Isospin effect in asymmetric nuclear matter (with QHD II model)

1. Isospin effect in asymmetric nuclear matter(with QHD II model) Kie sang JEONG

2. Effective mass splitting • from nucleon dirac eq. here energy-momentum relation • Scalar self energy • Vector self energy (0th )

3. Effective mass splitting • Schrodinger and dirac effective mass (symmetric case) • Now asymmetric case visit • Only rho meson coupling • + => proton, - => neutron

4. Effective mass splitting • Rho + delta meson coupling In this case, scalar-isovector effect appear • Transparent result for asymmetric case

5. Semi empirical mass formula • Formulated in 1935 by German physicist Carl Friedrich von Weizsäcker • 4th term gives asymmetric effect • This term has relation with isospin density

6. QHD model • Quantum hadrodynamics • Relativistic nuclear manybody theory • Detailed dynamics can be described by choosing a particular lagrangian density • Lorentz, Isospin symmetry • Parity conservation * • Spontaneous broken chiral symmetry *

7. QHD model • QHD-I (only contain isoscalar mesons) • Equation of motion follows

8. QHD model • We can expect coupling constant to be large, so perturbative method is not valid • Consider rest frame of nuclear system(baryon flux = 0 ) • As baryon density increases, source term becomes strong, so we take MF approximation

9. QHD model • Mean field lagrangian density • Equation of motion • We can see mass shift and energy shift

10. QHD model • QHD-II (QHD-I + isovector couple) • Here, lagrangian density contains isovector – scalar, vector couple

11. Delta meson • Delta meson channel considered in study • Isovector scalar meson

12. Delta meson • Quark contents • This channel has not been considered priori but appears automatically in HF approximation

13. RMF <–> HF • If there are many particle, we can assume one particle – external field(mean field) interaction • In mean field approximation, there is not fluctuation of meson field. Every meson field has classical expectation value.

14. RMF <–> HF • Basic hamiltonian

15. RMF <–> HF • Expectation value

16. HartreeFock approximation Classical interaction between one particle - sysytem Exchange contribution

17. H-F approximation • Each nucleon are assumed to be in a single particle potential which comes from average interaction • Basic approximation => neglect all meson fields containing derivatives with mass term

18. H-F approximation • Eq. of motion

19. Wigner transformation • Now we control meson couple with baryon field • To manage this quantum operator as statistical object, we perform wigner transformation

20. Transport equation with fock terms • Eq. of motion • Fock term appears as

21. Transport equation with fock terms • Following [PRC v64, 045203] we get kinetic equation • Isovector– scalar density • Isovector baryon current

22. Transport equation with fock terms • kinetic momenta and effective mass • Effective coupling function

23. Nuclear equation of state • below corresponds hartreeapproximation • Energy momentum tensor • Energy density

24. Symmetry energy • We expand energy of antisymmetric nuclear matter with parameter • In general

25. Symmetry energy • Following [PHYS.LETT.B 399, 191] we get Symmetry energy nuclear effective mass in symmetric case

26. Symmetry energy • vanish at low densities, and still very small up to baryon density • reaches the value 0.045 in this interested range • Here, transparent delta meson effect

27. Symmetry energy • Parameter set of QHD models

28. Symmetry energy • Empirical value a4 is symmetry energy term at saturation density, T=0 When delta meson contribution is not zero, rho meson coupling have to increase

29. Symmetry energy

30. Symmetry energy • Now symmetry energy at saturation density is formed with balance of scalar(attractive) and vector(repulsive) contribution • Isovectorcounterpart of saturation mechanism occurs in isoscalar channel

31. Symmetry energy • Below figure show total symmetry energy for the different models

32. Symmetry energy • When fock term considered, new effective couple acquires density dependence

33. Symmetry energy • For pure neutron matter (I=1) • Delta meson coupling leads to larger repulsion effect

34. Futher issue • Symmetry pressure, incompressibility • Finite temperature effects • Mechanical, chemical instabilities • Relativistic heavy ion collision • Low, intermediate energy RI beam

35. reference • Physics report 410, 335-466 • PRC V65 045201 • PRC V64 045203 • PRC V36 number1 • Physics letters B 191-195 • Arxiv:nucl-th/9701058v1