Nucleon Effective Mass in the DBHF

同位旋物理与原子核的相变

CCAST Workshop

2005 年 8月 19 日－ 8月 21 日

马中玉

中国原子能科学研究院

Introduction

Importance of isospin physics in many aspects:

exotic nuclei; astrophysics; heavy ion collision etc.

Isospin dependence of many quantities:

asymmetric energy as function of density

effective interaction; effective mass etc.

Effective mass characterizes the propagation of a

nucleon in the strongly interacting medium

reaction dynamics of nuclear collisions by unstable nuclei

neutron-proton differential collective flow, isospin equil.

neutron star properties

Introduction

Less knowledge of isospin dependence from experiments

Study from a fundamental theory

NN interaction + SR correlation

Many works in non-relativistic and relativistic approaches

RMF : success in describing g.s. properties

isospin dep. based on stable nuclei

Us, U0 are energy indep.

Our work Relativistic approach DBHF

Definition of effective mass

In non-relativistic approach

effective mass m* describe an independent quasi-particle

moving in the nucl. medium

characterizes the non-locality of the microscopic potential

in space (k-mass) and in time (E-mass)

Jeukenne, Lejeune, Mahuax ‘76

1 2

2 2

0*

( , ) ( , )

( , )2 2

V r r V k

k kV k V

m m

* ** k Em mm

m m m

Effective mas in non-relativistic appr.

1*

( )

1 ( , ( )) 1 ( ( ), )k k

m m d dV k k V k

m k dk d

Effective mass derived by the two equivalent expression

Jaminon & Mahaux’89

It can be determined from analyses of experimental data

in nonrel. Shell model or optical model

Typical value is m*/m ~ 0.70 ± 0.05 at E = 30 MeV

by phenomenological analyses of experimental

scattering data

Dirac mass

Relativistic approach

Effective mass is usually defined as

M* = M – Us Dirac mass (scalar mass)

describe a nucleon in the medium as a quasi nucleon

with effective mass and effective energy, which satisfies

Dirac equation.

M* and m* are different physical quantities

Can not be compared with each other

* *

*

*

( ) ( ) ( )

( )

( )

s

o

p M r E r

M M U r

E E U r

Dirac and Lorentz mass

RMF Us Uo are constant in energy

Us= 375±40 MeV Ring’96

~0.60±0.04 Dirac mass (scalar mass)

Schroedinger equivalent equation

Schroedinger equvalent potential

,

Lorentz mass

*

1 sM U

M M

2 21( ) ( )

2s o s oV U U U UM M

*

1 ( ( ), )m d

V km d

Isospin dep. of effective mass

Isospin dependence of effective mss in RMF

,

Energy dep. of nucleon self-energy is not considered

~0.7 Lorentz mass (vector mass)

Lorentz mass

not related to a non-locality of the rel. potentials,

can be compared with the effective mass in non-relativistic

appr.

Isospin dep. of Lorentz mass, compare with that in non-rel.

2 21( ) ( )

2s o s oV U U U UM M

*

1 oUm

m m

*

1 ( ( ), )m d

V km d

DBHF approach

Relativistic approaches

NN + DBHF

Success in NM saturation

properties

DBHF G Matrix ––-

Nucleon effective int.

Information of isospin dependence

Dirac structure of G Matrix

Bethe-Salpeter equation

3-dimensional reduction: (RBBG)

G=V+VgQG V NN int.(OBEP) g propergator

Q Pauli operator

Self-consistent calculations important

G ? Us, U0 Dirac eq. s.p. wf

G matrix --- do not keep the track of rel. structure

Extract the nucleon self-energy with proper isospin dep.

New decomposition of G

Decomposition of DBHF G matrix

V : OBEP

G a projection method

(1, ) (1, )

Short range

m (g/m)2 finite

E. Shiller, H. Muether,

E Phys. J. A11(2001)15

G V G

Nucleon self-energy

Nucleon self-energy for scattering (k is related with E)

Calculated in DBHF by G = V + G

Direct Exchange

: V OBEP G pseudo meson (1, ) (1, )

: vertex a,b : isospin index

single particle Green’s function

( )kG

Isospin dep. of Nucleon self-energy

Single particle Green’s function

T t : isospin operator

Direct terms isoscalar

isovector

Exchange terms isoscalar

isovector

Nucleon potential

The optical potential of a nucleon

the nucleon self-energy in the nuclear medium

Nucleon self-energy in the nuclear medium with E > 0

0

*0

2 2 20

[ ( )]

[1 ( )] [ ( )] ( )

k

v s

E M k

k k M k k

k – E E incident energy

1 10 1001

2

3

4

10 100

k(p)

k (

fm-1)

E (MeV)

= 0.0 = 0.3 = 0.6

kF = 1.36 fm-1 k(n)

E (MeV)

= 0.0 for free N

Self-Energy of proton and neutron

-350

-300

-250

0 100 200 300 400

200

250

300

100 200 300 400

(a)

Us

p

(MeV

)

(b)

Us

n

(c)U0

p

(MeV

)

Energy (MeV)

(d)

U0

n

Energy (MeV)

＝ 0, .3, .6, 1

Isospin dep. of effective mass in RMF

Dirac Lorentz

Neutron-rich asymmetric NM (RMF)

, ,

Taking account of isovector scalar meson

, , ,

Us Uo should be of momentum and energy dependence

* * * *

,

,

n p n ps s o o

n p n pM m

U U U

M m

U

* * * *

,

,

n p n ps s o o

n p n pM m

U U U

M m

U

*

1 sM U

M M

*

1 oUm

m m

Nucleon self-energy in DBHF

0 20 40 60 80 100

250

300

350

Nuc

leon

Sel

f-en

ergy

/MeV

Insident Energy /MeV

=0 =.3 Proton =.3 Neutron

Us

U0

0.0 0.2 0.4 0.6 0.8

250

300

350

Us

Nu

cle

on

Se

lf-e

nerg

y /M

eV

Proton E=0 MeV Neutron E=0 MeV Proton E=50 MeV Neutron E=50 MeV

U0

In neutron-rich matter

Us Uo of neutrons stronger than of protons

DBHF

0.0 0.2 0.4 0.6 0.80.60

0.65

0.70

0.75

M*/M

M*/

M

Proton

Neutron m*/m Dirac mass in DBHF :

Lorentz mass:

Isospin dep. of OMP is

consistent with Lane pot.

*0 0 011 1s sU dU U dm U U

M d M Mm M d

* *n pM M

* *n pm m

Ma, Rong, Chen et al., PLB604(04)170B.A.Li nucl-th/0404040

Summary

Isospin dependence of the nucleon effective mass is

studied in DBHF

New decomposition of G matrix is adopted

G=V+G

RMF approach with a constant self-energy can not

account the isospin dep. of m* properly

Isospin dep. of effective mass

* *n pm m

Thanks Thanks

G Us Uo

Single particle energy

R. Brockmann, R. Machleidt PRC 42(90)1965

Momentum dep. of Us & U0 are neglected

works well in SNM, inconsistent results in ASNM

wrong sign of the isospin dependence

2 20( ) [ ( )] ( )i i

i sk k M U k U k

2 20( ) [ ]i i

i sk k M U U

Asymmetric NM

Inconsequential results for asymmetric

nuclear matter

Us U0

isospin dep. with

a wrong sign

S. Ulrych, H. Muether,

Phys. Rev. C56(1997)1788

2 20( ) [ ]i i

i sk k M U U

Projection method

Projection method

F. Boersma, R. Malfliet,

PRC 49(94)233

Ambiguity results are

obtained

for with PS and PV

Shiller,Muether, EPJ. A11(2001)15

5 51, , , ,G 0,sU U

Asymmetry Energy

0.1 0.2 0.3 0.4

20

30

40

50

60

0.0 0.2 0.4 0.6 0.8 1.0

0

10

20

30

40

50

60

E/A

(,

) -

E/A

(,0

) [M

eV]

2

1.0 fm-1

1.2 fm-1

1.4 fm-1

1.6 fm-1

1.8 fm-1

DBHF BHF

a asym

()=

E/A

(,1

)-E

/A(

,0)

[MeV

]

[fm-3]

3-body force

Parabolic behavior increase as the density

Ma and Liu PRC66(2002)024321;Liu and Ma CPL 19 (2002)190

asya

Dirac and Lorentz mass

RMF

Us Uo are constant in energy

~0.60 Dirac mass (scalar mass)

Schroedinger equivalent potential

,

~0.70 Lorentz mass (vector mass)

Although not related to a non-locality of the rel. potentials,

comparable with the effective mass in non-relativistic appr.

*

1 sM U

M M

2 21( ) ( )

2s o s oV U U U UM M

*

1 oUm

m m

*

1 ( ( ), )m d

V km d

Nucleon effective mass