 1 2 µ I, µ = µ B 3 2 µ I µ B : Baryon chemical potenkal µ I : Isospin chemical potenkal 17/05/23 JAEA 27

28 Pressure, Density, Phase structure Pressure vs. µ B m N0 =700, m Δ0 =700 symmetric matter g ωδδ =g ωnn Delta and nucleon pressure H m N0 = m D0 = 700 MeV g ωδδ = g ωnn pressure A C D B µ B m Δ0 800 m Δ Phase structure D D G G G F m N0 =700 symmetric matter g ωδδ =g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase 2 µ B 3 4 E /05/23 JAEA 28 each density number Density vs. ρ B m N0 =700, m Δ0 =700 symmetric matter g ωδδ =g ωnn N(939) Δ(1232) D G G ρ N /ρ 0 ρ Δ /ρ N liquid-gas coexistence phase stable N phase coexistence phase of Delta-N ma(er and nuclear ma(er stable N-Delta phase 3 4

29 Dependence of the phase structure g ωδδ = g ωnn m Δ D D G G on ωδδ coupling m N0 =700 symmetric matter g ωδδ =g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase m N0 = m D0 = 700 MeV N liquid-gas coexistence phase stable N phase coexistence phase of Delta-N ma(er and nuclear ma(er stable N-Delta phase g ωδδ = 0.5 g ωnn g ωδδ = 0.8 g ωnn m N0 =700 symmetric matter g ωδδ =0.5g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase m N0 =700 symmetric matter g ωδδ =0.8g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase m Δ0 900 m Δ /05/23 JAEA 29

30 m Δ0 Dependence of phase structure on m N0 m N0 = 700 MeV g ωδδ = g ωnn g ωδδ = 0.8 g ωnn g ωδδ = 0.5 g ωnn D D G G m N0 =700 symmetric matter g ωδδ =g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase m Δ m N0 =700 symmetric matter g ωδδ =0.8g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase m Δ m N0 =700 symmetric matter g ωδδ =0.5g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase m N0 = 500 MeV m Δ m N0 =500 symmetric matter g ωδδ =g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase m Δ m N0 =500 symmetric matter g ωδδ =0.8g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase /05/23 JAEA 30 m Δ m N0 =500 symmetric matter g ωδδ =0.5g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase

31 m Δ0 Phase structure in Asymmetric ma(er symmetric ma(er (µ I =0) g ωδδ = g ωnn D D G G m N0 =700 symmetric matter g ωδδ =g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase asymmetric ma(er (µ I =-60MeV) neutron ma(er phase neutron-proton ma(er phase coexistence phase of Delta-N ma(er and nuclear ma(er stable N - Δ - phase stable N Δ - - Δ 0 phase stable N Δ - - Δ 0 Δ + phase stable N Δ phase m Δ m N0 = 700 MeV g ωδδ = 0.8 g ωnn m N0 =700 symmetric matter g ωδδ =0.8g ωnn N liquid gas coexist stable N phase N Δ-N coexist stable N-Δ phase g ωδδ = g ωnn ; g ρδδ = g ρnn g ωδδ = g ωnn ; g ρδδ = 0.8 g ρnn g ωδδ = 0.8 g ωnn ; g ρδδ = g ρnn m Δ m N0 =700 µ I =-60 g ωδδ =g ωnn g ρδδ =g ρnn neutron neutron + proton N + Δ N + Δ - + Δ 0 N + Δ - + Δ 0 + Δ + N + Δ unstable phase m Δ m N0 =700 µ I =-60 g ωδδ =g ωnn g ρδδ =0.8g ρnn neutron neutron + proton N + Δ N + Δ - + Δ 0 N + Δ - + Δ 0 + Δ + N + Δ unstable phase m Δ m N0 =700 µ I =-60 g ωδδ =0.8g ωnn g ρδδ =g ρnn neutron neutron + proton N + Δ N + Δ - + Δ 0 N + Δ - + Δ 0 + Δ + N + Δ unstable phase 17/05/23 JAEA 31

32 5. Predickons of Delta Ma(er 17/05/23 JAEA 32

33 Effeckve masses m N0 = m Δ0 = 700 MeV; g ωδδ = g ωnn nuclear ma(er m N0 =700, m Δ0 =700, µ I =0, g ωδδ =g ωnn nucleon effective mass Δ effective mass Delta ma(er each mass Coexistence phase of Delta-N ma(er and nuclear ma(er Both the effeckve masses of nucleon and Delta suddenly change their values associated with the appearance of the Delta baryons in ma(er. 17/05/23 JAEA 33

34 pressure[mevfm -3 ] Pressure symmetric ma(er m N0 = m Δ0 = 700 MeV; g ωδδ = g ωnn Delta nuclear ma(er ma(er m N0 =700, m Δ0 =700, µ I =0, g ωδδ =g ωnn N-Δ N ρ ρ B /ρ B /ρ ρ 0 B /ρ ρ B /ρ Coexistence phase of Delta-N ma(er and nuclear ma(er pressure[mevfm -3 ] asymmetric ma(er (µ I = - 60 MeV) m N0 = m Δ0 = 700 MeV; g ωδδ = g ωnn Delta nuclear ma(er ma(er 17/05/23 JAEA m N0 =700, m Δ0 =700, µ I =-60, g ωδδ =g ωnn, g ρδδ =g ρnn N-Δ N Coexistence phase of Delta-N ma(er and nuclear ma(er The pressure of N-Delta ma(er is smaller than that of the ordinary nuclear ma(er. The Delta sonens the equakon of state as expected.

35 Symmetry Energy m N0 = 700 MeV; g ωδδ = g ωnn ; g ρδδ = g ρnn m Δ0 = 670, 700, 730 MeV symmetry energy m N0 =700, g ωδδ =g ωnn, g ρδδ =g ρnn m Δ =670 m Δ =700 m Δ = ρ B /ρ coexistence phase for m Δ0 = 700 MeV coexistence phase for m Δ0 = 670 MeV The symmetry energy suddenly changes its value around the density where Delta enters the ma(er. 17/05/23 JAEA 35

36 Chiral Condensate symmetric ma(er m N0 = m Δ0 = 700 MeV; g ωδδ = g ωnn nuclear ma(er m N0 =700, m Δ0 =700, µ I =0, g ωδδ =g ωnn N-Δ N Delta ma(er 100 asymmetric ma(er (µ I = - 60 MeV) m N0 = m Δ0 = 700 MeV; g ωδδ = g ωnn 80 nuclear ma(er m N0 =700, m Δ0 =700, µ I =-60, g ωδδ =g ωnn, g ρδδ =g ρnn N-Δ N Delta ma(er σ - σ Coexistence phase of Delta-N ma(er and nuclear ma(er Coexistence phase of Delta-N ma(er and nuclear ma(er The appearance of Delta ma(er accelerates the restorakon of the chiral symmetry. 17/05/23 JAEA 36

37 6. Summary We include the Δ baryon based on the parity doublet structure, with the chiral invariant mass m Δ0. We study density dependence of Δ baryon masses from the mean field contribukons of sigma and omega mesons, relakng it with parkal chiral symmetry restorakon. We explore phase structure and find that Stable Delta ma(er exists for ρ B > 3 ρ 0. The onset density of Delta ma(er can be smaller than 2ρ 0. We also observed in symmetric dense ma(er that larger m N0 tends to lower the transikon density to the ``stable N-Δ phase and the phase structure changes significantly with the finite isospin chemical potenkals. We then calculated the in-medium chiral condensate, effeckve masses, pressure and symmetry energy to observe that the appearance of $ Delta$ ma(er accelerates the chiral symmetry restorakon, and that their density-dependence changes draskcally around the onset density of $ Delta$ ma(er. 17/05/23 JAEA 37

38 Discussion In the present study we do not consider any possibility of having hyperon ma(er because strangeness is conserved with strong interackons and terrestrial dense ma(er from low-intermediate heavy ion collisions sustains much short compared to the kme scale of weak interackons. It will be intereskng to see how the observakons made in this study such as the transikon to Δ ma(er affect the observables in heavy ion collisions such as neutron-proton colleckve flows and π + /π - rako at low and/or intermediate energy in a transport model simulakon, which is relegated to our future study. The fate of pion condensakon with Delta ma(er in heavy ion collisions will be inveskgated. 17/05/23 JAEA 38

39 17/05/23 JAEA 39

40 Density dependence of effeckve masses m (e ) ± = m (e ) ± = Iuse g! = g!nn =5.4 as a typical example. Masses (MeV) q (g 1 + g 2 ) m 2 0 (g 1 g 2 ) + g!! ( ) q (g 1 + g 2 ) m 2 0 (g 1 g 2 ) g!! (anti- ) m 0Δ = 700 MeV m 0Δ = 1400 MeV ( ) ( ) (+) (+) Increasing or decreasing of Δ(+) baryon mass only is not enough for measuring the chiral symmetry restorakon. 17/05/23 JAEA 40

41 Parkal chiral symmetry restorakon ( ) (+) (MeV) 500 m 0Δ = 700, 1400 MeV 2500 (+) + (+) (MeV) m 0Δ = 1400 MeV m 0Δ = 700 MeV Studying the mass difference of chiral partners gives a clue for parkal chiral symmetry restorakon, independently of the value of m 0Δ. Taking sum of parkcle and ank-parkcle will give a clue for chiral invariant mass. m (e ) ± = m (e ) ± = q (g 1 + g 2 ) m 2 0 (g 1 g 2 ) + g!! ( ) q (g 1 + g 2 ) m 2 0 (g 1 g 2 ) g!! (anti- ) 17/05/23 JAEA 41

42 4.Summary 1600 m 0Δ = 700 MeV We studied density dependence of Delta baryon masses from the mean field contribukons of sigma and omega mesons. Increasing or decreasing of Δ(+) baryon mass only is not enough for measuring the chiral symmetry restorakon. Studying the mass difference of chiral partners gives a clue for parkal chiral symmetry restorakon, independently of the value of m 0Δ. Taking sum of parkcle and ankparkcle will give a clue for the chiral invariant mass m 0D /05/23 JAEA m 0Δ = 1400 MeV m 0Δ = 700, 1400 MeV m 0Δ = 700 MeV m 0Δ = 1400 MeV

43 Discussions For more realiskc study of Δ mass in ma(er, we should check the effect of π-n loop, which is needed to study the in-medium spectral funckon. There seems skll a difference of the eskmated value of the chiral invariant mass of nucleon between the one at vacuum and the one in the ma(er. The analysis in the 3-flavor parity doublet model shows that the chiral partner of the posikve parity nucleon, N(939), is not N*(1535), but a mixture of N*(1440), N*(1535), N*(1650), [H.Nishihara and M.Harada, Phys. Rev. D92, (2015), and work in preparakon.] It is intereskng to include 2 parity doublets for nucleon into a model. 17/05/23 JAEA 43

44 In [Y. Motohiro, Y.Kim, M.Harada, Phys. Rev. C 92, (2015)], we studied nuclear matter using a parity doublet model, and showed some relations between the chiral invariant mass of nucleon and the phase structure. We also presented a density dependence of the nucleon mass, which changes reflecting the partial chiral symmetry restoration. What happens to the mass of Delta baryon? 17/05/23 JAEA 44

45 Change of Delta at RHIC? STAR: arxiv:nucl-ex/ In-medium spectral funcion of Δ by Hees-Rapp, PLB606, 59 (2005) These show that medium effects push up the Δ mass at high temperature. What happens at high density? What is the relakon to the parkal chiral symemtry restorakon? 17/05/23 JAEA 45

46 In this talk, I will show a preliminary study of the mass of Δ baryon at high density, based on a parity doublet structure. Outline 1. Introduckon 2. Nuclear ma(er from a parity doublet model 3. Density dependence of effeckve mass of Delta baryon 4. Summary 17/05/23 JAEA 46

47 Parity Doublet model for nucleon C.DeTar, T.Kunihiro, PRD39, 2805 (1989) D.Jido, M.Oka, A.Hosaka, PTP106, 873 (2001) An excited nucleon with negative parity such as N*(1535) is regarded as the chiral partner to the N(939) which has the positive parity. These nucleons have a chiral invariant mass in addition to the mass generated by the spontaneous chiral symmetry breaking which is caused by the existence of the sigma condensate, < σ > 0. 17/05/23 JAEA 47

48 La ce analysis The result in [G. Aarts, C. Allton, S. Hands, B. Jaeger, C. Praki, and J. I.Skullerud, Phys. Rev. D 92, (2015)] and [L.Y.Glozman, C.B.Lang, M. Schrock, Phys. Rev. D86, (2012)] seems to show the existence of large chiral invariant mass. G. Aarts et al., PRD92 G.Aarts et al., PRD92 Y.L. Glozman et al., PRD86 17/05/23 JAEA 48

49 Model parameters : Inputs from vacuum phenomenology We include sigma and omega mesons as well as the pion and rho meson in our model We have 11 parameters. 3 parameters in the scalar potential µ,, 6 : V = 1 2 µ masses for,,! : m, m, m! 5 parameters in the baryon sector m 0 : chiral invariant mass ; g 1, g 2 : Yukawa couplings g NN, g!nn : NN and!nn couplings We determine 10 parameters for a given value of the chiral invariant mass m 0. We use the following 6 inputs at vacuum (MeV) 49 17/05/23 JAEA

50 Dependence on the ωδδ coupling m g ωδδ = ( ) 0Δ = 700 MeV ( ) (+) (+) g ωδδ = 2.7 g ωδδ = /05/23 JAEA 50