A modified Nambu-Jona-Lasinio model for mesons and baryons

Nuclear Physics A551 ( 1993) 541-579 Nonh-Holland

A modified

Masayuki Dqartmenr

NUCLEAR PHYSICS A

Nambu-Jona-Lasinio mesons and baryons Katb,

Wolfgang

model for

Bentz and Koichi

Yazaki

of Phy.sics, Fucu//y oJ’.Wet~cq L:~~irer.sic~~yf Tok,w~, Hongo 7-3-l. Bunk~,o-ku, Tok,w 11.3, Jqan

Kazuhiro

Tanaka

Radiarion Laboraror~~, Rikcw, Hirosuw:a 2-l. Wake-shi, Sairunra 351-01, Japan Received

13 41a): 1992

Abstract: )\ baryon-like bound state of three \alcncc quarks in the NJL model is invcstigatcd. We find that in the flavor SU(2) cast thrrc exists no such state. Using the modified flavor Sti(3) model WC can obtain a baryon-like stale. The essential ingredient which stabilizes the system is the “insrantoninduced” six-fermion interaction describing the U,A( 1) anomaly in a phenomenoloyical way. The propertics of mesons arc albo investigated in this model.

1. Introduction

In traditional nuclear physics, the baryons and mesons are thought to be elementary particles. Nowadays it is believed that these are not elementary but composite particles, and the constituents of these two kinds of particles are essentially the same. Therefore a consistent description of mesons and baryons attracts much interest in hadron physics. For this purpose we need a theory which describes the constituents of hadrons and the strong interaction between them. It is commonly believed that QCD is the fundamental theory of the strong interaction and governs the hadron physics. In the energy scale of hadrons (low-energy region), QCD has two important

properties:

spontaneous

chiral-symmetry

breaking

and confinement.

Since it is very dilficult to handle QCD directly at low energy, effective theories are often used to study hadron physics. These models should incorporate the above properties of QCD as much as possible. For example the Skyrme model I-‘) is one of these effective theories. This model is based on the fact that in the limit of infinite color number and assuming confinement, QCD reduces to a mesonic theory. The Skyrme model includes only the lightest meson (rr-meson). Baryons are represented as solitons 01’ this mesonic field theory. In this model the mesons are treated as elementary particles, and it is therefore somewhat unsatisfactory as a model to investigate the structure of hadrons. The quark rr-model “) is a theory similar to the Correspondence to: Dr. K’. Bents, Dept. of Physics, Hongo, Bunkyo-ku, Tokyo 113, Japan. 037.5.9474/93/SO6.00 January

1993

~$3 1993 - Elsevicr

Science

Faculty

Publishers

of Science,

University

B.V. Ail rights resewed

of Tokyo,

7-3-l

M. Ku16 et al. / A

542

familiar nucleon states of quarks, are still treated effective

model.

modified

NJL model

o-model j). In this model the baryons are represented as bound which is more satisfactory than the Skyrme model, but the mesons as elementary It realizes

since the confinement

particles. both

The chit-al color dielectric

chiral-symmetry

is incorporated

breaking

via an infinite

model 6, is another

and confinement,

mass of quarks,

but

the negative-

energy quarks become also confined and therefore the space outside the baryon becomes completely empty (no sea quarks). The Nambu-Jona-Lasinio model (NJL model) ‘) is one of the effective theories, too. This model can describe the chiral-symmetry breaking, and the lagrangian of NJL-type is supposed to be derivable directly from QCD “). Due to the success of the non-relativistic quark model, most of the physical quantities of hadrons are supposed

to be carried by quarks,

so the quarks should

be relatively

more important

than the gluons to investigate hadron structure. In this respect the NJL model is well suited because as a starting point it includes only quark degrees of freedom. Also it is directly related to other effective models. For example, the bosonized NJL model corresponds to the chiral o-model ‘), and the skyrmion-type lagrangian is obtained by a derivative expansion of the bosonized action I”). The main drawback of the NJL model, however, is that it does not incorporate the confinement. We can expect, however, that the difference between binding and confinement has no serious consequences as long as we limit ourselves to the description of the ground state. This is partly supported by the QCD sum rule ‘I) which tells us that the baryon mass is mostly due to the quark condensate. The NJL model has been used to investigate many subjects of nuclear and particle physics, for example the properties of the pion ‘I) and other mesons 13-15), as well as the properties of nuclear and quark matter I”). As we said above, effective models are constructed to share the known properties of QCD. There is one more important property of QCD, namely the breaking I?). In hadron physics this is connected to the n’-mass modified SU(3) NJL model “) includes also this property. In the NJL model mesons can be described rather simply as bound states. So this model has often been used to study meson However, it is not so clear whether this model can give baryonic In recent

years,

Meissner

et al. suggested

U,( 1) symmetry problem In). The quark-antiquark properties “- 15). solutions or not.

that in the Hat-tree approximation

there

exists a baryon-like state as a solitonic three-quark bound state I’). In this paper we re-examine the existence of this soliton-like state including the vacuum-polarization effect using the same flavor SU(2) model as in ref. 19) and the modified flavor SU(3) NJL model proposed in ref. 14). This model will be explained in sect. 2. We will find that in the llavor SU(2) model there exists no stable baryon-like state, but due to the “instanton-induced” six-fermion interaction which breaks the U,(l) symmetry there exists a stable baryon-like state in the flavor SU(3) model (see sect. 3). The properties of this state as well as those of the mesonic solutions will be investigated. Conclusions are given in sect. 4.

M. Karij er al.

i A modjfied

ML

543

model

2. Model As a lagrangian

for the flavor SU(3) NJL model we use the one proposed Y=q(iB-m)q+G, + G,,[det

in ref. “)

i [(9h,q)‘+(qiA,,ySq)‘] n -0 !?,(I-

dq.,

+h.c.l

(2.1)

,

where q is the quark field q = (f) with $ the SU(2) part of q, II,= ($), and m is the matrices current mass matrix m = diag (nz,, md, m,). A, are the flavor Gell-Mann with AO= ~$1, and the indices i, j of the last term in eq. (2.1) represent the flavor. Gs and G,, are the four-fermion and six-fermion coupling constants, respectively. If m =O, the first two terms in eq. (2.1) are invariant under the chiral U(3) transformation exp (iy?A,,w,) leading to 9 Goldstone hosons (see sect. 2.2). The last term in eq. (2.1) (the “determinant interaction”) breaks the U,(l) symmetry corresponding to the generator A,) and reflects the chiral anomaly in QCD. The remaining symmetry under chiral SU(3) transformation with 8 associated Goldstone bosons can be further broken down to SU(2) by using a non-zero nz,.

2.1. MEAN-FIELD

AI’PROXIMA7‘10N

We treat the lagrangian (2.1) in the Hartree (mean-field) approximation. Using Wick’s theorem we can rewrite the four-fermion interaction term in the Hartree approximation

as

(gr’q)‘=2:clrq:(~f’q)+(~~Tl’q)~ + (terms

neglected

= 2(qrq)ly$ + (terms

Similarly

the six-fermion

in Hartree

approximation)

in Hartree

approximation)

- (@l’q)’

neglected

interaction

becomes

in Hartree

.

(2.2)

approximation

(Yf,Y)(~~~Y)(~~~q)~ql’,q(~~~q)(q~~q)+(~f,q)ql-~q(Y~‘~Y) (2.3)

+(qr,4)(4f24)q1;9-2(~TIq)(~r,q)(~r~q~. For simplicity, we first treat non-zero expectation values (uu)~-1y,

the vacuum

(Jd)o=P,

case (zero quark

density),

( .~s)o= y

are constants. In this case, when the determinant in eq. (2.1) is expanded, term 2G,(z?u)(&I)(Ss) contributes in the Hartree approximation. The vacuum lagrangian becomes in the Hartree approximation ,iP+~‘,H,,=~(i~-M)q-2Gs(a2+~2+y2)-4GDa~~,

where

the

(2.4) only the

(2.5)

544

where

M. Kurd cr al. / A modified MJL model

M

is the

constituent

mass

m,-4G,~u -2G,$y, Md= md-4Gsp to calculate (Y,/3, y, we must specify time regularization

“‘), which

be removed

in proper-time

eventually.

regularization

A4 = diag (M,,

is a Lorentz-invariant The expectation scheme

which holds also for finite systems. hamiltonian h with the eigenvalue

M,, =

Since

for the regularization

is given by (see appendix

Here q,, is the eigenfunction Pi, i.c., hq, = ~~q*, and

CY,,/3 and y by calculating

scheme.

value of a quark bilinear

orbit. where E,,, is the energy of the valence-quark incomplete I’-function defined by * em7rp-’ dr. I’( p, .u) = II We can obtain

Md, MS) with

regularization

the cut-off (.I ) introduced

the NJL model is not renormalizable, cannot

matrix

-2G,-,ya, M,= m,-4G,y-2G,,ap. In order the regularization scheme. We use the proper-

operator

A)

of the single-quark

In eq. (2.6),

eq. (2.6) in the vacuum

I’($, X) is the

(2.7) case:

(2.5) where ek =v’k’+

Mi and NC is the number

of colors.

From now on we assume the SU(2) symmetry (m,, = md), and therefore take /3 to be equal to cy. Now the gap equations are obtained from eq. (3.8) by demanding self-consistency,

i.e., where

Mu= m,,-4(Gs+iG,,y)a,

(2.9)

where

M,= m,-4G,y-2G,,a’.

(2.10)

Next we derive the form of the lagrangian appropriate for the mean-field calculation in the finite system. In this case, allowing for the possibility of a classical pion field, we have to take into account the r-dependent expectation values (&I) = I, (Ss) and (&iyjr$). Following again the prescription in eqs. (2.2) and (2.3), WC obtain

-(G,+

G,,(Ss))((~~)‘+((Liy5T~)2)-2GS(~~)2,

(2.11)

M. Knrri et al. / A modified

where

!VJL model

545

D, and D, are given by D,=iB--

M,+~LYG~(S.~)‘+~G~((~~)‘+~~~~~

D,= id-

M,+~G,(S~)‘+~L~G,(~~)‘+~G~,((~~)’~+(~~~~I~)’).

Here Gx is the r-dependent

coupling

constant

(t&y&)),

defined

GX = G,+ :G,,(ss)

The equations

of motion

(cJiy,s$)

for quarks

= %(r)

(2.13)

by

,

(2.14)

and the quantities with prime mean the deviation from the vacuum values, (.%)-y and (IJI,!J)‘=(&)-~~. We assume the hedgehog configuration ‘), for which the fields become (rL$) = S(r) ,

(2.12)

(Ss) = Y,(r) .

,

(.EF)‘=

(2.15)

are

[~~~p+/3{M~-2uG&(r)-2G,(S’(r)+P(r)iy,~~~))]~=~~. [a~p+~{M,-4GsY~(r)-2aG,,S’(r)-~G,(S’(r)’+P(r)~)}]s=~s,

(2.16) (2.17)

where S’(r)=S(r)-20, such that all fields S’(r), Y:(r),

K(r)

P(r) vanish

= YsSF,(r) - Y,

(2.18)

for r+ 3~.

It is important to note that in the Hartree approximation the four-fermion term does not induce flavor mixing. If we also included the Fock terms, the four-fermion term would induce flavor mixing, too. The total energy of the system consists, as usual, of the regularized vacuum energy [the unregularized form of which is -f N,- x, (IF,,] - 1~“~I), where the .Q* are the eigenvalues when S’(r) =.9’:(r) = P(r) = 0 in eqs. (2.16) and (2.17)], the energy of the valence quarks, and the contribution due to the mean-fields obtained from the second line of eq. (2.11):

+

dr4~r’((Gs+G,,~s(r))(S(r)Z+P(r)‘)-4(G,+GDy)~2} I

+2G,

dr 4m2(Ys(r)2-

y’) ,

where the sum over A runs over all the single-particle (2.17) and P~.,[ is the energy of the valence-quark eq. (2.19), see appendix A.) 2.2.

MESONIC

EXCITATION

(2.19) levels both of eqs. (2.16) and orbit. (For the derivation of-

MODES

In order to treat the mesonic (qq) excitation modes on the vacuum, we have to derive the residual effective four-fermion interaction from the lagrangian (2.1). For

34. Ku6

546

this purpose,

it is convenient

terms of A,,, and thereby

er al. i A modified A;JL model

to expand

express

the matrix

the determinant

Then by using

Wick’s theorem four-fermion

term in terms of the quantities (2.20)

K,, = WA,q.

S‘, = @,,9,

to the effective

M,, - q,( 1 - ys)q, in eq. (2.1) in

as before, interaction

the contribution

of the determinant

term

becomes

Gddet M + h.c.14-~cm,,on =$G,,

-2y

;(2~ + y):S;:

-;(~cI

;

;

:S;“:-2U

- y):S;:

:SZ.--‘!
-;\~(cu

- y):S,,S,:

+ y):Pi:

t:(4cu

3

+;“‘?((Y-y):P<,Pg:+2y In terms of S,, and P,,, the residual term in eq. (2.1) has the form

1 :Pf,:S2cu CI 1 four-fermion

7

c :P:‘,: 0 -4 interaction

- y):Pi:

1 .

(2.21)

due to the second

G!, i (:S::+:Pf,:). n- 0

(2.22)

From this and eq. (2.21) we can deduce the effective coupling constants for the mesonic modes. For example, the pionic mode corresponds to P,,2,3, and the effective quark-quark coupling constant in the pionic channel is (2.23)

G, = Gs+;G,,y. The K-meson

corresponds

to P.,fi.6,7, and the coupling GK=Gs+_tGI,~.

constant

is (2.24)

The RPA-type propagators for the pion (I-r,) and the kaon (d,j can be obtained from the qq scattering amplitude in the ring approximation (see fig. 1) as follows: 2iG, 1 - ZG_J’““‘(

p’)

= Cig,Y(i4,Cp’)),

2iGK ~=((ig~)~(id~(p~)). 1 - 2GJ7’““( p’)

(2.25)

(2.26)

Here the quark-meson coupling constants g,, g, are defined such that the propagators have unit strength at the respective poles. II’““’ and U”“’ correspond to the qq polarizations in the pseudoscalar channel, see fig. 1. Their explicit forms in the proper-time regularization are given in appendix B. The condition for the pion pole is 1 -2G,U’““‘(m;)=O,

(2.27)

M. Koto^ CI 01. / A modified

MJL model

547

+

Fig. 1. The relation

betueen

the Rf’.4-type

four-fcrmion

and for the K-meson

diagrams

interaction

for the quark

coupling

constant

M,,

=-4N,.(l,(m,,

due to the

7’ channel

(2.28)

=O.

g, (see fig. 1) is given by

M,)+m:I;(mS;,

where lI’““” (p’) = (r)/i),~‘)fl’“~!( p’) and I, given in appendix H. The n and

amplitude

is shown.

pole 1 -~G,@““(vI;)

The pion-quark

scattering

and the meson propagator

I;($,

is more complicated,

M,, Mu)),

(2.29)

M, M) = Cd/dp9)1?(pz,M, M) since the P,, and P8 parts

with

(i.e., the

n,) and ns mesons) mix with each other when the flavor SU(3) is broken by the mass term in eq. (2.1) and consequently LY-y#O in eq. (2.21). The relevant interaction

part of the lagrangian

is obtained

from eqs. (2.21) and (2.22) as (2.30)

where G,o=Gs-;(2a+y)Grj, G,,, = We define

G and 11 as follows:

G,,=G,+t(4a-y)G,,,

+d5(u

- y) G,, .

(2.31)

548

%I. Kar6 rf al. / .A mod[fied

where nii (i,j = 0,8) In terms expressed

is the pseudoscalar

of the polarizations

U”‘“‘,

XJL

model

qq polarization

n’“”

with flavor vertices

and 17”’ in appendix

A,, A,.

B, the n,,‘s

are

as follows: I7,,,( p’) = i(2P”“‘( LIss( p’) = l(P”“‘(

p’) + P’(

p’)) )

(2.33)

p’) + 2P’(

p’)) )

(2.34)

fl,,,( p’) = f&( ,‘Ilu)(p?) Then the qq scattering

amplitude 2iC

The n and n’ meson

- n’s”( p?)) .

in the nn’ channel

(2.35)

becomes

1

(2.36)

I-2II(p’)G’

poles are given by the solutions

of

det(l-2TI(p’)C)=O. Around these poles, the eigenvalues can be identified with (-igf,A?(p’))

(2.37)

of the matrix (2.36) (r”‘( p’) and T”“(p’)) and (-igt,.d,,(p’)). If n(p,‘) is assumed to be

real, the eigenvectors of eq. (2.36) are usually lJ) given in terms of a mixing angle H(-:sr<8<~;r)by77=77,~sinH+77,cos8and7l’-;rlocos8-77,sin8.Here770=(:) and vX = (y) are the eigenvectors in the absence of mixing corresponding to the quark contents v$ tiu + dd + Ss) and d:( lllr + &f - 2sS), respectively. When Gt, = 0, eq. (2.37) reduces to (1 -2GsU’““‘(p2))(1 -2G,II’““‘(p2)) =O. In this case the u-, d- and s-quark parts are separated, the n-meson is a tiu bound state and degenerate with the pion, and the n’ is a Ss state. The eigenvectors of u-, d- and s-quark parts are %d= A(Y),

d,=

j&j,

respectively, which corresponds to 0 = 54.7” (“ideal will represent the eigenvectors as rlK %I+ V,,nL,

mixing”).

7l’K 771‘i+v?$l:d ,

When

GL) # 0, we

(2.38)

where V,, V,,. are generally complex numbers. The values of V,,, V,. at the poles give the corresponding quark contents as n ~v$(Uu + lsd) + V,.Fs and n’s &tiu + &)+ V,,.$s. The case of no mixing between no and n8 corresponds to V,, = -v?, V,,. = d’i, and the ideal mixing corresponds to V,, = 0, V,,. = X. [For real KI( p’), we have V,,V,,, = -1, where V,,, V,,. are the values at the respective poles, and the relation to the mixing angle is given by tan 0 = (t/2+ V,)/(l -dV,,).] If pZa4M:(4Mz), Kf’““‘(II”“) has an imaginary part. These imaginary parts correspond to the unrealistic decay to the unbound quark-antiquark system. There are two methods to treat these imaginary parts. The first one “) is as follows: These imaginary parts should be neglected because their presence is due to the shortcoming

M.

of the model

(no confinement),

masses are determined the imaginary imaginary function. different

Kar6

part

NJL model

and this decay

process

as poles of the propagators of the self-energies.

part of the propagator If the model because

er al. / A modjfied

The second

the conlinement,

there would be no continuum

is unphysical.

which are obtained

can be physically

included

549

method

interpreted the strength

The meson by neglecting

is as follows: as a meson function

The

strength

would

look

states for the quarks and the strength

function would exhibit the discrete spectrum. One might expect, however, that the distribution of the strength is still similar to what is calculated in the present model. The meson masses are then determined as the peak positions of the imaginary part of the propagators. In the next section we will adopt the second treatment, but we will also investigate the propagators according to the first treatment. The pion decay constant f= is obtained from the relevant diagram for pion decay as I’))

= -4g,N,.M,IJmt,

M,,

M,)

where g, is the quark-pion coupling constant of I2 after regularization is given in appendix relation

($=m!_),

determined IL) Using

(2.39)

by eq. (2.29). (The form eq. (2.29), we obtain the

(2.40) For m, > 0, combining

the gap equation

eq. (2.39), we can obtain

-f Finally

in this section,

in the various

symmetry

(2.9), the pion-pole

the Gell-Mann-Oakes-Renner

irnt

condition

relation

(2.27) and

‘>) in this model,

.

(2.41)

we discuss the structure

of the pseudoscalar

limits of the lagrangian

(2.1). (Compare

mesonic

spectra

with the discussion

below eq. (2.1).) In the case m,,= m, = 0, GD = 0 the chiral U(3) invariance leads to the 9 massless Goldstone bosons rr, K, n0 and Q. This is easily seen by noting that in this case eqs. (2.27), (2.28) and (2.37) become identical, as is seen from eq. (2.8) and the formulae in appendix B. (Note that in this case cy = y and M, = M,.) If G,, f 0 the U,( 1) symmetry is broken and the no becomes massive. This can be seen by noting that, in this case, Goo< Gfi8 (see eq. (2.31), we assume G,
M. KarG EI a/. / A modified

550 2.3,

I-L.AVOli

SU(ZJ

n~ndcl

CASE

In the next section SU(2)

NJI.

case assuming

we will also carry out a mean-field chiral

from eq. (2.1) by replacing term then becomes

symmetry

calculation

(m,, = 0). The lagrangian

q + II, = (i),

for the flavor

can be obtained

and h,, + ‘;h with r0 = 1. The determinant

simply ~G,,[(5~)2+(ll;ly~T~/)~-_(~~y511/~2-(~~~)2].

With G,, = 2Gs = G the lagrangian

therefore

Ysr.,:) = iidti

mean-field

calculation

takes the standard

+ G[(&//)z + ( I&T$)~]

We note that in this case the restriction

(2.42) form

.

Gr, = 2G, does not affect the results of the

in any way, since there only the terms (&)*

contribute. To compare with previous following quantities:

(2.43)

works,

we will discuss

our results

and (&yir+‘)’ in terms

of the

(2.44) where S(r) and P(r) are the same as in the SU(3) case, see eq. (2.15). In terms of the quantities (2.44), the reduced lagrangian (2.11) reads ~~~1,,,,=~[iB-M-R(a'(~)+iyir.n(r))]rCr-f~LZ(~(i-)Z+5T(r)~)

with M = M, =gcr,

(see eq. (2.9)). Here CT= wv+ CT’(V)with uv the vacuum

(2.45) value.

At first sight it seems that in this formulation the number of parameters has increased from the original two (G, t) to three (g, CL*,.4 ). However, since g multiplies the pion Geld in eq. (2.45) it is subject to the condition of the pionic wavefunction renormalization. If we choose the renormalization condition such that g becomes the quark-pion coupling constant gir (defned as the residue of the qq scattering amplitude at the pion pole), eq. (2.29) gives the condition 4N,.g’L(O,

M, M) = -1 ,

(2.46)

which determines

g in terms of :1. (Eq. (2.46) can also be obtained directly by the has unit strength at the pole.) On requirement that 1\,( p’) = [p2 - g’n’““’ (I?)]account of eq. (2.46), eq. (2.39) becomes Uv =J;;. TV is determined

by the gap equation p’=

(2.47)

in terms of (g, pz, :1) as (see eq. (2.9)) 8N,g’l,(M)

(2.48)

with I, given by eq. (B.3) and M = gcr,. In the present chiral-symmetric case, eq. (2.48) coincides with the condition for the pion pole at p2 = 0, see eqs. (2.27) and (B.1).

551

M. Karl* er al. ! A modified NJL model

3. Results 3.1. FLAVOR

SL’(Z) CASE

First we discuss

our results

In the mean-field

treatment

for the flavor SU(2) case assuming based on the lagrangian

chiral symmetry.

(2.43) there are two parameters

(G, ,;I), and if one considers (2.39) as a condition to reproduce the experimental value of,fii = 93 MeV, one is left with one parameter which can be varied freely. An equivalent treatment, which we follow in this section, is to use the lagrangian (2.45). There are then three parameters (g, puz, .;I) and the additional renormalization condition (2.46). Again, if we consider eq. (2.39), or equivalently eq. (2.47): as a condition to reproduce fl, = 93 MeV, one can choose one parameter which can be varied freely. We use g as this free parameter. The numerical calculation was carried out as follows. At first we choose the initial CT-and a-fields as cT(r) -f;; where we choose

cos fl( r) ,

n(r) -_& sin 0(r) ,

O(r) = ‘in exp (-I./&,),

(3.1)

R. = 2/ M, and solve the Dirac equation (3.2)

by the method due to Kahana and Ripka “) which is explained in appendix C. Since this method amounts to diagonalizing the Dirac hamiltonian in a discrete and finite basis, it introduces two more parameters, the size of the box (D) and the size of the model space characterized by the maximum momentum (p,,,,,,) of the basis formed by the spherical wave solutions of the free Dirac equation (a’ = r = 0). The results, of course, should not depend on these two parameters. Next we calculate the new (T- and n-fields from eqs. (2.44) and (2.6) with f = 1 and I‘= iy,i. T, respectively. This process is continued to reach the self-consistent solution. The size of the box is taken as D = 20/M. The results do not change when I> is increased further. For plnax we use values between 5M and 10M as will be discussed in detail later. The calculation was done for various values of g. First we calculated from g : 3.5 to g =4

(corresponding

to a constituent

quark

mass between

325 and 372 MeV)

with an interval of dg = 0.1. We found that the calculations with g = 3.5, 3.6, 3.7, 3.8, 3.9 give a “spreading” solution and the calculation with g = 4 gives “collapsing” solution. Here “spreading” means that in the course of the iteration procedure the (T-, si-fields spread in space, while “collapsing” means that they become deeper and narrower in such a way that no convergence is obtained in either case. Since we can naturally think that if the mean fields spread (collapse) for some value of g, they also spread (collapse) for smaller (larger) g, next we wanted to determine the upper (lower) limit of g which gives the spreading (collapsing) solution. Performing a calculation with dg = 0.01 we found that g = 3.90 and g = 3.91 give the spreading

552

M. Kad

and collapsing give a stable

solutions, solution,

respectively.

(i) If g is small

the meson

is smaller

the meson

than 0.01. in space

a free three-quark

fields become

for g which

condition:

fields spread

and finally we obtain

(ii) If g is large (g-‘3.91), course (iii) limits We

if there exist values

we arrive at the following

(g <3.90),

is continued,

NJL model

Therefore

the range of these values

From these calculations, procedure

rf al. / A rnod$ed

deeper

as the iteration state.

and narrower

in the

of the iteration procedure, and the system collapses. There is no value of g which gives a stable soliton-like solution (within the of the present numerical calculation). now explain these results in more detail. The first case of the spreading mean

fields is easy to understand. It simply means that when the coupling constant is small the attractive force is too weak to form a bound state. The collapsing case needs more explanation. For example, we choose g = 4 as a typical value leading to the collapse (see fig. 2). At first we calculate in the model space with the P,,_, = 5M. In this case the fields apparently conv:erge towards a “solution” (dotted lint in fig. 2), but this is not a real solution because it collapses when we enlarge the model space to P,,,.~~= 7SM. During further iteration, the o-field becomes deeper at the origin, the range of the potential becomes smaller and the energy of the system decreases. The reason why the results change when the size of the model space is enlarged is thought as follows: The bases in the model space do not form a complete included in this model space is restricted. The problem set because the momentum

iteration -._.-.-.-. ------

1 = W FioC. 2. The w- and r-fields (see eq. (X44)), for the collapsin, 0 case (g = 4) of the flavor SL(Z) model obtained b! the calculation with the model space are shown. The dotted lines represent the “solution” is about 50.) The other lines are obtained ,I,,,., = SM. (The number ol‘ iterations to reach this “solution” arc roughly 70 by the calculation in the model space with p,;,.,, L 7SM The numbers of iterations (dash-dotted lines), 90 (dashed lines). 110 (solid lines).

553

M. Kaio^er cd. / A modjjied 1ziJ.Lmodel

is whether function the quark

the space is large enough in the potential, wave function

obtained

That is to say, when we expand the component speaking,

to adequately

especially

the valence

represent

by diagonalization

is accurate

the real wave function

out of the model

we need higher momenta

space

should

the real quark

state, or in other words,

wave

whether

enough

or not.

in terms of spherical

waves,

be sufficiently

small.

Generally

in the basis states when the range of the potential

becomes smaller (more localized). In our case the model space with P,,,;,~= 5M is insufficient. This can be checked directly as follows: We can obtain the valence-quark wave function also by solving the differential equation (3.2) directly. (In practice, however, it is very hard to include the sea-quark contribution in this way, and we did it only for the case of the valence orbit.) Comparing (the one obtained by diagonalization and the one obtained

these two wave functions by solving the differential

equation), we can see whether the model space is sufficient or not. As shown in fig. 3, the real wave function for the potential given by the dotted line in fig. 2 is more localized than the diagonalized one. Therefore, although our iteration procedure converged for the small model space, the result is insignificant because the Dirac equation has not been solved properly. When we use the larger model space P,,,~~= 7.5 M, the wave function obtained by diagonalization agrees very well with the full line in fig. 3. However, we then tind that the solution collapses and the valence energy becomes lower and lower, finally getting negative. If the charge-conjugation symmetry is conserved, this never occurs. The reason is as follows: In the case of charge-conjugation symmetry, there exists

differential

- - - - -

equation

diagonalization

0.76

1

r (fd big. 3. The wave functions and diagonalization

(dashed

of valence quark obtained lines) are shown.

by solving the diflcrential

/-Cr.) and G(r)

and lower component,

represent

respectively.

equation

(solid lines)

the radial parts of the upper

554

M. Ku16 er al. i A motl$ki

the charge-symmetric opposite

partner

of the valence

sign and the absolute

value

NJL model

level. The energy

is the same.

of the partner

In this case, when the valence

level crosses zero, it also has to cross the partner

level, but this is impossible

they have

in our case,

symmetry valence

the same is broken

energy

quantum

numbers.

due to the classical

becomes

Since

is of

charged-pion

however,

because

the charge

fields, it is possible

that the

negative.

In the flavor SU(3) calculation to be discussed in the next section we will use an even larger model space (p,,,, = 10M) in order to judge whether the obtained solution is accurate enough or not. In the present SU(2) calculation, however, it is not necessary to further increase the model space: As is clear from our above discussion, if the system collapses when a certain model space is used, it will also collapse when the size of the space is increased. Also, the spreading case occurs for any larger model space, too, because in this case the space with pmax = 5M is already sufficient. We conclude that our results obtained above do not change when the model space is increased further. The difference between our model and the non-linear one “) (where there is a restriction cr”+ r’=Sl), which is known to have stable solutions for a wide range of g, is the freedom to vary the depth of the a-field at the origin. (Note that since r vanishes at the origin, ~(0) =ji in the non-linear model.) As a consequence, if the non-linear condition is imposed, the strength of the potential is roughly determined by its range “). Then if the range of the potential becomes small, the binding becomes weak and the energy is expected to increase. But in our linear model, when the range of the potential becomes small, the binding can nevertheless become strong because the depth of the potential can increase. The reason why the smallspace calculation apparently gives a stable ‘*solution ” is related to this fact: If we restrict the model space, the depth of the potential is restricted by the highest momentum of the basis states. Therefore the potential cannot become unlimitedly strong as its range is reduced. As we explained in detail

above,

we found

that

the model

specified

by the

lagrangian (2.43) and the Hartree approximation does not permit stable soliton-like solutions. A natural extension of the model is to include a vector-type interaction of the form -( 1&‘4)~, which gives rise to a repulsive potential in the single-particle Dirac equation, and therefore one might expect that the collapse can be avoided by including this term. This term can be incorporated into the lagrangian either as a Fock term, or adding it by hand. In either case, one ends up with the following lagrangian to be treated in the Hartree approximation: (3.3) (If the vector interaction is derived as a Fock term of eq. (2.43), Gv becomes :G, and the G of the chiral part changes to :-iG, since the Fock terms not only give a vector term but also a scalar and a pseudo-scalar term.) In the hedgehog configuration (i.e., a field configuration which leads to a hamiltonian invariant under rotations

M. ICar6 et al. / A modijied NIL

with the generator In the Hartree

G =j+$),

approximation

only p = 0 contributes this lagrangian

555

model

in the last term of eq. (3.3).

therefore

becomes

(compare

with

eq. (2.11)) .PL&

= &[iiil-

-

M,,+2G((&b)‘+

iy,T.

($iy57b))

-2Gvyo((l;y”4)IG

G[( I+&)’+ ( tiiyjTJ/)l] + G,;( $y”&)’ .

(3.4)

If Gv> 0, the vector term acts repulsively for valence quarks. The numerical calculation was done by varying Gv for several values of G which gave the collapsing solution in the previous G,, = 0 calculation. (In terms of the original parameters (G, .4 1 with G treated as a free parameter, G 2 0.683 fm’ leads to the collapse, and G < 0.680 fm’ corresponds to the spreading case.) In this case the variation of the parameters was not done as systematically as in the previous G, = 0 case, nevertheless we found that the general features of the numerical results do not change essentially due to the inclusion of the vector term, at least as far as we calculated. When G, is small the solution collapses and if it is large the solution spreads. If there exists a G, which gives a stable solution for a fixed G, the range of such G,) is very small. This result was found for several values of G. As a result, there is no set (G, G,) which gives a stable solution within the limits of our numerical calculation.

3.2.

FLAVOR

SU(3)

CASE

In this section we discuss our results for the flavor SU(3) case. The model now contains the five parameters m,, m,, Gs, G,, and ~1. As we will explain in detail below, in this case we can obtain a self-consistent baryon-like solution due to the effect of flavor mixing induced by the determinant term in eq. (2.1). However, for this we will need large values of IGvJ, dnd ’ this is one of the reasons why it is not possible to reproduce the spectrum of the pseudoscalar mesons at the same time. Therefore, as constraints on the parameters we only use eq. (2.27) to reproduce the experimental m, = 138 MeV and eq. (2.39) to reproduce J_ = 93 MeV. Our final parameters will be chosen so as to reproduce also the kaon mass mK = 496 MeV as well as possible by using eq. (2.28). Since it is our aim to discuss the conditions under which a stable soliton solution can be found, the most important constraint we impose

on the parameters

is to give a stable

baryon-like

solution.

As we will

see, however, it is also possible to give a reasonable description of the pseudoscalarmeson spectrum at the same time. The calculation is carried out similarly to the flavor SU(2) case. The initial fields are chosen as (see eq. (2.15)) S(r) =2CY cos e(r) ) Y?_(r) = 7, where R,, = 2/M,, method as before.

P(r) = 2cu sin O(r) ,

O(r) = 5i exp (-r/&),

(3.5)

and solve the Dirac equations (2.16) and (2.17) by the same The new fields are then calculated from eq. (2.15) using (2.6),

556

M. Ku16 er rd. / A motl(fied

and this process

is continued

!NJL model

to reach the self-consistent

box is again taken as D = 20/ M,, and a further increase

solution.

The size of the

does not influence

the results.

We prepare three model spaces with P,,,;,~= SM,, 7.5 M,,, lOMU, and call them small, middle and large spaces, respectively. We use the criterion that the solution is a real one if it does not change sect. 3.1, the adequacy

when going to the large space.

of the model

space

can be also checked

As explained by solving

in the

differential equation directly for some special cases and comparing the solution with the result of the diagonalization. If the solution for the above fields spreads out in space, it is not necessary to do the calculation in a larger space, since also the small model space is sufficient to represent an unbound state, that is to say, even if we calculate in a larger space, the result will not change. The reason is that the unbound states do not include high-momentum components. If we obtain a stable solution for a certain model space, it is not clear whether this is the real one or not. In such a case, this provisional solution was checked by a calculation in a larger space. in fig. 4 we show a solution. This is obtained has been checked by the large-space calculation wave

functions

directly. 1. (Note masses, LY= (tiu)

The that the and

with the solution

obtained

in the middle-space calculation and and by comparing the valence-quark by solving

the differential

equation

parameters used in this calculation are shown by the first line in table Gr, < 0.) Also shown in table 1 are the results for the constituent quark masses of the pseudoscalar mesons as well as the asymptotic values y =(C). All mesons except the 77’emerge as bound states. The values

-23

0

1

2 r (fm)

3

4

Fig. 4. The mean fields (see eq. (2.15 I) for the stable baryon-like solution in the Ilabor S1!(3) model arc shown. They were obtained by gradually enlarging the model space up to y,,,,, L lOM! where convergence was found after a total number of iterations of roughly 70.

M. Kato^ rr al. 1 A mod$ed

557

NJL model

The parameter set which we used to obtain the stable baryon-like solution (upper line), and the resulting values for the constituent quark masses, meson properties and quark condcnsatcs. The parameters are chosen to reproduce the experimental values of _I, and rn,. The experimental values of the remaining quantities are mK = 496 Me\‘. m,, - 54Y Me\/, m,. = 958 MeV. For the n’-meson mass, we give the peak position of the imaginary part of the propagators (unbracketed) and the pole of the propagator obtained by neglecting the imaginary part of the self-energy (bracketed) ,)I,,

=

15.9McV.

m, = 753.2 Mev. ----

---_m, = 138 MeV,

GS - 0.463 fm’, ---

CT,=-

lfm’, ---

!M:, = 365.2 MeV, !M, - 580.0 Me\‘, f,: = 93 MeV r?lK= 407.4 MC\‘. m, 7 387.7 McV, rn,.-800 !5.s) = f - 150.4)” MC\‘? (liU;)-(-170.6)3 klev’.

for the mass of the 7’ meson in table previously, i.e., the one in the bracket the imaginary part of the self-energy, to the second method of looking for propagator, which we adopt in this detail below.

.I - 635.7 ZleV --.(1459.1) MC\’

1 are obtained by the two methods explained corresponds to the first method of neglecting while the one without the bracket corresponds the peaks of the imaginary part of the meson paper. This point will be discussed in more

In order to understand why we can obtain a stable solution in the flavor SU(3) model, we study the equations of motion (2.16) and (2.17) in more detail. The potential for the s-quark in eq. (2.17) can be rewritten as U,(r) = m, - 4GSY,( r) -;G,(S(

r)‘+ P(r)‘)

,

(3.6)

or U,(r) = MS-4G,Y:(r)-;G,,[(S(+(~CI)‘)+P(Y)’]. The form

(3.6) can be considered

as an r-dependent

(3.7) constituent

mass (compare

with the form of M, in eq. (2.10)) plus an additional pseudoscalar mean-field term which is not present in the vacuum. Near the origin, the dominant term in eq. (3.7) is the third one proportional to G,,, and it is repulsive there. (Note that G,,
for the u-quark

in eq. (2.16) is

U,(r)=m,,-2G,(r)(S(r)+P(r)iy,i.r),

(3.8)

U”(r)=

(3.9)

or M,,-2aG&(r)-2G,(S’(r)+P(r)iy,r^.~)

with G,(r) given by eq. (2.14). Again, eq. (3.5) has the form of an r-dependent constituent mass (see eq. (2.9)) plus the pseudoscalar mean-field contribution. The second term in eq. (3.9) is attractive near the origin (LYCO). The binding of the

558

):c) SS

GD

Fig, 5, ‘The contribution

of the six-fermion coupling constant

interaction, which of the four-fermion

leads to a position-dependent interaction.

effecti\:e

u-quark, however, is due to the third term proportional to GX in eq. (3.9), which has the same form as in the SU(2) case, eq. (3.2), except that the coupling constant GX is r-dependent. This r-dependence is due to the flavor mixing induced by the determinant term in eq. (2.1), see fig. 5. From the above discussion, since the s-quark feels a repulsive force, Y,(r) and therefore also the r-dependent coupling constant G, decreases near the origin, i.e., the potential felt by u-, d-quarks becomes weaker there. In fig. 6 we show the effect of the determinant term on the potential (3.9) for a valence quark. The importance of the r-dependence of GX is clearly seen. After all, the attractive force felt by u-, d-quarks becomes weaker near the origin due to the effect of flavor mixing induced by the determinant

term.

0

”

”

”

0.6

”

”

1

”

”

”

1.6

’ ”

1 2

r 04 Fig. 6. The potential felt by the u-quark (L’,,(r) - M,, of eq. (3.9)) is shown. The dashed line shows the contribution of the third term when the r-dependence of G, is neglected [i.e.. GX(r) + Gx Cr; xc)). If the r-depcndencc of G, is taken into account one obtains the dash-dotted line. The solid line is the total potential including also the second term in eq. (3.9).

559

M. Karol et a/. / A modified NJL model

The mechanism

discussed

not collapse

as in the SU(2)

four-fermion

coupling

density

dependent,

that the flavor cause radical

above provides

constant

the explanation

why the solution

case. It can also be interpreted of the original

flavor

SU(2)

see fig. 5. In order that this mechanism

mixing changes

is rather

strong

or inconsistencies

(large

IG,l).

by saying model works,

does

that the

has become it is necessary

But this large ICI,1 does not

for the physical

quantities

investigated

in

this paper, as can be seen from table 1 and our following discussions. Before discussing the detailed properties of our baryonic solution, we explain our values for the 77 and 77’ masses shown in table 1. In fig. 7a and 7b the r/77’ channel scattering amplitudes (i.e., the eigenvalues of the matrix the imaginary parts of self-energies are shown for the parameter 1. If we neglect the and T”’ will appear pole is below the iiu the imaginary parts

(2.36)) including set used in table

imaginary parts of the self-energies, the lowest poles of T”’ at x.4 = 387.7 and 1459.1 MeV (see table 1). The ~1: = 387.7 MeV threshold and therefore remains unchanged even if we include of the self-energies. (This pole appears as a singularity in the

full line of fig. 7a, which is shown on a larger scale in the insertion of that figure.) Therefore we can identify this pole uniquely as the q-meson mass. From figs. 7a and 7b we see that the imaginary part has peaks at Jr=800 and 1300 MeV, respectively. The broad peak in fig. 7b at &= 1300 MeV could be thought to correspond to the 1459.1 MeV pole mentioned above. However, this peak is already above the Ss threshold and therefore it cannot reflect the presence of a resonance pole. This is because, in terms of the non-relativistic constituent quark model, the pseudoscalar mesons are s-wave bound states. There is thus no centrifugal barrier, and a resonance state cannot exist above the Ss threshold. We therefore identify the well-pronounced peak at Y!:== 800 MeV in lig. 7a as corresponding to the 77’meson. In figs. 7a and 7b the ratios of the s-quark in eq. (2.38)) are also shown (dotted line). which corresponds to a ( qo, qJ mixing obtained here is thus predominantly Q. ) V,,,l= 2.2 at the 7’ peak means

component to the u-, d-component (1V,,,q Forthe q-meson we have V,,(mf,) = -1.54, angle (see sect. 2.2) of 6, = -2.27”. The r] At the 17’ peak, V is complex. The value

that the Ss component

The values in table 1 show that our presently account of the pseudoscalar-meson spectrum, to find the best parameter

is dominant.

used parameter set gives a reasonable although no attempt has been made

set. In order to compare

with the treatment

in ref. ‘“), we

show in table 2 a parameter set which reproduces the experimental values of Jr, parts neglected) to treat the qv’ In,, mK and m,, using the method 1 (imaginary channel. (The value m, = 5.5 MeV is fixed.) We cannot obtain a baryon-like solution with these parameters: Due to the small value of GF the solution spreads out in space and finally gives three free quarks. In figs. 8a and 8b we show plots similar to figs. 7a and 7b for the parameter set of table 2. Neglecting the imaginary parts, the lowest poles of T”’ and T”’ appear at &= 445.8 and 958.0 MeV, respectively (see table 2). Both are above the Uu threshold. peaks of the imaginary parts appear at

From figs. 8a and 8b we see that the &=400 and 780 MeV. The first

Fig. 7. The rent (solid tine) and im;qinary tdarhcd liner part of the two eigenvaiucs T”’ and T’” of the scattering amplitude tt.36f arc showy The parameters arc as in table 1, The dotted line represents i Vj nhich is the absolute x&x of’ the mixing parameter, 5ec eq. (2.38). (For cxpkmation see sect. 3.7.) The insertion in fig. 7a shows the behavior of the real part on a larger scaic.

M

Kurt

et nl. 1 A modified

NJL

561

modal

TABLE 2 The parameter set (upper line) determined by the frequently (bracketed numbers for 7 and 7’) by neglecting the imaginary values for the constituent quark masses. meson propertics and masses without brackets arc the peak positions of the m, - 5.5 MeV,

G, = 0.106 fm',

m, - 163.3 MeV.

used method of fitting the meson masses part of the self-energies, and the resulting quark condensates. The v- and q’-meson imaginary parts of the propagators G,,= -0.0163 fm’,

.I = 1054.X MeV -~.

-~ m, s 138 MeV,

one agrees

IM,, = 198.9 IMe\‘, mK = 496 MeV, (iid = ( -244.2)’

well with the location

Al, = 429.9 MeV, f, - 93 MeV m,. - 780 (958.0) MeV m,, - 400(445.83 MeV. (.Cj = (- 282.2)’ 42eV’ MeV’,

of the pole neglecting

the imaginary

parts

of’

the self-energies, while for the second one these two positions are different. Again we prefer to identify the peak positions of the imaginary part with the n- and n’-meson masses. The facts that both peaks are very sharp and the real parts vary rapidly around the peak positions indicate that the poles lie near the real axis. We now return to the discussion of our baryon-like solution. Its properties are listed in table 3. P,;,, is the energy of the valence quark and E,,,, is the total energy of the soliton given by eq. (2.19). Although the valence quark is bound, the total energy is larger than that of the free threc-quark state (3MJ. This is due to the energies of the classical meson fields and vacuum polarization. The root-mean-square radius is given by @)= [li r’drp( r)]“‘“, where p is the baryon density. Its value 0.775 fm is consistent with the experimental value for the isoscalar charge radius J(r’),+(r’),, = 0.79 fm. Fig. 9 shows the baryon density of the solution. Because of the regularization, the baryon number of the sea quarks is not total baryon number does not become 1 (it actually becomes the form factor F’( p’) =s,y’ r’drj,(pr)p(r.) (solid line), which isoscalar electric form factor Gf + Gy.. For comparison we dipole type one F(p) = [;,I;/( p’+ .I h)]‘. The parameter ,In which is consistent with the empirical value (842 MeV). Since is not exactly

1, F(0) # 1. We note that the problem

of non-zero

exactly zero and the 0.97). Fig. 10 shows corresponds to the also show the fitted becomes 863 MeV, the baryon number baryon

number

of

the sea quarks could be circumvented by using (without justification) the unregularized expression for p(r) which is Iinite. The quark contents of the baryon-like solution listed in table 3 are defined as (tin),==;

c

4d

dr S’( r) ,

(3.10)

J

where shown

S’( 7) = S(r) - 2cu, .Y:( r) = 9, (r) - y, and in fig. 4. We see that the ratio

S(r) = (I&J) and

Yc,(r) = (KY) are

(3.11)

6

l.+S

4

i”0

2

0

_..~.~._...r..*_____.

0

BOO

400

800

BOO

1000

QO 1200

M Kato^a al. / A mod$ed TABLF.

NJL model

563

3

Properties of the baryon-like solution. Shown arc the values for the valence-quark energy (F,,;,,), the total energy (E:,,), the r.m.s. radius, the values of the quark condensates (3.10). and the ;;N sigma term (3.12). The esperimental value of L,\ is taken from ref. “1

-_ (tiu),=

F,‘,I 7 221.8 MeV, 1.271, (Sj,, -- -0.043,

E,,, = 1232.8 MeV, PCu = 40.3 Me\‘,

I..‘:

0.775 fm x7’,\ (exp.) = 4s L 7 MMeV

extract information on JJ, namely the analysis of the TN sigma term Izs and the mass formula of the non-relativistic quark model 14). In the present case ,X7,, is given by 2”)

[ref. ‘“)I

I:_b = 2m,,( ziu),, .

(3.12)

Its value is listed in table 3 and is consistent with the empirical value (X,, - 45 I 7 MeV). For further discussion, we note that eq. (3.12) can be rewritten identically as

(3.13) where

U” = &2au

+ ss),

(NJu,IN)/(NIu,(N).

In terms of?!, A can be written

I I \ \ \ \ 1.5 \ \ \ \,

2.0

1.0

c, - v’:( M, - m,)

Us = 2v15 UU -a),

K

I-

I

I

,

I

I

I

I

I

as A =~,:(y+2)/(1

I

I

and A= -J:) and the

I

total - - - - - - “alnn”~ . . .. ..

c

quark

..u..

.-.._--. --.-

sea quark

0.5

0.0

-0.5

0

1

2

3

4

= (fm) Fig. 9. The baryon

density of the solution. The contributions due to the valence quarks and sea quarks (dotted line) and their sum (solid line) are shown.

(dashed

lint)

I-ig. 10. l‘hc solid line shows the form factor obtained by a Fourier-Bessel density in fig. 9. The dashed line is the littcd dip& form factor F(p) X63 Me\<.

transformation of the har>on =f.t&‘r p’ + .I{,)]’ with t,,:

TN sigma term becomes \‘ ASN -fi If we assume that the baryon to the quark mass difference,

&(N/&.

mass splitting the following

is calculated perturbatively hadron mass formula’“)

(N~c~u~~N)=M~-M,-(=-~O~M~V~.

(3.14) with respect holds: (3.15)

Then 3 I:__ = -x 1 -)!

13.5 MeV,

(3.16)

where we used our quark masses m, = 15.9 MeV and m, = 253.2 MeV from table 1. We see that in the present case the empirical value of the srN sigma term can be reproduced by _r= 0 which is consistent with our result for the baryon-like solution (y = -0.03). This is due to our relatively small ratio m,/m, between the current sand u-quark masses, and therefore the factor m,/( m,, - m,) becomes large in magnitude compared to the often used value. We note that our relatively small ratio m,/m,, does not cause apparent inconsistencies for the physical quantities, in particular the meson spectra, investigated in this work. For example, our value for m, in table 1 is large compared with the often used value (table 2). However, the effects due to different current quark masses and those due to different quark condensates tend

565

M. KutS et al. / A modified NJL model

to cancel for physical like eq. (2.41).

There

observables. the small

This is seen for example quark

condensate

mass and the 1.h.s. of eq. (2.41) continues A second

way to extract

of the non-relativistic

information

constituent

in low-energy

compensates

to reproduce

theorems

the large current

the (small)

prescribed

on 1’ is based on the baryon

value

mass formula

quark model using the Feynman-Hellman

theorem.

We describe this method in appendix E. The main points are summarized as follows: In order to extract values for the quark condensates in baryons, one has to know the dependence of the baryon mass on the current quark masses. For this, one assumes the dependence of the baryon mass on the constituent quark mass as given by the non-relativistic quark model, while the dependence of the constituent quark mass on the current quark mass is given in the case of the NJL model by eqs. (2.9) in table 1 we obtain a and (2.10). Following this procedure usin g our parameters value for (r?u),~ roughly consistent with that of table 3, but the value for (sS)r, is about 10 times larger than the “exact” one given in table 3. This discrepancy indicates that our present NJL soliton solution is not consistent with the non-relativistic baryon mass formula, i.e.: if we construct a baryon mass formula in our present treatment of the NJL it will be different from the simple non-relativistic one. In conclusion of our above discussion on the quark condensates in baryons, our small strange-quark content does not lead to apparent diliiculties or internal inconsistencies. Next we wish to calculate observables of the nucleon. We have to remember, however, that our baryonic solution still has no good spin and isospin because the hedgehog ansatz (2.15) breaks the angular momentum and isospin symmetries. To restore these symmetries we follow the cranking approximation which is described in detail in ref. ‘I). It amounts to introducing a slow time-dependent collective isospin rotation of the hedgehog fields (2.15) (i.e., (S+ Cr. EP)+ A( t)(S + ir . i?)A’( t) with A(t) = exp (-iii1 . it)), deriving the change in energy of the system due to this rotation (6E = 410’ where I is the moment of inertia),

and

then

quantizing

the rotational

(0 + - ?/I) thereby assigning good isospin quantum numbers T to the solution, Due to the original invariance of the hedgehog with respect to simultaneous isospin and spin rotations, this procedure leads also to good spin quantum the moment

of inertia

motion

numbers

in the proper-time

with J = T. The resulting regularization

expression

is (3.17)

1 = IV,, + I,,, , where I,,, and I,.:,, are the contributions polarization, respectively, given by

from

for

the valence

I,;,, = $N,. X J’(r*, ~V)(Plr3l~)(~lr~l~u), w’.)’

quarks

and

vacuum

(3.15)

(3.19)

M. Kati? ef ul. / A modijied NJL model

566

and f( F,+, E,.) is the regularization

function

(3.20) In ref. “)

formulas

for the nucleon

matrix

elements

of arbitrary

quark

bilinear

operators are given. There are two kinds of quark bilinear operators. The first kind has a contribution of zeroth order in R [called “type I” operators in ref. “)I, and the second kind (“type II”) does not have the zeroth-order contribution. For example, scalar and vector operators with respect to the grand spin are of first and second type, respectively. To lowest order in R, the expectation values of type I operators can be calculated by eq. (2.6), and those of type I1 are given by replacing one of T’S in eqs. (3.18) and (3.19) by the corresponding operator and dividing by -I. In the present case, the axial-vector coupling constant g,, (operator TJ;) and the isovector magnetic moment (( r x a);~ ,) are of type I, while the isoscalar magnetic moment ((rx~l)~) and the spin and angular-momentum operators are of type II. The type 11 operators have spurious vacuum contributions which are due to the boundary condition. The boundary condition (C.9) is imposed separately for each G and the orthogonality of the radial functions of basis states belonging to different values of G is not guaranteed. Due to this, the vacuum (no valence quark, S’ = P = .‘P: = 0) has non-vanishing expectation values. In this work we simply subtract these spurious contributions to the matrix elements. In ref. “) the difference between this simple subtraction method and the more complete calculation (changing the boundary condition of the nearest G bases) has been investigated and the difference is only about 0.5%. The results for the nucleon difference

observables

are shown

in table

4. The AN

mass

is given by J-N=3/(21),

(3.21)

TA HLE 4 ‘The moment of inertia, cq. (3.17). as well as physical properties of the nucleon calculated by using the crankin method. g,, is the asial\wtor coupling conhtant, and p, and pS arc the isovector and isoscalar magnetic momcnth. (1;:) is (twice) the spin expectation value, and (I.,) is the expectation value of the orbital an_gular momentum. All results are splitted into the contributions due to valence quarks and sea quarks Valence

Sea

I (fm) S* CL\ Ps i’.;? CL;,

1.098 0.709 1.761 0.679 0.557 0.137

Exp.

Total

------

----0.222 0.0526 0.559 0.0325 0.0014 0.083

1.320 0.761 2.320 0.711 o.s5s 0.121

1.25 4.7 1 0.88

A4 Karol er al. / A mod$ed

and with I from table value

A-N

modified

account

experimental

224.2 MeV compared

= 300 MeV. The axial-vector

due to the difference

the valence-quark into

4 it becomes

this

between

approximation modification,

567

NJL model

coupling

the cranking

constant

with the experimental g,

could

be further

and exact projection,

which in

gives rise to a factor ( NC + 2)/ NC [ref. “)I. Taking 1.27, in good agreement with the g, becomes

value. The isoscalar

magnetic

moment

pus is in good agreement

with

the experimental value, but the isovector magnetic moment pv is not. The reason could be similar to the case of g,, but for pv there are sizable sea-quark contributions. Concerning the expectation value of the spin operator the contributions of the sea quarks are rather small. The property (2L,) +(&) = 1 follows from (~~(~L+~(~)=(E.L)~~)v)=(cL~~G-T~~)=-((cL~T~v)

(3.22)

(CLf u)

(see the discussion following eq. (3.20)). The s-quark does not contribute to gA, p, 1; and L? because all these quantities involve vector operators in ordinary space, and the total angular momentum of the s-quark is zero since the s-quark is separated from the hedgehog configuration and the rotation of u-, d-quarks, and there are no valence s-quarks. In conclusion of our discussions in this section we can say that our present modified NJL model including the effect of the six-fermion determinant interaction allows a consistent and reasonable description of both the nucleon and the pseudoscalar mesons. As we explained in detail, in order to obtain a stable nucleon we have to use a large absolute value of the six-fermion coupling constant Gb. The values of Gu used by other authors are: G, = -0.0277 fm” [ref. I’)], Gb = -0.0108 fm’ [ref. 15)], G,_,= -0.0170 fm5 [ref. “)I, Gr, = -0.0418 fm’ [ref. ‘5)], and the value of G,, in table 2 is consistent with these values. [In ref. “), two values for CD based on two different estimates appear.] Our absolute value of G,, in table 1 is very large compared to these values. However, our discussions in this section have shown that there are no conspicuous inconsistencies for the physical quantities. One of the reasons for this is the compensation by our small values for the quark condensates.

Namely,

for quantities

having

the dimension

of energy,

G,

should

appear in the form G,,((qq))‘, and the difference between the values for the quark condensates in tables 1 and 2 tends to compensate the large difference of Gb. For example, the ratio of the s-quark condensates in tables 1 and 2 is (150.4/282.2)” = 0.15 and the square of this is about 0.02, which is sufficient to compensate the larger ratio of Gr, (about 61).

4. Conclusion In this paper we investigated baryon-like solutions in the modified NJL model including the effects of the “instanton induced” six-fermion interaction in the Hartree approximation. In the flavor SU(2) case we find no stable solution. The solutions spread or collapse according to the value of coupling constant, and this situation

M Kart er al. / A modified MJL model

568

does not change to stabilize model

even if we include

the system.

The physical

is not clear for the moment.

solutions.

The stabilization

the “instanton

induced”

effect of flavor mixing

comes six-fermion

the vector-type reason

interaction

for this instability

which

is expected

in the flavor SU(2)

In the flavor SU(3) case we can obtain from the flavor mixing interaction.

leads to a position-dependent

with the s-quark

As we explained

stable due to

in detail,

effective four-fermion

this

coupling

constant which decreases near the origin. In order that this mechanism prevents the collapse we need an appreciable flavor mixing (large IG,I). Indeed, the most conspicuous point of the parameter set which we used to obtain a baryon-like solution is the large lGr, (see table 1). However, we have shown that this large (GbI does not lead to inconsistencies for the physical quantities. One of the reasons for this is the compensation by our rather small values for the quark condensates. (These small values for the quark condensates also tend to cancel the effect of our relatively large current quark mass m, in physical quantities.) The quark contents of the baryon are very different from the values estimated by using the non-relativistic baryon mass formula. Especially the s-quark content is very small in spite of large JGt,l. The stable solutions thus obtained are of the hedgehog type and should be considered as the “deformed” intrinsic states. The nucleon and the d-isobar are obtained by quantizing the “rotational” motion of the hedgehog state. The physical observables of the nucleon have been calculated using the cranking approximation, and these are found to be roughly consistent with the experimental values. In the same framework we also investigated the properties of the pseudoscalar mesons. Since this model does not have conlinement, the mesons whose masses are above the cfq threshold pose a problem. In principle, w:e should search for resonance poles in the complex plane. In this work, however, we identified the peaks of the imaginary parts of the propagators with the meson masses above the threshold. In our actual calculation, all mesons except the n’ emerged as bound states (i.e., below the qq threshold). (For comparison with previous works we also discussed another method in which the imaginary parts of the meson self-energies are simply neglected.) We found that there exist parameter sets which give a stable baryon-like solution and at the same time allow also a reasonable

description

of the meson

spectrum.

However, we found it impossible to obtain a stable baryon-like solution using parameters which are strictly fixed to reproduce the masses of 8 pseudoscalar mesons (see table 2). In conclusion, the modified SU(3) NJL model is found to give a unified and consistent description of mesons and baryons with some emphasis on the role of the spontaneously broken chiral symmetry and the U,A(.l) anomaly. This work was supported by the Grant in Aid for Scientific Research of the Japanese Ministry of Education, project #03740135. The work of one of the authors (K.T.) was performed under the auspices of the Special Researchers’ Basic Science

M Karol et ul. / A modified NJL model

Program

in RI KEN. The numerical

computer

at the Computing

of Meson

Science

Center,

Laboratory,

calculation

was performed

University

University

569

of Tokyo,

on the HITAC

M880

and on the VAX6440-4

of Tokyo.

Appendix A The purpose of this appendix is to derive the relevant formulas in sect. 2, i.e., eqs. (2.6) and (2.19), and to show that these formulas in the proper-time regularization scheme agree to SU(3), even in In ref. I’)), where have been derived

with those obtained by the naive extension from the flavor SU(2) the presence of the six-fermion interaction term. the flavor SU(2) case is considered, the corresponding formulas by using the auxiliary-field method “). Clearly, the one-fermion

loop approximation in the auxiliary-field method is equivalent to the following standard formulation of the Hartree approximation: Compute the (free) energy using the effective (one-body) Hartree lagrangian obtained by introducing the relevant condensed composite fields, and then minimize the (free) energy with respect to those condensate fields. Here, we follow the second method since it has the advantage that it can be directly generalized to models including n-body interactions (n 3 3) as in the present case. [For the auxiliary-field method for models including many-body forces, see refs. 2*26).] We start from the Hartree lagrangian amplitude in the Hartree approximation

J

out(OIO>in = [dql IdL51 ev =e

-s,,,

(2.11). The ground-state-to-ground-state is given by

(I d4X19,” -

-My4ql

> iA.1)

>

where we have delined the effective action in the Hartree approximation Scff. The functional integral is defined in the euclidian space-time with .x4= ix’, y4= y’), y, = -iy’(x’,

y’, y’ are in the Minkowski

+(G,+

space-time),

and

G~,(.~s))((~I~)‘+((L~~~T~~)~)+~G~(SS)~,

DcF =d+ mu-(2G,+

G,(S.s))((&b)+iy,t

. (&y5irsl$)),

DsF=B+m,-4Gs(.Ts)-~G,((~1~)‘+(~iyj~~)~), which is obtained

from eqs. (2.11)-(2.14)

by continuing

(A.21 (A-3) (A.4)

from the Minkowski

euclidian space-time (B = y4d4 + y,di). We have introduced p in eq. (A.l) to treat the finite baryon number, following

to the

the chemical potential refs. “.“). Note that the

M. Ku16 et al. / A mod[fied h’JL

570

baryon

number

B is given by

/I

1

a - Sc,r NC a/J

=--

model

We first perform obtain SCtl= -TrIn

(A.9

dx,.

the Grassmann

D,.+

functional

J

d’x {(G,+

integral

for the fermion

fields and

G,(.~~)))((~~)‘+(1Criy,?llr)~)+2Gs(h)Z}

where we set Da

DE=

-

0

CLY~

0

Ds,-PY,

(A.7)

>’

and Tr refers to the space-time variables and Dirac, flavor and color indices. In order to regularize the divergent fermion loop contribution SLti, i.e., the first term on the r.h.s. of eq. (A.6), we separate it into valence and the Dirac sea contributions: s,:, = s;; + S$

)

(A.81

with s:;’ = s,:, - SQ, -0 , s>: = s:,j,, Note that the valence

(A.9) (A.10)

-0.

S\ri is finite while the Dirac sea contribution

contribution

S$ is quartically divergent and should be regularized. To compute the (free) energy of the system we can concentrate on the real part of the effective action in the following. Introducing the proper-time regularization “,“) with cut-off )I, one obtains Re S;; = ; Tr

i

dr - e-sL’;L’, . 5 1; l2 7 /I =o

(A.ll)

Note that DiDb, is positive semi-definite. In the case oftime-independent condensate fields, the trace of eq. (A.11) can be performed by the standard method 19.“) of rewriting D, as &,-I h,-p &=Y,

(A.12)

0

with h,=a.p+B{m,,-(2Gs+G,(~.~))((~~)+(~jrs?~). /I, = 01 . p + p{ m, - 4GJ.%> - :G,,(($$)‘+

i~~r)}, (t+%yjr+)2)} ,

(A.13) (A.14)

M. Ku6

and introducing

the complete W&(x)

571

et al. :’ .A modtjied NJL model

set of spinors

{lb,,,(x) e -iO”~,s*,(x) e-iw.Y4}satisfying (A.15)

k.Y,\(X) = F*,%,(.~) .

= c&%*.(x) 3

Then we obtain

Substituting

dx,.

: exp ( -TE:,) +C exp(-mf,) *,

Tr e eq. (A.16) into eq. (A.ll),

we obtain

for the Dirac-sea

(A.16)

contribution (A.17)

where the sum over A refers to both A, and A, of eq. (A.16). For the valence part (A.9), the functional trace in eq. (A.6) can be evaluated similarly

without

regularization,

giving

FL{hr(-iw+l-.*-p)-ln(-iw+E,,)} &T *

ReSsfl=-N,.Re

dx, I

=-:N~7_(lr,-iL,-,‘*,)ldr, h = N,.

Ii

dx,,

C (F~--)+~~~sgn(Fh) c* p h [ 0‘.C

(A.18)

where, again, h means both A, and A, of eq. (A.15). Thus, the total effective action to the free energy with fixed chemical potential p Re Se, which corresponds multiplied by the time interval j dx,, is given by substituting the sum of eqs. (A.17) and (A.18) for the first term on the r.h.s. of eq. (A.6). The equations of motion in the Hartree approximation can be obtained by minimizing Re Sz,r, which is a functional of (&$), (~+&~r(l,) and (CT), with respect to these condensate fields: 6

8

SC’ =G(&(x)iy,7rj,(x)) fi(&(xMx)) Re

Re &

6 = G(S(x).s(x))

(A.19)

RescR =O.

By using the relation

= and the similar (G,+

relation

A,,(x)’

6h,

G(“r(x)+(x))

for E*, , the first equation

Gu(S(x).s(x)>)(~(x)C//(x))

~&(.~)

(A.20)

9

of eq. (A.19) gives

- (G,+;G,,Wx).~(x)))p(x) -;Gr,(&(x)$(x)>~,(x)

= 0,

(A.21)

512

M. Karl

where we have introduced

er ul. ! A modified

ML

model

the quantities:

We note that eq. (A.21 ) can be rewritten (G,+;G,,{ss))((&//)-p)+;G,,(&J)((ss)-p,)=O. Similarly,

the second

and third equations

(A.24)

of eq. (A.19) respectively

(G,+~G,(b))((~iy,~rL)-p,)+!,G,,(~iy,~rL)((~.~)-p,) G,((ss) -

P,) + :G,,{(~~~M&b)

=O,

- PI

(I+E~~TIL) . ((&iy&)

i

give

(A.25)

- p,,)} = 0,

with

(A.26) Therefore, the Hartree equations of motion, given by eqs. (A.24) and (A.25), are coupled quadratic equations for the condensate fields at each space point. To seek the solution, we follow the hedgehog ansatz given by eq. (2.15) and obtain from eqs. (A.24) and (A.25): 0

Gs+iG,.V’,(r)

$G,,S(rj

Gs+$GI,Ys(r)

0

hG,P(r) 1

aG,P(r)

SGt+(rj with

do 4,

p,(r) =

p,(r)

>

(A.28) where integral

means

fore, the solution

the angular

average

r-space.

There-

is given by S(r)-p(r),

if the determinant

in the three-dimensional

P(r)

of the matrix

YJ 1.1= p,(r) ,

=pAr),

of eq. (A.27) does not vanish,

s

i.e., if (A.30)

l++$t/,(r)-s(S(r)‘+P(r)‘)

-

(A.29)

s

‘44. Kar6 er al. / A modified NJL model

513

As can be seen from tables 1 and 2, Gs > 0 and G,, < 0. Also, in the usual applications, the strange-quark

condensate

Y,(r)<0

on the 1.h.s. of eq. (A.30) is positive terms

and

principle,

the third

term

it is possible

of eq. (A.30)

are positive

that at certain

vanish,

there

(see also fig. 4). Therefore,

definite,

while in the second

and

points

negative,

the first factor

factor the first two

respectively.

Therefore,

in space, which make the second

exist additional

solutions

different

from

in

factor

eq. (A.29).

However, such a special treatment of radial points seems to have little physical meaning. Thus, in practice, the Hartree equations of motion are given by eq. (A.29), which coincide with the equations obtained by a naive extension from the flavor SU(2) case Iv) to the flavor SU(3) case. Finally, we derive the energy of the system. number

of the system using

eqs. (AS),

1

II

0 . c, /,I Then,

the energy

can be obtained

To this end, we compute

(A.8)-(A.lO)

fixed II, and we obtain from eqs. (A.6), (A.17), the hedgehog configuration (see eq. (2.15)) E,,, = Re %r

transformation

from fixed p to

(A.18) and (A.31) for the case of

/I

dx,+pN,R

r

=N,. OS-

+

(A.31)

1-it:sgn(F*). A

by a Legendre

the baryon

and (A.18):

J

+2G,

I-\

c

,I

d~4~~‘{(G,+G,,.C~~(r))(S(r)2+P(~)’)-4(G,+G,~)~2}

J

dr4&(Y>(r)‘-y’),

(A.32)

where in the final form we have subtracted an irrelevant constant to make E,,, = 0 in the R = 0 vacuum, i.e., when p = 0, S(r) = 2a, P(r) = 0, Ys(r) = y (u and y are the r-independent constants introduced in eq. (2.4)), and the F”,, denote the eigenvalues of eq. (A.15) in this case. In the case of the finite-system calculation to obtain the R = 1 soliton solution, P in the first term on the r.h.s. can be eliminated by taking k immediately above the lowest energy level down from the upper continuum (valence level) by

N,

>3

“-. c*.: P

E,+ -=

Ncqva,~,, ,

(A.33)

with nVal given below eq. (2.6) and E,,, is the energy of the valence level. Then, E,,, of eq. (2.19) is obtained. Also, due to eq. (A.33), the equations of motion (A.29)

574

M. KarS er ul. / A mod(/ied NIL model

are equivalent equations

to eq. (2.6) in the main

of motion

that our equations extensions presence

text. In the B =0

reduce to the gap equations of motion

of the formulas of the six-fermion

(2.8)-(2.10).

and the expression for flavor SU(2) interaction

case, i.e., k = 0, these Thus, we have shown

for the energy agree with the naive

to the flavor SU(3j

case, even in the

term.

Appendix B In this appendix we give the functions which enter into the meson calculated in the proper-time regularization.

self-energies

The regularized pseudo-scalar meson self-energies can be obtained by the following procedure: First calculate the effective action from the effective (one-body) Hartree lagrangian including the condensed fields corresponding to the composite pseudoscalar mesons P, = (ijiy,A,,q) (u = 0, 1, . . 8) in the proper-time regularization scheme. (The one-body effective lagrangian including these condensates is constructed in the same way as in sect. 2.1 for the cast of the condensed pion field.) Expanding inverse the result in powers of f’<,, the second order terms give the regularized propagators expressions

of the pseudo-scalar

mesons.

From

these

inverse

propagators,

the

for the bubble graphs of fig. 1 for the corresponding mesons can be of eq. (2.25) for the pion (corresponding to a = 1,2,3), II”“’ i.e., II’““’

extracted, of eq. (2.26) for the kaon no-n8 channel. The one-loop self-energy II’““‘($)

(cu = 4,5,6,7),

and

IIon, fl,,$, Lrxs of eq. (2.32) for the

for the pion becomes = 8N,.[I,(M,)

-$L(p?,

(B.1)

M,, IV,,)] )

and for the K-meson n’““(p’)=4N,.[I,(M,)+I,(M,)+((M,-M,)’-p.’)lz(p2,

Mu, MJ1

(B.2)

where the I, and I, are given by dJQ :,o’- VI.‘) (2Z)4 e (B.3)

(B.4)

(R.5) where the integrations in eq. (B.4).

with respect to Q are euclidian

and we set P’= -p* = -p”p*

M. KatG et al. / A modified NJL model

After integration,

we obtain,

for ME = M2,

for M, f l& and p’ = 0,

for M, # M2 and p” f 0,

fz(p’gM,

, Md =

When pz 2 (Ml + M?)“, I2 has an imaginary

The self-energy

II(“) in eqs. (2.33)-(2.36)

pan:

is obtained

from eq. (B.l)

by replacing

Mu* Ms.

Appendix C In this appendix, we explain the numerical method. To soive the Dirac equaticm, we use the method invented This method

consists

in diagonaiizing

the hamiitonian

by Kahana

and Ripka “).

in a spherical

base. We put

the system in a box of radius D. Since the hamiltonian of eqs. (3.2) or (2.16) does not mix the different grand spins and parities, we can diagonalize separately for each grand spin and parity. For quarks with mass m, the positive energy (E > 0) free spherical waves with grand spin G and parity (- 1)” are G,j= @f(l) = ?fr

-i$--j,+,(prjll=

G+$,

G, M) G-t l,j=

C+i7

G, M}

’

CC.11

516

M. Ku6

et al. / A modified NJL model

j,(pr)lI=G,j=G-i,G,M)

~~j,.,(~r)]l=G-l,j=G-I,G, and with parity

M)

(C.2)

’

’

(-I)“-

j~;,,(pr)ll=G+l,j=G+:,

G, M)

*I”‘= N

(C.3)

i&j,;(pr)l/=G,j=G+l,G,M)

i ,i(;.,(pr)lf=G-l,j=G-1,G,M)

4

The negative

(-0 =

-i&jc;(pr)il=

energy

G,j=

(E < 0) basis states for parity P

tir,i=

(c.4)

N

G-4,

(-1)”

G, M)

are

.

E_m.lr;(Pr)l~=G,,i=G+~,G,M)

N

-ij,,,(pr)JI=G+l,j=G+f,G,

,

(C.5)

M)

(c.6) &-,(pr)\I=G-l,j=G-:,G,M)

and for parity

(- l)ti -’ $,.:,=

f

EP,,j,..,(pr)lf=G+l,j=G+i,G,M)

N

&j(pr)lI=G,j=G+~,G,M)

(C.7)

(

~j,;_,(pr)lf=G-l,.i=G-!,G,M)

$o,EN

(C.8)

-~c;(pr)lf=G,j=G-:,G,M)

Following4)

we impose

the condition

that at the boundary (C.9)

jc;(W)=O. Then the momentum The normalization N=[P{,+(~~)j_.,(~~)]

p becomes

constant

discrete,

p = (zeros of spherical

Bessel function)/

II.

becomes I:‘(+

for (I/“.” and - for &(‘*“).

(C.10)

(The general condition for these states to form an orthogonal base is (upper component) = (lower component) x const. at the boundary, where the constant can be different for each rb”‘.)

&I. Koro^ et cd. / A modjfied

The number

of bases is still infinite.

517

h’JL model

Then we introduce

the maximum

momentum

also restricts the P IndYand take only the momenta p below this. This automatically possible values of G, since beyond some value G,,;,, eq. (C.9) has no solution in the region

p
the Dirac

equations.

parameters

Of course,

of bases is finite and we can use it to solve

the results

D and P,,,~,~.As explained

should

not depend

on these

in the main text, we actually

artificial

used D = 20/M,,

u-quark mass. and plnlr between 5 M,, and lOM,,, where Mu is the constituent If we use the base with the boundary condition (C.9), the scalar density shows a rapid variation (decrease) at the boundary. This occurs even in the free case. Therefore, if we take the ratio to the free scalar density, it becomes almost 1 at the boundary. We remove the remainin, 0 deviation from 1 by imposing the non-linear condition u2 + $ =,fa near the boundary. The similar prescription is applied in the SU(3) calculation.

Appendix In this appendix orbit.

D

we give the radial Dirac equations

If we write the wave function

with grand

which we used for the valence

spin G and parity

(-l)G

as

ig,(r)(l=G+l,j=G+:,G,M)+ig,(r)lI=G-l,j=G-i,G,M) CD.11 then the Dirac equation

(3.2) gives

g7T(r)(fi-2v’G(G+l).~‘r)-(F+gcr(r))g, =O,

g~(r)(2JG(G+l)gr+gJ+(&-gc(r)).f,=O. If G = 0, Jz and g, are absent form

and the differential

(-f+f >

equations

g,+gr(r)g,+(s-gu(r)).f;=O.

(D.2) reduce

to the simple

(D.3)

M. Kurd et ul. i A modjfietl

578

XJL

model

Appendix E In this appendix the non-relativistic the baryon

we explain quark

the method

based

on the baryon

mass formula

model used in ref. lJ). If we write the baryon

of

state as IH),

mass is

(E.1)

Mn =(BlhlW where h is the quark hamiltonian. Differentiating quark mass mi (i = u, d,s), we obtain

this with respect

to the current

(E.2) which

is the so-called

“Feynman-Hellman

non-relativistic quark model constituent quark masses:

Mu=Mo+

theorem.”

has been

used

In ref. ‘“), the result of the

for the dependence

C ,_“.d.I(Mi+eJ +bZjt$’

of MR on the

(E.3)

where a, b and M,, are constants. From table 1, M, = Md = 365 MeV and M, = 580 MeV, and the constants determined to reproduce the experimental masses MP = 938 MeV, Ml = 1232 MeV and Ml1 = 1672 MeV are a = (276.2 MeV)*, b = (186.9 MeV)3 and M,, = -324 MeV. The baryon masses calculated with these constants are shown in table E.l. They are in good agreement with the experimental values. The quark content of the proton is (E.4)

Rar>on masses calculated from the mass formula of the non-relativistic quark model, eq. Ct.3). The values for the constituent quark masses in eq. (E.3) are taken from table 1. The constants M,, u. h are determined to reproduce the N, A and R masses

.-----P (MeV) ‘I (MeV) 2’ (41eV) I* (WV) Z (MeV) E* (MeV) .I (blew R (Mew

Calc. _---938 1232 1157 1372 1334 1519 1169 1672

hp.

938 1232 1193 1385 1314 1530 1116 1672

579

M Kntci er al. / A modified N/L model

and in A the quark content

is (E.5)

If we assume quark

that the dependence

masses

dM,/dm,=

is described

of the constituent

by the NJL model,

quark

we obtain,

-0.178 from eqs. (2.91, (2.10) and the parameters

masses

in table

u- and s-quark contents of P and A, based on the non-relativistic relation (E.3) and the NJL relations (2.9), (2.10) are (tiU),-Comparing

1.97)

(Ss),, = -0.42,

these values with our “exact”

(UU)~ = 1.63,

on the current

aM,,/am,= 0.842 and 1. Then the quark-model

(ss),J = -0.35.

results in table 3, we see that the u-quark

content is roughly in agreement but the s-quark content IO. (The values in table 3 refer to both p and il.)

differs by a factor of about

References II 2) 3) 4) 5) 6) 7) XI 9) 101 11) 12)

13) 14) 15) 16) 17) 181 19) 20) 21) 22) 23) 241

25) 26) 27)

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