nucleon-nucleon scattering in the two-center mit bag model with nonlocality and removal of the...

Nuclear Physics A397 (1983) 413-446 @ North-Holland Publisbing Company

NUCLEON-NUCLEON SCA’LTERING IN THE TWO-CENTER MIT BAG MODEL WITH NONLOCALLY AND REMOVAL OF

THE SPURIOUS RELATIVE MOTION+

SADATAKA FURUI and AMAND FAESSLER

Institut fiir Theoretische Physik, Universitiit Tiibingen, Auf der MorgensteUe 14,0-7400 Tiibingen, FR Germany

Received 6 September 1982

Ahstraet: The short-range part of the nucleon-nucleon interaction is studied in the two-center MIT

bag model. We calculate the minimum of the bag energy for given distances between the two

centers. But contrary to DeTar we include both the [6] and [42] spatial symmetry even at distance

zero. Similar to the result of DeTar, our “local” potential has a soft core and in the intermediate

range it is attractive. We expand the relativistic quark wave function into a harmonic oscillator

basis and separate the wave function which describes the relative motion between the two centers

(two nucleons). This allows one to subtract the relative kinetic energy of the three-quark clusters.

The %r and ‘Se channel bag-bag scattering phase shifts are calcutated by the resonating group

method including the full nonIocaIity of the problem.

1. Introduction

Derivation of the nucleon-nucleon interaction from the quark model has been studied already by several authors lm3). In the non-relativistic quark model, where the constituent quark mass is assumed to have about 300 MeV, the nucleon-nucleon scattering phase shift was calculated in the resonating group method (RGM) and a phase shift similar to that given by a hard core of about 0.4 fm was reproduced. The local potential calculated with the same model showed however no core if the spatial [42] symmetry was also included at distance zero 4). The hard-core-like behavior of the phase shift was caused by an almost energy-independent node in the relative wave function of the spatial [42] symmetry I).

A fully relativistic derivation of the nucleon-nucleon interaction from the MIT bag model ‘) was attempted by DeTar 6). In this model, the nucleon-nucleon local potential was defined as the difference between the energy of the bag at a given distance and two non-interacting bags. This energy was calculated as a minimum under certain constraints on the shape of the bags. The potential has a soft core of about 300 MeV at short distances and is attractive in the intermediate range. In the evaluation of the energy of the two bags the kinetic energy of the quarks

Supported by DFG.

413

414 S. Furui, A. Faessler / Nucleon-nucleon scaftering

were added and the relative kinetic energy between the three-quark clusters was

not subtracted. Therefore it was impossible without double counting the relative

kinetic energy of the two nucleons to incorporate the potential into the Schriidinger

equation and to calculate the phase shifts.

In order to solve the scattering problem of two bags, Jaffe and Low 7, proposed

to use the P-matrix formalism which is an extension of the Wigner-Eisenbud

R-matrix theory. The model was applied to the nucleon-nucleon scattering by

Mulders ‘) with a special form for the quark-quark interaction potential. For a

realistic derivation of the residue of the P-matrix which characterizes the matching

between the internal six-quark state and the state with two separated hadrons it

is necessary to find first the internal wave function. On the other hand if the relative

wave function has an energy independent node as suggested by the non-relativistic

calculations ‘), the ansatz adopted by Mulders becomes inappropriate since he

modifies the lowest pole which corresponds to a solution which has the first node

at the matching radius r =b.

In this paper, we try to modify the model of DeTar and solve the bag-bag

scattering problem by the resonating group method. We use an approximation in

which the dynamics of the relative motion of the bags is treated non-relativistically

while the dynamics of the single quark is handled relativistically. This approximation

is a good one since we use the current quark model with the mass of the u- and

d-quarks equal to zero and each of the nucleons has a mass 938 MeV. The relativistic

quark wave function is described by the Dirac equation which consists of the upper

component and the lower component. It satisfies a special boundary condition

which leads to confinement. In the MIT bag model the amplitude of the lower

component is at the surface of the bag the same as the upper component, and so

in the region where two surfaces of the bags overlap, there will be large deviations

from non-relativistic model calculations. As the two bags overlap the relativistic

quark wave function loses the spherical symmetry around the center while in the

non-relativistic model, the quark wave function of a nucleon is always described

as a (OS&~ harmonic oscillator wave function. That means that the two nucleons

are not polarized during the collision process. In the limit of zero distance the wave

function of the (OS,,~)~(O~&’ configuration in addition to (OS&~ must be

present 1,2*4,9). Faessler et al. ‘) observed that the effective “hard-core radius” is

connected linearly with the root mean square radius of the quark content of the

nucleon being equal to the oscillator length b of the basis. Here in the relativistic

quark model, one can also ask whether the soft-core strength is connected with

the size of the bag.

We derive the local potential using the method of DeTar 6). First we give the

shape of the bag and calculate the relativistic quark wave function by the variational

method. Then we calculate the gluon exchange and self-energy for the 3Sr and ‘SO

channels using the coefficient of fractional parentage (c.f.p.) given by Harvey4).

The energy of the six-quark bag is given by the gluon exchange energy, the gluon

S. Furui, A. Faessler / Nucleon-nucleon scattering 41s

self-energy, the quark kinetic energy, the zero-point energy and the volume energy.

Subtracting the mass of the two nucleons, we obtain the local potential.

In order to derive the non-local part of the potential we expand the quark wave

function into a harmonic oscillator basis and calculate the overlap of the six-quark

wave function. We assume that the non-local part of the potential is described by

the norm-overlap lo). When the quark wave function is expanded in the harmonic

oscillator basis, one can subtract the relative kinetic energy and derive the Hill-

Wheeler equation ‘I). This integro-differential equation is then solved by the

variational method “).

In detail we go through the following points. In sect. 2 we derive the quark wave

function in a deformed bag. Using this single quark function, we calculate the

one-gluon exchange energy and the self-energy in sect. 3. The matrix elements for

the six-quark system are calculated using the c.f.p. method. We have to add

zero-point, volume and quark kinetic energy to obtain the energy of the total bag.

Subtracting the mass of the two nucleons we derive the local potential in sect. 4.

The application of the RGM to our bag-bag scattering problem is shown in sect.

5. Finally in sect. 6, we discuss the results and present the main conclusions.

2. Quark wave function in a deformed bag

In the static MIT bag model one neglects the oscillation of the boundary of the

bag and solves the Dirac equation for a given shape of the bag. We use the symmetry

basis of Harvey4) for the two quark wave functions (w.f.) and adopt the Cassini

curve, which has two centers, as a parametrization of the boundary of the bag.

This boundary is given by

(x2+y2+a2)*=4a2x2+b4, (2.1)

(2.2)

By varying a and b one obtains various kinds of shapes with centers at x = fa. We assume axial symmetry around the x-axis. The quark w.f.s which have their

centers at x = a and -a are written as

Ijl~tR) =

4WUm gL(R’)= (i,. Stx(R’)u,) ’

(2.3)

(2.4)

where V,,, is the Pauli spinor, S and S’ are vector fields which reduce to normal

unit vectors on the surface of the bag. The vector S should reduce to the radial

vector R = r in the limit of a = 0 and should not be zero except at the center x = a.

416 S. Fund, A. Faessler / Nucleon-nucleon scattering

Fig. 1. The parametrization of the coordinates for the Dirac wave function of a quark in a bag. The

boundary of the bag is described by the Cassini curve. In order to describe a quark wave function with

its center at the point x = a, we introduce for each point r(r, 0) in the bag, a vector R = r(r, 9) -a(a, 0). The angle between the vector R and a is defined as a. The distance from the center a to the point

where the extension of the vector R and the boundary crosses is defined as Rc(a). With the values of

Rc(a) and its maximum R,,, =R,(180”), we construct the vector S =SReR +S,e, as defined in eq.

(2.5), where en and e, are unit vectors in the R-direction and perpendicular to it. To describe a quark

wave function with its center at x = -a, we introduce a vector R’ = r(r, 4) +a(a, 0) and the distance Rb from -a to the surface of the bag in the R’-direction.

We adopt the following form:

S(R)= Rmax ( R dRo/da

JRk,,+R2-RE JRg+(dR,/da)2eR-JRi+(dRo/da)ie” ’ (2.5)

where Ro(a) is the distance from the center x = a to the point where the vector R

and the boundary crosses. R,,, is the maximum value of R in the bag and (Y is

the angle between the vector R and the x-axis. The definition of S’(R’) is obtained

by replacing a, R and R. by -a, R’ and Rb.

The quark w.f. should satisfy the boundary condition

in,y’* = 4, (2.6)

where n, = (0, -S) is the inward directed normal of the boundary of the bag. In

order to make the w.f.s $IR(R) and &_(R’) satisfy the boundary condition, we take

the ansatz suggested by DeTar 6):

4(R)=c 1+p(R2-R;)

l-PR; ’

R l+y(R2-R;) x(R)=cg

l-PR; ’

(2.7)

(2.8)

S. Furui, A. Faessler / Nucleon-nucleon scattering

where c is a normalization factor. p and y are fixed by minimizing

417

in the case of massless quarks. The ansatz is good for a spherical bag since the

exact w.f.,

4(R) = Cj&lRIRcJ , (2.10)

x(R) = cia * &(x_1R/Ro) ) x-1 = 2.04, (2.11)

can be well approximated by the above trial functions. For a deformed bag the

ansatz is not of such high quality. But it is a good compromise between numerical

economy and exactness. We choose 6 < l/R f that means that it also can be positive.

For 0 < a/b, the w.f.s @n and I,!J~ are not eigenstates of parity. For the calculation

of the color electromagnetic field energy we define

*s = &MR) + $l_(R) , (2.12)

(CIA=&h(~)-(LL(~)). (2.13)

The eigenvalues are calculated by

The w.f.s I&~ and $A are different from those of DeTar 6). DeTar’s bag has only

one center and the normalized w.f.s ljls and *A are introduced from the beginning.

They are taken there as 0~~~~ and 0~~~~ in the spherical limit. In our case the large

component of $A reduces to a p-wave but the small component reduces to a

complicated form near the center.

The shape of the quark w.f. changes drastically as the distance “a” increases.

The sign of p becomes positive for a/b ~0.3, i.e. when the distance between the

bags increases the lowest energy state is realized by the wave function which has

its minimum of the upper component near its center. One finds therefore two sets

of wave functions one with the maximum of the upper component near its center

which connects the a = 0 and the a = 00 solutions and the other with a minimum

of the upper component near the center. The latter has lower energy for u/b 2 0.3.

At this distance of the two nucleons clearly two centers develop. For u/b B 0.8 the

eigenvalue rises unphysically due to the formation of a neck in the bag. As the two

bags begin to overlap, the transition from one set to the other may occur but our

418 S. Furui, A. Faesder / Nucleon-nucleon scattering

model is not so refined as to describe the dynamics in this region. We discuss the problem later in sect. 5.

3. The one-gluon exchange interaction

We calculate the gluon exchange energy up to second order in the coupling. The hamiltonian of our system is

El= {:~+(-icw~V)q:+~[E”2+Ba2]-J(t~An}dV. f

(3.1)

In the second quantized form, we define the charge and current density as

P& = Z_ gp r.r_ 62 .,rm&L%zA L cfm 9 (3.2) (RR) (RI CR)

a;, = C gJ.L ibt c~~m~%Z*c~L~b L cfm 3 (3.3) (RR) icc’fmm’ (RR) CR) (RI

where b and b+ are quark creation and annihilation operators, the indices c, f, m are color, flavor and magnetic quantum numbers respectively. The coupling constant g is taken as g2/4m = (Y, = 2.2 following the MIT bag model ‘). A,“,, is the color SU(3) operator which satisfies the equation

and C& is the spin SU(2) operator. The electromagnetic fields are written as

(3.4)

(3.5)

(3.6)

The gluon exchange interaction can be separated into three categories: (i) static electric field; (ii) transition electromagnetic field; (iii) diagonal magnetic field. We calculate each contribution as follows. (i) Static electric field. Since the six-quark system is a color singlet, the gluon

exchange and self-energy contributions cancel if the orbits are the same. What remains is the gluon exchange between different orbits 6**3).

S. Furui, A. Faessler / Nucleon-nucleon scattering 419

The interaction can be written as pD& where

=lv( l+p(R*-R;) l+p(W2-RA2)

l-PR; l-PRI:

+s s, RR’ l+y(R*-R:) l+y(R’*-R&*)

R&h 1 -PR: (3.7)

and the scalar potential 4D is parametrized as

l+dl(R2-R:) l+&(R’*+Rb*) 4D=C1[ l-&R; 1 -dlRb*

+s s, RR’ l+eI(R2-Rg) l+el(R’*-Rb2) .- RoRb 1 -dlR: I l-dlR$ *

(3.8)

The parameters cl, dI and el are fixed by minimizing

WE= (~E*--PD~D)~V, s

(3.9)

where

E=-VC$D. (3.10)

In the limit a = 0, rjlR and l/lL become identical and the energy of the electric field

is canceled by the self-energy. This is different from DeTar 6), since (j/A=

G($R-GL) of DeTar is normalized and goes to the 0~3,~ w.f. in the limit of a = 0.

The difference is not important however since at a = 0 the main contribution comes

from the magnetic interaction.

(ii) Transition electromagnetic field. The transition charge density psA and current

density JSA are written as follows:

PSA=~((RI+(LI)JT(IR)-IL))=~((RIR) -(L/L))

+& l+r(R’-R:)

Ro l-@R; (3.11)

= (crxs)R l+r(R*-R;) 1+P(R2-Rg)

Ro 1 -pR; 1 -pR;

_(axS,)& l+r(R’*-Rb*) l+P(R’*-RR’)

Rb 1-@R;2 1 -pRb2


_ & 1 -r(RZ-R;) 1 +p(R’*-Rb*)

Ro 1-pRi 1 -$R;=

_is, R’ l+#“-Rb=) l+p(R*-R;) -7 Ro 1 -pR&= I l-@R: ’

(3.12)

The scalar potential (fts~ and vector potential ASA are parametrized as

+ /_ l+b~(R=-&)

Ro 1 -blR;

s, R’ l+bl{R’=-R;i2) -7 Ro 1 -blR&*

R l+ex(R’-RX) l+d~(R”--R&~j

1 -dlR; 1 -dlRb=

_is, R’ l+c~(R’~-Rb~) l+dl(R2-R:) -7 Ro

2 1 -dlRo 1 -dlR:

With B = V x A, we first minimize

Ws= (~~B~2-R~Js~&~)dV, I

and fix c?, c?, cl, dl and el. Then with

E= -V&A+i(O,y-O&&A,

(3.14)

(3.15)

(3.16)

we minimize

WE=; (#i2-ReP~~4~~)dV, s

and fix c$, c$, al, bl, bi and bf.

(3.17)

The transition field contributes to the g&on exchange and self-energy diagrams. When the two bags separate the contribution from the transition electric field becomes very important.

(iii) Diagonal magnetic field. In the case of magnetic field, the gluon exchange and self-energy do not cancel and in principle both should be taken into account.


In the MIT bag model however the self-energy from the magnetic field was not

taken into account and so we consider only the gluon exchange diagram.

The diagonal current is written as

J &

= :((LIJb)) +I((RIJIR))(+)~((LIJIR))(~)t((RtJIL))

+ l+r(&-RE) l-tp(R’2-R~2)

(-) 1-pR; 1 -PR;* I

R’ l+y(R’*+R8) l+p(R’*-R;*) +(oxS’)R,

0 1 l-PR;* 1 -OR::

+ l+r(R’*-R;*) l+P(R*-R;) (-) I--pR;* 13 1-pR: *

(3.18)

We parametrize the vector potential

A ss (AA)

=c1 (CM)% [

R l+bI(R2-R:) l+b#*-R:) + l+&(R’*--Rh*)

0 l-bsR; i 1 -bsR; (-) 1 - b3R&* I

R' l+b@‘*-R;*) +(uw~

l+b3(R’*-Rb*) + l+b&?*-R:)

1 -bsR;: 1 -b3R;* (-) l-b3R; II ’ (3.19)

and with B = V x A we minimize

($jBl*-J.A+$(J.A)‘)dV, (3.20)

where the last term was added to restore the current conservation 6), and fix cl, bl and b3 for SS and AA separately. For small “a” the color-magnetic interaction

becomes dominant and it gives a repulsion.

In the above expression the diagonal magnetic field is for example calculated as

follows: We write the vector potential A as

A = (a x S)f(R, R’) + (a x S’)f’(R, R’) .

The magnetic field is then given by

(3.21)

B = (u(V . S) -S(u . V))f(R, R’)

+ (u(V * S’) -$(a * V))f’(R, R’) . (3.22)

If a = 0, S =S’ = r and we get the exact form of the MIT bag. For general “a”,

422 S. Furui, A. Faessler / Nucleon-nucleon scattering

we calculate the first term in cylindrical coordinates (r, 19, z) as

flz (v * f%f(r, 8, a 1

+ ( u,(V * W-S, ( uri+u8+ &))e,f(r, 6, a)

+ u&V * S)--$9 ( ( 1 a ,;+,- - r a6 >> w?, 8, a) , (3.23)

and the same for the second term. The scalar product of the field (3.23) is calculated as

c~,a, (v * Wr, 6, a )I2

= UOUO(V - Sf(r, 6, ~2))~

+ (terms proportional to crl~l and (T_~u_~) ,

and similarly for the other terms.

(3:24)

The one-gluon exchange and self-energy diagrams which we consider are shown in fig. 2. The expectation value of the field energy is then written as

E field = EES + EEx + EBx + EBD . (3.25)

(i) EES = w2c2, (3.26)

(ii) EEX = W2&2p + W~P,~ZP-+ w1lc110+ wl2c121+ Wl3c130

+ w14c14, + w2p’s2p,+ Wl3S130 + Wl4S141 , (3.27)

EBX = WIP~IP + WIP’~IP’ + w9c90 + wloclo~ + w7c70

+ w&3, + wlP’slp’+ w7s70 + w9s!3, , (3.28)

(iii) EBD = w3c30 + w4c41+ w5c50 + w6c61 f WlSclSO + w16c161 , (3.29)

where Wi = l $E2 dV or -5 $B2 dV, dependent on whether it is electric or magnetic. Ci, Cio and Ci, are expectation values of the two-body operators 11, c~(TO and


11 c2

2a) c2p

C2p’

cllo+c12~

cl3o+c1Ll.

2 b) C,p

Clp'

c9o+clo~

c7o+c&

3) C3o+L

c50+c61

c150+c16~

Fig. 2. Specification of the gluon field energy. It consists of (1) the static electric field, (2a) the transition

electric field, (2b) the transition magnetic field, and (3) the diagonal magnetic field. The transition

electric and magnetic fields also give self-energy contributions. The symbol S and A mean spatially symmetrized and spatially antisymmetrized one-quark states, respectively. The symbols C;, Ci, mean

that operators are spin independent while CiO + Cj, means that the operator contains the term (~0 . ~0 and ~~(r-l +u-~(TI.

(T~(+-I +a-1~1. Si, Sio and Si, are expeMation values of the one-body operators 11,

(TO(TO and ~1~1 +~-l~l, respectively. In the basis of SS, AA, AS and SA the

two-body matrix elements are listed in table 1, where we define

Al = W3C30+ W4C41,

A2 = WlpCw+ W2&2~+ W&~O+ WloClo~+ Wl~Cllo+ W&m,

A3 = WsCso+ W&x ,

A4 = WwCw+ W2&‘2~,+ W7C70+ W&u+ W13C130+ Wdw,

As = W2C2 + W15C150+ W16Cm . (3.30)

In order to estimate Ci, C’io and C’il, we must transform from the two-quark basis

SS, AA, SA and AS to the basis adopted by Harvey 4), i.e. RR, LL, m=

&RL+ LR) and KL = &RL-LR). They are related by the unitary transfor-

mation

(3.31)


TABLE 1

Definitions of symbols used in eqs. (3.30) and table 2 for the one-gluon

exchange interaction between two quarks (BC/ W&DE)

) ISS) i-Q4 IsA) /AS)

Here B, C, D and E represent the labels S or A for a one-quark

state spatially symmetrized /S) = &I R) + 1 L)) or antisymmetrized 1 A) = JT(I R) - 1 L)) between the two bag centers.

-- In the RR, LL, RL, RL, basis, the interaction matrix elements are given in table 2.

TABLE 2

One-gluon exchange matrix elements between two quarks described in a basis where they belong to

the right 1 R) or to the left ) L) center of the quark bag

(RR/

u-a

(=I e-1

+(AI+A~) $(AI+.&) (Al-A,)/242 +$(A*+Az,+As) +&AZ-A‘,-As)

~AI+As) :(Al+Ad (A, -A,)/2& +;(A2-A4-As) +$(A,+A,+A5)

(A, -A&2& (A, -A,)/245 :(Al+A3-2Az)

(As-Ad

They are expressed in quantities defined in table 1 in the spatial1 symmetrized or antisymmetrized

basis. The two quark states I=) = Jf(l RL)+ILR)) and I E) = z(\RL)- ILR)) have already been Jr

used by Harvey 4).

The self-energy comes from the sum of the processes in which the intermediate state is the same and different from the initial state. Schematically they are written in fig. 3.

Using the relation (RLIWIRL) +(LR\WjLR) = (m\WIm) - (l?-LjWI&) we find the self-energy for a single quark reduces to the sum of the matrix elements of (RRlWIRR)(LLjW(LL), (EIWjE) minus (a1WIRI.J which in the SS, AA, AS and SA basis is equal to iA.+

Fig. 3. The self-energy contribution in the single-quark basis of IL) and IR) defined in eqs. (2.3) and (2.4).

S. Furui, A. Faessler / Nucleon-nucleon scattering

4. Local potential

In the previous section, the two-body and one-body matrix

one-gluon exchange was calculated. We now calculate the field

six-quark system using the symmetry basis adopted by Harvey 4). . -

425

elements from

energy for the

For the ‘So or ‘S1 channels the physical basis consists of NN, AA and CC where

CC is the hidden color channel. They are related by the unitary transformation to

the symmetry basis:

4NN = &5,(33) + hh421{33) - 2~~421~1 7

JIdd = &2$[6]{33) +4+[42]{33) + %42]{51}1 ,

‘kc = ‘&‘hm) - ~[421{51)1 9 (4.1)

where [f] and {f’} specify the symmetry in orbital space and in the SU(4) spin-isospin

or spin-flavor space. The norm of the radial wave function of [6] and [42] symmetry

for a bag with relative distance between the centers 2a is given by the overlap of

the quark wave function m = (RIL) by

(~~61I(l~6~)=1+9rn~+9rn~+rn~,

(1/1~42~l(lr~421)=1-m2-m4+m6. (4.2)

As the relative distance “a” goes to 0, ($‘[42114[42$ =Nc421(u) goes to 0. But the

matrix elements with operators Ah or hhuu do not go to 0 in this limit.

We consider the two-body matrix elements between the states of spin S and S’

and the projection A4 and M’. For the spin-dependent operator we use the equations

(TrJ(To = (-A[, xa]b2’+[fl x&“‘)/(-&) ,

(+1(+-l +(+_1(T1= ([(I. xu]~2’+x&T x(+]b0’)J6/3 . (4.3)

The matrix elements are

= 4&ll[a xu]‘2’lls;> (Z, ; _“M) [ ;; s’, s~,(-)s2+s4-M&w

x (S4m41S4mi )(S2m2S4m4lSM)&n 34mk IS’iW

= &&s,ll[cT x (r1’2’lls;> (L, ; _“M) { ;I s”, s;) (-)S~+s~-Mw

+$J3(s,ll[u xu]‘“‘lls;)(~, 8 _&)I ;: s”, s!](-)s2+s4-Ms~‘. (4.5)


Here SZ, S; are the sum of the spins of particles one and two and Sq, Sk are the sum of spins of particles three to six. In the present calculation for S-waves only, we omit the term [a x aIf’. The matrix element then becomes

$.for&=Sh=l

-1 for Sz=Si =0 I for moue and gr(+-l + (+_l~l .

The antisymmetrization of the six-quark wave function can be taken into account by using the c.f.p. The two-body matrix elements are calculated by the formula 14)

(~S[PlR{B’}wl~l~S[flR{g}w)

=77 -“2({gk}w)rl-1’2({gs}W)

xxcpw (~g4Iw~g*~wo”4S4, T*S*&TkIWw

x cpW({g,)W{g2)W(T4s4, T2s2){g6}WTs)

x CpR([f41R[f21R& [fk 1Ra i)

X CPR.([~~IR[~~IR~~~ if61~a2)

x K ([f6 lR[f4lR[f2lRs kk)Wk4)WkZ}W)

x K ([f6lR[f4lR(fZlR, k6}Wk4}Wk2)W)

X u4([fA IRC~~IR, [f41~aka4)

x UR([f21Ra b2) uW(T2s2) ~C([f*lRkZ)W)

x77(k4Iw)15 7 (4.6)

where cpw and cpR are the c.f.p. in the W(4) symmetry basis and in coordinate space, respectively, as defined in ref. 14). K is the matrix that relates CG coefficients

for Se to those for S4; U4 is the overlap of 4-particle clusters. UR, VW and UC are two-body matrix elements in orbit, spin-isospin and color space respectively and n is the dimension of the indicated representation of the symmetric group.

Similarly the one-body matrix elements are

(TS[flRb’}WlCTITS[flR{g}W)

427

(4.7)

S. Funk, A. Faessler / Nucleon-nucleon scattering

x K ([f6lR[f5lR{g6}W{gS)w)

x ud[fb lR[f61R, [fS]Ra iaS)

X URUW UC77 (-kslwF .

The c.f.p., matrices K, and the overlap U4, Us are given in Harvey’s paper 4).

The bag energy consists of the one-gluon exchange two-body matrix element

EZ, the self-energy El, the zero-point energy Eo, the free quark energy 6w and

the volume energy E,. The zero-point energy E0 comes from the change of the zero point eigenmode

when the vacuum is separated into a small bag region and the rest. It cannot be

calculated for an arbitrary shape of the bag. So we choose an ad hoc form which

satisfies the following conditions:

(i) &=-$, with 20 = 0.365 GeV * fm at a = 0,

w (ii) Eo = -22,-

2.04 ’ in the limit of two separated bags.

Our choice is

E 0

Z-Z w+m(l-(I&Q) n

2.04 ’ (4.8)

where w = 2.04/R in the spherical limit.

The volume energy comes from the assumption that there are two phases in the

vacuum, one is the normal vacuum and the other is the confined phase vacuum.

The vacuum inside the bag is assumed to have an energy density which is by

57.55 MeV - fmw3 higher than in the normal vacuum 9). The volume energy is

E,=BV, B = 57.55 MeV/fm3.

So far the scale of the bag was not fixed. We determine it by minimizing the energy

E(T) = 6w +Eo+E1+Ez+

77 Cq3, (4.9)

where n is the scale parameter. In the limit of the two bags, the energy to be

minimized is

iE(d = 3w +;Eo+EZ

++Evq3. (4.10) 77

The numerical result of the bag energy for various distances R = 2a is shown in

fig. 4a for the 3S1 and ‘So channels. The local potential for the 3S1 channel is plotted

in fig. 5a. Although the local potential in the NN channel for 3S1 and ‘So are not

so large, there are large differences in the transition and CC channel potential. In

N-N

ch

anne

l ba

g’s

ener

gy

1.5

- >

0.5-

E

v

‘5 B

W o-

oh--

-- -

-0.5

” E

Q

24

I.’

0.l

6 c3

W

x/6w

5-

\_/E

v 5-

10,

&I

--

Eg

aIfm

l E

2

-0.5

t

Fig.

4a.

T

he q

uark

kin

etic

ene

rgy

6~0,

the

zero

-poi

nt

ener

gy E

c, t

he

Fig.

4b.

T

he s

ame

as f

ig. 4

a bu

t in

clud

ing

the

cent

er-o

f-m

ass

corr

ec-

self

-ene

rgy

Et,

the

gluo

n ex

chan

ge e

nerg

y E

2 a

nd t

he v

olum

e en

ergy

tio

n of

Liu

and

W

ong

“).

The

pa

ram

eter

s of

th

e ba

g m

odel

ha

ve

E,

for

the

NN

cha

nnel

. T

he s

olid

lin

es a

re f

or t

he s

S, c

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el a

nd t

he

been

rea

djus

ted

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epro

duce

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e m

asse

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the

bar

yons

af

ter

corr

ec-

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ed

lines

are

for

the

‘SO

chan

nel.

tion

for

the

cent

er-o

f-m

ass

mot

ion

“1.


Fig. 5a. The 3S1 NN channel local potential in the adiabatic approach. The solid line is the result of our model and the dashed line is the result of DeTar.

0.5 -

5 t3 w

04l

-0.5-

Fig. 5b. The 3Sr NN channel local potential being a linear combination of the results in the adiabatic and in the sudden approach. The relative kinetic energy between the two quark clusters multiplied by the factor (-1) is given as a dashed line. The relative kinetic energy has to be subtracted from the NN potential to prevent double counting if it is used in a Sehr~ing~r equation to describe NN scattering.

fig. 5a, we also show the result of DeTar. He plotted the energy as a function of 8 which is defined as

(4.11)

In the two-bag limit @ = 1 and qs = f(qR+qL), qA = i(qR-qL) and

$(q;:+qt%(qR-q& dV=$ k&R-qtqL)z dV. (4.12)

Hence 6 corresponds to our “a”, or half of the distance between the two centers. The shape of the bag for 2a = 0,0.72, 1.22, 1.92 and 2.66 fm is shown in fig. 6.


2a=O fr, 2a=0.72 fm

2a=l.22 fm 2a=1.92 fm

2a=2.66 f m 2a=2fm

Fig. 6. The shape of the bag in the adiabatic approach for 2a = 0, 0.72, 1.22, 1.92 and 2.66 fm and in the sudden approach for 2a = 2 fm.

For a spherical bag with quarks occupying the same orbit, there is no color

electric contribution and the magnetic field energy yields for a nucleon of radius 1 fm,

Ez=-; 1 iIBk1’dVa (

(Y, C Ufc+fhJj >

= - 157 MeV, (4.13) k=l i#j

where we used

( 1 UiajAiAj ~16. i*j >

(4.14)

ij= 1,2,3

Adding the zero-point energy E,J = -365 MeV, the volume energy E, = 241 MeV

and the quark kinetic energy 3w = 1210 MeV we obtain E = 928 MeV for a nucleon.

For a spherical six-quark system with [6]{33} symmetry of radius 1 fm one

obtains:

where we used

c UiUjAiAj = > I

-Y for S=l, T=O

i#j -16forS=O,T=l. (4.16)

ij=1,...,6

The exact MIT bag model gives E2 = -148 MeV for three-quark and E2 = 49.3 MeV

S. Furui, A. Faessler f Nucleon-nucleon scattering 431

and 148 MeV for the six-quark S = 1, T = 0 and S = 0, T = 1 system, respectively 9). As compared to DeTar where the [42] symmetry is absent in the spherical limit, the color magnetic repulsion is decreased, but in the total bag energy Ebag, the correction is small and the height of the soft core is almost the same as in DeTar’s calculation.

The zero-point energy of the MIT bag, however, includes the effect of spurious center-of-mass motion of the three quarks of a hadron. Liu and Wong 15) recently showed that the zero-point energy and the effective quark gluon coupling constant can be reduced if the center-of-mass effect is taken into account. The equilibrium condition is modified to

E(v) = 6w +Eo+Et+Ez-T,.,.

+Evq3. (4.17) 77

If E, is assumed to be proportional to R3, the new parameters are Z0 = 0.027 GeV * fm, (Y, = 1.532 and B = 77.10 MeV/fm3. The zero-point energy decreases considerably and the size of the bag shrinks. The soft core increases by about 70 MeV. In the model of Liu and Wong, [6] symmetry is assumed in the limit of complete overlap, and the soft core increased by 90 MeV. The local potential in this model for 3S1 and ‘So channel is shown in fig. 4b. In the following c~culation we will use the original MIT bag model,

In the NN and AA channels the two-body color electric interaction is attractive, but the self-energy contribution from the color electric interaction is repulsive. The strong attraction near a = 0.7 fm is mainly due to the low quark energy eigenvalue. The quark energy is different from DeTar since DeTar’s w.f. has only one center at the origin and a linear combination of the eigenvalues of the symmetric w.f. and the antisymmetric w-f. is taken. In our case we have two centers. They are shifted by *CX from the origin and all the six quarks are assumed to have the same eigenvalues. We included also the w.f. with the spatial [42] symmetry even in the limit of complete overlap. The effect was however not so drastic as that of the non~relativistic calculation of Harvey 4). In contrast to Harvey’s and to Faessler et al. i)‘s calculations where eigenvalues of the six-quark system were calculated in the potential model with properly normalized wave function, in our model matrix elements for ‘free’ quarks in the bag not perturbed by the two-body interaction were calculated and the relative weight of the [6] and [42] spatial symmetry are fixed by that of asymptotic channels. The dynamics which mixes the relative wave function for the [6]{33}, [42]{33} and [42]{51} channels are handled with the generator coordinate method which we discuss in the next section,

5. Resonating group method for the bag-bag scattering

In the non-relativistic quark model, nucleon-nucleon scattering was successfully solved by the resonating group method 1-3), In our case, however, there is a difficulty


in deriving the resonating group kernel since firstly we cannot choose Jacobi

coordinates for the quark wave function and secondly the wave functions are not

spherical. The kernel can, however, be calculated via the generator coordinate

method (GCM) 10*11V16). The idea is to approximate the six-quark wave function as

q(R) = W(Rr, . . . , &)

= dSf(S)@s(R), I (5.1)

where S is the generator coordinate representing the relative distance between the

center of three-quark clusters. Hereafter the vector S in the Dirac wave function

not to be mixed up with the above generator coordinate is written as SReR +S,e,.

The Schrodinger equation,

HW(R) =EP(R) , (5.2)

then reduces to the Hill-Wheeler equation,

J [H(SIS') -EN(SIS')]f(S') dS’ = 0 , (5.3)

where

H(SIS’) = (@Sl~l@S~) 9 (5.4)

N(SIS’) = (@sl@s*> . (5.5)

In our case, the wave function @Q(R) is the product of a three-quark wave function

with the center at x = $S and a three-quark wave function with a center at x = -$S.

The single-quark wave function was written as:

i

1+p(R2-R$ u

l+R; m =N

iaRSR& l+r(R2-R@+ia s _R_ l+r(R*-Ri)

Ro l-PR; a uRo 1-PR; (5.6)

The relative motion between the two clusters is described by f(S) and we assume

that this follows non-relativistic kinematics. The wave function &(R) includes the

whole kinetic energy of the quarks and so the relative kinetic energy for the motion

along the axis parallel to S must be subtracted for describing the internal energy.

Since our quark is massless we approximate the square of the relative momentum

as

(p2>

s

jm(-fu. vs)cf- * Vs)I@sv?)) (@s(~)l@s(Jw ’ (5.7)

where Vs is the gradient operator in the direction of S.


In order to evaluate the numerator, and also to evaluate the overlap, we expand the components of the single-quark wave function @k(R), @z(R) and @G(R) of eq. (5.6) in a harmonic oscillator (HO) basis.

Q4v?)= f. (C,o~o”(R)+C”l~l”(R)+C,2~2,(R)), n=O

@a?) = i: ~C,R,~o”~~~+~~l~l”~~~+~~*92”~~~~ 7 (5.8) n=O

where

qbln (R) = i’C,l e-f(R’b)Z@niO p 0 , (5.9)

Cd = t-l)“[ n!(21+2n +l)!! 1’2 1 7r3’2P21+n ’ (-l)nP2f+”

~“‘“(r)=(21+2n+1)!!rL” JG

’ ‘+1’2(r2)Ylm(&$) --j-.

(5.10)

(5.11)

L!,+“2(r2) is the Laguerre polynomial. For simplicity, we truncated the angular momentum at I= 2 and the radial

quantum number at a = 3. Since R = r +a, the wave function &,,(R/b) = &,,((r +a)/b) can be separated into an u-dependent part and the rest “):

(5.12)

The wave functions @g(R), C = (1, R, a) are now written in the following form

= cDc(r, a). (5.13)

We describe the oscillation of the center of the cluster by the OS HOWF:

F(Pu _S) = (F)3’4 e-w-~) (5.14)


with Y = l/26’. The Hill-Wheeler equation is then written in the form

~~l-~Vi+(H+~Vf)l~,,)-E(~~lh,)]f(S’)dS’

=jF(S-2u)(@(r,u)~-$+~P(r,u))~F(S’-2u)duf(S)dS’

+ H+LV$, @(r, a) 2cL I >

dF(S’-2u) daf(S’) dS’

+

F(S+2u)(@(r,u)l@(r,u))F(S’-2u)duf(S’)dS’

= 0, (5.15)

where the superscript d indicates that the part without exchange of quarks between

the clusters is kept 16).

Here we took into account the antisymmetrization of the two clusters. We

introduce an inverse operator F1(S - 2~) that satisfies

F-‘(S-2u)F(S-2u’) dS =S(2u -2u’). (5.16)

Explicit construction of the operator will be shown later. We define the relative

wave function

,y(2u) = 1 F(S’-2a)f(S’) dS' . (5.17)

Multiplying 8(S-S”)F-‘(S”-2a) dS” from the left of (5.15) and inserting F’(S’-

2u’)F(S’- 2~‘) d(2u’) before f(S’), we obtain,

I 8(2.‘-2u)(-$‘+(2u)d(2u)

+ I

S(2u’-2u) @(r,u) H+$V:, ( I 1 >

W, u) dxC2uJ 42ul

S. Fund, A. Faessler f nucleon-nucleon scattering 435

xF-‘(S’-2u’)x(2u’) d(2a’)

-E cS(2a’-2a)x(2u) d(2a)

-E F-‘(S-24 li(~i_2u)(~(r,a)fcP(r,u))F(S’-2u)d(2u) J J xF-“(S’-2u’)x(2af)d(2u’)

= 0. (5.18)

The energy consists of the quark kinetic energy 6w, the volume energy E,, the zero-point energy Eo, the gluon exchange energy Ez and the self-energy El. The first three energy terms do not depend on the configuration. The last two terms and the expectation value of (P’)~/~P are different for different configurations.

The expectation value of the internal hamiltonian is given for a 2u > 2 fm by two free bags and for 2a < 2 fm by two overlapping bags. The quarks may change their orbit. In order to connect the two solutions, we choose the potential as a superposition of the two expressions with weights w = max (0,l -a) for that of the overlapping bags and 1 - w for that of the free bags. One problem is the threshold for the CC channel. The potential (El + Ez)(a) in the hidden color (CC) channel increases with the distance from 0 to 0.3 fm in the same way as in the non-relativistic quark model. It falls off, however, in our model for a bO.4 fm. The solution for small “u ” extrapolated to 2a = 2 fm should have a very large expectation value of (El + E&a). We therefore choose the threshold for the CC channel to be 2 GeV. Justification of the choice is done a posteriori by checking that the CC channel wave function outside the bag is small.

With the mass of hadron in the asymptotic channel M[cj, the RGM equation reduces to

+21M[clu -w)krc!1W)

+g, zqs-24 F(S+2u) J J x(El+ Ez)rc.c~(a )N~c,c'1(a)wF(S'-2a)d(2a)

xF-~(S’-~U’)X~~~,(~U) d(2u’)

+ F-*(S-2~) F(S-2u)K~c,(a)wF(S’-2u)d(2a) J J ~F-‘(S’-2u’)~~c,(Za’) d(2u’)


+I F-‘(S-2u) I F(S+2a)K;c,(a)wF(S’-2~) d(2a)

x F-l(S’ - 2a)xEc,(2u’) d(2u’)

- (2m +~)xccm)

--I? F-‘(S-24 F(S+2u)Ncc,,,(a)wF(S’-2u)d(2u) I I

x F-‘(S’ - 2u’)xcc,(2u’) d(2u’)

+ V(2u, 2~‘) +K(2u, 2~‘) -EiV(2u, 2~‘) I

,Y(~u’) d(2u’)

= 0, (5.19)

where the suffix [C] specifies the symmetry of the channels NN, AA and CC. The

values of 6w, E,, I$, Er and EZ are given by the local potential calculated in the

previous section.

The overlap Ncf,n (a) can be written as

N~6.6](a)=1+9~(a)2+9rit(u)4+rii(u)6, (5.20)

N~42,427(u)=1-~(u)2-rii(a)4+~(u)6, (5.21)

where rii (a) is the overlap of the quark wave function (c 18). Using the relations

SReR + Seea = R max

JR:,, +R*-R;dR: +(dRolda) ,-(ReR -dRo/dae,)

R mall

=~R;,.+R2-R:JR;+(dRo/da)Z

X K

dRo a sin 6 r--acos8+-----

dRo r-u cos6

dcu R a sina--

da R > I es ,

(5.22)

RR’(&eR +S,e,)(SkeR, +S&e,,)

RL

=JR;, +R*-R;JR:+(dR,,/da)‘JR:, +R’*+Rb*JR;* +(dRb/dc#

x[(r*-u2)(RR’+~$$)+2rucos4(R’~-Rs)], (5.23)

S. Futui, A. Faessler / N~~ie~~-~u~ie~~ scattering 437

x i i c;,+; i (c:+c;: I I -1

hW2R2 dR . (5.24) I=0 n=O I=0 n=O

In the calculation of rfi (a) we make an approximation that we keep only the S-wave for the angle dependence of a. For the axially symmetric w.f. the term proportional to ru sin 6 reduces to zero in this approximation (see appendix A). This approximation is good for small a. The overlap before and after the HO expansion is shown in fig. 7.

For the transition potential E2(a)NC6,421 (a), we take

N/6,42] (a) = N[4,4fl[2,2] = I+ 5!?r (a)2 + 56 (a I4 + rii (a J6 1 (5.25)

We took into account that the coupling occurs only when the systems are decom- posed into [4] symmetry w.f.‘s and [2] symmetry w.f.‘s and N[4,&2) = 1+4+qa)2-tni(a)4.

(LIR)

0 0.5 10 a[fml

Fig. 7. The overlap (L/R) between a single-quark wave function IL) of the left center at -a and IR) of the right center at a (solid line) calculated numerically. The dashed line represents the same overlap after the numerical expansion of the single-quark wave function in [ns), (np) and /nd) oscillator states

with n = 0, 1, 2, 3.

438 S. Fund, A. Faessiet / Nucleon-m&eon scattering

The kinetic energy kernel is given by inserting the operator i 3

between the antisymmetrized w.f.s. We obtain

Ki.l(a)=6n3($n)+36n3(~n)m2+18n(~n)m4,

K;a,.(a) = 18n4 (~m)m+36n2(~m)m3+6(~m)m5, and

K 1421 (U ) = -h 4(f,),-4n2($m)m’+6(~m)m”,

(5.26)

(5.27)

(5.28)

(5.29)

(5.30)

where

V2 --p1=-1_2 2P 2EL

a * eRRcDR +v,, * e,R@” + ia a v2. X eRRQR +ia - VZa xcJWQ~

x IVza .eRR~R+Vz,.e,RQz”+iu.‘tr2a~e~R~R+i~.Vz,xe,RcD”)

+53v2.9’lv2d~, IE=l, (5.31)

and

V2 -m =$(V 2P

zaeR’R’~R’tV2,e,lR’~,*‘+icr*Vz,xeR~R’rSR’Si(f’V2aXe,~R’~“’l

X /V,,eRR@ +V;?,e,R@” + iu - Vza x effRQIR + irr * ‘l;lza X e,R@“)

+$ O’&J’W)~V+WR 1). (5.32)

For the mass CL, we choose half of the mass of a nucleon. The correction of the relative kinetic energy can be seen in fig. 5b. Here we plotted

as well as

{@w + E, + Eo)(a ) + (Ei + E2)[N.N]@ > - 2&q}W .

S. Fur& A. Faessler / Nucleon-nucleon scattering 439

The height of the soft core reduces by about 0.2 GeV in the limit of a = 0. The reduction of the soft core was also remarked by Wong is).

The operation of F-’ on both sides of F(S +IZa)N(a)F(S’-2~) is performed as

follows. First we calculate the Fourier transform

s(k, k’) = j T;(S +2a)N(a)F(S’-2~) da eik’S e-ik”s’ dS dS’ .

We divide it by the Fourier transform of F(S)

@(k)=lF(S)eik’SdS.

(5.33)

(5.34)

Then we perform the inverse Fourier transformation

I *(k, k’) ik.2a e-ik’.2a’ dk dkt (5.35)

The coupled eq. (5.18) can be solved by the Kohn-Hulthen-Kato’s variational method in the way as given by Kamimura r2). In this method we write the NN channel wave function as the su~rposition of locahy peaked gaussian functions:

with

xvi(Ry) = PyiXI? WY) for R,<R:

Xt_‘(k$y)-SyiXI+) (kJ?,) for R, > R ‘, . (5.37)

xl”‘tW 3

R, A =p (5.38)

The peak positions Si are properly chosen mesh points. xr’ and xl;” are incoming and outgoing spherical Hankel functions. The AA and CC channel are treated as closed and described by the wave function X$?)&, where L$,, is again a locally peaked gaussian function. The complex constants p+ and S+ are given by the matching condition at R, = RG,. They are

w[x;-‘9 XY'I Pri = w~ci;‘, xy] ’

w@;), &‘] Syi = w[xCiy), xl’)] ’

with the definition of the wronskian:

(5.39)

(5.40)


The coefficients C$’ can be obtained by solving the linear equation

1 5 2&,,C&’ +~2Z;~,<,*bjf) =.M$‘, 1 F 5fip,*jC$’ +CcF”,bz’ =IMI” f 6 j=l II 6 j=l Y

Z’-yisj =K+sj -Kyosj -Kyiao +K~o~so 3 Tv,j =Kvaj -K~so 9

.re,, = K,, , M”’ = -Ky+o +K Yl voco, &fcc) = -K ” vco 3

K yisj = I

Xyi-%aj d V t Kvsj = ~Z%~j dV, I

(5.41)

where

d=-~V:,+V(2a)-(2m+E)+V(2a,20’)+K(2a,2ar)-EN(2a,2a’).

(5.42)

The S-matrix is given by

Spasr = SRn,t +i I 1 2 I(BoyiC$) +CKoopbf’ y i=O P 3 9 (5.43)

(5.44)

In fig. 6, we show the result of the numerical calculation of the phase shift. Since our potential does not contain the influence of the mesons on the interaction the model is unreahstic for intermediate and longer ranges. We want only to demon- strate that the short-range part can be understood in bag model and the GCM approach. The local potential in the iSo channel in our model is more repulsive than that of the 3S1 channel, In order to show the effect of the subtraction of the relative kinetic energy of the bag, we calculate the phase shift with setting K (2a, 2a’) = 0. From fig. 8 we observe that the subtraction of the relative kinetic

Centre of Mass Energy IMeVl

Fig. 8. The nucleon-nucleon scattering phase shift in our version of the two-center MIT bag model for ‘S1 and ‘SO channels. The dashed curve corresponds to the ‘SO channel phase shift calculated without

the subtraction of the relative kinetic energy.

S. Furui, A. Faessier / Nucleon-nucleon scattering 441

energy makes the potential more attractive and the effect is as large as the difference between the 3Si and ‘So channel potentials.

6. Discussion and conclusion

In this paper we showed that the nucleon-nucleon scattering in the relativistic quark model can be solved with the resonating group method. As was shown by DeTar, the local potential has a soft core of about 300 MeV. In the intermediate range one obtains an attractive potential which is due to the low eigenvalue of the single-quark wave function at this distance 2a. The height of soft core increases by about 70 MeV if one takes into account the spurious center-of-mass motion of the three-quark system and readjusts the parameters of the MIT bag. “}. In order to solve the scattering problem, the relative kinetic energy of the bag was subtracted. This increases the attraction and is a major effect in the phase shift.

Comparing with the result of the non-relativistic quark model, we find two main differences. First, in the non-relativistic quark model the core disappeared due to the strong configuration mixing with the spatial [42] symmetry if the Born-Oppen- heimer approximation is used 4). The mixing occurred mainly through the color dependent confinement and Coulomb interaction, while in our relativistic bag model the Coulomb energy is canceled by the self-energy and the mixing occurred only through the magnetic interaction at distance zero. Secondly there appears attraction in the relativistic quark model at an intermediate range due to the low quark eigenvalue. Although our trial wave function for the variational calculation is crude, this can be understood due to the lower eigenvalue of the quarks for motion in the direction of the symmetry axis of the six-quark bag at intermediate distances.

The dynamics of the collision effects the shape of the two quark bags. In a sudden approach the two quark bags would keep their shape while in an adiabatic collision the shape is determined by minimizing the energy for each distance 2~. In the calculation presented here we took a linear combination between both extremes with weights discussed in the text. In the intermediate region, the lower component of the Dirac wave function is large and there the non-relativistic wave function is not justified.

We did not take into account the influence of pions and other mesons. When one includes pions and restores the chiral symmetry of the bag hamiltonian, a bag with smaller radius of about 0.4 fm was found to reproduce the mass of a nucleon 20~21~22). The increase of the quark kinetic energy was compensated by the decrease of the pion self-energy in this model. In order to apply the little bag model to our method, it is necessary to readjust the parameters for the pressure balance. It is also necessary to calculate the pion self-energy for an arbitrary shape of the bag.

The authors thank Dr. Fernandez for supplying us the computer program for the calculation of the phase shift, Dr. Kiippel and Dr. Nishimura for help in the


use of the computer. The numerical calculations were done on an UNIVAC at the Rechenzentrum der Universitlt Tiibingen.

Appendix A

In this appendix we present useful formulas for the calculation of the overlap of HOWF which have different centers.

(A-1)

c ntn’t’ = (_l)nsn’ffS22Sft+t’-L) SL (:, b’ 3

XC (-1)“+“(1+I’+L+2@ -t2vi-l)!!(a(l+z’-L)+~ +Y)!

ILY (&f-I’-L)+~ +v-S)!/L!V!(n -/.&)!(n’--v)

(A.2) A derivation is given in ref. I’).

The overlap can be calculated also by using the formula (5.12). After some angular momentum calculation, we obtain

j- h%( ~),wi( y) dr

= c Cn&(k :, ;)[I L’, f]wY+A ZLL’Uh

x tl ( “0’ ~)(~ “0 xi ^d 3

L”f2fcv+L) ~,f$(N’+L’)

Xl (z(N --I,))! (lv +L + l)!! &iv-L’))! (iv’+L’+ l)!!

x(-l)N f 0

NtN’

e -a2fh2C;; J&G Yypo(ri), (A.3)

where C,, is given in sect. 5. We checked the equality of (A.l) and (A.3) by


comparing the coefficient of the powers of (a/b). The last form was used for the calculation of the kinetic energy kernel.

2”‘S’a! (2A + 2v + l)!!

2 u+*a’! (2n’+ 2a’+ l)!!

xc 1 (-l)“(A+A’+2v+3)!!(${A-A’)+l+v)!

v (y(A’-A)+v+1-rr’)!u!(c+-v)!(2A+2~+1)!!

221”2f(N+L)

Xl

~,2ft2f(N’+L’,

(z(N-L))!(iv+L+l)!! (~(N’-L’))!(N’+L’+l)!!

N+N' (_l)L+h e-a*,‘b2

xp2 1 L A

I L’ 1’ 3 ) JGY&W ( f, ; =f)(f 8 ;). 64.4)

( 2”‘/C2/C2a! (2h + 2rr + l)!! 1’2 x 2”+‘o’!(2h’+2cF’+l)!! I

(-l)“(k+~‘+h+2~+1)!!(&+~-A’)+~)! xc 1 y (5(k+A-h’)+v-cr’)!v!(v-v)!(2A+2v+1)!!

L”2f2;(N+LY L”r2f$(N’+L’)

Xl (&v -L))!$V,+L + l)!! &iv-L’))! (N%L’I l)!!

(_l)L+h e-a+/&*


(A.3

For $8 = 0 the magnetic quantum number m should be 0. Hence the matrix element

of r sin 8 = Y1,i(8, 0) vanishes.

Appendix B

OVERLAP OF THE SIX-QUARK WAVE FUNCTION

In this appendix we present a method to calculate the overlap of different

six-quark wave functions. For the six-quark wave function we first write the Young

diagram of the S3 symmetry for three quarks in the left orbit, which we label by

L. Then we add three quarks in the right orbit and construct the Young tableaux

of the Sg symmetry.

In coordinate space and for the S-wave interaction between two nucleons only

the [6] and [42] symmetry are allowed. In the case of the [6] symmetry the tableau

is

LlLlLjRlRlR.

We consider the overlap with no L-R pair interchanged one, two and three L-R

pairs interchanged. All terms have positive sign since all the quarks are spatially

symmetric. Hence we obtain as the overlap

03.1)

where m is the overlap (L] R) and NC61 is the normalization factor.

In the case of [42] symmetry we first assign

Since for the wave function equals to zero, the assignment is unique.

If one L-R pair is interchanged we obtain

-

S. Furui, A. Faessier / Nucleon-nucleon scattering 445

If two L-R pairs are interchanged we have

and if three L-R pairs are interchanged we have

The sum of the all contribution reduces to

03.2)

In spin space, the spin-l state is characterized by a (42)s symmetry. In order to calculate the overlap in spin space, we must take into account that the quantization axis is chosen in the body fixed frame. The overlap of a single spin wave function is given by

G cos $9 -sin $8

sin $8 1 u

cos$8 m’

where U,,, is the Pauli spinor and is the angle between the quantization axis. We consider the overlap for the diagram

03.3)

If no pair is interchanged the overlap is

(b ( $)“(-$I -$)’ . 03.3)

If one pair is interchanged it is

-{ii _$“{f@“{-$j_$)” , CR.41

if two pairs are interchanged

-<21-b”<314>“, tB.3

and if three pairs are interchanged

<Bj--4)” . (B.61

Adding all contributions we obtain

cos6 $9 - sin2 $8 cos4 $8 - sin4 49 cos* $9 -sin” $9 = cos? $9 . 03.7)

In the calcuIation of the overlap with the gaussian function F(S -2a) and F(S’-2~) one should take into account the angle dependence of the overlap of


the spin wave function, if the axes S and S’ are not parallel. In the present calculation

for s-wave only, we can ignore this angle dependence.

References

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April 1981

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