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Finite-size pion effect on its self-energy
Nuclear Physics @ North-Holland
A437 (1985) 619-629 Publishing Company
FINITE-SIZE SADATAKA
PION EFFECT ON ITS SELF-ENERGY’
FURUI,
S.B. KHADKIKAR*
and
AMAND
FAESSLER
Institut fir Theoretische Physik, Universitiit Tiibingen, Auf der Morgenstelle 14, D-7400 Tiibingen, F. R. Germany Received 12 July 1984 (Revised 18 October 1984) Effects of finite size of a pion on its self-energy in the bag models are studied. With a scale-dependent pion-quark coupling the system can become stable if the pion has a finite size. If one adopts the Nambu-Jona-Lasinio-type lagrangian and relates the pion-quark coupling to the confinement potential the system becomes unstable even if the pion has a finite size.
Abstract:
1. Introduction
In models of quark bags coupled to a pion field the finite size of the pion is usually ignored. A difficulty associated with these models is the lack of stability of the quark bag due to the rapid increase of the pion pressure on the bag ‘). De Kam and Pirner ‘) investigated the effects of the finite size of the q4 pion on the pion-quark bag coupling by a nonlocal pion-quark interaction. We perform a similar procedure for our bag model with a cr2-dependent confining potential. The quark satisfies the Dirac equation [y&+iy*V-
U(r)]+(r)=0
(1.1)
with a phenomenological confinement potential U(r) = cr*. Weise et al. ‘) introduced the potential U(r) of the mass term in analogy with the Nambu-Jona-Lasinio (NJL) model 9): u(r) =%(r)
tr
[(ICl(r)i(r))l,
(1.2)
where
and tr means that a trace of the vacuum expectation value is taken. They assumed a chiral invariant NJL lagrangian of the type &,, = i(x)
@V(x) +
W6(xMb)12+ [~(x)irsMx)12~ .
(1.3)
Hence if the confinemtnt potential U(r) is determined the pseudoscalar isovector coupling gr( r) is also determined in this model and both have the same dependence + Supported * Permanent
by the Deutsche Forschungsgemeinschaft. address: Physical Research Laboratory, 619
Ahmedabad
380009, India.
S. Furui et al. / Finite-size
620 on
the scale. They described
The chiral model
symmetry
of pion-quark
a pion as a collective
is, however, coupling
broken
effect
qq mode with a size (r*)“* = 0.4 fm.
in the real world.
with a radial
We have constructed
a
dependence c2??
g,fl( r) =
9ln(A+r,/r)’
where A and r. are fitted by the p-meson nucleon form factor and the charge form factor of a proton “). These parameters fix the scale of the radial dependence which is quite different from those of Weise et al. “). The function g&r) is singular at r = 3.0 fm but we are interested in the region r~ 1 fm. We compare the pionic self-energy
for different
radial
dependence
2. The pion-quark In analogy
of the pion-quark
coupling.
bag coupling
to the chiral bag model 5.6), we determine
an effective pion current
= _---M(r)f’fJr)&(r)r5T$(r) > fr
as
(2.1)
“6
where M(r) is a function which has a peak near the surface of the bag not necessarily equal to the mass term U(r) = cr*. The function P( 7) defines the size of the pion. We parameterize:
(2.2) The ratio feff(r)/fc is given by
P(lh - r21)~(rl)y5T~c’(r2)8(rl + r2-2r) d3r,d3r2 >
fedr)
S,
-=
(G(rh5MrNst
fc
P(v)$(r+rOw+(r-q) d3T> (G(r)w$(r))sr ’ St
=
The suffix st means the expectation functions are
value in the spin-flavour
(2.3) space. The quark wave
(2.4)
S. Funk
et al. / Finite-size
621
efect
We define
(2.5)
F=ia
. sf(lr-ql)g(lr+ql)+
ia - *lr+dMlr-rll)
.
(2.6)
Since the bag is spherical, we can expand the wave functions g(r) and f( r)/r in s-wave harmonic oscillator functions h,(r) [ref. ‘)I, which is defined as
=N,,e-
(r’b)2’2 @,&r/b)
@,&r/b) =
,
(2.7)
(ii:;;,.,.L!/(r2/ b2),
(2.8)
where L:“(x) is a Laguerre polynomial. Using the formula that separates &,o(lr+ql/ b) into a sum of products of a part dependent on n and the function @+, (r/b) and performing the angle integral over dq, we obtain in the case of no spin flip F(r,q,a)=ia*r
C GnF,fN,,N,f [ nn’
1 D”(N,A)D”‘(N’,h)(~/b)N+N’ CAP u’
x@,,,(r/b)@,.,,,(r/b)
e-(r2+T2)‘b2~2
1,
(2.9)
where G, and F,. are the expansion coefficients of g(r) and f(r)/r, respectively. Next we make an average over the angles in C, @,A,@(T,APand find .!A( r) -=
_L
C G,F,.N,N,D’(N,
[nn' 00 h
aN+N’ XT-312
-
(,&+
A)D”‘(N’, A)
b3
(2N+2N’+ l)!! & bZ)N+N’+3 2((N+N’)/2+ 1) 2 2h
'(2A
+2u+
C
nn’ mr’
l)!! (2A +2a’+
G,F,fN,N”‘(-)“+“’
A+I/2($/b2)L;+l/2
l)!!(r’b) L” 2
(r’lb’)]
PI+“’
(2n + l)!! (2n’+ l)!!
L!,“( r2/ b2) LL12(r2/ b2)
-I 1 ,
(2.10)
622
S. Furui et al. / Finite-size effect b = 0.6L fm
0 = 0.2 fm 1.2
zv 1.0 -L3 0.8Ia 0.
0.2
0.L
06
0.8
r [fml Fig. I. The radial dependence
of the effective pion-quark coupling for the pion size parameter and the quark oscillator length b = 0.64 fm.
a = 0.2 fm
where N=2n-2a-A,
i=diTi, 2
D"(N,h)=D(N,A)=
I/2(
N+h
b
497 -
i .
($(N-A))!(N+A+l)!!
The parameter b is given by the size of the bag. If the size of the pion a is kept constant and the bag size or the parameter b becomes smaller then the effective coupling constant becomes smaller. For numerical calculation, we expand the wave function g and f/r by superpositions of OS to 3s wave functions. The oscillator parameter b was chosen to be the rms radius ( r’)“2 = 0.64 fm. f& r)/f, was found to be almost constant for r s b (fig. 1). For larger r this value rises. But this increase is due to an insufficient number of the basis wave functions. When the wave function is a single gaussian, then f&( r) is a constant. De Kam and Pirner ‘) also found that for a characteristic pion size f&(r)
is almost
constant. 1.0
0.
Fig.2 0.2
0.4
0.6
0.8
b Ifml Fig. 2. The bag size or the quark
oscillator
length
dependence
of the effective
pion-quark
coupling.
S. Furui et al. / Finite-size The
coupling
constant
f&r)
for different
623
efict
sizes of a bag can be obtained
by a
scale transformation; the Dirac equation ( 1.1) for a massless quark is scale-invariant. If one changes the size by a factor K and changes the potential strength c to ( l/~~)c then the wave function
has the same form. Since the size of the bag is characterized
by b, we plot in fig. 2 the &(O) as a function of b for a = 0.2 fm. In the model of Weise et al. ‘) the pion-quark coupling has the same r dependence as the potential M(r) = cr*. In this model, the b3 dependence of thefeff( r) is cancelled by l/ b3 dependence of M(r) of eq. (2.1) and the effective pion-quark bag coupling remains constant. This leads to the collapse, of the bag due to the pion self-energy. 6
3. Pionic self-energy In a naive pion-quark interaction l/( q* + p’). The self-energy becomes AE2 = $q.i
picture
one can take the pion propagator
as
M(r)
~kz(r)_f(r)~g(r’)_f(r’) 77 7r
xj,(ipy,)h$“(
iFy,)r*
dr r’* dr C (ai i#j
f * *uj~i -T~),~ eff , () _L
(3.1)
where (ai * UjTi - Tj)st is the expectation value for the 3-quark system. The summation is over different quark pairs (fig. 3). This energy is scale-dependent due to the mass of the pion and it goes to 0 as the bag size becomes 0, since the p-wave pion wave function is small for r < l/p. In addition to this static pion exchange energy there is another self-energy due to the pion field outside the bag. It contains both one-body and two-body processes as shown in fig. 3. The pressure on the bag,due to the poinic field is usually calculated by approximating the pion field by the classical massless pion field which satisfies the equation *)
M(r) LB f,+(r)-w+(r)
V’+(r) = -if, We approximate
jefi( r)/f,
by a constant
and choose
.
M(r)
(3.2) as a scale-dependent
function: M(r) =
2 7Glfm 9 In (A+ ro/r) ’
A Fig. 3. The pion self-energy
B diagram.
(3.3)
624
where
S. Furui et al. / Finite-size
A and
r. can be fitted from the information
The product Therefore
of M(r)
and the quark M(r)
we approximate
wave function
2&Rfrr Cefi has a dimension
V2$( *) = -i
different
form factor “).
have a peak near the surface.
S(R -
r) ,
(3.4)
from that of C,,. The classical
pion field is
2& 9 In (A+ r,/R)
1
2&fi
4~ 9ln(A+r,/R) A solution
of the nucleon
by
9 In (A+ r,/R) where
effect
f
- r)?.
2a f iTf(R)g(R)S(R
(3.5)
c
is --R feff
+(r)
=&
9 In ;;$Rf-(R)s(R)
C (a
* 64~
7
0
(3.6)
;
--
R2 Tcff
The energy
if rZ=R if r=SR
shift is given by 2
dr{V+(r)}‘=-& .zr
5R3
f(R)2g(R)2
fs 2 fc ' (1
C (ui ij
* uj7i
*~ji>
(3.7)
which has a l/R3 dependence We identify the field strength
as expected. at r = R with that of the dipole field “) - 01 * V( 1/47rr):
(3.8) or
$91 c The self-energy
n $,R)
r.
fUOdR)=$$
(3.9) N
for R = 0.64 fm is given by (3.10)
S. Furui et al. / Finite-size effect
625
where are used C, (Ui - UjTi * TV)= 48.4 for the average
of A and N, and g, = 13.6.
When the bag size becomes becomes becomes
small.
smaller then (f,,JjJ*
becomes
If the scale of the system is changed
smaller and the self-energy
from R to
KR,
the self-energy
1% (A-+ ro/~R) fetr for the M(r) given by (3.3). Here f$ for R = 0.64 fm. When obtain
one chooses
is the value
= cr* with c =
M(r)
c,/K~
of the rrNN coupling
where
K
(3.11) constant
is 1 for R = 0.64 fm, we
(3.12)
The self-energy diverges in this case for KR = 0. We estimate now the static pion-exchange contribution Cefi by the physical coupling constant of TNN: g,(O) = YMN
AE2. We adjust the value
f,9 -dr=13.6, 9 ln (A+ ro/r) f,
27Gtl
f(r)g(rEr3
(3.13)
or
9 ln (A+ ro/r)
(3.14)
fC 2
g(rlf(rk(r')f(r')
X
In (A+ rO/Kr) In (A+ rO/Kr’)
x j,( ipKr,)h(l’)( ipKr,)r* dr r’* dr’
xx
(Ui
’
UjTi
. rr fC
f ’
Tj)sr
ij
(
*
(3.15)
>
When the system becomes small and K goes to 0, j,( ipKr<) goes to 0. Physically it means that the p-wave pion has a small amplitude near its origin. When the mean distance of the quarks in the bag is small, pion exchange is ineffective. When we choose M(r) = cr* with c = co/~3
I
g,(o) =!?MN af(r)g(r)fcr5f+dr= 0
13.6, c
626
S. Furui et al. / Finite-size
efect
Or
xxg(Ui *
*
Tj)sr
fc
Fig. 4 shows the self-energy
for M(r)
In the static pion exchange
(s >2. f
UjTi
2~Clif~
=
9ln(A+r,/r)’
we multiply C (Ui
’
the sum over different UjTi
’
Tj)
=
quark pairs:
30.
i#j
0.
01
02
I
I
0.3
0.L
I
I
05
06
I
I
0.7
0.8
09
R [fml Fig. 4. The pion self-energy for the $r In (A + r,,/ r)-dependent pion-quark coupling. The solid line is due to the classical pion field and the dash-dot line is due to the static pion exchange. The quark kinetic energy 2w is plotted by the dashed line.
S. Fur-G et al. / Finite-size eflect
621
7
0.
0.1
0.2
0 3
0.c
0.5
0.6
0.7
0.8
0.9
R Ifml Fig. 5. The same as fig. 4 for the c(K)?-dependent
pion-quark
coupling.
The dashed curve corresponds to the kinetic energy with the c.m. energy subtracted fx3w=2w. Fig. 5 shows the self-energy for M(r) = cr2. The energy of the classical pion field increases monotonously. In order to compare with the result of de Kam and Pirner ‘) we plot in fig. 6 the result for M(r) = const. We observe that the absolute value and the peaking behavior of the self-energy depends sensitively on the radial dependence of the pion-quark coupling. In analogy with the MIT bag model, we assume the energy of the system to be E=2w-AEl-AE,-Zo/R+$rBR3,
where the 4th term is the zero-point energy and the last term is the volume energy. We neglected the one-gluon-exchange interaction energy. The kinetic energy can be written as 0.691 R GeV for R in fm. With Z,, = - 1.3 GeV fm and B = 0.60 GeV/fm3 and AE, and AEz given by the values of (3.11) and (3.15) we obtain the minimum of the energy 0.9 GeV at R = 0.67 fm (fig. 7). We do not have any justification for
428
21
i ~~_---‘----___ ,,i\
/ ,J ._t
0.
01
-
.--
0.2
0.3
0.L
0.5
0.6
0 7
08
09
R lfml Fig. 6. The same as fig. 4 for constant
0.
0.2
0.4
06
&m-quark
coupiing.
0.8
R Ifml Fig, 7. The energy
of the system
for the $r In (A + rO/ r)-dependent
@on-quark
coupting.
S. Fund et al. / Finite-size efict
629
the choice of 2, and B. We intended only to illustrate that the system we considered in ref. “) can in fact be stable. The static pion energy AE2 which is absent in the little bag “) but which could exist in the cloudy bag has the effect of making the size of the bag large in this model. 4. Conclusion We studied the finite-size pion effect on its self-energy by taking into account the radial dependence of the pion-quark coupling. We showed that if one adopts the Nambu-Jona-Lesinio-type lagrangian and relates the confinement potential and the pion-quark coupling constant ‘) the system collapses even if one takes into account the finite size of the pion with a non-local quark-pion coupling. We assumed that the quark-pion coupling has a scale-dependent form given by (3.3). Although there is ambiguity in the evaluation of the zero-point energy and the volume energy we showed that the system can be stable.
References 1) 2) 3) 4) 5) 6) 7) 8)
V. Vento, M. Rho, E. Nyman, J.H. Jun and G.E. Brown, Nucl. Phys. A345 (1980) 413 J. de Kam and H.J. Pirner, Nucl. Phys. A3I39 (1982) 640 V. Bernard, R. Brockmann, M. Shaden, W. Weise and E. Werner, Nucl. Phys. A412 (1984) 349 S. Furui, A. Faessler and S.B. Khadkikar, Nucl. Phys. A424 (1984) 495 A. Chodos and C.B. Thorn, Phys. Rev. D12 (1975) 2733 G.E. Brown and M. Rho, Phys. Lett. 82B (1979) 177 S. Furui and A. Faessler, Nucl. Phys. A397 (1983) 413 T.E.O. Ericson, Lectures given at the Int. School of Nucl. Phys., ErIce, April 1983, CERN report TH-3641 9) Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345
A437 (1985) 619-629 Publishing Company
FINITE-SIZE SADATAKA
PION EFFECT ON ITS SELF-ENERGY’
FURUI,
S.B. KHADKIKAR*
and
AMAND
FAESSLER
Institut fir Theoretische Physik, Universitiit Tiibingen, Auf der Morgenstelle 14, D-7400 Tiibingen, F. R. Germany Received 12 July 1984 (Revised 18 October 1984) Effects of finite size of a pion on its self-energy in the bag models are studied. With a scale-dependent pion-quark coupling the system can become stable if the pion has a finite size. If one adopts the Nambu-Jona-Lasinio-type lagrangian and relates the pion-quark coupling to the confinement potential the system becomes unstable even if the pion has a finite size.
Abstract:
1. Introduction
In models of quark bags coupled to a pion field the finite size of the pion is usually ignored. A difficulty associated with these models is the lack of stability of the quark bag due to the rapid increase of the pion pressure on the bag ‘). De Kam and Pirner ‘) investigated the effects of the finite size of the q4 pion on the pion-quark bag coupling by a nonlocal pion-quark interaction. We perform a similar procedure for our bag model with a cr2-dependent confining potential. The quark satisfies the Dirac equation [y&+iy*V-
U(r)]+(r)=0
(1.1)
with a phenomenological confinement potential U(r) = cr*. Weise et al. ‘) introduced the potential U(r) of the mass term in analogy with the Nambu-Jona-Lasinio (NJL) model 9): u(r) =%(r)
tr
[(ICl(r)i(r))l,
(1.2)
where
and tr means that a trace of the vacuum expectation value is taken. They assumed a chiral invariant NJL lagrangian of the type &,, = i(x)
@V(x) +
W6(xMb)12+ [~(x)irsMx)12~ .
(1.3)
Hence if the confinemtnt potential U(r) is determined the pseudoscalar isovector coupling gr( r) is also determined in this model and both have the same dependence + Supported * Permanent
by the Deutsche Forschungsgemeinschaft. address: Physical Research Laboratory, 619
Ahmedabad
380009, India.
S. Furui et al. / Finite-size
620 on
the scale. They described
The chiral model
symmetry
of pion-quark
a pion as a collective
is, however, coupling
broken
effect
qq mode with a size (r*)“* = 0.4 fm.
in the real world.
with a radial
We have constructed
a
dependence c2??
g,fl( r) =
9ln(A+r,/r)’
where A and r. are fitted by the p-meson nucleon form factor and the charge form factor of a proton “). These parameters fix the scale of the radial dependence which is quite different from those of Weise et al. “). The function g&r) is singular at r = 3.0 fm but we are interested in the region r~ 1 fm. We compare the pionic self-energy
for different
radial
dependence
2. The pion-quark In analogy
of the pion-quark
coupling.
bag coupling
to the chiral bag model 5.6), we determine
an effective pion current
= _---M(r)f’fJr)&(r)r5T$(r) > fr
as
(2.1)
“6
where M(r) is a function which has a peak near the surface of the bag not necessarily equal to the mass term U(r) = cr*. The function P( 7) defines the size of the pion. We parameterize:
(2.2) The ratio feff(r)/fc is given by
P(lh - r21)~(rl)y5T~c’(r2)8(rl + r2-2r) d3r,d3r2 >
fedr)
S,
-=
(G(rh5MrNst
fc
P(v)$(r+rOw+(r-q) d3T> (G(r)w$(r))sr ’ St
=
The suffix st means the expectation functions are
value in the spin-flavour
(2.3) space. The quark wave
(2.4)
S. Funk
et al. / Finite-size
621
efect
We define
(2.5)
F=ia
. sf(lr-ql)g(lr+ql)+
ia - *lr+dMlr-rll)
.
(2.6)
Since the bag is spherical, we can expand the wave functions g(r) and f( r)/r in s-wave harmonic oscillator functions h,(r) [ref. ‘)I, which is defined as
=N,,e-
(r’b)2’2 @,&r/b)
@,&r/b) =
,
(2.7)
(ii:;;,.,.L!/(r2/ b2),
(2.8)
where L:“(x) is a Laguerre polynomial. Using the formula that separates &,o(lr+ql/ b) into a sum of products of a part dependent on n and the function @+, (r/b) and performing the angle integral over dq, we obtain in the case of no spin flip F(r,q,a)=ia*r
C GnF,fN,,N,f [ nn’
1 D”(N,A)D”‘(N’,h)(~/b)N+N’ CAP u’
x@,,,(r/b)@,.,,,(r/b)
e-(r2+T2)‘b2~2
1,
(2.9)
where G, and F,. are the expansion coefficients of g(r) and f(r)/r, respectively. Next we make an average over the angles in C, @,A,@(T,APand find .!A( r) -=
_L
C G,F,.N,N,D’(N,
[nn' 00 h
aN+N’ XT-312
-
(,&+
A)D”‘(N’, A)
b3
(2N+2N’+ l)!! & bZ)N+N’+3 2((N+N’)/2+ 1) 2 2h
'(2A
+2u+
C
nn’ mr’
l)!! (2A +2a’+
G,F,fN,N”‘(-)“+“’
A+I/2($/b2)L;+l/2
l)!!(r’b) L” 2
(r’lb’)]
PI+“’
(2n + l)!! (2n’+ l)!!
L!,“( r2/ b2) LL12(r2/ b2)
-I 1 ,
(2.10)
622
S. Furui et al. / Finite-size effect b = 0.6L fm
0 = 0.2 fm 1.2
zv 1.0 -L3 0.8Ia 0.
0.2
0.L
06
0.8
r [fml Fig. I. The radial dependence
of the effective pion-quark coupling for the pion size parameter and the quark oscillator length b = 0.64 fm.
a = 0.2 fm
where N=2n-2a-A,
i=diTi, 2
D"(N,h)=D(N,A)=
I/2(
N+h
b
497 -
i .
($(N-A))!(N+A+l)!!
The parameter b is given by the size of the bag. If the size of the pion a is kept constant and the bag size or the parameter b becomes smaller then the effective coupling constant becomes smaller. For numerical calculation, we expand the wave function g and f/r by superpositions of OS to 3s wave functions. The oscillator parameter b was chosen to be the rms radius ( r’)“2 = 0.64 fm. f& r)/f, was found to be almost constant for r s b (fig. 1). For larger r this value rises. But this increase is due to an insufficient number of the basis wave functions. When the wave function is a single gaussian, then f&( r) is a constant. De Kam and Pirner ‘) also found that for a characteristic pion size f&(r)
is almost
constant. 1.0
0.
Fig.2 0.2
0.4
0.6
0.8
b Ifml Fig. 2. The bag size or the quark
oscillator
length
dependence
of the effective
pion-quark
coupling.
S. Furui et al. / Finite-size The
coupling
constant
f&r)
for different
623
efict
sizes of a bag can be obtained
by a
scale transformation; the Dirac equation ( 1.1) for a massless quark is scale-invariant. If one changes the size by a factor K and changes the potential strength c to ( l/~~)c then the wave function
has the same form. Since the size of the bag is characterized
by b, we plot in fig. 2 the &(O) as a function of b for a = 0.2 fm. In the model of Weise et al. ‘) the pion-quark coupling has the same r dependence as the potential M(r) = cr*. In this model, the b3 dependence of thefeff( r) is cancelled by l/ b3 dependence of M(r) of eq. (2.1) and the effective pion-quark bag coupling remains constant. This leads to the collapse, of the bag due to the pion self-energy. 6
3. Pionic self-energy In a naive pion-quark interaction l/( q* + p’). The self-energy becomes AE2 = $q.i
picture
one can take the pion propagator
as
M(r)
~kz(r)_f(r)~g(r’)_f(r’) 77 7r
xj,(ipy,)h$“(
iFy,)r*
dr r’* dr C (ai i#j
f * *uj~i -T~),~ eff , () _L
(3.1)
where (ai * UjTi - Tj)st is the expectation value for the 3-quark system. The summation is over different quark pairs (fig. 3). This energy is scale-dependent due to the mass of the pion and it goes to 0 as the bag size becomes 0, since the p-wave pion wave function is small for r < l/p. In addition to this static pion exchange energy there is another self-energy due to the pion field outside the bag. It contains both one-body and two-body processes as shown in fig. 3. The pressure on the bag,due to the poinic field is usually calculated by approximating the pion field by the classical massless pion field which satisfies the equation *)
M(r) LB f,+(r)-w+(r)
V’+(r) = -if, We approximate
jefi( r)/f,
by a constant
and choose
.
M(r)
(3.2) as a scale-dependent
function: M(r) =
2 7Glfm 9 In (A+ ro/r) ’
A Fig. 3. The pion self-energy
B diagram.
(3.3)
624
where
S. Furui et al. / Finite-size
A and
r. can be fitted from the information
The product Therefore
of M(r)
and the quark M(r)
we approximate
wave function
2&Rfrr Cefi has a dimension
V2$( *) = -i
different
form factor “).
have a peak near the surface.
S(R -
r) ,
(3.4)
from that of C,,. The classical
pion field is
2& 9 In (A+ r,/R)
1
2&fi
4~ 9ln(A+r,/R) A solution
of the nucleon
by
9 In (A+ r,/R) where
effect
f
- r)?.
2a f iTf(R)g(R)S(R
(3.5)
c
is --R feff
+(r)
=&
9 In ;;$Rf-(R)s(R)
C (a
* 64~
7
0
(3.6)
;
--
R2 Tcff
The energy
if rZ=R if r=SR
shift is given by 2
dr{V+(r)}‘=-& .zr
5R3
f(R)2g(R)2
fs 2 fc ' (1
C (ui ij
* uj7i
*~ji>
(3.7)
which has a l/R3 dependence We identify the field strength
as expected. at r = R with that of the dipole field “) - 01 * V( 1/47rr):
(3.8) or
$91 c The self-energy
n $,R)
r.
fUOdR)=$$
(3.9) N
for R = 0.64 fm is given by (3.10)
S. Furui et al. / Finite-size effect
625
where are used C, (Ui - UjTi * TV)= 48.4 for the average
of A and N, and g, = 13.6.
When the bag size becomes becomes becomes
small.
smaller then (f,,JjJ*
becomes
If the scale of the system is changed
smaller and the self-energy
from R to
KR,
the self-energy
1% (A-+ ro/~R) fetr for the M(r) given by (3.3). Here f$ for R = 0.64 fm. When obtain
one chooses
is the value
= cr* with c =
M(r)
c,/K~
of the rrNN coupling
where
K
(3.11) constant
is 1 for R = 0.64 fm, we
(3.12)
The self-energy diverges in this case for KR = 0. We estimate now the static pion-exchange contribution Cefi by the physical coupling constant of TNN: g,(O) = YMN
AE2. We adjust the value
f,9 -dr=13.6, 9 ln (A+ ro/r) f,
27Gtl
f(r)g(rEr3
(3.13)
or
9 ln (A+ ro/r)
(3.14)
fC 2
g(rlf(rk(r')f(r')
X
In (A+ rO/Kr) In (A+ rO/Kr’)
x j,( ipKr,)h(l’)( ipKr,)r* dr r’* dr’
xx
(Ui
’
UjTi
. rr fC
f ’
Tj)sr
ij
(
*
(3.15)
>
When the system becomes small and K goes to 0, j,( ipKr<) goes to 0. Physically it means that the p-wave pion has a small amplitude near its origin. When the mean distance of the quarks in the bag is small, pion exchange is ineffective. When we choose M(r) = cr* with c = co/~3
I
g,(o) =!?MN af(r)g(r)fcr5f+dr= 0
13.6, c
626
S. Furui et al. / Finite-size
efect
Or
xxg(Ui *
*
Tj)sr
fc
Fig. 4 shows the self-energy
for M(r)
In the static pion exchange
(s >2. f
UjTi
2~Clif~
=
9ln(A+r,/r)’
we multiply C (Ui
’
the sum over different UjTi
’
Tj)
=
quark pairs:
30.
i#j
0.
01
02
I
I
0.3
0.L
I
I
05
06
I
I
0.7
0.8
09
R [fml Fig. 4. The pion self-energy for the $r In (A + r,,/ r)-dependent pion-quark coupling. The solid line is due to the classical pion field and the dash-dot line is due to the static pion exchange. The quark kinetic energy 2w is plotted by the dashed line.
S. Fur-G et al. / Finite-size eflect
621
7
0.
0.1
0.2
0 3
0.c
0.5
0.6
0.7
0.8
0.9
R Ifml Fig. 5. The same as fig. 4 for the c(K)?-dependent
pion-quark
coupling.
The dashed curve corresponds to the kinetic energy with the c.m. energy subtracted fx3w=2w. Fig. 5 shows the self-energy for M(r) = cr2. The energy of the classical pion field increases monotonously. In order to compare with the result of de Kam and Pirner ‘) we plot in fig. 6 the result for M(r) = const. We observe that the absolute value and the peaking behavior of the self-energy depends sensitively on the radial dependence of the pion-quark coupling. In analogy with the MIT bag model, we assume the energy of the system to be E=2w-AEl-AE,-Zo/R+$rBR3,
where the 4th term is the zero-point energy and the last term is the volume energy. We neglected the one-gluon-exchange interaction energy. The kinetic energy can be written as 0.691 R GeV for R in fm. With Z,, = - 1.3 GeV fm and B = 0.60 GeV/fm3 and AE, and AEz given by the values of (3.11) and (3.15) we obtain the minimum of the energy 0.9 GeV at R = 0.67 fm (fig. 7). We do not have any justification for
428
21
i ~~_---‘----___ ,,i\
/ ,J ._t
0.
01
-
.--
0.2
0.3
0.L
0.5
0.6
0 7
08
09
R lfml Fig. 6. The same as fig. 4 for constant
0.
0.2
0.4
06
&m-quark
coupiing.
0.8
R Ifml Fig, 7. The energy
of the system
for the $r In (A + rO/ r)-dependent
@on-quark
coupting.
S. Fund et al. / Finite-size efict
629
the choice of 2, and B. We intended only to illustrate that the system we considered in ref. “) can in fact be stable. The static pion energy AE2 which is absent in the little bag “) but which could exist in the cloudy bag has the effect of making the size of the bag large in this model. 4. Conclusion We studied the finite-size pion effect on its self-energy by taking into account the radial dependence of the pion-quark coupling. We showed that if one adopts the Nambu-Jona-Lesinio-type lagrangian and relates the confinement potential and the pion-quark coupling constant ‘) the system collapses even if one takes into account the finite size of the pion with a non-local quark-pion coupling. We assumed that the quark-pion coupling has a scale-dependent form given by (3.3). Although there is ambiguity in the evaluation of the zero-point energy and the volume energy we showed that the system can be stable.
References 1) 2) 3) 4) 5) 6) 7) 8)
V. Vento, M. Rho, E. Nyman, J.H. Jun and G.E. Brown, Nucl. Phys. A345 (1980) 413 J. de Kam and H.J. Pirner, Nucl. Phys. A3I39 (1982) 640 V. Bernard, R. Brockmann, M. Shaden, W. Weise and E. Werner, Nucl. Phys. A412 (1984) 349 S. Furui, A. Faessler and S.B. Khadkikar, Nucl. Phys. A424 (1984) 495 A. Chodos and C.B. Thorn, Phys. Rev. D12 (1975) 2733 G.E. Brown and M. Rho, Phys. Lett. 82B (1979) 177 S. Furui and A. Faessler, Nucl. Phys. A397 (1983) 413 T.E.O. Ericson, Lectures given at the Int. School of Nucl. Phys., ErIce, April 1983, CERN report TH-3641 9) Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345