A method of cut-off regularization is proposed to evaluate vacuum corrections
in nuclear matter in the framework of the Hartree approximation. Bulk
properties of nuclear matter calculated by this method are a good agreement
with results analyzed by empirical values. The vacuum effect is quantitatively
evaluated through a cut-off parameter and its role for saturation property and
compressional properties is clarified.Comment: PACS numbers, 21.65.+f, 21.30.+

Hasegawa, Akira

Koide, Kazuharu

Kouno, Hiroaki

Nakano, Masahiro

English

arXiv

A method of cut-off regularization is proposed to evaluate vacuum corrections in nuclear matter in the framework of the Hartree approximation. Bulk properties of nuclear matter calculated by this method are a good agreement with results analyzed by empirical values. The vacuum effect is quantitatively evaluated through a cut-off parameter and its role for saturation property and compressional properties is clarified

Koide, K

Kouno, H

Hasegawa, A

Nakano, M

CERN Document Server

SAGA-HE-85-95
June 1995
Bulk properties of nuclear matter in the relativistic Hartree approximation with
cut-o regularization
Kazuharu Koide, Hiroaki Kouno* and Akira Hasegawa
Department of Physics, Saga University, Saga 840, Japan
and
Masahiro Nakano
University of Occupational and Environmental Health, Kitakyushu 807, Japan
(PACS numbers: 21.65.+f, 21.30.+y )
Abstract
A method of cut-o regularization is proposed to evaluate vacuum correc-
tions in nuclear matter in the framework of the Hartree approximation. Bulk
properties of nuclear matter calculated by this method are a good agreement
with results analyzed by empirical values. The vacuum eect is quantitatively
evaluated through a cut-o parameter and its role for saturation property and
compressional properties is claried.
* e-mail address: kounoh@himiko.cc.saga-u.ac.JP
1
In a relativistic approach for the study of nuclear matter vacuum eects
are very important corrections which can not be taken into account in any
non-relativistic formalism. About twenty years ago, using a simple -! model
(Walecka model), the mean eld theory (MFT) claried the saturation mech-
anism of nuclear matter relativistically[1] and in succession vacuum eects to
the saturation property were evaluated by a method of dimensional regulariza-
tion in the framework of the Hartree approximation (RHA)[2]. It was shown
that vacuum corrections made an eective nucleon mass larger and the incom-
pressibility of nuclear matter smaller. A small incompressibility is desirable for
experimental ndings of today[3,4]. Recently it has been reported that there
is a strong correlation between an increase of eective mass and a decrease in
incompressibility[5].
Nearly ten years after a proposal of the Walecka model, corrections due to
vacuum polarization in quantum uctuation around the mean eld (the Hartree
eld) were evaluated[6]. The results were undesirable. Corrections were larger
by far than the magnitude of mean eld and furthermore brought forth insta-
ble ghost poles (the Landou ghost) in meson propagators which made nuclear
matter unstable. One of ad hoc but powerful recipes to escape from these dis-
asters was to introduce some form factors at each vertex[7]. There was another
idea that the form factors should be derived from vertex corrections[8]. These
recipes are grounded on existence of internal structure of hadron. We are afraid
that vacuum corrections evaluated by the method of dimensional regularization
may be overestimated because the size of hadron is not considered in such a
regularization as the one used well in the elementary particle physics.
When we use a conventional form factor in a calculation of polarization in-
sertion of vector meson, however, we need a safety device to assure the baryon
current conservation. In this report, then, we show another recipe to esti-
mate vacuum corrections in the framework of Hartree approximation in simple
Walecka model. This is the rst request to a method of cut-o regularization
stated in the following discussion.
A nucleon propagator G(k) in the relativistic Hartree approximation has the
following standard form
G(k) = G
F
(k) +G
D
(k)
=
 1
i

k


+M

  i
+
 
  i

k


+M


i
E

k
(k
F
  j
~
kj)(k

0
  E

k
); (1)
E

k
=
q
~
k
2
+M

2
; (2)
M

= M +
S
; (3)
k


= (
~
k; k
4
+ 
4
) = (
~
k; ik
0
+ i
0
); (4)
where the subscripts "F " and "D" of G(k) denote the Feynman part and the
density part, respectively, M and k
F
denote the physical nucleon mass and Fermi
2
momentum, respectively. In the simple Walecka model the nucleon self-energies
are given as follows,

S
= 
SD
+
SF
= i
 
g
s
m
s

2
Z
d
4
q
(2)
4
tr

G
D
(q) + G
F
(q)
	
; (5)

0
= i
 
g
v
m
v

2
Z
d
4
q
(2)
4
tr


0
G(q)
	
=  
 
g
v
m
v

2

B
; (6)
where 
B
denotes the baryon density and m
s
, m
v
, g
s
and g
v
denote the -meson
mass, the !-meson mass, the -nucleon coupling and the !-nucleon coupling,
respectively, and  denotes the degeneracy,  = 2 for nuclear matter and 
= 1 for neutron matter. The Feynman part of self-energy 
SF
is a divergent
integral in the 4-dimensional momentum space while 
SD
and 
0
have nite
values. In the Hartree approximation there appears another divergent integral
in the Feynman part of baryon energy density dened as follows,
"
B
= "
BD
+ "
BF
=


2
Z
k
F
0
k
2
p
k
2
+M
2
dk  
Z
d
4
k
(2)
4
tr


4
G
F
(k)


4
: (7)
We discuss how we introduce a cut-o parameter into the two divergent
integrals in the formalism of the Hartree approximation. We start discussion
with the nucleon eective mass M

. The eective mass is given by using the
scalar self-energy with a cut-o parameter ,
M

= M
0
+ 
SD
(M

; 
B
) + 
SF
(M

;)
= M +
SD
(M

; 
B
) + 
SF
(M

;) 
SF
(M;); (8)
M
0
=M   
SF
(M;); (9)
where M
0
denotes the bare mass of nucleon. The last term in eq.(8) is the
self-energy at zero density and is introduced for the eective mass M

to be the
physical mass M at zero density. We require the bare nucleon mass M
0
to have
cut-o dependence so that the physical nucleon mass M dose not depend on
any cut-o parameter . We note that the bare nucleon mass M
0
() becomes
the physical mass as ! 0.
To make progress we write the energy density as follows,
" =
1
2

2

V

2
B
+
1
2
2
1

S
 
M

 M

2
+ "
B
; (10)
where 
S
and 
V
are dened,

S
=
 
g
s
m
s

2
; 
V
=
 
g
v
m
v

2
: (11)
As the second request we require Hugenholtz-von Hove theorem[9],
("+ P )=
B
= E
N
(k
F
)  E
F
; (12)
3
where P denotes pressure and E
N
(k
F
) denotes a nucleon energy at Fermi mo-
mentum. The condition to satisfy this identity is given by
@"
@M

= 0()M

=M   
2

S
@"
B
@M

() 
S
=  
2

S
@"
B
@M

: (13)
The relation Eq.(13) between 
S
and "
B
is exactly satised both in the
density dependent part and in the Feynman part, respectively, in RHA[2]. In
the method of cut-o regularization, however, the same relation is not satised
in Feynman part although it is satised in the density part. Then we require
that 
SF
is combined with "
BF
by Eq. (13) and obtain the expression for "
BF
as follows,
"
BF
=  
1

2

S
Z
M

M
h

SF
(x;)  
SF
(M;)
i
dx
=

4
1
4
2
h
  
4
ln


2
+M

2

2
+M
2
	
 
2
 
M

2
 M
2

+M

4
ln


2
+M

2
M

2
	
 M
4
ln


2
+M
2
M
2
	
+4
 
M

 M

M
n

2
 M
2
ln


2
+M
2
M
2
	
oi
;
(14)
where

SF
(M

;) =

4

S
M

h

2
 M

2
ln


2
+M

2
M

2
	
i
; (15)
and the lower limit M in the integral is chosen so that "
BF
vanishes at zero
density. The scalar self-energy and the baryon energy density in the density
dependent part are given by

SD
=  

2

S
M

h
k
F
E

F
 M

2
ln

k
F
+ E

F
M

	
i
; (16)
"
BD
=

8
2
h
2k
F
E

F
3
 M

2
k
F
E

F
 M

4
ln

k
F
+ E

F
M

	
i
; (17)
respectively, where E

F
=
p
k
2
F
+M
2
.
We make use of the saturation condition to determine the cut-o parameter
at the normal density because the cut-o parameter is independent of density.
From Hugenholtz-von Hove theorem the saturation condition is expressed by
P = 
B

E

F
+ 
2

V

B
  e

= 0; (18)
e = M   15:75 [MeV ]; (19)
at the normal density. Then we have a following relation between M

and 
V
at the normal density.
E

F
+ 
2

V

B
= M   15:75: (20)
4
On the other hand, the coupling strength 
S
is given by

S
=
M

 M


s
; (21)
where


s
is dened as 
s
= 
S



s
. Putting these 
S
and 
V
into energy
density equation, Eq. (10), we have a kind of relation between the eective
mass M

and the cut-o parameter  which satises the saturation condition
at the normal density.
In Fig. 1, we show the saturation curves for several sets of M

and . The
other bulk properties of nuclear matter can be calculated for each set and are
summarized in Table.
Fig. 1 and Table
First of all we discuss the magnitude of cut-o parameter  which looks very
small at rst glance. We try to calculate 
SF
again by introducing the following
conventional monopole type of form factor into vertex,
F
NN
(p; q) =

2
N
 M
2
p
2
+
2
N


2
N
 M
2
(p + q)
2
+ 
2
N


2

q
2
+
2

; (22)
Fig. 2
We note that we use this form factor for 
SF
(M;) at zero density while we
modify it for 
SF
(M

;) at nite density by replacing M with M

in Eq.(22).
We make a comparison between the new result and 
SF
(M

;), and obtain
the following relation between two kind of cut-o parameter,

2
N
= M
2
+ 
2
;
 


N
2
= M

2
+
2

; (23)
where another cut-o parameter 

dose not participate in Hartree calculation
because q
2
= 0. When we take about a half of nucleon mass as the cut-o
parameter , for example, we have familiar values for the cut-o 
N
of the
monopole type of form factor.
Next, we are very interested in the saturation curves in Fig. 1. The sat-
uration curves are softer as an increase of . Starting from the most sti
curve (MFT) obtained without vacuum eect ( = 0), there exist the curves
with smaller values of incompressibility less than 200 MeV. We can understand
the saturation mechanism as a stable balance of three force, i.e., the attractive
-meson, the repulsive !-meson and another repulsive vacuum eect. Contribu-
tions of meson to energy density depend strongly on the baryon density whereas
contributions of vacuum eect depend weakly on the baryon density. So, since
the energy density is xed at the normal density, if the vacuum contribution to
the energy density is larger, the !-meson contribution is smaller. Thus, the in-
crease of  dulls the density dependence of energy density in the neighbourhood
of the normal density. This is the reason that the incompressibility K becomes
small if  increase, as shown in Table.
5
Also we make a remark on the relation between M

and . The eective
mass depends on the attractive -nucleon coupling g
s
and the repulsive vacuum
parameter . The former makes M

small and the latter dose M

larger. The
eective mass continues to increase if 0 < =M < 0:4, reaches to the maximum
at =M  0:4 and decreases if 0:5 < =M < 0:56. Then, the vacuum eect for
the eective mass is largest at =M  0:4. It is reasonable that the repulsive
!-nucleon coupling strength g
v
shows a minimum value at =M  0:4 ( the
maximum eect of vacuum ). It can be observed through the eective mass
that the parameter  and g
v
are complement each other as the two repulsive
eects.
The eective mass M

decreases as the incompressibility K decreases less
than 400 MeV. The small M

is rather desirable since the empirical spin-orbit
splitting in light nuclei supports M

= 0:6M [10]. The skewness K
0
( the third
order derivative of saturation curve at the normal density in Ref. [5]) can also
be calculated. Using this K
0
, the Coulomb coecient K
c
is given by
K
c
=  
3q
2
e
5r
0

9K
0
K
+ 8

; r
0
=
 
3
4
0

1=3
; (24)
based on the scaling model[11], where g
e
denotes the proton electric charge. K
c
is the coecient of leptdermous expansion[11],
K(A;Z) = K +K
sf
A
 1=3
+K
vs
I
2
+K
c
Z
2
A
 4=3
+    ; I = 1  2
Z
A
; (25)
where K
sf
and K
vs
are the surface-term coecient and the volume-symmetry-
term coecient, respectively. These coecients are determined from the giant
monopole resonance(GMR) data of many nuclei. In g. 3, we show the K  K
c
relation together with results analyzed by empirical values in Table 3 of [3] and
in Table IV of [4]. We have a ne agreement with results analyzed empirically
if 200 MeV < K < 350 MeV.
Fig. 3
The symmetry energy a
4
in Table includes the -meson contribution which
depends on the square of the ratio of coupling strength to mass of -meson and
makes a
4
increase to about 30 MeV, i.e., without this contribution a
4
becomes
20.6 MeV at K = 300 MeV, in Table.
The K  K
vs
relation is shown in Fig. 4. The quantity K
vs
is given by
K
vs
= K
sym
  L

9
K
0
K
+ 6

; (26)
in the scaling-model, where L andK
sym
are the rst and second order derivative
of asymmetry energy, respectively ( see the detailed denitions in Ref. [5]). Also
we have another ne agreement with empirical values[3,4] if 250 MeV < K <
400 MeV.
In summary, we proposed the method of cut-o regularization to evalu-
ate the vacuum corrections in nuclear matter in the framework of the Hartree
6
approximation. We found that this method, RHAC( Relativistic Hartree Ap-
proximation with Cut-o regularization), can prepare the values from 200 MeV
to 546 MeV for nuclear incompressibility in spite of a few adjustable parame-
ters. So we note that the RHAC method is a very useful phenomenological one
under the present situation that there is much uncertainty in the experimental
determination of compressional properties.
We made quantitative analysis of the vacuum correction by the cut-o pa-
rameter. The results are summarized as follows.
(1) An increase of  means an increase of vacuum correction. The parameter
 can be connected with the cut-o parameter of the conventional monopole
type of form factor.
(2) The vacuum correction gives the repulsive eect both to the eective
nucleon mass and to the baryon energy density. The saturation property is
yielded by the interplay among the attractive -meson, the repulsive !-meson
and the repulsive vacuum eect. The repulsive vacuum eect makes the nucleon
incompressibility small because of its weak dependence on the baryon density.
(3) The calculated asymmetry energies in Table agree well with the empirical
values if C
2

= (g

M=m

)
2
= 54.71 is used as the -meson coupling strength[5].
(4) The calculated curve on the K  K
c
plane is a ne agreement with the
empirical candidates in region 200 MeV < K < 350 MeV and also the curve
on the K   K
vs
plane is a good agreement with the empirical candidates in
region 250 MeV < K < 400 MeV. Therefore, to account for K
c
;K
vs
and a
4
simultaneously, the RHAC method is valid if 250 MeV < K < 350 MeV.
Acknowledgment
The authors would like to acknowledge instructive discussions with Dr. T.
Harada. They would like to thank T. Mitsumori and N. Noda for useful dis-
cussions. The authors are grateful to the members of nuclear theorist group in
Kyushyu district in Japan for continuous encouragement.
References
[1]. J. D. Walecka, Ann. of Phys. 83, 491(1974).
[2]. S. A. Chin, Ann. of Phys. 108, 301(1977).
[3]. J. M. Pearson, Phys. Lett. B271, 12(1991).
[4]. S. Shlomo and D. H. Youngblood, Phys. Rev. C47, 529(1993).
[5]. H. Kouno, N. Kakuta, N. Noda, K. Koide, T. Mitsumori, A. Hasegawa
and M. Nakano, Phys. Rev. C51, 1754(1995),
H. Kouno, K. Koide, T. Mitsumori, N. Noda, A. Hasegawa and M. Nakano,
to be published in Phys. Rev. C.
[6]. R. J. Furnstahl, R. J. Perry and B. D. Serot, Phys. Rev. C40, 321(1989),
K. Lim, Ph.D. Thesis in the Dept. of Phys., Indiana Univ., (1990) ,
K. Lim and C. J. Horowitz, Nucl. Phys. A501, 729(1989).
[7]. M. Prakash, P. J. Ellis and J. I. Kapusta, Phys. Rev. C45, 2518(1992).
V. K. Mishra, G. Fai, P. C. Tandy and M. R. Frank, Phys. Rev. C46,
1143(1992),
J. A. McNeil, C. E. Price and J. R. Shepard, Phys. Rev. C47, 1534(1993).
7
[8]. M. P. Allendes and B. D. Serot, Phys. Rev. C45, 2975(1992),
B. D. Serot and H.-B. Tang, Phys. Rev. C51, 969(1995).
[9]. N. M. Hugenholtz and L. van Hove, Physica 24, 363(1958).
[10]. A. R. Bodmer and C. E. Price, Nucl. Phys. A505, 123(1989).
[11]. J. P. Blaizot, Phys. Rep. 64, 171(1980).
8
Table and Figure Captions
Table The numerical results for parameter sets of (M

, ). K, K
0
, K
c
, a
4
,
K
vs
, K
sym
and L are shown in MeV. The a
4
, K
vs
, K
sym
and L depend on value
of C

= g

M=m

where C
2

= 54.71[5].
Fig.1 The k
F
-dependence of binding energy. All curves take a minimum value(
-15.75 MeV ) at the normal Fermi momentum ( 1.35 fm
 1
). The bold solid line,
the bold dash-dotted line, the bold dashed line, the bold dotted line, the dash-
dotted line, the dashed line, the dotted line and the solid line are the results
for K = 546, 500, 450, 400, 350, 300, 250 and 200 MeV in Table , respectively.
The bold solid line is the result of MFT without the vacuum eect.
Fig. 2 Diagrammatic representation of the form factor at NN vertex. The
solid ( wavy ) lines denote nucleon ( -meson ) propagators. The form factor is
shown as the circle at vertex.
Fig.3 The K  K
c
relation. The crosses with error bars are results in ref. [3]
and the solid squares are the data from the table IV in ref. [4].
Fig.4 The K  K
vs
relation. The crosses with error bars are results in ref. [3]
and the solid squares are the data from the table IV in ref. [4].
9
=M K K
0
K
c
a
4
K
vs
K
sym
L M

=M g
s
g
v
0.0000 546 226.39 -8.971 30.2 -890.4 86.3 100.36 0.5470 10.438 12.899
0.2827 500 160.77 -8.330 29.5 -775.3 72.0 95.27 0.5722 10.447 12.498
0.3884 450 101.96 -7.676 29.0 -671.5 65.6 91.69 0.5924 10.779 12.165
0.4892 400 54.71 -7.058 29.0 -587.6 79.8 92.29 0.5919 11.903 12.173
0.5407 350 0.05 -6.118 29.7 -463.3 127.9 98.50 0.5650 13.467 12.615
0.5559 300 -78.33 -4.320 30.5 -197.4 186.0 105.02 0.5398 14.535 13.011
0.5609 250 -177.10 -1.242 31.1 288.0 246.4 110.74 0.5202 15.262 13.308
0.5624 200 -293.20 3.972 31.6 1142.4 308.9 115.86 0.5044 15.801 13.542
Table
10


Bulk properties of nuclear matter in the relativistic Hartree approximation with cut-off regularization

Bulk properties of nuclear matter in the relativistic Hartree
  approximation with cut-off regularization

Bulk properties of nuclear matter in the relativistic Hartree approximation with cut-off regularization

Abstract

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