247

SOFT QCD LIGHT PHYSICS WITH CHROMODYNAMICS

Nathan Isgur Department of Physics University of Toronto, MSS 1A7

The addition of dynamical ingredients suggested by QCD dramatically improves the ability of quark models to describe soft hadronic phenomena . This approach is discussed in terms of a particularly simple non-relativistic potential model which has been used to study the spectroscopy, decays, and static properties of and . The application of the model to such QCD exotica as multiquark states and glueballs is also mentioned. In addition, an attempt is made to at least partially rationalise the success of the model in terms of more fundamental physics. 248

I. Introduction to the Model

Why Potential Models? A. The successes of quarkonium models for the cc and bb systems , based on a non-relativistic potential model with flavour and spin independent con­ finement and one exchange, at least raise the question of where , as a function of quark mass, such models become useless. �he models I will discuss here are,in response to this question, based on an optimistic extension of the physics expected from QCD for heavy quark sys tems into the light quark domain . We shall see that the models not only provide a useful framework within which to describe the physics of such systems , but moreover that this extension has, at least at some level , been successful. The reasons for this success are only partly understood; a possible beginning for an interpretation is given below.

While there are many possible variants, I will concentrate on the l,Z,3) version of such models that I know best It has three main ingredients: confinement , "point-like" constituent , and one gluon exchange .

B. How Do Potentials Confine Colour?

It is non-trivial to construct a potential which will, for example, 4) confine qqq but not qq. One solution is to assume---b ased on a colour electric flux tube mechanism for confinement---a colour-dependent two body potential between quarks i and j of the form

(1)

with V(r ) a flavour and spin independent confining potential and with the ij prescription that antiquark potentials follow by the replacement c * _,_ (--t ) . The resulting model allows only colour singlet hadrons to t t = 1) exist , but has the property that wh ile the confinement of qq and qqq is automatic, the existence of more complicated colour singlet bound states (like qqqq) becomes a dynamical issue . The model is also very economical in that it relates the physics of baryons , mesons , and multiquark systems ; this feature will be discussed below. 249

C. What Are the Constituent Quarks?

The quarks of the model are not the current quarks of the QCD Lagrangian, but rather a set of "dressed" constituent quarks appropriate to the distance scale of light hadrons , These quarks (which are assumed to be approximately Pointlike) have masses which are (in part by their nature and in . part because they are at this stage also probably repositories of some of our ignorance about the potential, relativistic corrections, etc.) ill-known, but the values

m m 0.33 GeV (2) u d m - m MeV (3) d u 6 m 0.55 GeV (4) s m 1. 75 GeV (5) c etc. , are typical . These masses play a crucial role in the model because their differences are the principal origin of flavour symmetry breaking. Note that if we assume that the quarks in a proton are in a Gaussian wave function with 2 the proton 's observed charge radius , then with these masses � 100 MeV so that these light quark models are probably not hopelessly relativistic . �

D. Where Are the ?

The last principal ingredient of the model is one gluon exchange . The mo st important effects of this type are due to colour hyperfine (magnetic dipole-magr,etic dipole) interactions which in lowest order in " s are given by

ij ll --hyp - s.·s.J } l J (6)

where 2a s in 3m m a i j (7) 4a s in a 3m m i j The first term of ' called the contact term, is , in electromagnetism, l\iyp responsible for the 21 centimetre line of hydrogen; the second (tensor) term is the usual interaction of a magnetic dipole with a magnetic dipole field. The most important characteristics of H are : 1) it is short range in h y §__ 2) it gives a fixed relation ( �) etween the strengths of the contact 3 S.l ·S. J , and tensor terms , 3) it has a strength inversely proportional to the product � 250

of the quark masses, and it violates SU (6) and in particular automatically 4) gives m m and m p > n � > �· One gluon exchange also gives ris·e to spin-orbit interactions from the interaction of a moving quark colour magnetic moment with colour electric fields . As in the case of electromagnetism, this effect tends to be masked by Thomas precession (the relativistic precession of an accelerated spin) , but in this case the suppression of spin-orbit effects is (in the case of qq with m ) by a factor of (1-l;z- f) where the and the "f" come from Thomas mq >> q "'>" precession in the Coulomb-like and confinement potentials, respectively. In the light quark hadrons , the quark wave function overlaps very strongly with the confinement region so that f is very large and spin-orbit effects are 5) suppressed ,

Finally, one gluon exchange gives rise to a variety of spin­ independent effects which can normally be absorbed into the unknown confinement potential . Most prominent among these is, of course, the Coulomb-like l/r potential itself .

II. Baryons First

A. Why Baryons First?

We begin our discussion of the model with the qqq baryon sector even though the qq meson sector is simpler to treat theoretically . There are several reasons for doing this; probably the most important of these is that the baryons are much better known experimentally than the corresponding mesons (a consequence of their accessibility as s-channel resonances) . Not only are more resonances known , but their masses and widths are better known and there is a wealth of information on their (signed) decay amplitudes . Specifically, the P-wave mesons remain very poorly known (think of the A and the while 1 E) , all seven expected S=O P-wave baryons are known and have reasonably well measured amplitudes for both electromagnetic and strong channels (typically 5 to 10 amplitudes per resonance). Aside from their being more extensively known, baryons probably have other advantages in practice . For one thing, the quarks in a baryon appear to be somewhat more non-relativistic than those in a meson, making their treatment more reliable. It is also possible that three bodies are sufficient to make the effective potential seen by a quark in a baryon significantly "smoother" than in a meson, thereby making baryons less sensitive to precise knowledge of the potentials. Finally, baryons do not have the isoscalar mixing problem (to be described below) which , while interesting, renders I=O meson data unreliable for spectroscopic studies and leaves very few 251 mesons indeed to compare with a potential mo del.

B. What is the Model for Baryons and Its Solution?

The application of the model to baryons is quite straightforward in

6 7) 8 9 10) . its simplest form ' . Building on the pre-QCD analyses ' ' (which were on their own terms already very successful) we take the simple Hamiltonian p ,2 H 3 = E ( m E i 2 . . . (8) i=l + i > + i

and treating � and the anharmonicity U as perturbations . Note that V here yp contains both the "true" confinement potential and pieces of the one gluon exchange potential (so that U may contain l/r terms , linear terms , etc.) ; also note that spin orbit effects have been completely dropped . We have already mentioned that Thomas precession is expected to make spin-orbit effects smaller, but the adequacy of completely neglecting them in baryons is not well understood. Evidence from me sons , where the suppression is more easily studied, 5) does , however, tend to support this approach .

The approximate solution of this model Hamiltonian is simple. In the harmonic limit with m = m m and m m' (the most general case required for 1 2 = = 3 u, d, and s quarks in the approximation m = m ), u d p 2 p 2 p A 2 2 H +--+ -- + - k ( p + A (10) 2m 2m;i. 23 ) p where

_,. _,. _,. p = l r - r (11) /2 ( l 2

_,. 1 _,. _,. _,. A r r 2r (12) = /6 ( l + 2 3 and mp m (13)

m (14) A 2m+m'3mm1 so that the unperturbed spectra with up to two units of excitation are those of Figure I. One interesting -- and important -- feature is that the solutions of 6) the confinement problem maximally violate SU(6) in excited baryons . For + example, the p-A basis = 2 eig�nfunction p! is a 45° mixture of the LP . 12 2 + 1 2 2 [56,2+ ) wave function (p + A ) and the [70,2 ) wave function /2 + + rz (p A ) + + 252

(r S=n,-J I: Jigure the unperturbed solutions to the harmonic confinement problem (with arbitrary scales set to give the same spacing in each sector) .

of SU(6) . Clearly, to the extent that this phenomenon is important , an SU(6) (or even SU (3)) analysis of excited baryons will fail.

We next take these states and perturb them with the anharmonic term U. It turns out to be unnecessary to actually specify U since one can show that ll) II. in first order every U gives the same pattern , shown in Figure

[20,1 ' J (70, 2']

I 56,2'1 II: 70,0' l Figure the positions of [ the SU(3)-symmetric super­ multiplets under the influence of an arbitrary anharmonic perturbation.

(56,0']

One can therefore take the parameters '1 and t, of this Figure as describing the potential ; the phenomenologically required sign of t, is, however, consistent with the expected existence of the -1/r anharmonicity. To deal with the case 1i m F m ' the U-perturbed A excitations are scaled by a factor (m /m ) A p p A appropriate to the harmonic limit.

The next step in the approximate solution of (8) is to turn on the hyperfine percurbations . These interactions are of crucial importance: they create huge spectroscopic splittings and very strong mixings that destroy almost all vestiges of SU(6) symmetry (except in the ground states) . The final step in the solution Pis the simplest: one takes the sectors of the Hamiltonian of fixed flavour and J and diagonalises the resulting matrices. 253

C. How Can the Baryon Model Be Compared to Expe riment?

There are two crucial elements in the comparison of a model like this with experiment: one is to compare with spectroscopic evidence and the other is to check , via an analysis of decay amplitudes , the predicted internal structure of the eigenstates . The former check is relatively simple ; the latter requires the construction of a decay model . We have used for this lZ) 13) decay model a slightly generalised form of the single quark emission mode1 14) that has much in common with more algebraic approaches . It is based on the "elementary" emission processes shown in Figure III.

Fii:1ure III : (a) photon coup�ing in the single quark emission B' y B' M model. (b)meson coupling in the single quark emission model .

B B (a) r (b) r

As one example of the ways in which a decay analysis can reveal the internal structure of a state, consider the coupling of a uds state with some excitation in the variable to the channel . As shown in Figure IV, since p KN

N K

u d d Ci s

Figure IV : the decoupling of Y * 's _ with excitation from the p NK channel.

u d s * :ry the (ud) spectator pair remains excited, it cannot overlap with a nucleonic (ud) pair; the amplitude for this process is therefore zero and the model 6) predicts that such states should not be seen in partial wave analyses . KN

The full comparison of the baryon model to experiment thus involves comparing not only to an observed spectrum, but also comparing against hundreds of measured decay amplitudes. The flavour of this comparison -- but not its extent -- is reflected in Figures V(a) to V(d) and in Table I. 254

'"' � N"Jl2-

Figure V(a)_: the predicted S=O negative Figure V(b): the predicted S=O parity baryons compared to experiment ; positive parity baryons compared the regions in which the masses of the to experiment; states that are resonances probably lie are denoted by predicted to decouple from nN shaded boxes. are shown as stubs.

-�------;,

- ·-- . .r-;-;-/1 t2Z!/2.'./_dL/j � � 1eoo t£JfJ£i£� _ IGOOr L�· �· - � � 170le;,, �. . - 1_·· r-- j ' �r- o· n I _ __J______l ______l_ _ ___ J____ _ A0!1l- r"ll�- J____ 1:'311- A'512- ,.. ;· . I'S/�- r=b_- .-- �- " q· _ � - i Figure V(c) : as in V(a ) for the S=-1 Figure V(d) : as in V(b) for S=-1 =cnegative parity baryons ; states that positive parity baryons ; states are predicted to decouple from are predicted to decouple from NK NK shown as stubs. are shown as stubs .

Table The Decay Amplitudes of 811(1535) and Sll(l650) I.

Sll (l535) Sll(l650) theory experiment theory experiment

5 8±3 9 9±2 Nn Nn + 5 + 2 2±1 ZK 9±2 - - no no - 2 ± 2±1 J\K no no - 3 - 4±1 2 (- 1±1 - 8 - 4±2 i'ITI ) yp +145 + 80±20 +90 +50±15 yn -120 -110±35 -35 -45±25 255

The Figures show a comparison between observed baryons and those predicted to be observable in S=O and -1 partial wave analyses . (The ground state baryons are not shown since there agreement is to within 10 MeV for all states . ) The Table gives the full comparison between predicted and measured decay amplitudes for just two of the observed states . The model is clearly crude and of limited numerical accuracy for a few of these hundreds of amplitudes it seems to simply fail -- but it appears nevertheless to have captured the principal features of the physics of baryons .

In particular, the ordering of the multiplets appears to be dominated by a simple anharmonic term, with the splitting effect playing a p-A 5 - 5 - crucial role in S = -1 and -2 (consider, e.g., the A ; - L / splitting in 2 2 the P-waves) . Within a given mode, the contact part of the hyperfine interaction produces spin splittings that are comparable to orbital splittings; these forces in turn cause large mixings between SU(6) multiplets like + + [56,2 ] and [70,2 ]. Finally , in certain key places the tensor force produces strong S = 1/2 S = 3/2 mixings : the amplitudes of Table I would , for ++ example , be completely wrong were it not for tensor forces .

The net effect of the model is to resolve many old problems with the quark model of baryons . Perhaps the most crucial of these is that the violent SU (6) breaking of the model has the effect , as seen from Figures V, of decoupling large numbers of predicted resonances from s-channel phase shift lZ) analyses , thereby resolving the problem of the "missing baryon resonances" .

D. What Else Can It Do ?

This simple baryon model has also had success in other areas which we will briefly mention:

15) baryon isomultiplet splittings : After the model succeeds in making the S=O to S=-1 transition (e .g. , uuu uus) , it is natural to let it make the + transitions through an isospin multiplet (e.g. , uuu uud ddu ddd) With + + + . m - m � 6 MeV, but no other new parameters , the model gives the isomultiplet d u splittings of Table II. 256

Table II. Baryon Isomultiplet Splittings

difference theory(MeV) experiment(MeV)

- n -1. 3 -1. 3 P+ o E - E -3.3 -3.1 ± 0.1 - o E - E +4.9 +4.9 ± 0.1 o - - � +6.8 +6.4 0.6 o ± L\� L\ -3.0 -2.6 ± 0.4 6.9 L\�*+ L\ - -5.9 ± 3.1 E - E *- -5.8 -5.1 0.7 *o ± *+ E +3. 7 +5.4 2.6 E *- *o ± - - := +3.8 +3.2 ± 0.6

configuration mixing in the nucleon16 •17) : The hyperfine interaction also has the effect of distorting the nucleonic wave function: it pushes the two parallel spin quarks toward the periphery and pulls the anti-parallel spin quark into the centre of the nucleon (i.e., it mixes [70,0+] into the pure [56,0+] nucleon) . This has many observable effects including: a) it gives the neutron a charge radius16•17) , b) it leads to violations of the Moorehouse 7) p *5 p 5 1 (N -+Ny) = (N * -+ Ny) leads selection rules A312 ;2 A112 ;2 = O, and c) it to violations of the Faiman Plane selection rulel7) A(A5; - = The - 2 + NK) J. predicted effects are in each case in agreement with experiment.

_baryon magnetic moments: The constituent quark masses of Section I lead, via their assumed Dirac magnetic moments (compare to equations (6) and (7)), to values for the baryon magnetic moments in good (though not perfect) 18) agreement with the observed values • charmed baryons: The model may easily be extended to the charmed19) and charmed-strange baryons20). The predicted C=l, S=O ground states seem to be in accord with experiment; there is as yet no information on other sectors.

One interesting prediction of the model, however,is that the p-A and E 1/ + - A 1; + splittings will have become sufficiently large to make the c 2 c 2 1 - lowest-lying orbital excitation A c ; 2 stable (or nearly stable) against strong decay. 257

III. Mesons , Too

What 's New in Mesons? A. The mesons have several new features beyond the change from qqq to qq constituents . One is that all colour factors in the Hamiltonian change 21) from 2/3 to 4/3 . There are many consequences of this stronger colour for example, the splitting is predicted (and observed) to be coupling: p-n ab out twice the l-N splitting. low pion mass is thus natural in this model A and the naive relation m (meson) 3? m (baryon) is satisfied by the 2 1 1 hyperfine-unperturbed masses: .!.m m) ( m ) 4 l4 l . TI + p 3 2 + 2 Mesons can, in addition to checking meson-baryon connections� , play a role all on their own in checking various assumptions of the model . Since the two body problem is easily solved numericall�, mesons can provide explicit information on the potential V(r) as well as checks on phenomena like the (near) ) cancellation of spin-orbit effects via Thomas precession5 . There is already evidence from the cc system for the efficacy of Thomas precession in reducing the strength of the spin-orbit interaction to give the observed ) x(2++) - x(l-H) - x(O++) spacings5 ; we shall see below that there may be additional evidence for this mechanism in light mesons .

What Is the Model for Mesons and Its Solution? B. For mesons we take the same Hamiltonian as for baryons , except that spin-orbit terms are shown explicitly, colour factors are changed, and qq 22) annihilation may now occur, so that p 2 p 2 _l _2_ l2 l2 l2 H = _ V H H H (1 ) 2m + 2m hyp so + 5 1 2 + + + A where the potentials are all twice as large as in baryons, where the spin-orbit 12 interaction H is given by so

l2 12 H H (16) so (CM) + so (TP)

(CM stands for "c olour magnetic" and TP for "Thomas precession") where 4a l2 s H = so (CM) 3 3r

Hl2 so (TP) 258

and where HA is the annihilation interaction of Figure VI which must be tacked on to any model of mesons.

Figure VI: the origin of the annihilation term H via gluon intermediate states.A

HA is in principle calculable, but in practice it requires the introduction of a new parameter A(nJPC) for each meson multiplet. This amplitude causes mixing between the uu, dd, and ss sectors so that after H-HA is used to calculate the masses of the excitations with quantum numbers nJPC in these sectors, HA creates.a mixing matrix of the form (we neglect SU(3) breaking and radial excitations here for simplicity)

m(uu)+A A A A m(dd)+A A (19) A A m(ss)+A

On diagonalisation this matrix gives one eigenvalue (assuming also mu = md for simplicity) m = m = m = m = m and two I=O eigenvectors and l=l,I3=0 uu- dd u- d d-u eigenvalues that depend on A. Phenomenologically, A(nJ PC) is normally small (corresponding to nearly "ideal" mixing to (u-;:;+ dd) and s;, and leading to a PC nonet with one isoscalar mass just above (below) m = if 6(nJ ) is just ° )z- I l below (above) e ideal � 35 ), though in the pseudoscalar nonet A is very large and leads to nearly "perfect" rnixing23) to the states

- - n (n') 1 1 + + = /2 [12 (uu dd) ss (20) - ° ° ° and to ep � 35 - 45 � -10 . While this understanding of I=O mesons is at least partially satisfactory, '·it makes I=O mesons much less useful for spectroscopic analyses and makes it clear that it is necessary to focus on the limited number of established I=l and 1/2 states. The exact solutions of the meson problem (for IfO) are most readily 259

obtained by numerical integration in a given (J,L,S ) sector of the Hamiltonian; tensor mixing (e.g., between 3 D and 3 s ) and spin orbit mixing for m i!m can 1 1 1 2 then be treated perturbatively in a (rapidly converging) nearby neighbour mixing expansion. A candidate fit to the I=l spectrum is shown in Figure VII;

Figure a fit to the I=l mesonVI I:spec trum (pre­ liminary) .

6 =

�)

work is in progress ZS) on this problem and our final solution (which may look very different from this early one!) will be reported once we have completed a decay analysis of mesons along the lines of the baryon analysis reported above.

C. What Else Can It Do? As with the baryon model, the meson model has had success in other areas: transition magnetic moments: The constituent masses and model wave functions lead (with a simple ansatz for dealing with relativistic ambiguities) to meson magnetic dipole decay rates in quite good (though not 6 perfect) agreement with experiment2 ) .

+ - o yy annihilation decays: Decays like � + µv , p + e e , and � + proceed through qq annihilation and so are sensitive to �(O), the qq wave function at zero relative coordinate. These processes (once again with a simple ansatz for dealing with relativistic ambiguities) are in reasonable accord with experiment. 260

charmed mesons : The dominant physical effects in light mesons remain apparent in charmed mesons . For example, the splittings K* -K, and D * -D n, m m p- are in roughly the ratios 1 : d/m : d/m as expected from (7) . s c isospin violation: As with the baryons , the model can be applied to the lS) breaking of isospin symmetry . In addition to ordering the observed isomultiplet mass differences , the model predicts dramatic violations of 27) isospin symmetry in certain hadronic decays .

IV . Comments on QCD Exotica

A. Do Multiquark States Exist?

It is absolutely certain that multiquark states exist : consider the deuteron, or to be more extreme , a uranium nucleus . This comment is not made flippantly: in potential models there is an analogy between possible "novel" multiquark states and nuclei . 28) In the bag model the existeftce of multiquark states is in some sense automatic: the static bag model has stationary states corresponding to any colour singlet combination of quarks and antiquarks . In this approach the "existence" of multiquark states is certain but their widths (and ��ence observabilities) are determined by the (presently uncalculablc) rate at which the bag undergoes fission into a "fall apart mode" (e.g. , ->- q + q ). '· q q Since the bag model was designed to confine , and in view of this problem, it would be prudent (perhaps I should say it would have been prudent) to be wary of drawing the conclusion from the bag model that novel multiquark states 29) (baryonia, five quark baryons , etc.) exist , at least without support from other models.

In the potential models the existence of multiquark states is an 30) intrinsically dynamical question . To examine the question one mus t, to take the simplest example of qqqq, set up a Schrodinger equation for the four body problem and seek states which exhib it binding in all three relative coordinates . Such states , if they exist, will necessarily be below threshold for falling apart into two q mesons (just as the deuteron is below q NN threshold) . It is actually straightforward to show that· the possible colour and spin recouplings of two pseudoscalar mesons do lead to an attractive 28) potential in certain channels (the cryptoexotic channels) ; calculations we 31) have just completed in fact have now proved that this effect ca� lead to a fully bound qqqq system under certain conditions . Thie may well be the reason * 28) why the s (980) and 6(980) -- two prime candidates for s;dd cryptoexotics -- 261

32> are found just below threshold . t is also clear that, as with the KK I deuteron, the binding may be SU(3) asymmetric so that there needn't be a full nonet of such cryptoexotic bound states . We hope to report more fully on these investigations shortly, including a report on the quark mass dependence of the effect important for the question of the possible existence of stable charm­ 3o , 3z> strange exotics just below DK threshold .

These questions are , as already stressed , closely related to the problem of the nucleon-nucleon force and preliminary results of an 33) investigation of this old problem along rather new lines look promising .

B. Do Glueballs Exist?

with multiquark states, the existence of glueballs is automatic As 34) in the bag , and so might be regarded once again with caution. On the other hand , the constituent quark model offers little guidance in this case (apart 35) from suggesting the idea of a massive constituent gluon mode1 with the corresponding sorts of colour dependent potentials) and I can only offer some comments here on possible alternatives .

One alternative to the (basically reasonable!) bag model picture is that glueballs exist but with widths that are very large so that the entire glueball spectrum is smeared into a continuum. A less drastic (but related) possibility is that the low-lying glueballs (assuming that they have non-exotic quantum numbers) have mixed with ordinary qq mesons and thereby become "diluted" in the spectrum. We certainly believe (see Section III B and Figure VI) that ordinary qq states with I=O do have a glue component and it is possible that at least the low-lying glueballs can only be disentangled qq spectroscopy from 36) once this experimentally difficult subject is itself more clearly resolved .

V. Conclusions

37) A. Does It Work At All? Why Despite the optimism of Section IA, it is surprising that the non­ relati vis tic potential mo dels work so well. Their phenomenological successes certainly lead one to try to understand the relation of such models to the quark model in its other guises : relativistic quark models like the bag, the quark-parton model, and current quarks .

We have recently examined -- in a very rudimentary exercise which we believe is still revealing -- the question of "relativisation" of the non- _ 13) relativistic quark mode1 . We took the similarity of the bag model ground 262

state phenomenology with that of the potential models as a clue that relativistic effects are, for the most part at least, absorbed into the parameters of non-relativistic descriptions, and that as relativistic effects are added to the model, they can mostly be eliminated by a process of "renormalisation" of masses, couplings, etc. (e.g., in the bag the non­ = ...§.__ relativistic result µp 2m becomes µ p - ...§.__2E ). To test this idea we took some typical momentum space wave functions for mesons and baryons (which are, as previously mentioned, actually rather relativistic) and calculated various properties (static and transition magnetic moments, GA/GV, annihilation amplitudes like f , f , fA, etc.) using full Dirac matrix elements instead of the usual static 7f(i. e.p , noni -relativistic) approximations. We found , as hoped, that most of the results of the non-relativistic quark model are practically unchanged with a modest renormalisation of its parameters. The few significant changes that did occur were, in fact, welcome : e.g. , GA/GV moved from its naive value of 5/3 to near}y its observed value of 1. 25, and fAj_ (the amplitude for the transition A +-+ w± via the axial current operative in , e.g. , A v ) changed from its naive value of zero to near the current T + 1 T algebra prediction. � The consistency between the constituent quark model and current algebra, exemplified by the cases of GA/GV and fA just mentioned, leads one to suspect that the massive constituent quark model1 may be a basis, appropriate for discussing soft phenomena, which is actually equivalent to the current quark picture. The mechanism of this equivalence might be that a) confinement occurs at r - A-l (where A is the QCD scale parameter) , b) as at this scale is (by definition) very large, as - 1, c) m ff - A at this scale (via confinement and/or the dressing of the quarkse 37) ), and a d) residual interactions of strength- - A occur. is to "conspire" to make m2 0, etc., so that the picture is The net effect -!- physically equivalent to the current algebra7f approach. Of course one picture or the other may certainly be more convenient for discussing particular phenomena: e.g., one would use constituent quarks to discuss baryons, and current quarks for discussing the effects of chiral symmetry. Such speculations aside, I believe it is now abundantly clear that one reason that the non-relativistic models work is that they provide a simple, calculable framework on which it is possible to hang the dominant physics of the quark model. Most all of the successes of the picture correspond to using 263

this framework to describe simple physical effects like the repulsion of parallel spins , the smaller chromomagnetic interactions of heavier quarks, and the slower frequencies of heavy quark excitations .

What Have We Learned and What Are Some Outstanding Problems? B. As just stated, it seems clear that the models , while crude, reflect the dominant physics of the qqq and qq states. In particular, we have learned from studying these systems that a) quark colour hyperfine interactions with the expected properties probably exist , b) the confinement potential indeed seems to be flavour independent , c) flavour symmetries are broken (apart from small electroweak contributions) by quark masses, and d) colour factors relate mesons and baryons.

There are, of course, many outstanding questions . The multiquark and glueball sectors remain largely unresolved (including the nucleon-nucleon problem) . The many "missing" baryon and meson resonances need to be found by looking at processes where they do not decouple. Finally , but by no means exhaustively, much theoretical work needs to be done on the foundations of the models, and especially on the elucidation of the relations between partons , current quarks, and constituent quarks .

In spite of these outstanding problems , I would conclude that QCD ingredients have dramatically improved the ab ility of quark models to describe soft phenomena.

Acknowledgements

The work reported here was almost all done in collaboration with others, especially Gabriel Karl , and including Kuang-Ta Chao, Les Copley, Stephen Godfrey, Cameron Hayne , Roman Koniuk, H.J. Lipkin , Kim Maltman, Hector Rubinstein, D.W.L. Sprung, Adam Schwimmer, and John Weinstein. 264

References

1. For a review , see my lectures at Erice in 197B, in "The New Aspects of Subnuclear Physics", edited by A. Zichichi, Proceedings of the XVI International School of Subnuclear Physics, Erice, 197B (Plenum, New York, 19BO) , p. 107. See also refs . (2) . This general approach to soft hadron physics flowed from the seminal papers of refs . (3) . 2. Gabriel Karl , in Proceedings of the XIX International Conference on High Energy Physics , Tokyo , 197B, edited by S. Homma, M. Kawaguchi, and H. Miyazawa (Phys . Soc. of Japan, Tokyo , 1979) , p. 135 ; O.W. Greenberg , Ann. Rev. of Nucl . and Part. Phys , 2B, 327 (1978) ; A.J.G. Hey in Proceedings of the 1979 EPS Conference on High--ilnergy Physics , Geneva, and in Proceedings of Baryon 19BO, Toronto, 1980, edited by Nathan Isgur (, 1981) , p'. 223; Jonathan Rosner, in Proceedings of the Advanced Studies Instl.tute on Techniques and Concepts of High Energy Physics , Virgin Islands, July , 1980 ; Nathan Isgur, in Proceedings of the XX International Conference on High Energy Physics, Madison, 19BO, edited by Loyal Durand and Lee Pondrom (AIP , New York, 19Bl) , p. 30 . 3. A. de Ruj ula, H. Georgi, and S.L. Glashow, Phys . Rev. Dl2 , 147 (1975) ; T. deGrand , R.L. Jaffe, K. Johnson, and J. Kiskis , Phys. ReV:- Dl2 , 2060(1975) . 4. This solution is very similar to some pre-confinement modelS-considered by Y. Nambu in "Preludes in Theoretical Physics': edited by A. de Shalit, H. Feshbach, and L. van Hove (North Holland, Ai/isterdam, 1966) and H.J. Lipkin, Phys . Lett. B45, 267(1973) . The dynamical basis for the restriction to colour singlets is, however, very different : see ref. (1) . 5. See the discussion of this point by Howard J, Schnitzer, in these Proceedings , and the references therein . See also A.B. Henriques, B.H. Kellet, and R.G. Moorhouse , Phys . Lett. 64B, B5(1976) ; H.J. Schnitzer, Phys . Lett . 65B, 239 (1976) ; 69B, 477(1977) ; Phys . Rev. DlB , 34B3(19 7B) ; Lai-Him Chan , Phys . Lett. 71B, 422 (1977); L.J. Reinders in Proceedings of Baryon 19BO , Toronto , 19BO, edited by Nathan Isgur (University of Toronto, 1981) , p. 203; F.E. Close and R.H. Dalitz in a paper presented to the Workshop on Low and Intermediate Energy Kaon-Nucleon Physics , University of Rome , 19BO . 6. Nathan Isgur and Gabriel Karl, Phys. Lett. 72B, 109 (1977) ; 74B, 353(197B) ; Phys . Rev. DlB, 41B7(197B) ; Dl9 , 2653 (1979) and-rJ23 , Bl7 (E) (l9BJ::"f; D20 , 1191 � (1979) . For-Y:elated work on baryons , see as examples refs . (7) . 7. D. Gromes and I.O. Stamatescu, Nucl. Phys . Bll2, 213(1976); W. Celmaster, Phys . Rev, Dl5 , 1391(1977) ; D. Gromes, Nucl. Phy�Bl30-, 1B(l977) ; L.J. Reinders , J. of Phys� G4 , 1241(197B) . B. O.W. Greenberg, Phys . Rev. Lett. 13, 59B(l964) ; O.W. Greenberg and M. Res • .koff, Phys . Rev. 163, 1B44 (1967) ; D.R. Divgi and O.W. Greenberg, Phys . Rev. �15, 2024 (196B) ; H. Resnikoff, Phys . Rev. DB, 199 (1971) . 9. R-:-li. Dalitz , in "High Energy Physics", edited by C. deWitt and M. Jacob (Gordon and Breach, New York, 1966) ; R.R. Horgan and R.H. Dalitz , Nucl. Phys . B66, 135 (1973) ; R.R. Horgan, Nucl. Pliys . B71, 514(1974) . 10. G. Morpurgo, Physics 2, 95(1965) , reprinted in J.J.J. Kokkedee , "The Quark

Model" , W.A. Benjamin, New-York, 1969 • 11. A general derivation of this rule was given in refs. (1) and (6) , but its origin was not understood. Recently K. C. Bowler, P.J. Corri, A.J.G. Hey and P.D. Jarvis, Phys . Rev. Lett . 45 , 97(19BO) , have shown that the rule follows from the Sp(l2,R ) spectrum generating algebra of the three body oscillator problem and have extended its application to higher excitations . 12. Roman Koniuk and Nathan Isgur, Phys . Rev. Lett. 44 , B45 (19BO) ; Phys . Rev. - D21, 1B6B(l9BO) and D23, BlB (E) (l9Bl) ; Roman Koniuk i; Proceedings of Baryon 19BO , Toronto, 19BO , edited by Nathan Isgur (University of Toronto, 1981) , p. 217. 13. C. Becchi and G. Morgurgo, Phys . Rev. 149, 12B4 (1966) ; 140B, 6B7(1965) ; Phys . Lett. 17, 352 (1965); A.N. Mitra and M-:--iloss , Phys . Rev. 15B,1630(1967) ; D. Faiman an�A.W. Hendry , ibid. 173,1720 (196B) ; H.J. Lipkin , Phys . Rep . BC , 265

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29. The nature of the qqqq bag states in view of their being unbound against bag fission has been considerably clarified recently by the introduction of a P-matrix analysis of the bag model predictions. See R.L. Jaffe and F.E. Low, Phys. Rev. Dl9, 2105(1979) . 30. For dis-;;;-ssion of these dynamics see H.J. Lipkin, Phys. Lett. 74B, 399 (1978) and H.J. Lipkin in "The Whys of Subnuclear Physics", edited b"Y°"A. Zichichi, Proceedings of the 1977 International School of Subnuclear Physics, Erice, Italy, 1977 (Plenum, New York) , p. 11. 31. John Weinstein and Nathan Isgur, in preparation. 32. Nathan Isgur and H.J. Lipkin, Phys . Lett. 99B, 151(1981). 33. Kim Maltman and Nathan Isgur, work in progress. 34. R.L. Jaffe and K. Johnson, Phys. Lett. 34, 1645(1976). 35 . See the talk by Ted Barnes in these Proceedings, and also D. Robson, Nucl. Phys . Bl30, 328(1977); J.J. Coyne, P.M. Fishbane, and S. Meshkov, Phys. Lett. 91B, 259(1980) . 36. F;;-;=-a recent look at such a possibility , see Jonathan L. Rosner, "Tests for Gluonium or Other Non-q Admixtures in the f(l270)", Minnesota preprint , February 1981. q 37. For a discussion of these issues see H.J. Lipkin's summary talk in the Proceedings of Baryon 1980, Toronto, 1980, edited by Nathan Isgur (University of Toronto, 1981), p. 461; see also I. Co�2n and H.J. Lipkin, Phys. Lett. 93B, 5b(l980) , and references therein. Cameron Hayne and Nathan Isgur, "Beyond the Wave Function at the Origin: JS-:-Some Momentum Dependent Effects in the Non-Relativistic Quark Model", University of Toronto report, March 1981 .