Nuclear Physics A438 (1985) 620-630 @ North-Bolland Publishing Company

QUARK MODEL OF NUCLEON-NUCLEON SPIN-ORBIT POTENTIALS

FAN WANG* and CHUN WA WONG

Depu~~enf of Physics, University of &~l~orn~~ Los Angeles, CA 90024, USA

Received 22 October 1984

Abstract: The lowest-order spin-orbit potentials between two nucleons are calculated by using nonrela- tivistic two-quark spin-orbit interactions obtained from meson masses. The resulting scattering matrices (calculated in the Born approximation) are compared with those from phenomenological nuclear forces. They are found to have the correct signs and roughly the correct strengths in both the T = 1 and T= 0 channels. The many uncertainties and difficulties inherent in the quark picture of nucleon-nucleon spin-orbit forces are briefly discussed.

1. Introduction

Quark effects are expected to become manifest when nucleons of finite size are forced to overlap each other. Two aspects of the resulting short-range nuclear forces have been studied in the past: short-range repulsion and the spin-orbit potential. Of these, the spin-orbit problem by virtue of its spin dependence might provide a more specific and possibly clearer window into quark effects in nuclear forces.

Studies of quark models of short-range repulsion have been quite numerous. On the spin-orbit problem, we are aware of only a few detailed studies I-‘). They can be separated into two groups which differ in how the quark coordinates are eliminated in favor of nucleon coordinates.

Refs. ‘-‘) study only the interaction between the two quarks i and j which are being exchanged between the nucleon clusters a and b. By separating the isospin exchange operator P; from the spin exchange and interactions, the transition to nucleon operators can be achieved by a simple summation

(1.1)

In our opinion, this appears to be a dangerous procedure of undetermined validity. First, the spin-isospin functions in each nucleon are not independent of each other. Rather they must be correlated properly to produce a totally symmetric wave function. Secondly, the independent sum over isospin and spin quark pairs means that the isospin exchange and the spin exchange and interactions often involve different quark pairs. This does not correspond to the physical picture that the exchanged quarks are the ones which interact.

* Permanent address: Department of Physics, Nanjing University, Nanjing, China.

620

F. Wang, C. W. Wong / Spin-or&it pofenrinls 621

In contrast, Warke and Shanker “) (WS) and Morimatsu et al. “) (MOSY) have included all possible quark-quark interactions, and have treated them correctly by using a resonating-group (RG) method “). The usual RG scattering equation can be written formally as

(H-E)tCt=(H--E)QJI, (1.2)

where Q is the nonlocal exchange operator, and H = T+ V is the hamiltonian. Thus the RG scattering potential is

VRo= W-Q)--(T--WQ, (1.3)

where the spin-orbit part comes from V( 1 - Q) in NR models. This is the nonlocal potential discussed in this paper.

Alternatively, one can define an energy-independent nonlocal potential for the wave function (1 - Q)“*$. The spin-orbit force then comes from the term (l- Q)-“*V( 1 - Q)“‘. This is the form used in refs. 475). We should point out in this connection that WS actually used a certain local approximation to the normalization 1 -Q which can be shown to be incorrect. Fortunately, a jacobian factor ($)’ was accidentally left out in calculating matrix elements of Q, so that the calculated numbers are off by only about 20%. The problem of missing jacobians in ref. “> has already been noted by Robson ‘).

After allowing for this technical difference, our RG spin-orbit kernels still appear to differ from those of WS in several places. For this reason, it appears useful to report our results (calculated in the same NR quark model) in some detail. This will also supplement the work of MOSY where no detail is given on the kernels.

The calculated nonlocal kernels can next be used to obtain scattering phase shifts in different NN partial waves. Because of the complexity of nuclear forces, it is often desirable to compare the calculated results directly with phenomenological spin-orbit potentials. Since the latter are usually local, it is not clear how the comparison should be made. In MOSY, the diagonal, or adiabatic, part of the kernel is used for this purpose. [To be more precise, their kernel is not a RG kernel, but a CC kernel “) in which each nucleon has been allowed to oscillate about a mean position.] This allows for only a very rough comparison, since information on the off-diagonal matrix elements is not used.

WS does better in handling the nonlocality by using a Taylor expansion of the wave function. If quarks 3 and 4 are exchanged between nucleon clusters (123) and (456) separated by R, then

This contains the leading terms of both even and odd parity in R. The second term contributes to the spin-orbit potential, while the first is involved in all the other components of the NN potential (which are even in R). It is obvious that this

622 F. Wang, C. W Wong / Spin-orbit potentials

approximation is best at the lowest energies under circumstances in which the scattering wave function does not change appreciably over the nucleon size.

We would like to point out here that perhaps a better way of comparing a nonlocal RC kernel with a local potential (short of actually calculating phase shifts) is via the Born approximation to the scattering f-matrix. This has the additional virtue that the accuracy of the WS Taylor expansion can be tested.

The paper is organized as follows. Sect. 2 defines our quark model and describes how our quark-quark interaction is chosen and what the resulting NN spin-orbit kernel looks like. Explicit expressions are then given in sect. 3 for the resulting t-matrices in the Born approximation. These are then compared numerically with those from the Reid soft-core potentials “) and from the phenomenological spin-orbit t-matrices of Love and Franey9). In agreement with MOSY, we find that the quark-model result agrees both in signs and also roughly in magnitude with those of phenomenological t-matrices. Sect. 4 contains brief concluding remarks, especially on the limitations and uncertainties of the present study. In particular, caution is urged in interpreting the observed agreement between calculated and phenomeno- logical potentials.

2. The NN spin-orbit kernel

An isolated nucleon is here taken to be a cluster of three quarks:

tfi~(l23)=xc(l23)77~~(123)&,(123), (2.1)

where xc is the color singlet wave function in SU;, rjsr is a symmetric SUY 2 SUF X

SU; spin-isospin function, and c&, is a symmetric gaussian spatial wave function

~5,,(123)=(3A2/p2)3’4exp[-~h(r:,+r2,,+r2,,)], (2.2)

where rii = ri - rj, and the parameter h can be fitted to the proton charge radius (r,, = 0.83 fm) if it takes the value of

&=$=0.484fm-2. P

(2.3)

The two-nucleon system will be described by the simplest resonating-group wave function:

?P(l * * * 6) = ~{[1cIN(123)l(lN(456)lsrI=~rN(r)}, (2.4)

where SQ is an antisymmetrization operator for quarks. This wave function has been coupled to a total spin S and isospin T, and it depends on the relative coordinate r=r -rNz. NI

In the NR first-order perturbation theory, the NN spin-orbit force comes from the quark-quark spin-orbit interaction

vy=xi- Xjf(qj)(Pii xp,,) * (Sii-Sj), (2.5)

F. Wang, C. W. Wong / Spin-orbit potentials 623

where we shall use a Breit-Fermi or gaussian radial form:

(2.6)

fG( r) = V, e-@‘* . (2.7)

Here as= g2/4n is the color coupling constant, and m is the quark mass. The

gaussian form is not only easier to use, but it also simulates to some extent the

effect of gluon confinement in the nucleon. For the same reason, the mean square

radius of the interaction is chosen to be the same as the mean square radius of a

quark pair in the nucleon, i.e.

/J=;I\. (2.8)

The strengths CX, and V, will be determined by fitting the spin-orbit splittings of

p-wave mesons in the following way. If the p-wave qq wave function is proportional

to r exp (-br*), then

(sd Vdr)kFdr = 2 -& (2b)"'W. S>, . (2.9)

The size parameter b can be deduced from the rms radius r, of the (3S,) vector

meson. We take the calculated value r, = 0.72 fm for the p-meson from the NR

potential model of Liu and Wong lo) to get 2b = 0.72 fm-*. Eq. (2.9) then yields

3m2 JG -AM, as = 32 (2(,)3/=

where

AM =&M(3P2) -$M(‘P,) -$M(3Po) (2.11)

is a combination of p-wave meson masses in which only the spin-orbit contribution

survives in first-order perturbation theory. We use AM = 92 MeV from model b of

ref. lo). which gives a global fit to light meson masses in a NR quark potential model.

For a quark mass m = 0.34 GeV, (Y, turns out to be 0.37. This is the value used in

sect. 3.

A similar development gives the gaussian potential the strength

V, = --A( 1 + &2b)“*AM. (2.12)

The result also depends on the inverse range parameter p, which has been chosen

according to eq. (2.8). We should note that these choices of p and V,, like those

of (Y, and m, are not very precise. In addition, other radial forms (e.g. those containing

spin-orbit interactions coming from the confinement potential) could also be inter-

esting, but they will not be studied here. Three-quark spin-orbit interactions have

not been included in the present study.

624 F. Wang, C. W. Wong / Spin-orbit potentials

In the quark model, the direct contribution to the NN spin-orbit force vanishes

because the color operators C Ai. Aj are evaluated between color-singlet nucleons.

Of the 15 distinct pairs of qq interactions in the exchange kernel, only 5 give nonzero . .

contributions: 2)34 and t+,, vZs, v16, vZ6 if quarks 3 and 4 are exchanged between the

nucleons each in a (1~)~ spatial state. The nonlocal spin-orbit kernel,

XLs(r, r’) = -9(+( 123)(cr(456);rSMsTM,I C vf”l$( 124)$(356);r’S&&TM,) q

x[l -(-)s+rPx]) (2.13)

where P” is the space exchange operator between nucleons, is nonzero only for

S = S = 1. It can be written in the form

X$‘(r, r’) =2&iC:$,,(-)“F,.(r, r’)[l -(-)s+rPx], (2.14)

containing a Clebsch-Gordan coefficient and a vector in space along the normal

direction ii = i x i’:

F(r, r’) = -~+#A/T) 3’2$h 1

A,3(rxr’)exp[-$A(r’+r’*)+TAr* r’]f&r-r’))

+B~,(3A/fl)~‘~exp[-~A(r~+r’*)](r-3r’)

X I

r15exp[-3hr:,+qA(r+r’) * r1Jf(r15) d3r,, . I

(2.15)

Here 9 is a color factor, and

AT=&= 1jls3+s4)1S= l)T=$(-&) for T = l(0) ,

B,=-J~(S=llls,$s,IIS=l),=-~(-~) for T= l(O), (2.16)

are isospin-dependent reduced matrix elements. They have been calculated in a

straightforward if tedious way. Eqs. (2.16) have been checked independently by Y.

He (using a computer) and by L.J. Deng (using Racah algebra). We note that AT is proportional to 9+57, * T* rather than the value 9+7, * T* suggested by the

approximation of ref. ‘). The vector F can be evaluated in closed form for both

Breit-Fermi and gaussian form factors.

3. Comparison with phenomenological potentials

Although the Taylor expansion (1.4) appears well suited to the short-range

spin-orbit problem, the RG kernel calculated here has significant contributions of

longer range, since it is linear in both r and r’ at small distances. The reason is that

at small distances the chosen cluster wave function (2.4) approaches the (1~)~

configuration where the spin-orbit potential has zero matrix elements.

A more flexible way of comparing nonlocal with local potentials, especially at

the higher energies, is to use the t-matrix in the Born approximation. For our

F. Wang, C. W. Wong / Spin-orbit potentials 625

spin-orbit kernel, this is

t;t,“(E

9

q) Jk lMs~wIw~‘, lMs7w-) i(lMs((a,+a2). n^(lMs) ’

(3.1)

where n* = ff’ x I;, while

E=2k2/M=2kr2/M, q=k’-k (3.2)

are the lab scattering energy and the momentum transfer of nucleons of mass M.

Except for the Born approximation, this tLS is the same as that of eq. (16~) of ref. ‘).

The Born result for the BF form is

tf”,( E, q) = -(y, 97r m2

+4J3BT5e-k2f9A q [l-pkk)l

where

(3.3)

P(x) = i epx2 I x

ey2 dy , 0

K = (k2-fq2)‘j2.

The Born result for the gaussian form is

(3.4)

(3.5)

t&‘( E, q) = $T”~ V,qK { [Ar($)-s’2exp [-$--(q2+$k2)]

+ BT(3h)-5’2 exp [

-& (q2+4k2) 11 +(-)s+r[AT(y)-5’2exp [-&-$I

+ BT(3A)-5’2 exp [

$-(q2-8k2) . III (3.6)

The WS Taylor approximation to eq. (3.6), calculated under the approximation

exp [ ik’ * r'] = [ 1 + i( r' 7 r) * k’] exp [ ik’ . r] , (3.7)

626

is

F. Wang, C. W. Wong / Spin-orbit poientials

t&( E, q) = $?r3” V,qK 5h --s/2

i 0 AT 1 ,-llqy90A + B&h)-5’2 e-3qz/‘oA

+(-)s+T[AT(~)-5’2enp [-$(q2-4/c’)]

+ BT(3A)-5’2 exp & (q*-4k2) II} . For comparison, the Born result for the Reid soft-core (RSC) potential

V”“(r)=C &emRr’/Rir I

is

t&-(E,q)=llrrqKC V,R;{(lfq2Rf)-2+(-)S+T(1+4K2R~)-2}. I

(3.8)

(3.9)

(3.10)

The RSC spin-orbit potentials are fitted independently in different partial waves.

Because of various approximations, only semi-quantitative comparisons appear

justified at this stage. For this purpose, it appears adequate to use the 3P2 -3F2

(T = 1) and the 3SI - 3DI ( T = 0) spin-orbit potentials in all the other relevant partial

waves in eq. (3.10). Reid has given two different versions of the T = 0 potential,

with one or two terms in the sum. We shall refer to them as RSCI and RSCZ,

respectively.

These Born results are shown in figs. 1 and 2 for T = 1 and T = 0, respectively,

at a lab energy of 800 MeV for different choices of the nucleon size. The quark-model

results are shown as x (A = A,), l (A =2&J, and + (A = 4Ao) for the Breit-Fermi

potential and as broken curves for the gaussian potential. The areas between these

two sets of curves for A = A0 and A = 4A. have been shaded to help the eye. They

suggest that there is little difference between BF and gaussian shapes, but a stronger

dependence on the nucleon size. A size parameter of A = 3A. seems to be best for

T = 1, and A = A0 for T = 0. This difference comes about because the phenomenologi-

cal spin-orbit potentials have different ranges for different T-values.

Using A = 2ho as a compromise choice, we show in fig. 3 how the results of the

gaussian quark model (broken curves) compared with those of the RSC potentials

(solid curves) at lab energies 100, 425, and 800 MeV. We see that the qualitative

features of the RSC results are roughly reproduced in both T = 1 and T = 0 channels.

The quark-model strengths are weaker by a factor of about 2, but not by the large

factor of 100 found by WS. Part of the difference with WS comes from the fact that

(1) our Breit-Fermi potential strength (which is roughly equivalent to our gaussian

potential) is almost three times stronger than that of WS, and (2) our nucleon inverse

range parameter A0 is about eight times smaller than that used by WS, so that the

C

- -2oc 2 z 3

e w, - -4oc

-6oc

F. Wang, C. W. Wang / Spin-odd porenfials

q (fm-‘) 1 2

I

621

Fig. I. Comparison of the T = I spin-orbit r-matrices at 800 MeV calculated in the Born approximation for the gaussian quark model (GQM: dash curves) and the Breit-Fermi quark model (BFQM: X, l and

+) at three different values of the inverse nucleon size parameter A = A,,, ?A0 and 41,) with the results

for the Reid soft-core (RX) potential.

T =O E = BOO MeV

Fig. 2. Comparison of the T=O spin-orbit r-matrices at 800 MeV in the two quark models with the

results for two versions of the Reid soft-core potentials. The notation is that of fig. 1.

628 F. Wang, C. W. Wong / Spin-orbit potentials

0 T=l

\ \

E = 100 MeV

-100 \ L-l \

\ \

, GQM

‘, / RSC

-200

E = 800 MeV

9 W-f’)

.,L T=O I

q W-9

Fig. 3. Comparison of spin-orbit t-matrices at 100,425 and 800 MeV calculated in the Born approximation for the gaussian quark model (GQM: dash curves) at an inverse nucleon size parameter of A = 2A\, with

the results for Reid soft-core (RX) potentials. The dash-dot curves give (I) the Warke-Shanker (WS)

approximation to GQM at 425 MeV, and (2) the uj4 contribution (labeled QMI) to GQM at 800 MeV.

t-matrix is stronger at small momentum transfers. These features account for most,

but not all, of the discrepancy between us and WS.

Our results can also be compared with the empirical Love-Franey spin-orbit

f-matrices ‘). The latter (not plotted here) are complex numbers whose absolute

value reaches a maximum of 135 (165, 136) MeV . fm3 at q = 1.5 (2.6, 1.9) fm-’ and

E = 100 (425, 800) MeV in the T = 1 channel, and a maximum of 68 (91, 90)

MeV . fm3 at q = 0.8 (1.5, 1.8) fm-’ in the T = 0 channel. Hence our Born t-matrices

have similar q-dependences, but are somewhat stronger than those of Love and

Franey.

At 425 MeV, the gaussian results of eq. (3.8) calculated in the Taylor approxima-

tion used by WS are also given in fig. 3 as dash-dot curves labeled WS. We see that

this approximation can be used for qualitative work, but it overestimates the absolute

value of 1 by about 35%. It is likely to be better for kernels which are shorter in range.

At 800 MeV, the gaussian results from 2134 only (calculated with AT # 0, BT = 0) are also given in fig. 3 as dash-dot curves labeled QMl. We see that the neglected

E Wang, C. W. Wang / Spin-orbit potentials 629

interactions of the v,~ type account for only about 10% of the total. Thus the neglect of uf5 interactions in refs. le3) is not a serious shortcoming.

4. Discussion

In agreement with MOSY [ref. 5)], we find that the simple quark model used in this paper gives not only the correct signs, but also roughly the correct strengths of empirical NN spin-orbit potentials in bark T = 1 and T = 0 channels. The calculated results should be considered to be only qualitative because of various uncertainties. These include ( 1) the question of nucleon size, (2) the effect of configuration mixing or channel coupling, (3) the influence of gluon, or string, degrees of freedom, (4) the choice of quark-quark interaction form and strength, and (5) the limitations of NR quark models, especially in their description of spin effects. In contrast, the limitations of the Born approximation may be considered minor, especially because it can be avoided altogether by calculating partial-wave phase shifts.

Much more serious than the uncertainties listed above is the possibility that the quark-quark spin-orbit interaction responsible for the calculated kernel might be partially or totally suppressed in baryons. This possibility arises from the observa- tion “,“) that the corresponding spin-orbit splittings in P-wave baryonic excited states are not seen in experimental baryon spectra. The origin of this suppression is not understood, however. It could be due to a contribution of the opposite sign from the Thomas spin-orbit term of the confinement potential 12), or from the effects of confined gluons and relativistic quarks 13). Before this question has been properly addressed, the significance of the observed agreement between calculated and empirical results cannot be determined.

In addition, one cannot rule out the possibility that the traditional boson-exchange picture of NN spin-orbit potential retains some validity in the quark model in the form of higher-order contributions involving sea quarks from both outside and inside the interacting nucleons. Interesting additional effects arising from boson exchanges at the quark level include (1) quark-exchange kernels from quark-quark boson exchange potentials 14), and (2) multiquark or “exchange” contributions to the meson-quark-quark vertex 15).

Finally, we must mention the possibility that part of the baryon-baryon spin-orbit potential could be of relativistic origin r6). The successes of Dirac phenomenology I’) in accounting for spin observables in proton-nucleus scattering must be taken into account in reconsidering the extent pair suppression might operate to reduce these relativistic effects.

Thus the NN spin-orbit problem, though more specific than short-range repulsion, has not yet materialized as a clearer window into quark effects in nuclei.

This work is supported in part by NSF grant PHY-82-08439. F. Wang wishes to thank the Department of Physics, University of California, Los Angeles, for its hospitaiity.

630 E Wang, C. W Wong / Spin-orbit potentids

References

1) M.B. Kislinger, Phys. Lett. 798 (1978) 474

2) H.J. Pirner, Phys. Lett. 85B (1979) 190

3) G.E. Brown, Prog. Part. Nucl. Phys. 8 (1982) 147

4) C.S. Warke and R. Shanker, Phys. Rev. C21 (1980) 2643

5) 0. Mo~matsu, S. Ohta, K. Shimizu and K. Yazaki, NueI. Phys. A420 (1984) 573;

0. Morimatsu, K. Yazaki and M. Oka, Nuci. Pbys. A424 (1984) 412

6) J.A. Wheeler, Phys. Rev. 52 (1937) 1083;

C.W. Wang, Phys. Reports 15C (1975) 283

7) D. Robson, Prog. Part. Nucl. Phys. 8 (1982) 257

8) R.V. Reid, Jr., Ann. of Phys. JO (1968) 411

9) W.G. Love and M.A. Franey, Phys. Rev. C24 (1981) 1073

IO) K.F. Liu and C.W. Wong, Phys. Rev. D21 (1980) 1350

11) R.H. Dali&, Prog. Part. Nucl. Phys. 8 (1982) 7;

D. Gromes and 1.0. Stamatescu, NucI. Phys. B112 (1976) 213

12) N. Isgur and G. Karl, Phys. Rev. Dl8 (1978) 4187

13) H.R. Fiebig and-B. Schwesinger, Nucl. Phys. A393 (1983) 349 14) K. BrLuer, A. Faessler, F. Fernandez and K. Shimizu, On one-pion exchange potential with quark

exchange. . , fuebingen preprint (1984)

15) M. Heyrat, Ph.D. thesis, UCLA (1983)

16) L.D. Miller and A.E.S. Green, Phys. Rev. C5 (1972) 241;

J.D. Walecka, Ann. of Phys. 83 (1974) 491;

L.D. Miller, Ann. of Phys. 91 (1975) 40

17) B. Clark, S. Hama and R. Mercer, in Interactions between medium energy nucleons in nuclei, ed.

H. Meyer, AIP Conf. Proc. no. 97 (1982) 260