Isospin effect in asymmetric nuclear matter

(with QHD II model)

Kie sang JEONG

Effective mass splitting

• from nucleon dirac eq. here energy-momentum relation

• Scalar self energy• Vector self energy (0th )

Effective mass splitting

• Schrodinger and dirac effective mass (symmetric case)

• Now asymmetric case visit• Only rho meson coupling

• + => proton, - => neutron

Effective mass splitting

• Rho + delta meson coupling

• In this case, scalar-isovector effect appear

• Transparent result for asymmetric case

Semi empirical mass for-mula

• Formulated in 1935 by German physicist Carl Friedrich von Weizsäcker

• 4th term gives asymmetric effect

• This term has relation with isospin density

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

*

QHD model

• QHD-I (only contain isoscalar mesons)

• Equation of motion follows

QHD model

• We can expect coupling constant to be large, so perturbative method is not valid

• Consider rest frame of nuclear sys-tem (baryon flux = 0 )

• As baryon density increases, source term becomes strong, so we take MF approximation

QHD model

• Mean field lagrangian density

• Equation of motion

• We can see mass shift and energy shift

QHD model

• QHD-II (QHD-I + isovector couple)

• Here, lagrangian density contains isovector – scalar, vector couple

Delta meson

• Delta meson channel considered in study

• Isovector scalar meson

Delta meson

• Quark contents

• This channel has not been consid-ered priori but appears automatically in HF approximation

RMF <–> HF

• If there are many particle, we can as-sume 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.

RMF <–> HF

• Basic hamiltonian

RMF <–> HF

• Expectation value

Hartree Fock approximation

Classical interaction be-tween one particle - sysytem

Exchange contribution

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

H-F approximation

• Eq. of motion

Wigner transformation

• Now we control meson couple with baryon field

• To manage this quantum operator as statistical object, we perform wigner transformation

Transport equation with fock terms

• Eq. of motion

• Fock term appears as

Transport equation with fock terms

• Following [PRC v64, 045203] we get kinetic equation

• Isovector – scalar density• Isovector baryon current

Transport equation with fock terms

• kinetic momenta and effective mass

• Effective coupling function

Nuclear equation of state

• below corresponds hartree approximation• Energy momentum tensor

• Energy density

Symmetry energy

• We expand energy of antisymmetric nuclear matter with parameter

• In general

Symmetry energy

• Following [PHYS.LETT.B 399, 191] we get Symmetry energy

nuclear effective mass in symmetric case

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

Symmetry energy

• Parameter set of QHD models

Symmetry energy

• Empirical value a4 is symmetry energy term at saturation density, T=0

When delta meson contribution is not zero, rho meson cou-pling have to increase

Symmetry energy

Symmetry energy

• Now symmetry energy at saturation density is formed with balance of scalar(attractive) and vector(repulsive) contribution

• Isovector counterpart of saturation mechanism occurs in isoscalar chan-nel

Symmetry energy

• Below figure show total symmetry energy for the different models

Symmetry energy

• When fock term considered, new effective couple acquires density dependence

Symmetry energy

• For pure neutron matter (I=1)

• Delta meson coupling leads to larger re-pulsion effect

Futher issue

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

reference

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