Few-Body Systems, Suppl. 6, 1-8 (1992) F..ew­tlOav

Systems © by Springer-Verlag 1992

mE NAivE QUARK MODEL AND BEYOND

G.Kari

Department of Physics, University of Guelph,

Guelph, Ontario N1G 2Wl, Canada

(E-mail: PHYGKARL@VM.UOGUELPH.CA)

Abstract: I comment on the (lack of) connection between the Naive Quark Model and

Quantum Chromodynamics. The relation between magnetic moments of baryons and the

problem of the spin of the proton is discussed in more detail; in particular it is emphasized

that data on the magnetic moments and axial couplings of baryons are now much more

accurate than in the past, and are less encouraging for the Naive Quark Model than in the

past.

The world of baryons and mesons has been studied in ever increasing detail for nearly

fifty years. Although there is now wide consensus that these systems are composite, there is

much discussion about details, especially the relevant degrees of freedom and so the situation

is still not satisfactory. The most optimistic viewpoint is that we know the relevant

Lagrangian (at short distances), called QCD, and only mathematical details are missing. The

most extreme version of this view is that only the backwardness of present day computers

prevents a complete numerical solution of all questions experimenters might be able to

measure in the future. Since the present author believes that the future tends to resemble

the past, he does not subscribe to such an extreme view. The past way of development in

physics consisted in finding (usually painstakingly slowly) the right approximation to use, (even

when the Lagrangian was known) by using some help from experiment. It is likely that

hadron physics will follow the same pattern, even if we know the Lagrangian of QCD.

Part of the specific difficulty with hadron physics, is that the observed hadrons

themselves are not the degrees of freedom at short distances. So we have to rely on

approximate descriptions of bound states. I will try to list some of the approximations used,

with a short description to them.

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I should first list the Naive Quark Model, (NOM) since it is the main topic of my talk.

It is indeed one of the models widely used to descn'be hadrons. It is hard to say, in a

mathematical sense, that the NOM is a well defined approximation. The best derivation we

have of the NOM comes from the limit of QCD as the number of colors Nc becomes large.

Witten [1] showed that in this limit baryons are well descnbed by NOM wavefunctions. To

be more precise the limit of QCD as Nc is large becomes identical to the limit of the NOM

as Nc is large. In the real world Nc equals three and the NOM is an approximation; the

usefulness of this approximation is non-uniform. In some applications such as spectroscopy,

the NOM works very wel~ while in others, such as the "spin of the proton" it fails. Aside

from the large Nc limit we have no justification for the Naive Ouark Mode~ and so its use

tends to get mixed reviews [2]. Generally, the Bag Model is considered as a different "ad hoc"

model, but I take it to be just another version of NOM, which only differs by using relativistic

quarks. There are many other relativistic potential models, harmonic oscillators, etc, but they

all can be thought of as examples of NOM. Their common feature is the fact that they have

qq and qqq as allowed states.

These models should be distinguished sharply from QCD on the lattice, which is a well

defined and justified theory of hadrons which however suffers from the problem of being still

in search of reasonable approximations. The practitioners of lattice QCD are crying for more

computing power, while probably good insight into the physics of QCD would be equally

welcome. The gap between data and lattices will be bridged not entirely by brute force, one

suspects.

An entirely different and rigorous approach to hadron physics at low energies is based

on chiral symmetry breaking, and it consists of deriving properties of hadrons which follow

from the smallness of light quark masses, relative to the QCD scale. These properties are

supposed to follow rigorously from the QCD Lagrangian, although there is some ambiguity

in relating the parameters of the fundamental Lagrangian to the parameters of the chiral

Lagrangian [3]. However chiral models are quite limited to the very low energy sector of the

hadronic spectrum, pion physics and perhaps ground state baryons. In these models the

nucleon is mimicked by so called chiral solitons - the skyrmion. The application of

skyrmions to the whole baryon octet is much more problematic than that to the nucleon itself.

In particular the claim that skynnions quantized as SU3 flavor octets correspond in some way

to the large Nc limit of OCD [4], is simply wrong, since in the large Nc limit the baryons are

not members of SU3 octets [5].

Another approach in hadron physics is based on the importance of instantons in the

ground state of QCD. This approach concentrates on the computation of correlation

functions in hadronic matter [10].

On the theoretical side there is a great deal of support for the ideas of Quantum Field

Theory. In hadron physics the relevant field theory is Q.C.D. which has been studied in a

variety of ways, but primarily on the lattice. This work is certainly ongoing, and it is certainly

making progress, although there are still many difficulties, both conceptual and computational.

This work is important because the nature of the vacuum of Q.C.D. is still not fully

understood and different from the vacuum of QED, so that the discussion of bound states in

QCD is still unclear.

The numerical work is already at the stage of providing fits of hadron properties

which are almost competitive with phenomenological models, especially for nonstrange

hadrons [9]. Of course the major strength of lattice QCD computations is that they contain

no arbitrary parameters, so that even if the results are still imperfect they nevertheless can

be said to reOect the properties of QCD. Aside from this, much conceptual progress has

been provided by lattice calculations which should form the basis of newer and better

phenomenological models. In particular chiral symmetry breaking can be studied in numerical

models in addition to so called chiral Lagrangians. It also seems probable, at the present

time, that the chiral phase transition is independent of the existence confmement, and it

accounts for the constituent quark mode~ along lines suggested by Georgi and Manohar [6].

To be more explicit, although quarks have small masses at short distances, due to CSB they

acquire a "constituent" quark mass, while remaining confined. There is also recent work on

using QCD on the light cone to obtain hadron wave functions [7].

In summary then, we have many approximations in hadronic physics, but the

connection between them and their connection to the underlying theory (QCD3+l) is still not

clear. In particular the Naive Quark MOdel for 3 colors is not an approximation to QCD.

Thus hadron physics still lacks a secure foundation. Therefore, this is a good subject for

further study, with lots of room for progress.

In the remainder of this review, I shall be much more specific and deal with the topic

of baryon magnetic moments and their connection to the distribution of the proton spin. This

is an important topic because the data contradicts our expectations based on NQM.

Therefore we are given information which we should use to construct models which improve

NQM. This has not yet been done - but it is important to know the weaknesses of a model

if we wish to improve it. I now discuss problems connected with the spin of the proton. In

particular, I wish to discuss how, starting from magnetic moments, one can show that the

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contnbution of the helicity of the quarks and antiquarks to the helicity of the proton is but

a small fraction. This conclusion contradicts the NQM where the entire proton helicity is

carried by the quarks. This conclusion is based on magnetic moments of baryons in the octet,

which are by now known experimentally reasonably well

The argument is based on a set of equations which I call the Generalized Sehgal

Equations (GSE) which connect the magnetic moments to the contnbutions of the quark

spins:

(1)

where 'J.&." lid' "" are effective quark magnetic moments, and As for example is the contnbution

,of the u-quark helicity to the proton helicity:

(2)

Equation (1) holds [8] in a class of models in which all quarks and antiquarks of a given flavor

are in the same shell of j = 1/2 of some cavity or potential, and the proof involves a simple

transfonnation between the linear combination natural to magnetic moments [n(u t )-n(u1)­

n(ut )+n(u1)] and the quantity Au. In other words, equation (1) is not generally valid, but

only in a specific class of models which generalize the NQM, where Au = 4/3, Ad = -1/3 and

As = o. Ifwe assume SU3 flavor symmetry, we can obtain the magnetic moments of all baryons

in the octet from equation (1). For example, for the neutron Nt

etc.

<N,IiiY,y,aJN,> = <P,IdY,Y,dIP,> • ul

<N,'tdyzY,t4N,> = <p,liiY,y,aIP,> ;; 611

where we used only isospin, so that the analogous equation to (1) is

Similar application of SU3 symmetry, for example to the I+ implies:

(3)

(4)

so that

<ll;liY,v,slll;> = Ml

<ll;!d'v,v,dlll;> = bS

and similar equations for the other baryons in the octet.

(5)

(6)

Therefore we have a set of equations for the magnetic moments of P, N, I+, Z", A,

EO, E", and the transition moment IO .. A parametrized in terms of Au, Ad, As and p.", lid' 1-'."

In fact the GSE are quite symmetric, and they do not depend on six parameters but only five,

as the linear combinations (Au + Ad + As) and (Ii_ + lid + Ii.) only appear as a product with

each other. Moreover, we have to prevent the relative rescaling of Au, Ad, As w.r.t. Ii., lid' and

1-'., which we can do by choosing as a normalization, for example:

All - bil - 1.26 (7)

though this choice is arbitrary - from the point of view of the G.S.E.

Therefore there are four parameters left: (Ii. + lid + 1-'.) (Au + Ad + As), Au + Ad"

2As, p." " lid' p." + lid "21i.. If we make the substitution:

(8)

there is no loss of generality, as the number of parameters remains four, but now we can

choose as parameters lid' 1-'., (Au + Ad + As) and (Au + Ad - 2As), which are more interesting.

So we can parametrize eight magnetic moments in terms of four parameters. A special set

of our parameters corresponds to the NOM which has three parameters p.", lid' Ii.. We can

now go to the data and see what it tells us. We fmd that the NOM parametrization fits at

the level of 15% whereas the full GSE parametrization fits at the level of about 7%.

Moreover the best GSE fit corresponds to (Au + Ad + As) = 0.2 ± 0.2, therefore a small

fraction of the proton helicity is carried by quark helicities.

I should explain a little better the way one obtains these fits. Since these equations

(GSE) do not fit the magnetic moments perfectly, we want to fit them without favoring any

particular baryon in the octet. This can be accomplished most simply by pretending that each

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Best Fits to Ma~etic Moments (in Nucl. Ma~etons)

Particle Magn. Fit 1 Fit 2 Moment GSE NQM

P 2.79 ± 0.10 2.69 2.68

N -1.91 ± 0.10 -1.85 -1.92

};+ 2.48 ± 0.11 2.59 2.55

X- -1.16 ± 0.10 -1.22 -1.13

EO -1.25 ± 0.10 -1.33 -1.40

- -0.68 ± 0.10 -0.61 -0.48

A -0.61 ± 0.10 -0.59 -0.59

A}; -1.6 ± 0.13 -1.53 -1.60

X2 4.42/4df 7.53/5df

a(l) = Au+Ad+As: 0.12 ± 0.17 0.28 1.00 (input)

a(8) = Au+Ad-2As: 0.60 ± 0.05 0.86 1.00 (input)

gA = Au-Ad: 1.26 ± O.ot 1.26 (input) 1.67 (input)

p.u -2.42 1.76

I'd: -1.21 -1.00

1'.: -0.71 -0.61

magnetic moment has a "theoretical" error of to. 1 n.m. and doing a r fit to all of them,

minimizing the total X2• Such a fit is presented in Table I. The main conclusion is that the

data on magnetic moments prefers quantitatively a set of parameters Au, Ad, As which is

nowhere near the NOM solution. The NOM fit to magnetic moments is worse, and we

should try to rethink our views about the NOM on the basis of these fits. We should try to

construct quark models which correspond to these fits. Such models would have many qq

pairs in the proton including strange pairs, and these pairs would be polarized. Many simple

questions are raised by such models; in particular how come that these pairs do not contnbute

to the spectrum of excitations of the proton at low energies. In other words, how come the

proton seems to be a 3-body system when kicked, but has many more degrees of freedom

when analyzed with a magnet?

Acknowled&ements: This report was written at the Aspen Center for Physics and the author

is very grateful for the opportunity to visit the Center. I also thank for conversations and

lectures on these topics E. Shuryak, J. Negele, S. Brodsky, K Wilson, and other participants

at Aspen workshops in the summer of 1991.

References:

[1] E. Witten, Nucl. Phys. Bl60, 57 (1979).

[2] for recent work on the NOM see e.g.

S. Capstick and N. Isgur, Phys. Rev. D34, 2809 (1986).

S. Godfrey and N. Isgur, Phys. Rev. D32, 189 (1985).

[3] see e.g. J. Gasser and H. Leutwyler, Phys. Reports a:z. 77 (1982).

D.B. Kaplan and A V. Manohar, Phys. Rev. Let. 2.Q, 2004 (1986).

K Choi et aI, Phys. Rev. Letts. 21. 794 (1988).

K Maltman, T. Goldman and G.L Stephenson Jr., Phys. Letters B234, 158 (1990).

[4] see e.g. S.J. Brodsky, J. Ellis and M. Karliner, Phys. Letters B206, 309 (1988).

J. Bijnens, H. Sonoda and M. Wise, Phys. Letters BI40, 421 (1984).

[5] J. Bijnens, H. Sonoda and M. Wise, Can. J. Phys. M. 1 (1986).

G. Karl, G. Patera and S. Perantonis, Phys. Letters BI72, 49 (1986).

G. Karl and H.J. Lipkin (to be published).

[6] H. Georgi and A Manohar, Nucl. Phys. B234, 189 (1984).

[7] S. Pinsky, KG. Wilson et aI: work described at the Aspen workshop on light cone

physics.

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[8] G. Karl, Guelph preprint GWP2-PP-91-02. Earlier references includes:

- LM. Sehgal, Phys. Rev. Illi!. 1663 (1974). - J. Bartelski and R. Rodenberg, Phys. Rev. 041. 2800 (1990). - R. Decker, M. Nowakowski and J. Stahov, Nucl Phys. A512, 626 (1990). - S.M. Gerasimov, Int. Conf. at Alushta (1987) Dubna preprint E2-88-122. - G. Karl and M.D. Scadron, Proceedings of MRST Conf. (1990).

[9] See for example M.e. Chu, M. Lissia and J.W. Negele, to appear in Nucl. Phys. B

(1991), who discuss extensive computations of correlation functions in the w, p and

nucleon.

[10] See for example E. V. Shuryak, Nucl Phys. m28. 85 (1989) and references therein.