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