quark effects in nuclear physics

Quark Effects in Nuclear Physics

CHUN WA WONG

Department of Physics, University of California, Los Angeles, CA 90024, U.S.A.

ABSTRACT

Since QCD is not well understood, we are forced to use phenomenological quark models fitted to known hadronic properties to learn about possible quark effects in nuclei and other multiquark systems. These "extrapolations" help us see how well the fundamental theory must be understood before we can obtain reasonable results. In this spirit, I discuss certain quark effects associated with valence quarks, sea quarks and valence antiquarks, using the M.I.T. bag model and potential models of quark dynamics. The contributions of valence quarks to many-nucleon forces turn out to be model-dependent, and not well understood. They contain large configuration-mixing effects which appear to arise from certain space-color-spin correlations. These correlations could sometimes cause an abnormal quark matter to collapse. The masses of multiquark states are also sensitive to the confinement mechanism. Sea-quark effects from qq fluctuations are shown to be important in nucleon properties. They also give rise to meson-nucleon coupling constants. The ~NN result calculated from the M.I.T. bag model agrees with experiment, while the 0NN results are onl~ roughly reproduced. Valence antiquark effects are discussed in the context of q ~2 mesonic spectra. A cluster treatment using a nonrelativist- ic quark potential gives a detailed description of various rotational excitations of low L which cannot be obtained readily in other models.

KEYWORDS

Quarks; valence quarks; sea quarks; bag model; potential models; nuclear forces; abnormal quark matter; nucleon properties; meson-nucleon coupling constants; multiquark states.

INTRODUCTION

Many exciting developments have occurred in particle physics during the last fif- teen or twenty years (Gell-Mann, 1964; Ting, 1977; Richter, 1977). Perhaps the most significant result, as seen by a casual observer, is the very considerable conceptual simplification which has occurred in the theory of interactions (Wein- berg, 1980; Salam, 1980; Glashow, 1980). One ingredient of this simplification is the theory of quarks as the constituents of strongly-interacting hadrons, including the nucleons of nuclear physics (Kokkedee, 1969; Marciano and Pagels, 1978).

223

224 Chun Wa Wong

Such a theory of nucleon structure is obviously relevant to nuclear physics. We recall that the traditional nuclear physics of point nucleons has been surprisingly successful in the description of nuclear properties, especially when supple- mented by the additional degrees of freedom due to mesons and isobars (Rho and Wilkinson, 1979). The successes of this traditional picture must be considered surprising to some extent, because it has been known experimentally for some time that nucleons are actually rather large objects: The experimental proton "charge" radius (Borkowski and others, 1974) of 0.88 fm implies a "matter" radius of about 0.81 fm, or an equivalent uniform-density radius of 1.05 fm. This hypothetical nucleon matter has a density of about 0.21 nucleons/fm 3, as compared to a density of 0.15 nucleons/fm 3 in normal nuclear matter. Thus the nuclear interior may be thought of as roughly 70% nucleon matter and 30% empty space. This is very different from the picture of point nucleons used in "classical" nuclear physics. The quark model of hadrons now gives us an opportunity to study the effects of nucleon structure on nuclear properties.

Before proceeding further, I would like to review very briefly the present picture of quark dynamics. Quarks are spin 1/2 fermions of fractional charges (2/3 or -1/3) and fractional baryon number (1/3). They come in different flavors. Five (u, d, s, c, b) have been seen experimentally. The flavors u (for "up") and d (for "down") are of direct interest in low-energy nuclear physics, because they form the isospin SU(2) group which is responsible for the isospin degree of freedom in nuclei. In the simplest quark model, each baryon contains three quarks, while a meson is a quark-antiquark pair.

Each quark of a given flavor comes in three colors, which form a color SU(3) group Quarks are assumed to interact through the exchange of colors. They do so by emitting or absorbing color-anticolor combinations. There are only eight color- changing combinations called $1uons, which are described by the eight generators %a of SU(3). Thus quark-quark interactions due to one-gluon exchange are proportional to %.'%.. The dynamics of color, or quantum chromodynamics (QCD), gives rise to a ~I ~j new fundamental interaction of great strength. Shielded remnants of this interaction among nucleons appear in the familiar form of nuclear forces.

Gluons are assumed to be massless, and to be associated with a gauge, or phase, transformation of the wave function, like photons in QED. Unlike photons, gluons do not commute. Such non-Abelian gauge fields (Yang and Mills, 1954) are intrin- sically nonlinear, and show many unusual properties. A particularly important property is that they are asymptotically free (Buras, 1980) in the high-frequency limit. That is, the coupling constant vanishes as the momentum of the system goes to infinity, or equivalently as the dimension of the system goes to zero.

The converse situation (infrared slavery) is also true--the coupling constant becomes very strong at small momenta or large distances. It then becomes more favorable for quarks to bind together into color-singlet bound states, rather than to separate into distinct colored objects. The binding effect might become so strong that quarks can exist only in such bound states but never separately, at least at zero temperature (Gross, Pisarski, and Yaffe, 1981). If this should happen, the quarks are said to be permanently confined (Kogut, 1979; Callan, Dashen, and Gross, 1979; Creutz, 1980).

The confinement of quarks is supposed to take place rather abruptly in the manner of a phase transition as the gluon-quark-quark color coupling constant g2 increases through a value of = 2 (Creutz, 1980). Since this coupling increases with distance, this leads to the simple picture that quarks interact increasingly strongly up to a certain distance; beyond that they cannot go because of confinement.

Quark Effects in Nuclear Physics 225

The mechanism of confinement is usually associated with gluons rather than quarks. The outside forbidden region is a state of zero color flux. When quarks interact, color flux exists between them. For strong coupling, the most favorable situation is a narrow flux tube of the shortest length, which is the distance r between the interacting quarks. As a result, the interaction energy grows linearly with r:

Vconf(rl2) = E1 " ~2 krl2 (i)

The confinement potential between quarks is expected to vanish rather abruptly as the coupling constant decreases below the critical value mentioned earlier. The flux between charges then spreads out suddenly into the well-known dipole pattern. The quark-quark interaction may then be described perturbatively, the leading term being the one-gluon exchange potential (OGEP):

VoGEp(rl2,Pl2) = ~i" ~2 (g2/4)I~i12 + fBF(rl2,Pl2 )] , (2)

where ~BF is the Breit-Fermi relativistic correction to the Coulomb potential, valid for quarks with large masses.

Thus much has been learned about QCD. It seems timely to consider its signifi- cance in nucleon structure and in nuclear properties. The basic problem is that QCD is not sufficiently simple, and still not sufficiently well understood to provide detailed guidance. We are forced to use phenomenological models of quark dynamics to extrapolate from known hadronic systems. The educated guesses so obtained on hadronic properties are not necessarily correct, but they might serve as signposts into the unknown.

However, different phenomenological models often make quite different assumptions on (i) quark and gluon confinements, (2) residual quark-quark interactions, and (3) the treatment of kinematics. Such variability raises the question as to whether the signposts they provide do point along the same general direction, or whether they simply add to the confusion of the explorer.

In these lectures, I would like to elucidate the connection between model assumptions and model results. This will help us appreciate how much our understanding of the fundamental theory must be improved in order to achieve some confidence in our description of QCD effects in nuclear properties.

For convenience of presentation, I shall discuss separately examples of nuclear properties connected with (i) valence quarks, (2) sea quarks, and (3) valence anti-quarks. A rather naive view of the quark model is given here in the hope that some simple insights into the properties of hadrons and nuclei can be gained without too much effort. In particular, I have avoided any discussion of effects arising from (i) the so-called PCAC (for the partial conservation of the axial vector current) phase of QCD, (ii) the possibility that not only the light mesons of importance in nuclear dynamics, but also all light baryons including the nucleons themselves are collective excitations of a color singlet condensate, and (iii) gluons in hadrons.

In writing these lectures, I refer with unusual frequency to the work in progress at UCLA with which I am naturally most familiar. This appears permissible for the purpose of illuminating my personal view on the problems associated with the use of simple phenomenological quark models. It will not do in an objective review of the state-of-the-art. This serious job is not attempted here. However, the dis- cerning reader should have no trouble forming a more balanced perspective by reading other accounts of localized research activities, including some which might have been presented in this School.

226 Chun Wa Wong

PHENOMENOLOGICAL QUARK MODELS

Two types of phenomenological quark models have been used frequently in discuss- ing quark effects in nuclear physics--the MIT (Massachusetts Institute of Tech- nology) bag model and potential models.

In the MIT bag model (De Grand and others, 1975; Johnson, 1975, 1977), quarks and gluons are confined to within the hadron bag by boundary conditions imposed on the bag surface. Inside the bag, quarks interact relatively weakly and presum- ably perturbatively. The pressure generated by the quarks and gluons inside the bag is counterbalanced by an assumed constant pressure B provided by the bag surface. This pressure B may also be interpreted as the positive difference in energy density between the perturbative vacuum inside the bag and the true vacuum outside. Thus the energy of a spherical bag of radius R may be written in the form

47 C (3) E(R) = -~ R3B + ~ ,

where C contains the contributions from the confined quarks and gluons, including their interaction energies.

The u,d quarks of special interest to nuclear physics turn out to be essentially massless. (These quarks, together with the electron and the electronic neutrino, form the first "generation" of elementary particles. In this language, nuclear physics is the physics of first-generation quarks and leptons.) The expression for C then simplifies considerably. In particular, for n massless quarks in is spatial orbitals, it has the simple form

C = Cn(S,T) = n~01s - Zo + 0.177 ~s an" (S,T) (4)

for a bag of intrinsic spin S and isospin T. Here Wls = 2.04 is the dimensionless eigenenergy of a is quark orbit, Z0 (= 1.84) is the adjusted zero-point energy of gluons, ~ = g2/4 (= 0.55) is the coupling constant (needed to fit hadron mass-

s es) , and

a M(S,T) 4 n = ~ [n(n-6) + S(S+I) + 3T(T+I)] (5)

is the weight of the reduced matrix element (of value 0.177) arising from the color magnetic interaction (proportional to ~.~.').a.) between quarks in Is orbits.

The MIT bag model is basically a shell model. The c.m. of the system oscillates inside the bag. Corrections for c.m. motion must in general be made on hadronic properties. Some of these corrections are obviously important. In the mean- square radius, for example, they are expected to be of the order n -I. This is huge, since n is only 2 for mesons and only 3 for baryons. Unfortunately, c.m. corrections are not easy to make accurately, because the quark motion is relativistic. In any case, they require a refitting of bag parameters to hadron masses. In one such attempt (Wong and Liu, 1978), the end result is to shift roughly 1.7 dimensionless units of the kinetic energy appearing in Eq. (4) to the -Z0 term. One advantage of this shift is that the new value of -Z0 (= -0.15) is much closer to the theoretical value (Milton, 1980) of -Z0 = 0.51.

For many qualitative discussions, the following rough treatment of the c.m. motion is often adequate. The total kinetic energy term for massless quarks is simply reduced by a factor l-i/n since the c.m. motion represents one out of n degrees of freedom. No refitting of the bag model parameters is then needed since we can simply shift the approximate c.m. kinetic energy of ~is/R for S-wave hadrons to the zero-point energy terms; the latter is then just


Zo' = Zo - mls = - 0.2 (6)

The second type of useful models consists of potential models (Kokkedee, 1969; Dalitz, 1970a, 1970b). They are characterized by the simultaneous use of a confinement potential for quark confinement, and a one-gluon exchange potential for mass splitting:

v = ~ [Vconf(rij) + )] (7) i<j VoGEP(!ij' ~ij

Typically, no provision is made to distinguish between strong and weak couplings. The color magnetic term in VOGEP is not identical to that in the MIT bag model, but it has roughly the same effect, since it is fitted to the same hadron mass splittings.

Such potential models have been used quite successfully to describe the heavy- quark systems of charmonium and upsilon mesons (Gottfried, 1981). The quarks in these systems are sufficiently heavy to permit the use of a NR kinetic energy operator. Even so, relativistic corrections to the kinetic energies are not completely negligible.

The quarks in the light hadrons of interest to nuclear physics are practically massless. Their kinetic energies are ultra relativistic as a result. In spite of this difference, the excitation spectra of light mesons bear an uncanny resem- blance to those of heavy mesons containing massive quarks, as shown in Fig. 1. It is then perhaps not unexpected that this similarity might extend to other hadronic

1500 systems.

co

i

<3

MeV

(3F4) h

(3D 3) ~ g w L?

~," ?7

(3p2) A2 K W~ f f ' - - X z

. . . . K ? . . . . Xo

(3P O) 8 s "x-

P ~ K ~" @ J/V ~ T 773 783 892 t020 3098 9400 [:t I:O U~ S~ C~ O~

Fig. i. Excitation spectra of spin triplet mesons above the n = 1 3SI-3DI states. Thick lines denote 3SI-3DI mesons while broken lines give the centers of mass of 3pj mesons. Figure is from Liu and Wong (1978).

One is therefore tempted to use a NR potential model also for the light hadrons. The choice of quark masses then becomes rather arbitrary, and the resulting kinetic energies may not be realistic. One can, however, fit such a NR potential to the total excitation energies of simple hadrons, and use it as a procedure for mass extrapolation to more complicated hadronic systems. Used this way, it is rather similar in spirit to algebraic mass formulas, but is perhaps more inclusive and flexible. It could give us some simple insights into the nature of multiquark systems under favorable circumstances.

228 Chun Wa Wong

Finally we note that the confinement potential between heavy quarks is expected to be independent of flavor. The situation is not so clear for practically massless quarks for which the potential concept itself is not so well defined. In any case, a simulation of light hadrons by a NR potential model may destroy the flavor independence of the fundamental interaction, because potential and kinetic energies are in general mixed up to some extent. The important thing is to learn what we can by using the model in a sensible way.

VALENCE QUARKS

If each nucleon in the nucleus is made up mostly of three quarks, called valence quarks, they could play important roles in many nuclear properties. One of the most interesting properties is that of nuclear forces, since they control the dynamical properties of nuclear systems. I would like to suggest below that the quark contributions to nuclear forces outside those arising from the exchanges of mesons are rather dependent on model assumptions on quark and gluon confinements and on the strength of the residual quark-quark interaction.

It is useful to recall first that in the meson-exchange theory of nuclear forces it is most convenient to go from large to small distances in the nucleon separations. This is because the strength and complexity of the interactions, including the number of exchange virtual mesons, increase as the distances decrease. The situation is just the opposite in QCD. Here the fundamental interaction is weak- est at the shortest distances. It is now more convenient to go from small to large distances. For this reason, we first discuss nuclear forces at the origin of scattering coordinates, with all the nucleons superposed completely on top of one another.

Nuclear Forces at the Origin of Scattering Coordinates

A naive picture of the A-nucleon poential v A at the origin of scattering coordinates can be given in three simple steps (Wong, 1981a, 1981c). In the first step, A nucleons are superposed on top of each other with no change in radius, and with the quarks in the (is) n orbital configurations. Here n = 3A is the number of valence quarks. This will be called a sudden approximation. In the second step, the multiquark system will be allowed to expand to an optimal radius. It will gain some energy, and give us an adiabatic approximation. Finally, we must allow for the fact that colliding nucleons can cause quark excitations. The resulting contributions are then said to be due to >onfisuration mixing.

Sudden approximation. The sudden approximation turns out to be very simple in those potential descriptions of quark dynamics which have the form

V = ~ [li-~2jf(rij) + ~2ioi.~jdj fM(rij)] (8) i<j

Here f(r) contains both a confinement term and a color Coulombic term, where fM(r) is called a color-magnetic interaction. In a color singlet (ic) state which is also completely symmetric in space (i.e., having the Young symmetry [n]x) , the operators in Eq. (8) have nhe simple expectation values

< Z ~ i ' ~ J > ! c - 8 i<j 3 n , (9)

<- ~ ~.O.'~.d.>r i = anM(S,T) (i0) i<j ~ z ~ l ~3~3 [nJ x


m Since only terms proportional to n contribute to m-body forces, the terms linear in n do not contribute to nuclear forces at all. The terms proportional to n 2, and terms dependent on S and T, give rise to the two-nucleon contribution of

V~ udden = v2M~A2 (Ii)

of color magnetic origin. Here

v2M = g2[a6M(s,T ) - 2a3M(s,T)] I M , (12)

where I M is an appropriate radial integral. It turns out that the S,T-dependent terms do not contribute to many-nucleon forces under normal circumstances. Hence Eq. (ii) gives the entire contribution from these potential models in the sudden approximation.

Equation (12) can be simplified by eliminating g2 in favor of the A-N mass difference, as is usually done in phenomenological quark models:

V2 M = b(S,T)(MA_MN ) = - { 340 3SI state 440 MeV in the iS0 (13)

Here we have used the constants b(l,0) = 7/6 and b(0,1) = 3/2. This approximation is not necessarily very accurate, since the contributions of virtual pions to MA- ~ have not been allowed for, but it might serve as a starting point for more realistic discussions.

The same color magnetic contribution appears in the MIT bag model also. Other important contributions arise from special features of the bag model. One such feature is associated with the fact that the terms involving the bag pressure B and the zero-point energy Z0 in the bag energy of Eqs. (3) and (4) are both independent of n. As a result, they appear only once in A superposed nucleons in the same bag, but A times in A separated nucleons. This immediately gives rise to many-nucleon potentials, which have particularly simple forms in the sudden approximation because all bag radii are constrained to the nucleon value. These are found by simple counting to be

VA(B, Zo) = (_)A+I BR 3 - , (14)

where (_)A comes from the x A term in the binomial expansion of (l+x) A for x = -i.

A less obvious, but nevertheless important, source of many-body potentials resides in the kinetic energy (K.E.) term nwl of Eq. (4). This term contains contributions from the A-I scattering degrees ±s of freedom. These scattering K.E. may appear as parts of the scattering energy, or as K.E. operators in the many-body scattering equation, but they should not appear in the potential energy. Each of these K.E. in the scattering coordinates can be estimated very roughly by counting degrees of freedom as ~is/R. Their exclusion from the potential energy gives rise to the many-body potential

VA(K.E.) = (_)A+I ~is/R (15)

If one should subtract out the K.E. in the scattering degrees of freedom, shouldn't one also subtract the K.E. of center-of-mass (c.m.) motion of all the quarks in the bag? The answer is that this does not matter.

To see this point, it is convenient to use the rough approximation for c.m. correction described by Eq. (6), i.e., by replacing Z0 with Z0' = Z0-Wls. The K.E. of the nonscattering coordinates is now proportional to 2A. It is therefore cancelled out exactly by the internal K.E. of A separated nucleons. This means

230 Chun Wa Wong

that the correction of Eq. (15) does not appear at all in this treatment.

The many-nucleon potential is now given by Eq. (14), but with Z0 replaced by Z0' However, this is just the sum of the original Eqs. (14) and (15). Hence the two treatments are essentially identical, and the c.m. correction is already taken care of.

Thus the bag model gives the many-nucleon potential

VA = (_)A+I I~__~ BR 3 - (Zo_~01s)/R) + v2M6A 2 (16)

in the sudden approximation. In this expression, the B term is exactly Mp/4 = 235 MeV in strength, while the -(Z0-~01s)/R term is only 40 MeV.

Adiabatic approximation. A multiquark system like that giving rise to V A may expand to an optimal size when allowed to do so. This happens when the colli- sion time is long compared to the relaxation time of system. Since this adiaba- ticity condition is most likely satisfied at low energies, the associated adiabatic lowering of energy may be interpreted as an energy-dependent potential.

This adiabatic lowering of energy is given roughly by the usual expression

AE = -(E')2/K , (17)

and is controlled by two factors. One is the compression modulus

K = H2E/dR 2 (18)

A

in the dimensionless radius R. It turns out to be roughly 3 GeV/A or 1 GeV/quark, for both potential and bag models.

The second factor is the energy slope

E' = dE/dR , (19)

which describes how far the system is from equilibrium. This turns out to be quite model-dependent.

In the simple NR potentials described by Eq. (8), the %..~. terms give contributions proportional to A, like that from the K.E. of ~l ~j non-scattering coordinates. If these are the only contributions, the (is) n multiquark systems for V A will all saturate at the same size, but of course at different densities. This turns out to be exactly the case for Liberman's potential (1977) for which the color-magnetic potential does not affect the saturation problem because its radial integral is a simple constant. More generally, the color magnetic energy, which gets increasingly repulsive as A decreases, provides a mechanism for adiabatic expansion in overlapping nucleons. This is indeed seen for the NR potential of Liu and Wong (1980), although the effect is not very strong because the color- magnetic interaction used has a rather long range.

The situation is quite different in the MIT bag model. The outward pressure of quarks and gluons inside a bag of overlapping nucleons increases with A, but the bag pressure which contains the system remains independent of A. The bag therefore blows up like (actually more easily than) a common balloon until the outward pressure has been sufficiently reduced. As is known, equilibrium will be re- established when the average mass density inside the bag is reduced to the universal value of 4B. This means, for example, that the Z = 2 bag for VA, which has a


mass of = 2M, will have roughly twice the nucleon volume.

When this happens, the B contribution to V A shown in Eq. (14) will be lost. In its place we find a great reduction in the energies of quarks and gluons inside the bag due to the increase in its radius. The large adiabatic expansion which is taking place ensures that there is a very significant adiabatic decrease in energy. The actual effects of this large adiabatic expansion on V A are shown in Table i. We see that the two-body repulsion of the sudden approximation has almost entirely disappeared. Its effect on many-nucleon potentials is to reduce their values by roughly one half.

TABLE 1 Estimates of the A-Body Potential V A (in MeV) in the Sudden and Adiabatic Approximations for the Energies of Overlapping Nucleons

Sudden Approximation

V 2

Adiabatic

I Potential Models MIT Bag Model Liu-Wong Liberman

i 0 370 350 90 0 1 470 450 190

i 1 V 3 ~ ~ 0 280

V 4 0 0 0 -280

Approximation

V 2 1 0 340 350 -30 0 i 430 450 50

I i V 3 ~ ~ -90 0 140

V 4 0 0 20 0 -150

It is thus clear from our discussion and from Table 1 that quark contributions to nuclear forces are rather dependent on the confinement mechanism, and on the details of the quark-quark color-magnetic interaction.

Configuration mixing. Colliding nucleons will also excite and polarize one another as they collide. Such configuration-mixing effects have been studied to some extent in the literature.

Obukhovsky and others (1979) have found that for A = 2, the (is) 6 shell model bag state appears at 2.17 GeV, while the (is)~(ip) 2 bag state appears at 2.38 GeV. For the Liberman potential, the (is)4(ip) 2 state appears at 2.16 GeV, even below the (is) 6 state now at 2.38 GeV. Thus the (is)4(ip) 2 state appears much lower in mass than expected when compared to the K.E. excitation of 0.4-0.6 GeV. This mass reduction is caused by a strong color magnetic attraction in the orbitally excited state.

A rather similar situation is found by Harvey (1980) in his calculation of the two-nucleon potential from NR potential models of quark dyanmics. He obtains a repulsive potential at r = 0 of 0.3 GeV (or 2.2 GeV in total mass) in both the (is) 6 and the (is) 4(ip) 2 channels. Configuration mixing, i.e., channel coupling, reduces the potential to roughly zero. Unlike Obukhovsky and others, Harvey finds that his effect comes from the color confinement potential rather than the color- magnetic interaction.

232 Chun Wa Wong

Thus the effects of configuration mixing are also model dependent. They can be very large, and appear to have a variety of dynamical origins. Let us try to un- derstand the situation.

Color Instabilities

I would like to suggest that these large configuration mixing effects might be precursory to possible collapses of quark matter via color correlations in unusual configurations (Wong, 1981d). Indeed under certain circumstances to be discussed below, the color magnetic (CM) energy turns attractive, or simply vanishes. This makes it possible for quark matter to collapse. This quark matter is an abnormal Fermi gas in which the single quark orbital states are occupied singly rather than with its full color-spin-flavor degeneracy. To put ~he matter into proper perspective, I must also point out that no such collapse has been seen experimentally. Thus the conditions for collapse are not normally satisfied.

Color masnetic instability. The physical origin of the effects described below lies in the extraordinary sensitivity of the CM potential energy to the permutation symmetry of the wave function. This can be seen readily by examining the permutation operator of a quark pair

p.. = p. C p S p. I P x = -i (20) ij ij ij lj lj

where the flavor dependence of the first-generation quarks (u, d) is expressed by the isospin I. The other degrees of freedom are color (C), spin (S), and space (x). By writing Eq. (20) as

p C p. . . . . X = -P I p. x , ij 13 13 13

we find the useful expression (Johnson, 1975)

2 2 .. (21) -)[i~i'~j~j~ ~ ~ = ~ + . . . . . li'~ j + ~ Oi'O j + 2(1 + T.I'T,)~j PI3x

Equation (21) shows that the expectation value of the CM operator can be evaluated easily in two special cases: (i) when all pairs are spatially symmetric, i.e.,

x p..X = i, and (ii) when all pairs are spatially antisymmetric so that P.. = -i. lj 13

To get p..X = 1 for all pairs, the spatial wave function must be completely symmetric w~h the Young symmetry [n]$. The matrix element of the CM operator is then given by the familiar expresslon

j!j>[n]x 4 [n(n-6) + S(S+I) + 3T(T+I)] (22) < -

i< j

T h i s o p e r a t o r i s r e s p o n s i b l e f o r t h e ~-N mass d i f f e r e n c e i n t h e ( l s ) 3 c o n f i g u r a - t i o n . Hence the associated radial integral is positive. This shows that the CM potential energy in (Is) n turns repulsive for n ~ 6. Such a repulsion prevents the system from collapse.

To get p. x = -i for all pairs, the spatial wave function must be completely anti- .13 [in]x" symmetrlc with the Young symmetry The CM matrix element is then

<_Z~.o..ho> 4 in2 ~l~i ~j~j n = 3 [- ~ + s(s+l) - 3T(T+I)] (23)

i<j [i ]x

The CST wave function must now be completely symmetric. Since the color wave function has the Young symmetry [A3]c, the spin-isospin wave function must also


have the Young symmetry [A3]s T. The supermultiplet structure of [A3]sT is identical to that of [A]sT, i.e., with S = T and A/2 > T > 0 or 1/2. As a result

2 2 _ 4__ 27 n(5n+3) _< <- ~ ~i~i'~jOj> ~ - ~ n (24)

i<j

This gives rise to an attractive CM potential energy if the associated average radial integral I M remains positive. However, there is no good reason why this should be so. Hence the situation is model dependent.

In Liberman's potential model, the radial integral I M is simply a positive constant. CM collapses are then unavoidable. Since no such collapses have been seen experimentally, we may conclude that this feature of the Liberman potential is un- realistic and should not be taken literally.

As another example, we note that the CM interaction in the Breit-Fermi potential (Bethe and Salpeter, 1957) is a 6-function. This is an S-state interaction which does not contribute at all in the antisymmetrie spatial pair states in [in] x. The CM potential energy is then identically zero in [i n] . Since there is no CM energy, there is automatically no CM collapse, x

The absence of a CM energy also means that there is no CM repulsion either. Other possibilities of color collapses might then appear, e.g., from the attractive color Coulombie exchange energy.

Color..electrqstatic instabilit Y . In potential models of quark dynamics, there is a confinement potential in addition to the color Coulombic interaction. Though normally repulsive, it is not necessarily unfavorable to collapses. The reason is that in the color wave function of Young symmetry [A3]c, there are

n 3 = n(n+3)/6 pairs of color states. (25)

n 6 = n(n-3)/3

Hence

8 4 < ~ l i'lj f(rij)> : - ~ n313 + ~ n616 , (26) i<j

where 13,6 are the average radial integrals. In the absence of space-color correlations I3 = I6 = I, so that the confinement energy is -(8/3)ni. This is of course repulsive (because I is negative). However, if space-color correlations are present such that Is < 13, we find immediately the attractive contribution of (4/9)nZ(I6-13). One possible scenario for such space-color correlations occurs when the color 6 pairs are further apart, while the color 3 pairs are closer together. This might appear in a finite multiquark system when there is surface color polarization. This very qualitative discussion does not make clear, however, if the inequality 16 < I3 can be preserved on a macroscopic scale in the quark- matter limit to permit a collapse through the confinement potential alone.

The situation is actually more speculative and uncertain than the above discussion might suggest. We do not really know what the confinement potential might be like beween color 6 objects because no particles made up of these objects have been identified experimentally. What is clear is that the confinement energy must be suitably reduced if collapse due to the exchange color Coulomb energy is to appear.

The model dependence of the situation can readily be appreciated by recalling that

234 Chun Wa Wong

the confinement energy in the MIT bag model is described not by an expression like Eq. (26), but by the bag pressure term (4~Ra/3)B, which is independent of n for constant R. As a result, a color Coulombic instability is more likely to develop.

Indeed, if the CM energy vanishes, the energy density of quark matter to order = g2 is

s

E/V = B + (kF4/8~2)[i - (2/3~)as(l+a)] , (27) where

k = (6~2n/V) I/3 (28) F

is the Fermi momentum of the abnormal quark gas of singly occupied spatial orbits. The parameter

a = 2 ~n 2 (29)

appearing in Eq. (27) represents a relativistic correction (Chin, 1981) to the usual attractive exchange energy of a Coulomb gas (Fetters and Walecka, 1971). Thus a color electrostatic collapse appears in the model when ~ exceeds the criti cal value of s

erit 3~ = = 2 (30)

s 2 (l+a)

This is dangerously close to the empirical value of a = 2.2 deduced from the &-N s

mass difference in the MIT bag model.

it must be emphasized that color collapses like these may not actually occur even when they are present in a model. For one thing, we do not know for sure if the CM energy indeed vanishes. There is also no reason to trust a theory only linear in ~ when something drastic is about to happen. The neglected terms of higher orderSin g might change the situation here more than their effects might be in the potential energies of normal configurations like those of observed hadrons used to fit the parameters of the bag model.

What our discussion seems to suggest is that there might be strong color correlations even in finite multiquark systems, but that estimates of these effects are likely to be model-dependent and uncertain.

Nuclear Forces at Finite Distances

Three aspects of two-nucleon forces at finite distances will be discussed below. As usual, we shall avoid formalisms where possible, and concentrate instead on the conceptual features.

Antisymmetrization and nonlocality. Before antisymmetrization, the quark spatial configuration for two nucleons at finite separations may be characterized as L3R 3, where L(R) refers to a localization to the left (right) of the c.m. In the resonating-group method (Wheeler, 1937), L 3 has its center at (say) -r/2, where r is the relative coordinate, while in the generator-coordinate method "(Hill and ~ Wheeler, 1953; Griffin and Wheeler, 1957) L is a is spatial orbital centered at an average distance -x/2. As another example, DeTar (1978) uses in his bag model calculation the second quantized operators

= b t g ~ - b t t = b j b t bL s - p ' bR s + ~ p '

where s (p) denotes the Sl/2 (P3/2) single-quark state centered at the origin.

(31)


(This leads to a deformed wave function which is possibly less accurate than the usual cluster wave function, especially at large separations.)

The wave function must also be antisymmetrized, thus giving rise to both direct and exchange contributions. The direct potential is local in the relative coordinate; it turns out to be exactly zero in the present situation because each nucleon is a color singlet. (Indeed, the confinement potential if taken literally as a two-body potential might give rise to a strong long-range nuclear force if it does not have the %.'%. form.)

~i ~j

This leaves the exchange potential, which is known (Wheeler, 1937) to be nonlocal. To see this nonlocality, we first note that a Pauli exchange is best visualized as an exchange of particle labels, leaving all dynamical variables unchanged. For example, a quark in a left nucleon with the dynamical variables (x, C, S, I) = (L, ~, +, u) when exchanged with a quark in the right nucleon with the variables (R, 6, +, d) will simply assume the latter's identity, i.e., (R, ~, +, d). How- ever, it still considers itself part of the left nucleon. Hence there is a change in the numerical value of the relative coordinate between the bra and the ket in the Dirac bracket expression which defines a contribution to the exchange potential.

Furthermore, its new color $ might differ from its original color ~. Hence the left nucleon after exchange might be in a color-octet state, thus giving rise to the so-called hidden-color channels in nucleon-nucleon scattering.

Many calculations of quark contributions to the two-nucleon potential give only an average local potential (e.g., Liberman, 1977; DeTar, 1978, 1979; Warke and Shanker, 1980; Harvey, 1981). Others have discussed or calculated the fully nonlocal potential (Robson, 1978; Ribeiro, 1980; Oka and Yazaki, 1980; Wang and He, 1980).

Configuration mixing. Configuration-mixing effects have been considered by DeTar (1978, 1979) and by Harvey (1981) in the local picture mentioned earlier. Since different models of quark dynamics are used, their results are quite instructive even though the nonlocality has been averaged out. Figure 2 gives sketches of their 3S I results, together with those of Liberman (1977) and of Wong (1981c) obtained with no configuration mixing.

The difference between the curves of Harvey and Liberman describes the effect of configuration mixing at finite distances in the potential model of quark dynamics. We see that the effect decreases in a rather smooth and monotonic manner. This is perhaps what we might intuitively expect.

The results obtained by DeTar in the MIT bag model appear very different from the others. There is a strong central repulsion, which gives way rapidly to a strong attraction at 0.7 fm. The central repulsion differs from the result obtained in a previous section, because DeTar has neglected to make the K.E. correction described by Eq. (15). The strong attraction is due partially to configuration mixing from the p-state admixtures in Eq. (31). However, its effect seems to be unusually strong. K. F. Liu (unpublished) has suggested that this may be due to the possibllity" that the P^~2J/ eigenvalue in the MIT bag model is just too low. One likely mechanism is that these single quark states have self energies due to the emission and absorption of virtual gluons (Chin, Kerman, and Yang, 1981). It would be interesting to see what the improved result might look like.

I have not discussed the situation in the "little bag" model (Brown and Rho, 1979; Brown, Rho, and Vento, 1979), since the presence of the pion cloud outside the bag

236 Chun Wa Wong

Fig. 2. Comparison of quark contributions to the two- nucleon potential at various average nucleon separations r. Solid curvescontain configuration-mixing effects. The broken curve does not. Underlined authors use the MIT bag model; the others use potential models.

4oo x ~_.~.~ Wong

200~ \\\\k Won \ LibermQn

J I ~ adiobatic ~

-200 ~ k / M : I ~

makes the situation more complex. The reader should read the lecture notes of Brown (1981) and of Rho (1981) given in this School to find out where the matter stands.

Color Van der Waals force. In charged systems, the effects of configuration mixing persist at very large distances in the form of a Van der Waals (VdW) force of induced dipole-induced dipole interaction. This possibility arises from the fact that the Coulomb interaction is long-ranged and that the virtual photons are spread out all over space in the familiar dipole pattern.

Very much the same effect appears between nucleons from the color Coulombic interaction. There is in addition a stronger contribution from the confinement potential. The situation has been clarified by several authors (Matsuyama and Miyazawa, 1979; Gavela and others, 1979). It is useful to give a brief account of their results.

A color VdW force is here defined as the second-order potential

<lllqB188> <88rVABrll> V2(VdW ) : - (32)

E88(r) - Ell(r)

at very large separations. Here 188> is the hidden-color channel obtained from the original NN channel Iii> through one-gluon exchange, and VAB is the sum of in- ternucleon quark-quark interactions v... At large distances, the matrix element

13

<881VABIII> = r "r. ! d v (33) ~i ~] r dr

has the usual induced dipole form. Equation (33) is interesting because it shows


that the configuration (is)~(ip) 2 dominates configuration mixing even at large distances.

For a nonconfining power potential r -m (m > 0), <88 V _Iii> is proportional to A~'

r -m-2, while the energy denominator is dominated by the constant difference in kinetic energies. As a result, V2(VdW) is proportional to r -2m-4. For example, for the Coulomb interaction (m = i) a long-range r -6 potential results.

For a confining power potential r n (n > 0) which is color-dependent, the energy denominator is dominated by the difference in potential energy, which is proportional to r n. Hence V2(VdW) is proportional to r n-~. This works out to be r -3 for the familiar linear potential of quark confinement, and r -I for the r -3 potential from gluon condensation.

It should be noted that if the confinement potential is color-independent, it will not contribute substantially to the energy denominator. An even stronger VdW force (~ r 2n-~) then results.

The color VdW force in NN scattering has been estimated (Matsuyama and Miyazama, 1979) to be

VeVdW(r) = - 28 MeV/r 3 (fm) (34)

for a linear confinement potential. This is comparable to the one-pion-exchange potential at nuclear distances. It is larger than the Coulombic potential for r < 4.5 fm, and larger than the gravitational potential for r ~ 1 km.

The presence of this long-range potential appears to be incompatible with experimental data in NN scattering and in the Cavendish experiment (Feinberg and Sucher, 1979; Matsuyama and Miyazawa, 1979; Gavela and others, 1979).

This discrepancy suggests that the model used in the discussion is not quite realistic. First of all, a second-order calculation may not be adequate for the strong potential at large separations. The confinement potential itself may not be made up entirely of pair interactions; it may have significant, and even dominant, many-body contributions in multiquark systems. These difficulties may still be present even if the confinement potential were color independent.

There is probably a simpler and more basic reason for the difficulty. Confinement is supposed to be a property of gluons, which differ from the photons of QED in that they do interact strongly with one another. As a result, the virtual gluons around each hadron do not spread out all over space, but are also confined to the hadron volume. Thus they cannot jump between two well-separated hadrons under normal circumstances. This effect of gluon confinement is not included in the potential model of Eq. (8).

Gluon confinement is already built into the MIT bag model. As a result, the color VdW force disappears entirely as soon as the nucleon bags separate completely from each other. This cutoff effect can probably be simulated in potential models by simply multiplying the naive result such as Eq. (34) by a cutoff factor which is related to the overlap between the two interacting nucleons.

Multiquark State__~s

Quark dynamics might also be accessible through the masses of multiquark shell- model states. The calculated masses are shown in Table 2 for the MIT bag model with c.m. correction (Wong and Liu, 1978), and for two NR potential models using Gaussian variational wave functions. We see that both potential models give

238 Chun Wa Wong

roughly the same masses, and that these are substantially higher than the corre- sponding masses of MIT bags. This seems to suggest that the difference might arise from some simple defects of NR kinematics.

TABLE 2 Calculated masses (in GeV) of multiquark shell-model states. The masses in parentheses have been used to fit model parameters.

Model MI_ T Bag Model Potential Models Liberman Liu-Wong

c.m. correction (original) Yes Yes Yes

n S T

3 1/2 1/2 (0.93) (0.94) (0.94) (0.95) 3/2 3/2 (1.23) (1.23) (1.24) 1.23

6 1 0 2.15 2.18 2.38 2.54 0 1 2.23 2.25 2.48 2.61 2 1 } 3 0 2 .34 2 . 3 4 2 .63 2 . 7 2

1 2 2 . 5 0 2 .47 2 . 8 3 2 . 8 5 0 3 2 .79 2 .71 3 . 2 2 3 . 0 8

9 1/2 1/2 3.50 3.46 4.31 4.38 3/2 3/2 3.70 3.50 4.61 4.56

12 0 0 4.90 4.73 6.59 6.35

A more careful examination of the situation reveals a rather different picture. Consider first the MIT bag model. If the c.m. correction is treated by the approximation of Eq. (6), the equilibrium bag mass has the form (Johnson, 1975) of the original MIT bag model

M n = MN[Cn(S,T)/C3(I/2, 1/2)] 3/4 (35)

The C parameters are exactly those of Eq. (4), but with the interpretation that Z0' of Eq. (6), not Z0, is the zero-point energy constant. We can easily verify that it is primarily the repulsive color-magnetic potential energy which drives up the bag masses faster than the nucleon number A, although the increase in the internal K.E. is also a contributing factor.

The Liberman (1977) model is the next simplest, because equilibrium is determined by just the K.E. (proportional to R -2, where R is a size parameter) and the quadratic confinement potential (proportional to R2). At equilibrium each term contributes equally, leading to the equilibrium mass

M = 2~[3n(n-l)] I/2 + nmc 2 + £~(S,T) , (36) n n

where the CM potential energy

AMM(s,T) = BaM(S,T) , (37) n n

is expressible in terms of the operator matrix element shown in Eq. (5). Equation (36) can be simplified by eliminating the constants ~ and 8 in favor of the experimental nucleon mass MN, and the model CM attraction

The result is


A~ = A~ 3 (2 ' i] =- 0.15 GeV (38)

Mn = AM N - A~ [A + aMn(S,T)/8]

+ A(M N - A~ - 3 mc 2) {[(3 - I/A)/2] I/2 - i} (39)

The quark mass m of the Liberman model is 0.15 GeV, so that the last term gives a positive contribution which does not exceed about 0.14 A GeV. [Equation (39) can also be used with other values of m, but it is clear that the m dependence is weak. The reason is that the other parameters must be readjusted in every case to give the same nucleon mass.]

It might appear from Eq. (39) that the additional mass increase with n comes primarily from the second, or CM, term. This is not the entire story, since it accounts for only 0.7 GeV of the 1.7 GeV difference for n = 12, as shown in Table 3. The rest is distributed among the K.E., quark mass and confinement potential terms, depending on the choice of m. For m = 0.15 GeV, the confinement term is larger than the MIT bag result (of 1.20 GeV for n = 12) by 0.3 GeV, while the K.E. plus rest masses account for the remaining 0.7 GeV.

TABLE 3 Selected contributions (in GeV) to the calculated masses of mu!tiquark shell-model states.

Original Model MIT bag Liberman Liu-Wong

n S T K.E. AMMn K.E. ~M Mn K.E. Mconf ~MMn

3 1/2 1/2 0.82 -0.16 0.32 -0.15 0.78 0.20 -0.16

6 i 0 1.55 0.04 0.71 0.05 1.35 0.91 0.04

9 i/2 1/2 2.12 0.51 i.i0 0.75 1.71 1.85 0.45

12 0 0 2.63 1.09 1.49 1.79 2.01 2.92 0.92

The situation is different in the more realistic Liu-Wong (1980) model. Its CM energies turn out to be surprisingly close to the MIT bag results. Its K.E.'s increase less rapidly with increasing n, because NR kinetic energies cool down more rapidly in expanding systems. To be more comparable with the bag model, we should add the contribution of the effective quark mass

mef f = (m + b/2) = 0.043 GeV (40)

made up of a quark mass of 0.12 GeV and a contribution b = -0.157 GeV from a color-dependent additive constant. [The definition of b is the same here as in the meson potential of Liu and Wong (1980), but its numerical value has been readjusted slightly from -0.08 GeV in order to fit the nucleon mass, as explained in Liu and Wong (unpublished).] The negative constant b reduces the effective quark mass to a rather small value. It is then not surprising that even the relativistic quark K.E.'s of the bag model are roughly reproduced, especially for the larger n's.

The remaining contributions (color Coulombic plus linear confinement terms) are shown in Table 3 under Meonf (for the total confinement energy). In comparison,

240 Chun Wa Wong

the remaining terms in the bag model give

M bag = (4~/3)BR 3 - Z '/R (41) conf o

This works out to be 0.28 GeV for the nucleon and 1.20 GeV for n = 12. Thus the difference lies in this confinement energy.

The confinement energy in the bag model does not depend explicitly on n. Its

value increases with n only because the bag volume has been blown up by the pressure generated mainly by the CM repulsion. In contrast, the ~.'~. color operator in potential models gives an extra factor of n. The assoclated Jradlal integral may also increase when the size parameter R increases. This second increas~ turns out to be rather modest in the Liberman mod~l, because the radial expansion

(RI2/R N = 1.08) is driven only by the increased K.E. In the Liu-Wong model, the system expands considerably more (RI2/R N = 1.46) under a strong CM pressure. A

rapid increase of Mconf appears as a reNult.

It is possible to imagine other potential models in which the effective quark mass is much larger than that used in the Liu-Wong model• Equation (39) suggests, however, that the main effect is a redistribution of masses from the confinement term to the mass term rather than a drastic change in the total mass of the multiquark

state. In a sense the physics has not materially changed, since the effective quark mass itself is an attribute of the confinement mechanism. Indeed, the quark mass term has the same n dependence as the confinement term in the potential model

of Eq. (8)•

Thus we see that in the Liu-Wong model, larger masses are generated by roughly the

same CM pressure pushing n quarks, rather than only one bag surface, up a spheri- cally symmetric "confinement" hill. The interesting question is whether the confinement mechanism acts through these n quarks, or in a collective manner independent of n. The present understanding of QCD seems to favor the MIT picture that

there might be a collective phase transition between the true vacuum outside the hadron and the perturbative vacuum inside (Callan, Dashen, and Gross, 1979; Creutz, 1980)• If this is the case, the confinement term in potential models is not

described correctly in Eq. (8).

It is also profitable to look at the situation from a different direction. If the masses of these multiquark states are somehow known, something can be learned about the confinement mechanism. These states are excited above the nucleon breakup thresholds, like nuclear shell-model states in the continuum. Also like nuclear shell-model states, they can give rise to resonances. They may also appear in scattering phase shifts as primitives (Jaffe and Low, 1979), since they are actually the internal parts of selected scattering states• Unfortunately, it has not been possible to decide if they are present in nucleon-nucleon scattering.

A more promising situation occurs when the strange quark is included, as Jaffe (1977c) has pointed out. This is because the CM operator matrix element, in

flavor SU3 is

hiOi').O.>rnlj = n(n-lO) + (4/3)[S(S+I) + 3C[]j , (42) <-

i<j ~ ~ ~3~J [ x

where C[ is the flavor SU3 Casimir operator. Thus the lowest mass state is always a flavor singlet. The mass of the strangeness S = -2 dihyperon in the MIT bag model turns out to be 2.15 GeV, about 0.i GeV below the AA threshold. Hence this dihyperon may be stable. Attempts have been made to detect this object, but it

has not yet been seen.


SEA QUARKS

To illustrate quark effects associated with quarks and antiquarks in the fermion "sea," I would like to discuss their contributions to (i) nucleon structure and nucleon properties, and (2) meson-baryon coupling constants. Again I shall use the simplest possible quark model. The idea is to try to learn from its failures as well as from its successes.

Nucleon Structure and Nucleon Properties

The empirical coupling constant ~ = g2 = 2.2 determined in the MIT bag model defines an effective quark-quark interaction

2 fl d3r d3r' [i~Y~(~A/2)~]r G(r' r)[i~y~(~A/2)~] r (43) Veff = geff ' ~ '

which gives the potential energy when used only in the lowest-order perturbation theory. It cannot be interpreted as the actual coupling constant, because such a strong coupling constant will have caused a phase transition to the OCD vacuum even inside the bag.

The effective interaction (43) contains both quark-quark interactions and qq crea- tions and annihilations. It has been used by Donoghue and Golowich (1977) to study sea-quark effects on nucleon properties.

Now the exact nucleon wave function may be written formally as

IN> = ]No> + U]N> , (44)

where No denotes the unperturbed q3 configuration, and

U = (E - H + ix) "I V (45) o

is the correlation operator. Since we have only an empirical effective operator Veff, it might appear that Eq. (44) should also be calculated only to order g2 to

give

IN > = INo> + Ii ci]*i > , (46)

where i is a q4~ state and c. is proportional to g2. This procedure is the one used by Donoghue and Golowi~h (DG) who also calculate operator matrix elements

<0> = <NI0[N>/<N]N> (47)

to order ge. Since there is no change in the normalization <NIN> to order gZ, thi means that

<0>DG = <0>o + 2 Re li c i <No]01~i> ' (48)

where <0> ° = <Nol01No > (49)

is the unperturbed value. DG include only the j = 1/2 single-particle states for sea quarks in the i sum, because they are the only states satisfying the MIT boundary condition for quarks on the bag surface. All the results presented here are obtained with only these spatial states. It is likely that other spatial states may also contribute, so the results must be considered very preliminary. We shall show that much can be learned even under this limitation.

Table 4 shows the results <0>o and <0>DG for the axial charge gA' the proton mag-

242 Chun Wa Wong

netic moment Dp, em p

of the axial charge <r 2 >. They are single-particle operators of the form ax

a = f d3r ~*(r) A(a) ~(r) with

A(g A) = ~3T3 , A(~) = ~r×~~ Q

A(<r 2 >) = r20 A(<r~x> ) = r2~3T3 em • '

the proton mean-square radius <r 2 > , and the mean-square radius

(50)

(51)

Q being the charge operator.

TABLE 4 Preliminary results on sea-quark effects on nucleon properties as calculated (i) in the original MIT bag model, (ii) to order g~ (Donoghue and Golowich (1977), (iii) to order g4, and (iv) to all orders of g~ in the correlated wave function of Eq. (4_~. Calculations use ~ = g2 = 2.2, except as noted in the table. The results in parts (iii) and (iv) include onl X u,d flavors in sea quarks_.

<r 2 > <r 2 > ProRerty Psea k(q)/k(q) . . . . gA ~p e m~ ax

Unit % % 1 GeV -I fm 2 fm 2

R-dependence 1 1 1 R R 2 R 2

(i) <0> 0 0 1.09 1.01 0.52 0.48 O

(ii) <0>DG 1.53 1.33 0.64 0.59

(iii) <0> 4 40 >15 1.22 i.i0 0.64 0.49 g

(iv) <0>

~=2.2 22 >ii 1.24 i.ii 0.60 0.50

1.5 16 > 8 1.23 i.ii 0.59 0.51

i.i ii > 5 1.22 i.i0 0.57 0.51

Experimental ii a 1 .25 b 1 .49 b 0 .66 b 0 .67 b

(a) Field and Feynman (1977). (b) Donoghue and Golowich (1977).

Of these four calculated quantities, gA is independent of the bag radius R, ~ is linear in R, while the mean-square radii are quadratic in R. The P R-dependent quantities are of course sensitive to c.m. corrections, and these have not been made on the results of Table 4. In part (ii) of Table 4, we have repeated DG's calculation to order g2, but have increased the i sum from 4 to 5 "blocks" of states. (These blocks are arranged in the order of increasing ener- gier:.) Only <r 2 > still seems to be sensitive to the truncation in block in the i sum, but the ax result is unchanged when the 78 states of the sixth block are included.

Table 4 shows that the sea-quark contribution to <0>DG is huge in ga' large in ~ , moderate in <r 2 > and small in <r 2 > . Some of these effects are so P

ax ' em p large because the excitation parameter for sea quarks is


< = ~i ci 2 ~ 0.86 (52)

This gives a normalized probability of sea-quark configurations of

< - 46% (53) Psea i + <

These results are obtained by summing over the flavors u, d, s, and c of the sea.

If only u and d flavors are included, the results are ~ = 0.66 and Psea ~ 40%. In these quantities, the convergence in the i sum is not quite achieved. The results are probably too small by 5-10%.

A moment's reflection will convince us that these large sea-quark effects cannot be trusted. The very strong effective coupling constant induces so much sea-quark excitations that the normalization <NIN> is not i, but 1.86. It is then clear that it makes no sense to do calculations only to order g2.

H. Borhan and I (work in progress) have studied how sea-quark effects can be treated more realistically within the context of the DG model. We first use the DG wave function, but calculate <0> to order g~. The results, including only contributions from 5 blocks of i states with flavors u and d only for sea quarks are given in Table 4. We see that their contributions have been reduced by 2/3

or more in gA' ~ ' and <r 2 >. It is then not clear if the treatment is really adequate. P ax

Another nucleon property which is also sensitive to sea-quark configurations is the ratio of antiquark momentum k(q) to the quark momentum k(q). We find that the i sum for this momentum ratio, calculated to the fifth block, begins to converge. Thus the result of 15% shown in Table 4 represents a fairly good lower limit for contributions from the flavors u and d in the sea. With the addition of s and c flavors, the value increases to 19%. This lower limit substantially exceeds the experimental value of 11% deduced by Field and Feynman (1977).

These results indicate that there are at least two related difficulties in the model--a perturbative treatment of the wave function in the presence of a strong

potential, and the very strength of the potential itself. The strength of the effective interaction is to a good extent unavoidable if the A-N mass difference is to be fitted. The perturbative treatment is avoidable, however. It should also be avoided, since a perturbative treatment of a strong potential does not make sense anyway.

These considerations suggest that once a model is constructed, the model wave

functions and operator matrix elements should be calculated exactly, rather than to order g2, or even g4.

Such an exact calculation can be done readily in a model which is closely related to that of DG. In their second-order calculations, DG have effectively constructed a model Hamiltonian in the shell-model space of q3 and q4~ configurations which

is made up of non-zero elements only in the first tow, the first column, and the diagonal. Such a matrix can be diagonalized exactly without too much trouble to give wave functions good to all orders of g2.

The preliminary results for this better treatment are shown in Table 4 for three

arbitrarily chosen values of ~. (These results are preliminary because only u and d flavors have been included and because certain contributions have been neglect-

ed.) We find significant reductions in Psea and in the momentum ratio. In particular, the value of ~ needed to fit the experimental momentum ratio appears

244 Chun Wa Wong

to be smaller than that deduced from the A-N mass difference. This might be a desirable feature, since part of MA-~ ~ is expected to come from the coupling to the pion cloud (Cottingham, Tsu, and Richard, 1981).

In an operator matrix element, the change from the unperturbed value <O> must be

linear in ~ for small ~. However, for the strong coupling constants used°here, the remaining four nucleon properties shown in Table 4 have become roughly independent of ~. Their values are roughly the same as those obtained in the simpler g4 model,

so that the latter can be used for a property which is not sensitive to the pre-

cise value of Psea"

With the addition of sea-quark contributions, the baryon mass contains an attractive correlational contribution Acsea(~) to be added to the mass parameter C of

Eqs. (3) and (4). Preliminary results for this correlation parameter are shown in Table 5 for p, A, and ~ using the coupling constant of the original MIT bag model. However, ~cSea(~) also depends on ~ nonlinearly. Thus the new model

parameters should actually be refitted to the experimental hadron masses in a self-constant manner. Unfortunately AC sea turns out to be the least convergent

of all the quantities calculated with respect to the i sum in both blocks and

flavors. In spite of this, it is interesting to see how the model is changed as a result. We therefore show in the second column of Table 5 the results from such a self-consistent model when the calculation contains only 5 blocks of sea-quark configurations involving only u and d flavors.

TABLE 5 The contribution &C sea from sea quarks for two choices of the model coupling constant.

~/4 0.55 0.38

£C sea -3.9 -2.2 P

AC2 ea -4 .3 -2 .4

&C sea -2.7 -1.4 W

The essential feature brought in by sea quarks is not just the presence of correlation energies. It is rather that they decrease the A-N mass difference. Since

the model parameters are fitted as before to the experimental MA-MN, the system must either increase ~ or decrease the bag radii to achieve this, depending on the nature of the third hadron mass fitted (here ~). The solution shown in Table 5 corresponds to reduced radii, with R = 2.6 GeV -I, R~ : 2.9 CeV -I, and R

2.5 GeV -~. This shrinkage is so dominant p that even the addition of the quark self energies of Chin, Kerman, and Yang (1981) does not appear to change

the situation significantly.

Such a large change in the model does not appear very comforting. Before we take the results too literally, however, we must remind ourselves of the limitations of these preliminary results: (i) The i sum in AC sea is not yet convergent both in blocks and especially in flavors. (2) There is no good reason to exclude the many j # 1/2 single-particle states for sea quarks. (3) No c.m. correction has yet been applied to the calculated hadron masses. Thus it is not clear yet if this model will turn out to be useful and physically appealing. (Again we have exclud- ed the pion cloud outside the bag from our model for the sake of simplicity.)

Of the calculated nucleon properties, the axial charge gA is probably least sensitive to the details of the model. It does indeed agree best with experiment.


However, it is very sensitive to the outside pion cloud which increases (gA)o by 50% (Brown, 1981; Rho, 1981). This large increase must next be eliminated by allowing the outside pions to induce a D-state component in the quark wave function inside the bag (Brown, 1981 ; Rho, 1981). The net result appears to be that only the valence- and sea-quark contributions remain. This suggests that in the simplest model it is better not to mention the pion at all.

The magnetic moment might be more sensitive to the neglected j ~ 1/2 states and c.m. corrections. The outside pion cloud is known to contribute significantly, hut in such a way as to reduce the overall dependence on the bag radius R (Brown, Rho, and Vento, 1981). The mean square radii are not sensitive to configuration mixing, but are of course dependent on R and on c.m. corrections. Thus they are strongly model dependent.

Meson-Nucleon Coupling Constants

The effective interaction of Eq. (43) also induces color-octet qq excitations on hadrons which can be recoupled to meson-nucleon structures, as shown in Fig. 3. This diagram may be equated to the field-theoretical meson-nucleon Hamiltonian density to give a model of the meson-nucleon coupling constants. We describe below calculations of the ~NN and pNN coupling constants based on this model.

Fig. 3. A quark descriptio of meson-nucleon coupling.

meson

N

N I

The pion-nucleon qoupling constant. This calculation was first done by Duck (1978). It has been repeated by M. Heyrat and me (work in progress). One starts by writing down the operator equation

H~N N = (f~/m~)~iY5Y~ ~" I = Vef f (~) . (54)

Since Vef f refers to quark coordinates, while H N N refers to hadron coordinates,

the coupling constant can be extracted only after calculating matrix elements.

To do this, Duck uses the Peierls-Yoccoz (1957) projected plane-wave states:

Ip > = N / exp[ip .x ] d3x Ix > , and (55)

IpN> = N N f exp[iPN'XN] d3XNlXN > ,

where Ix> is a hadron state centered at x. The resulting ~NN matrix element for Vef f involves a 15-dimensional integratio~ over the vectors x , xN, xN, , r, and

r'. Another unpleasant feature is that the confined gluon

246 Chun Wa Wong

in the Green's function G(r',r) in Eq. (43) is located in a complicated region of space with three (~, N, a~id ~N') overlapping bags.

Duck proposes an approximation containing the following features: (i) Approximate G(r',r) by the contact interaction

G(r',r) = A6(r-r') (56)

(ii) Approximate the quark spinor by the "Caussian"

u ( 9 ) = N e x p ( - ~ ~22) , ~ = r / R (57) ~ - i$~'~ ~ o '

so that the integration can be carried out analytically. (iii) Eliminate Ag 2 in favor of

AM = MA-M N (58)

The result can be expressed in the form (Duck, 1978)

fvNN = C (B) (AMRo)(m Ro)3/2 , (59)

, m 3/2 where £M comes from Ag 2 ~ comes from the matrix element of H NN, and the size parameter R0 must appear as shown since it is the only remaining size parameter in the problem. The dimensionless proportionality constant comes from the 15-dimensional integration, and may be written as

C (~) = cdir(~)~ + c~X(~) , (60)

where the direct (exchange) contribution appears when the quark in the ~ is (is not) involved in . The value of the constant depends on whether M is obtained from the veff unprojected MIT bags, or whether projected bag states are used for better accuracy, as is done in Duck's calculation. According to Duck, the exchange contribution gives less than 10% of the total. So we shall neglect all exchange effects in the following discussion.

The size parameter R 0 is determined by Duck by fitting the proton charge radius of 0.88 fm, but without including any c.m. correction. The results are then calculated for a range of 82 . Table 6 shows our results and Duck's; all these numbers are in rough agreement with the experimental value of (0.08) I/2 = 0.28. We note that the theoretically better results using projected states are significantly

TABLE 6 The pion-nucleon coupling constant f NI, calculated with and without momentum projection t

Heyrat-Wong Duck 82 R (fm) No proj. With proj. With proj.

1 0.61 0.32 0.20 0.18 1/2 0.63 0.26 0.19 0.14 1/4 0.66 0.23 0.18 0.16

smaller than the unprojected results. This is because £M is proportionally large~ with projection. Since momentum projection is equivalent to a correction for c.m. motion, our results suggest that c.m. corrections may also have important effects


on the fitted parameters of the MIT bag model. A satisfactory treatment of these c.m. corrections is not easy to make in the MIT bag model, however. This is partly because the bag volume itself carries a quarter of the hadron mass, and partly because it is not clear how gluons are confined in overlapping bags. An applica- tion of the Peierls-Yoccoz projection method to the c.m. kinetic energy of quarks in bags has been made by Wong (1981b).

Our projected results are in rough agreement with Duck's. However, we are unable to reproduce Duck's 82 dependence to within the 10% expected from our neglect of the exchange contribution. The source of this discrepancy is now being investi- gated. We also note for completeness that the approximate quark wave function (57) reproduces the original MIT results best for 82 = 0.25.

The most noteworthy feature/ of the result shown in Eq. (59) is that it varies with the hadron size as R~ 2. Since a c.m. correction has not yet been when R0 was originally chosen, the actual ~ value after this c.m. correction should be larger by roughly a factor of (1.5) sI~ = 1.7. This brings the projected result to roughly the experimental value. Thus the simple model seems to work surprisingly well. It is also clear that in the "little-bag" model, the quarks alone cannot

account for fzNN"

Another correction arises if the pion size parameter differs from the nucleon parameter. The pion radius is actually smaller than the nucleon radius by about 40%. However, its c.m. correction is 15% larger, so that the ratio r of pion to nucleon size parameters is roughly 0.8. This gives rise to a correction factor of roughly

(5-3r)/r 3/2 = 1.8. 2

On the other hand, it is not clear that the entire AM should arise from the color magnetic interaction. So the present calculation might have overestimated f~NN on this score. It is hard to pin down the theoretical result to within a factor of 2 or so. Within this uncertainty, there is rough agreement between theory and experiment.

Rho-nucleon c0uplin $ constants. Similar calculations have been done for the oNN coupling constant (Heyrat and Wong, unpublished). Here the Hamiltonian density is

H = g0~iy~O "~ + (f0/2M)~i(~ ~F "~ where oNN

= (y~y -y~y )/(2i)

FU~ = ~up ~ - 3 p~

Proceeding as before by equating matrix elements of H pNN

dir c ~3/2 gP 3 Im~ ~ 1+ (732/8) fdir -~ 5 2 , ~NN i + (2382/8)

where the factor 3/5 comes from the spin-isospin. For B 2 = 0.25, this ratio is 8, while the experimental ratio is 2 to 3. The exchange contributions have not

been included in this comparison, but they are not expected to cause any appre- ciable change. Thus the calculated vector coupling constant appears to be too large by roughly a factor of 3.

, (61)

(62)

and Veff, we find that

(63)

248 Chun Wa Wong

There is another way to extract go" If 00 completely dominates the electromagnet- ic isovector form factor, it will couple equally strongly to any charged hadron with isospin because of charge conservation. That is

gpNN = gp~ ..... gp (64)

The universal coupling constant go is also related to the 0-Y coupling constant

yyp = mp2/gp (65)

annihilation.(Sakurai' 1969). Thus gp can be calculated from yyp, which involves a simple qq

This calculation has been done by Duck (1976) in an earlier paper. He uses the projected MIT bag states and finds a result of the form

3/2 (66) = const. (mpR) gD

where R is the bag radius. The calculated result turns out to be smaller than the experimental value by a factor of about 1.5, just as in the calculation for f~NN" There is again a strong R dependence in the result• Again a c.m. correction should be applied. Since the MIT bag model then gives much too small charge radii after this correction, we may be tempted to increase the bag radius for p proportionally. This correction then increases gp to roughly the experimental value•

However, this estimate differs from that given by Eq. (63) both in value and in R-dependence. One might think that the latter estimate might be better because it does not depend purely on the wave function• (In the same way a plane wave, which shows no scattering, can be used with a potential to give a Born approximation for scattering.) The situation is more obscure here because the effective interaction itself is not well known, and must be deduced from AM, a procedure which adds R03 to the R 0 dependence in Eq. (59).

The reason for the discrepancy is of course that the model wave functions and interactions are not exact. In the exact theory, both calculations should lead to the same answer. Judging from the discrepancy, we may say that the model is not too good, but perhaps not too bad either.

A rather similar situation appears also for f . . For pion decay, the relation ~NN

analogous to Eq. (65) is the Goldberger-Treiman relation

M (67) F = gA

f~NN

where the pion decay constant F can also be calculated from qq annihilation• The result (Duck, 1976; Wong, 1981b) ~ in the MIT bag model is

F = const./(m R3) I/2 (68)

Duck's calculated value of F is 4.5 times the experimental value of 94 MeV •

(Sakurai, 1969) when a plon bag radius of 3.34 GeV -l is used. For the bag radius of 3.75 ~ GeV -I used by Wong, where the factor ~/2 allows for the pion motion inside the bag, Duck's result will be reduced to only 2.3 times the experimental value. This agrees with Wong's result (2.5 times the experimental value) obtained with the more approximate Gaussian wave function of Eq. (57).

The tensor coupling constant f can also be calculated in the same way as gp from

Vef f. We obtain the result P


fdir f=~ i+~i 62 Kdir = O ~ ,~, M ~ . _ _ ~ 1.4 , (69)

dir ~7 m 4 7 Bz go 0 i+~

where the factor (5/3) is the ratio of spin-isospin factors. We have not yet computed the exchange contributions, but if they are negligibly small, the calculated ratio K of coupling constants appears to be a factor 2-3 smaller than the experimental ratio of 3~{9 We should note, however, that the trouble appears to be in the calculated go which is too large by a factor of 3. That is, the cal-

culated fdir agrees roughly with experiment. 0

These results give a naive quark picture of meson-nucleon coupling constants. They must eventually be reconciled with other pictures, e.g., with the chiral bag picture where the O couples to the nucleon only through a pair of pions (Ferchlander, 1980; Brown, 1981).

VALENCE ANTIQUARKS

The final topic is that of valence antiquarks in q2~2 mesonic states. These states are interesting because they are the simplest multiquark states, and because they might be involving in meson-meson and in hadron-antihadron reactions.

Let us first recall that the four-quark states of orbital angular momentum L = 0 have been described by Jaffe (1977a, 1977b) in the MIT bag model. They are states in which the diquark is in the color 6 (7) representation either 66 (34)% or 34 (66)% of the time (Wong and Liu, 1980). The lowest state C o (0.65 GeV) is in state with 66% color 6 configurations.

States of high orbital excitations of the diquark against the antidiquark have been discussed by a number of authors (Chan and H~gaasen, 1978; Aerts, Mulders, and de Swart, 1980) under the name of baryonium states. These are pure color states with the diquark in either the color 6 or 3 representation, because the centrifugal barrier of orbital motion hinders color mixing. The stiffness against orbital excitations increases with the dimension of the color representation of the diquark. As a result, these states of high L excitations fall into two distinct rotational bands, with the color 3 band having smaller rotational energies or a larger Regge slope.

Since the color 6 configurations dominate the lowest L = 0 bag state, it is clear that as L decreases the color 6 band crosses over the color 3 band in the Chew- Frautschi plot. The rotational bands mix as they cross, so that the dominant color nature of the lower-mass state of the same L changes from 3 to 6 as L decreases. This is an example of the band-crossing phenomenon familiar to nuclear physicists.

This band-crossing region of moderate L turns out to be of considerable experimental interest, because some of these states might have been seen experimentally as Liu and I (unpublished) have discussed. In this lecture, I would like to concentrate on how Liu and I describe the interesting features of these color rotational excitations by using the NR potential of Liu and Wong (1980).

We use a cluster model constructed in the following manner: (i) First a rotational coordinate is chosen from among the interal coordinates. (2) The remaining internal coordinates are treated variationally, with Gaussian wave functions in the present calculation. (3) An effective, in generally coupled-channel, potential

250 Chun Wa Wong

in the rotational coordinate is then constructed. The potential contains specta- tor effects from the remaining internal coordinates. (4) The coupled-channel

Schrodinger equation is solved for different orbital angular momenta. (5) The variational parameters describing the remaining internal coordinates are varied, so that the energies and wave functions of the resulting rotational states contain the effect of centrifugal stretching.

This procedure is first checked out by calculating the rotational excitations in

various B = 1 baryons (including those with c and b quarks). The rotational coordinate is either that of the diquark (the p mode of Isgur and Karl, 1978), or of the third quark against the diquark (their % mode). We find that the calculated baryon mass differences come out quite well for all the baryon masses of the Data Table, but the absolute masses themselves depend somewhat on the treatment

of a flavor-independent additive constant

b ( q ~ ) = - 0 . 0 8 CeV ( 7 0 )

appearing in the meson-potential of Liu and Wong (1980).

If b has the same value between two quarks, and if it is a color-independent pair

interaction, we get the rather satisfactory results of 3% = 0.95 GeV and M A = 1.22 GeV. The trouble is that b is negative and contributes a term proportional to n 2 in a system with n quarks. Thus the prescription is useless in

really large multiquark systems.

If b is the strength of a color-dependent pair interaction proportional to ).-~., its contribution will be only linear in n. A negative b will simply serve ~i ~3 to reduce the quark mass, as indicated in Eq. (40). This is a nice feature, but the problem then is that the n = 3 baryon masses turn out to be too high by ~ 0.ii GeV. We decide to bring these masses down to roughly their experimental values by

using a different constant

b(qq) = b(qq) = -0.12 GeV (71)

in diquarks and antidiquarks.

Two types of rotational excitations are studied: (i) diquark-antidiquark excitations, and (2) excitations in a diquark (or antidiquark). These excitations are called H mode and e mode, respectively. They are shown in Fig. 4 where the solid

line represents the rotational coordinate, while the broken lines denote the "variational" coordinates. The same variational parameter is used for both variational coordinates in order to simplify the variational calculation.

For L = O, the H and ~ modes are just two different approximations of the same states. The calculated energies, given in Table 7, show that the H mode gives

better wave functions and energies (usually lower by 0.06 GeV). The L = 0 e states turn out to be very close in energy to those of the oscillator shell model, being lower in energy by only 1 MeV. The gain in energy in the H mode arises from a clustering effect in the diquarks. This effect is particularly strong for the H state at 1.62 GeV, and is related to the fact that the minor component (which is color 3×3) has one more radial node than the major component. This possibility has not been included in the variational Gaussian wave function used in the

mode.

Table 7 shows that our mixture probabilities agree well with Jaffe's, but our masses are higher by roughly 0.2 GeV. Mass differences in our model are typically larger by about 0. i GeV, but some differences are reduced by the improvement in

our radial wave functions.


I I

I I

L

(a)

~ 3 I £ r

I I

(b)

Fig. 4. The rotational coordinate (solid line) in (a) the H mode (diquark-antidiquark excitations), and (b) the e mode (excitations in a diquark). The broken lines denote the "variational" coordinates.

TABLE 7 Comparison of L = 0 four-quark states.

e mode H mode Jaffe

Mass (GeV) 0.85 0.79 0.65

p(6X6)% 68 65 66 a

Mass

p(6×6)% 1.36 1.30 1.15

37 36 34 a

Mass 1.82 1.62 b 1.45

p(6X6)% 43 24 34 a

Mass 2.11 b 2.03 1.80

p(6X6)% 68 48 66 a

(a) From Wong and Liu (1980). (b) Minor component contains one additional radial node.

The crossing of color rotational bands found in the coupled-channel calculation is illustrated in the Chew-Frautschi plots of Fig. 5 for the I = 0 H-states of natural parity P = (-)J. The two color-spin configurations being mixed are I(3,1)x(3,1);ii> and I(6,3)x(6,3);i,i>, where the labels are color or spin degen- eracies. Each state is denoted by a number giving the percentage of 6x6 color configuration. The positions of the unmixed, unperturbed, pure color states are indicated by the dashed curves for the n = 1 states. Circled numbers represent states with the n = 2 radial excitation in their major components. Only the three lowest states of each J have been shown.

252 Chun Wa Wang

_

4 -

2 -

O_6s 0

I I

I=0 H states

/ 3/ /.i 41

4

s/

/ 0 /

/0

/ 0 / (~) .-- .~-gq--

® L

8 12

M 2 (GeV 2)

Fig. 5. The mixed-color, I = 0 H-states of natural parity P = J

(-) . The position of each state is denoted by a number giving the percentage of 6x6 color configuration. The positions of the unperturbed pure color states are shown by the dashed lines. Circled num-

bers denote states with radial excitations in their major components.

We can easily recognize in Fig. 5 the almost pure color states of different Regge

slopes at J > 3. As J decreases, these color rotational bands approach each other and become substantially mixed. The 6x6 state crosses above the 3x3 state

(crosses below in mass) as J decreases below J = i. This is why the 6×6 configura tion dominates the lowest mass J = 0 state.

Figure 5 illustrates rather graphically the usefulness of the NR potential model

in describing these rotational excitations.

In Figs. 6 and 7 we show the M 2 of selected states of low orbital excitations. Masses of natural-parity states are obtained with the full effective potential, while those for unnatural-parity states are calculated after dropping the spin- orbit and tensor terms. In drawing the figures we have lowered all @ masses by the small constant amount needed to make the L = 0 e-state coincide with the cor-

responding H states. This makes the resulting mass plot a little clearer to the eye, and will also correct the relatively greater limitations of our approximate

variational procedure for the e states.

We see from these figures that the @ and H states of low L have comparable masses At larger L values, the H states of 3×3 color structure eventually appear lowest in mass because of their steep Regge slope. These are the so-called "true" baryonium states (Chan and H~gaasen, 1978). They are expected to be more readily

formed in antibaryon reactions than other four-quark states. The probability of production on the other hand is probably less sensitive to their detailed structure.

One interesting result of our calculations is that the yrast states (states of

lowest mass) in this low L region are not the simple rotational bands of maximum L (= J), but "stretched" states of minimum L (= J-2) and maximum S (= 2). The


I I

p=(_),T ~ ~ _

6 - e

t6~

(a) I =0 ~ H ~

2 -

0 I

4 - ( h i -

2 - - - L - - J - 2

P'O I j '

/M / / ' 4 (c) I = 0 , 1 , 2 / /

- - _ _ . . . . /H/. e ~ "

2- ~

O _ _ I 0 4 8 12

MZ(GeV z)

Fig. 6. Selected co lor r o t a t i o n a l bands of n a t u r a l - p a r i t y states with the diquark-antiquark isospins of (a) (0,0), (b) (0,i) or (i,0), and (lqq, lq )

(c) ( l , 1).

pure color 3x3 stretched H states of I = 0,1,2 and JP = 3-, 4 +, and 5- shown in Fig. 6c with calculated masses 2.11, 2.40, and 2.65 GeV are the most promising candidstes for the three broad structures seen by Carter and others (1977) in the pp -> ~ ~ reactions. These structures are T (mass = 2.15 GeV) width = 0.20 GeV,

U (2.31).0.21, and V (2.48) 0.28 with the quantum numbers jPCIG = 3--1 + , 4++0 - , ---- t

and 5 i , respectively.

The lowest mass I = 0 four-quark state at 0.80 GeV (marked E in Fig. 6a) is consistent with the possible experimental scalar c (0.80) found by Estabrook (1979). The lowest-mass I = I, JP = i- state is an L = 1 ~ excitation at 1.4 GeV, which is well separated from other states of the same quantum numbers, as shown in Fig. 6b. We tentatively associate this with the possible experimsntal vector meson p' (1.25) seen in photoproduction (Richard, 1979) and pp -~ e e (Bassompierre and others, 1977) experiments. It should be noted that none of these experimental states are established resonances.

254 Chun Wa Wong

3-

_

4 -

2 -

0 0

I 1

p=(_)a.l ~,.,

H E ) 8 , / /J s:l

e _ H I I 4 8

M2 (GeV 2 )

12

Fig. 7. Selected color rotational bands of unnatural- parity states. The isospins of these states have not been specified because several states of the same J but different I occur close together.

The point I want to make here is that in spite of its many limitations, a NR potential model can be used in a sensible way to illuminate certain features of the structure of multiquark states.

Finally I should mention that it has become very controversial to even mention multiquark states as possible resonances. I would like to indicate very briefly my preference in this controversy.

First, it seems to me that resonances should appear whenever confined states are coupled to open channels. A multiquark-state typically contains confined components involving "hadronic" constituents of integral baryon number (e.g., qq or q3) which are not color singlets. There is also a centrifugal barrier if L # 0. On the other hand, there are also components containing unconfined color-singlet hadronic constituents which appear as a result of both dynamical mixing and Pauli antisymmetrization. Thus multiquark states may not give rise to sharp resonances, but they might give rise to broad structures.

The difficulty of seeing these multiquark states might well be due to the small- ness of their production or formation cross sections, perhaps as a result of the weakness of their coupling to the common reaction channels.

CONCLUSION

We have discussed certain quark effects on hadron properties in a simple-minded way with the help of the MIT bag model and of NR potential models of quark dynamics. Such a naive approach has its limitations, but it appears to give us an appreciation of the ways by which quark degrees of freedom might show up in nuclei. For example, we find that the qualitative descriptions of color rotational excitations of q2~2 states are rather similar in both types of phenomenologica] quark models. Yet these models can differ substantially in their detailed


descriptions of many quark effects, including those contributing to nuclear forces. The difference appears to arise primarily from the different assumptions made concerning quark and gluon confinements. This result suggests that we should not take strongly model-dependent descriptions too literally until we have a better understanding of the actual confinement mechanisms.

ACKNOWLEDGMENT

This work is supported in part by NSF Contract No. PHY 78-15811.

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