Quark interactions and colour chemistry

Quark Interactions and Colour Chemistry CHAN HONG-MO Rutherford and Appleton Laboratories, Chilton, Didcot, Oxon OXl l OQX, U.K.

ABSTRACT The interaction between quarks, according to the current theory of quantum chromodynamics, is formally rather similar to the electromagnetic interaction between electrons and nucleons, both being governed by locally gauge-invariant field theories. It is tempting therefore to discuss the spectroscopy of hadrons, which are quark composites bound by colour forces, in the same language as the spectroscopy of atoms and molecules which are bound states of electrons and nucleons held together by e.m. forces. Because of the difference in gauge groups, however, the dynamics are very different. Nonetheless, it appears likely that metastable multiquark hadron states can exist which are analogous to atoms and molecules in QED. In these lectures, tentative steps are taken in developing the rudiments of a new 'colour chemistry'of these 'atoms' and 'molecules' KEYWORDS Gauge-invariance, iums.

colour,

confinement,

bags, colour atoms and molecules,

baryon-

The days are already long past when what we still sometimes call 'elementary' particles such as nucleons were indeed considered as elementary. The established existence of several hundred of these particles with varying properties has made the assumption of their elementarity highly unpalatable. Today, it is generally accepted that hadrons are made up of quark constituents. Quarks are believed to come in several flavours u, d, s, c, b, ..., and each in three colours. I shall not bore you with yet another reminder of how these conclusions were laboriously pieced together over the last decades through both theoretical and phenomenological considerations. I Together with the leptons e, ~, ~, ... and their associated neutrinos, quarks form, at our present stage of understanding, the fundamental building blocks of matter. Indeed, since quarks and leptons in the last few years have also started to proliferate, theoretical schemes are already being constructed in which quarks and leptons themselves are no longer fundamental but are in turn made out of some smaller number of even more fundamental objects. Such schemes as

I See e.g. the review at this meeting by Professor R.H. Dalitz

301

302

Chan Hong-Ho

exist, however, are still highly speculative, and need not as yet be considered seriously. In these lectures, therefore, I propose to regard quarks as elementary. What concerns us here in this school is how quarks interact among themselves, and in what way their interaction is reflected in the properties of nuclei. Being a particle physicist, I shall be discussing the first topic rather than the second, since even my limited understanding of the former is superior to my profound ignorance in the latter field. Two years ago, at about this time of year, I was lecturing here on a related subject in the school on exotic atoms which was held in parallel to Sir Denys' school on heavy ions, and I was honoured to be asked by Sir Denys to give an evening lecture for the entertainment of his nuclear physics participants. I remembel introducing the topic by remarking as a light relief how Nature seems to delight in the art-form of numerous repetitions with variations as that favoured by ancient Chinese poets (Anonymous, c. 500 B.C.). Molecules-atoms -nucleih a d r o n s - a n d now quarks. Every time we unwrap one of Nature's puzzles, we find another layer of substructure bearing some similarity to the last. Every time, the scientist hesitates and says: "Surely, this must be the end~" Thus, the Greeks thought that molecules were indivisible, while at the hadron level, Smatrix theorists (including short-sighted individuals like myself) spent years trying to foreclose Nature's hierarchical pranks by establishing a nuclear democracy, but so far, it has not worked. Nature repeated herself every time constituents are made of subconstituents - but each time there is a twist. Her variations are so ingenious that it has taken the concerted effort of all physicists plus some daring flights of imagination of a selected few to see the similarity at all, and when we finally see the light, we are all astonished. In my following three lectures, I wish further to enlarge upon this theme through hadron spectroscopy. Indeed, rather unfortunately for me, ever since Sir Denys first enlisted me nearly two years ago to give these lectures, I have stopped working in hadron physics altogether and started on something entirely different. Therefore, I not only have nothing new here to report, but have not even been able to keep up-to-date on the work of our active colleagues here. For this reason, I am hoping that the choice of a somewhat poetic and entertaining theme, may alleviate to some extent the obvious lack of material in my lectures. How do quarks interact? The only likely theory we have at present is quantum chromodynamics, which was constructed in close analogy to electromagnetism. Let me therefore first remind you how electromagnetic interactions between charged particles come about. A charged particle is represented in quantum theory by a complex wave function. The phase of the wave function is unphysical and cannot be measured. This means that if we change the phase of the wave function thus: ÷ 4' = exp i e ~ @

(I)

all measurable quantities remain unchanged. However, the wave function ~ depends on the space-time point x. Should physics remain invariant only if one makes the same phase change at all space-time points? It seems somewhat contradictory to the concept of locality to make a wave function in the laboratory on earth dependent on a phase change at some distant point, say behind the moon. It would appeaz more reasonable to require that physics be unaffected by phase changes at each space-time point independently. The result is the assumption of local gauge invariance. At every point in space-time then we have a phase factor, otherwise called an element of the gauge group U(1). Such a geometrical structure is called by the

Quark Interactions

and Colour Chemistry

303

m a t h e m a t i c i a n s a U(1) fibre bundle. In this structure we have the freedom to associate each phase $(x) at any point x with a phase ~(x + dx) at a n e i g h b o u r i n g point x + dx by specifying that: ~(x + dx) = ~(x) + e A

dx

(2)

where A. is called a connection on the fibre bundle by the mathematicians, and a gauge potentlal by us physicists. Under a local change of phase, i.e. a local gauge transformation,

$(x) + $'(x) = $(x) + e~(x), from which we deduce

(3)

that ~'(x + dx) = $'(x) + e ( A

or in other words, A

+ ~U ~)

dxp

(4)

transforms as A' = A

+ ~

~

(5)

Suppose now we wish to construct a dynamical theory of the system by specifying a Lagrangian, then to preserve local gauge invariance the Lagrangian must be a function only of gauge covariant quantities. The gauge potential AU transforming according to (5), is itself not gauge covariant but from it one can construct the following gauge invariant tensor: f

~

= ~

p

A

v

- ~

~

A

(6)

p

w h i c h is called the curvature tensor by the mathematicians and the field tensor by the physicists. Geometrically, the connection A~ specifies a change of phase alon~ any curve, and fu~ dxp dxv is the total change in phase at x obtained from the connection (2) after traversing the closed circuit x ÷ x + dx~ + x + dxu + dx~ ÷ x + dx~ ÷ x on the ~ plane as depicted in Fig. i. Being a difference between two phases at the same space-time point x, f ~ is obviously invariant under local

x+

dxy

x+dx)~+dxy

t

×

x ÷ dx}~ FIG

1

304

Chan Hong-Mo

gauge transformations. Similarly, given any wave function ~, its simple derivativ~ 8D @, being a difference of two phases at two neighbouring points x and x + dx~, i~ not invariant under independent phase changes at the two points, but its 'covariant derivative' D~ ~ = ( ~

- i e A ) ~

(7)

which is the difference between the phase of $ at x + dx~ and the phase, also at x + dx~, obtained from the phase of $ at x via the connection (2), is invariant under local gauge transformations. From such gauge invariant quantities f~v and D ~ one can now construct a Lagrangian for a dynamical theory. The simplest such Lagrangian -¼ f~v f ~ is the Lagrangian

- m ~ ~4i~

(8)

D~ Y~

for electrodynaraics.

Thus, one sees that by pursuing what are essentially geometrical considerations, one is led quite naturally to the Lagrangian (8). We are not suggesting of course that electrodynamics can be 'derived' in this way - it clearly cannot - but only that such a procedure allows one to appreciate the geometrical significance of the various quantities involved which may lead eventually to their further generalisations. We note that the Lagrangian (8) contains via D~ $ a term of the form e ~ A~ y~ ~ which represents in physicists' language the interaction between the electron fields $ via the exchange of photons as represented by the gauge potential A . We now turn to the interaction between quarks. Quarks are supposed to occur in three colours which are however completely indistinguishable. This means that if we represent the wave function of a quark by a colour triplet, then physics is completely unchanged if we choose to rotate the wave function by a SU(3) transformation, thus @ ÷ @' = exp (ig~ a la/2 ) @.

(9)

Again, the same question arises as in the case of electromagnetism whether physics should remain invariant only under the same SU(3) rotation for all space-time points, or under SU(3) rotations at each point independently, As before, considerations of locality would suggest the latter, i.e. our theory should have loca] SU(3) gauge invariance. In parallel to electromagnetism, we have then an SU(3) fibre bundle on which we can define connections or gauge potentials A~. Through such a connection each element of the gauge group at a point x is associated with an element at a neighbouring point x + dx. As in electromagnetism, one can then deduce that A~ transforms as A~(Ia/2)

~ Aa'(la/2)

= ~ [A~(Za/2)]

~-i + ! ~ ~ g

~-i

(i0)

for any local gauge transformation with ~ belonging to the gauge group SU(3). Further, from the connection A a, the curvature tensor (field intensity) U F a = $~ A a D Aa + fabc A b A c (ii) and the covariant

derivative

D

of any wave function

~ = [~. - i g A a~ (~a12)]~

,,

(12)

Quark Interactions

305

and Colour Chemistry

can be constructed which carry the same geometrical significance as the corresponding quantities in the U(1) theory of electromagnetism. Hence we can build gauge invariant Lagrangians for which the simplest example in analogy to (8) is ¼ F

a

F

~a - m ~ ~ ÷i ~ D

y~ ~

(13)

namely the Lagrangian for chromodynamics. Again, the Lagrangian contains through the covariant derivative D~ ~ a term of the form g ~ A~ y~ (~a/2) ~ which one may interpret as the interaction between quarks via the exchange of ~luons' as represented by the Yang-Mills gauge potential A~. The formal similarity between this colour theory of quarks to electrodynamics is then quite transparent.2 If the theory of quark interactions is indeed conceptually so similar to electrodynamics, how is it that physicists took so long over its discovery? The reason I think is the following. In electromagnetism, the test charges such as electrons or protons can exist isolated, so that one can perform actual experiments to study their interactions. Thus the basic equations of electrodynamics were constructed by Maxwell essentially as a phenomenological theory of Faraday's experimental results. All the philosophical concepts associated with the equations such as local gauge invariance by which we now set so much store came only afterwards as theoretical hindsight. For chromodynamies, on the other hand, the process is more or less reversed. As far as we know, there are no isolated colour charges with which we may perform experiments. Non-Abelian gauge theories were a product of the mind ~pecifically of Yang and Mills, 1954), and only when it was realised afterwards that the particular theory as embodied in the Lagrangian (13) may actually correspond to quark dynamics that it takes on a physical significance. However, writing down a Lagrangian is one thing, understanding what the Lagrangian implies is another. Indeed, the equations for non-Abelian gauge theories are so complicated we are still far from a solution. In spite of the enormous effort spent on QCD in the last few years, our knowledge is still rather limited. Nonetheless, we know enough to surmise that in spite of the formal similarity with electrodynamics, QCD has vastly different dynamical properties. This is where Nature has taken another of her ingenious twists. We know that the theory is asymptotically free, namely that at short distances the (renormalised) effective coupling becomes weak so that perturbation theory may be applied. 3 This has led to a large body of theoretical and phenomenological work on so-called hard QCD processes which have had some successes. Concerning the large distance behaviour on the other hand, we know that the theory can exist in one of three phases: ('t H o o f t , 1979). (I) Confinement phase, where colour-electric charges are confined and colourmagnetic charges interact via short-ranged forces. (II) Higgs phase, where colour magnetic charges are confined and colour electric charges interact via short-ranged forces. (III) Coulomb phase, where both colour-electric free but interact via long-ranged interactions.

and colour-magnetic

charges are

Now, quarks carry what were referred to above as colour-electric charges. Then, given the fact that no well-established experimental evidence as yet exist for

2 3

See also the lectures of Professor K. Gottfried in this school. See e.g. the lectures of Professors K. Gottfried and G. Baym

in this school.

306

Chan Hong-Mo

free quarks in Nature in spite of intensive searches conducted over the last decade, it is natural to assume that we are living in the confined phase (I). However, there is no proof that this is the case, nor do we clearly understand what way this phase can be realised.

in

Ironically, we do understand quite explicitly how the Higgs phase (II) can be realised, i.e. how monopoles are confined in certain gauge theories. Although this may not be of direct relevance to the problem of real interest in phase (I), it may possibly afford some insight on confinement mechanisms in general, and Sir Denys has asked me to include a brief summary of the present knowledge here for your entertainment. The Higgs phase is realised when the gauge symmetry is completely broken by a spontaneous Higgs mechanism. It is sufficient to illustrat~ this in a U(]) theory for simplicity. Given that there is a magnetic monopole somewhere in space, then it can be shown purely by topological arguments that on any sphere 4 surrounding the monopole, any complex wave function * (x) must at least have one zero. This is a special case of a quite general theorem in the theory of fibre bundles which was first proved in this context by Wu and Yang (1956). Since this is true on every sphere surrounding the monopole, it follows that the monopole must be attached to a string of such zeros. Now, if the symmetry is broken, i.e. if there exist in the theory some complex scalar (Higgs) field ~(x) whose v a c u u m expectation value<~} # O, then ~ must depart from its vacuum values by a finite amount along its string of zeros. In other words, the string must carry a finite amount of energy per unit length. Such a string will continue indefinitely until it meets another monopole of the opposite magnetic charge. Hence a monopole-antimonopole system will carry by virtue of the string connecting them an energy proportional to their separation. They will thus experience something like a linearly-rising potential and will remain permanently attached to each other. This is what we mean by confinement. Such considerations can be extended to non-Abelian gauge theories. To break the symmetry completely now, one may need to introduce more than one Higgs field . The statement above about zeros is now replaced by a statement on the Higgs fields' linear dependence. Nonetheless, for completely broken symmetries it can still be shown that monopoles are permanently confined by strings just as in the Abelian case. Now, the string confining monopoles in the theories described above are closely related to, and were indeed originally modelled by Nielsen and Olesen (1973) on the magnetic vortex lines in superconductors. 5 The fact that the magnetic flux emanating from a monopole is concentrated along its attached string can be seen as follows. (Again, we shall illustrate only with the Abelian case as the simplest example - the extension to non-Abelian theories is possible though not trivial). The Lagrangian of the theory contains a term (D~ ~)2 To minimise the energy therefore, we want ~ ~ = O. This implies that

[ ~ , Dv] ~ ~ F

# = 0

(14)

so that for ~ # O, F ~ must . . vanish. . . But F ~ v cannot be zero everywhere surrounding a monopole slnce by deflnltlon, the magnetlc charge g

= 4-~-

Fij d cij # 0 s

4 5

lectures

(15)

re

By sphere here we mean any closed surface topologically See also Professor G. Baym's

on a sphere

in the school.

equivalent

to a sphere.

Quark Interactions

and Colour Chemistry

307

Thus, the most energetically economical arrangement is to have all the magnetic flux directed along the confining string where ~ = O. The understanding of monopole ('magnetic' charge) confinement above suggests perhaps a similar picture for quark ('electric' charge) confinement. Instead of the vacuum being a superconductor we want a vacuum with dielectric constant e = O, and instead of 'magnetic' vortices as confining strings, we have 'electric' vortices, etc. 6 However, whether such a situation is attained is a dynamical question which can only be answered by solving the dynamical equations, which is far from being settled although the recent efforts in lattice gauge theories in this direction is slowly getting more hopeful. 7 Beyond this, purely logical deductions directly from the field theory lead no further at present, and it is up to us, using phenomenological or other means, to convince ourselves that QCD does describe hadron physics and is not just a figment of our imagination. One direction in which we may hope to verify the theory of chromodynamics is hadron spectroscopy. Hadrons are believed to be made up of quark constituents interacting via the exchange of colour (Yang-Mills) quanta, in much the same way that atoms and molecules are constructed out of electrons and nuclei interacting via photon exchange. It is obviously tempting to draw a parallel between the two cases so that hadron physics, which is the study of these new "atoms" and "molecules" becomes just the colour equivalent of chemistry. And just as the spectroscopy of atoms can teach us about quantum electrodynamics, so hadron spectroscopy can be made instrumental in probing the validity of QCD. If the analogy suggested above is indeed valid, then hadron spectroscopy could potentially be as rich in complexity, and develop into a branch of study as full of interest as ordinary (electrical) chemistry is today. However, the physicist's interest is in fundamentals, and in testing any fundamental theory, our standard tactic is to pick up a simple system and study that in detail. Thus in quantum electrodynamics, most of our attention is directed towards the hydrogen atom. Seldom are physicists interested in complexities just for complexities' sake. Now the simplest (electrical) atom H, is obtained by combining two constituents of opposite charges, namely e and p. So in hadron spectroscopy, our first concern should be the simplest colour atoms, combining a quark Q wif.h colour charge 3 with an antiquark ~ with the opposite colour charge 3, thus (Q3~)I, where Q = u, d, s, c, b, . . . . Because of the non-Abelian nature of the colour group SU(3), another simple atom can be obtained from three quarks as follows; ((Q3Q3)3Q3)I. These are the familiar mesons and baryons which account for nearly all entries in the Rosenfeld tables. However, there is in colour chemistry one good reason why we ought to be interested even for fundamental purposes in atoms and molecules more complicated than just these hydrogenlike mesons and baryons. In QED, one knows how to calculate accurately the properties of hydrogen so that by comparing experiment with theory, one has a direct and thorough check of their consistency. For chromodynamics, on the other hand, no such calculational methods exist. Instead, one has only make-shift model simulations, such as the potential model, the bag model etc., whose qualitative verification in some special cases is no serious test of the basic theory. Under these circumstances the best tactic would seem to be to use the ordinary hydrogen-like mesons and baryons to refine our QCD-inspired picture of quark interactions, and apply the result to more complicated colour atoms and molecules. If the application continues to hold, we gain confidence in our surmises.

6,7

See e.g. Professor G. Baym's lectures in this school

308

Chan ttong-Mo

Take f i r s t the concept of colour itself. I t was i n t r o d u c e d i n i t i a l l y as a d e v i c e t o o v e r c o m e an a p p a r e n t c o n t r a d i c t i o n b e t w e e n s p i n and s t a t i s t i c s for quarks in g r o u n d s t a t e b a r y o n s . 8 H o w e v e r , b e i n g now a new d e g r e e o f f r e e d o m , c o l o u r s h o u l d in principle l a r g e l y i n c r e a s e t h e c o m p l e x i t y of the h a d r o n s p e c t r u m , j u s t as the introduction o f t h e e l e c t r o n s p i n made t h e a t o m i c s p e c t r u m much r i c h e r , b u t t h i s i s n o t o b s e r v e d i n t h e s p e c t r o s c o p y o f t h e o r d i n a r y Q~ m e s o n s and QQQ b a r y o n s . To a v o i d c o n t r a d i c t i o n w i t h e x p e r i m e n t , we t h e n p o s t u l a t e d that colour is confined, meaning that only colour singlets can exist isolated. A l t h o u g h by now we h a v e grown u s e d t o t h e i d e a a n d m a y b e e v e n b e g i n t o u n d e r s t a n d t h e o r e t i c a l l y how t h i s comes a b o u t , i t r e p r e s e n t e d n o n e t h e l e s s a d e p a r t u r e f r o m o u r n o r m a l e x p e r ience. Can we verify that there is indeed such a hidden colour degree of freedom in hadron spectroscopy? This is possible only if we consider hadrons more complicated than the ordinary QQ mesons and QQQ baryons. Thus, two quarks each of colour 3 can in principle exist in two colour states: QQ:

3 x 3 = ~ + 6 ,

(16)

but,_by hypothesis only colour singlet hadrons exist, so that only the colour state (QQ)3 can combine with a third quark Q3 to form a baryon ((QQ)HQ3)!, since 6 x 3 does not contain a singlet. However, the colour confinement hypothesis will not hide the effect of colour in more complicated cases. For example, for a hadron containing QQQQ, both the colour QQ states in (16) can form hadrons, thus ( ( Q Q ) ~ ( ~ ) 3 ) I and !!QQ)6(QQ)~) I, since both ~ x 3 and 6 x 6 contain a singlet. The spectrum of QQQQ hadrons should thus be doubled because of colour. Therefore, if we were able to verify experimentally this doubling of the spectrum in this case, we would have confirmed the existence of colour as a degree of freedom. Next,consider the confining mechanism, for which we have at present only vague theoretical understanding. In practical application to spectroscopy, we employ certain model simulations which are based on, but by no means deduced from quantum chromodynamics. Although such simulations may be able to describe a certain portion of experimental data, it does not follow that one can then conclude that they give us already a correct picture of the confinement mechanism. For example, a potential model with a linear or logarithmic potential gives an adequate description of the ordinary QQ and QQQ hadrons including in particular those with heavy quark constituents c and b. 9 However, if one takes these potentials and assumes naively that they operate between any quark pair including quarks belonging to different colour singlet hadrons as genuine potentials should, then it has been shown (Matsuyama and Miyazawa, 1978) that they lead to colour Van der Waal forces between hadrons stronger than gravitation even up to distances of ~ 1 km which is clearly inadmissable. I0 In other words one would need a more sophisticated simulation of the confinement mechanism when dealing with multi-quark systems in general. More promising are models of the bag type in which confinement is simulated by a confinement pressure. ]I Here colour forces are by definition zero outside a hadron

8

9 I0 II

See e.g. Professor R. Dalitz's lectures in this school. See e.g. the lectures of Professors R. Dalitz and K. Gottfried in this school. See also lectures of Professor Robson in this school. See e.g. lectures of Professor G.E. Brown in this school

Quark Interactions

and Colour Chemistry

309

so that one has no problem with the unrealistic Van der Waals interactions encountered above. Even here, however, there are many basic aspects which can only be tested by multi-quark spectroscopy. For example, in the original MIT version of the bag model, Regge trajectories are asymptotically linear, namely for s large (s) ~ ~

o

+ ~' s

(17)

where ~ is the spin and M = /s is the mass of resonances belonging to the same Regge family. This fact can be understood as follows. In the bag model, high angular momentum hadrons can be pictured as an elongated object with its constituents separated into two clusters located at the ends and connected by a colour flux tube, as illustrated in Fig. 2. For sufficiently high orbital angular momentum ~, both the mass and the angular momentum of the system are

:'9 FIG 2

carried mainly in the colour flux tube, which has however uniform thickness in the bag model because of the postulated universal confinement pressure. 12 The mass M of the hadron is then proportional to the length L of the flux tube, and its angular momentum ~ to L2, and hence we have (17). The value of the Regge slope ~' has been calculated in the MIT bag model by Johnson and Thorn (1976) who give c~' =

I

4/2 73/2

i

I

I

f'~ x

/~

/B

(18)

s

where B is the bag pressure, a s the strong coupling constant and ~ x is the quadratic Casimir operator for the colour x carried by the flux tube. For ordinary QQ mesons and QQQ baryons, the colour x = 3, and ~ x = 16/3. Inserting the value of B obtained by fitting the masses of the lowest baryon states, one obtains ~' = .85 GeV -2. Now, experimentally it has been known for a long time already that Regge trajectories of ordinary meson and baryon resonances are linear to a

12

See also lectures of Professor G. Baym in this school.

310

Chan Hong-Mo

good accuracy with a common slope a' ~.9 GeV-2 Hence the prediction (17) and (18) can justly be claimed as a triumph of the bag model. One aspect of the formula (18) however remains untested, namely the dependence of ~' on the colour x, which is after all the aspect which relies most on chromodynamics. So long as we remain with only QQ mesons and QQQ baryons however, this variation with x cannot be seen since x = 3 always. It can be observed only in hadrons containing more constituents. For example, as explained above, for a h!dron with QQQQ, the diquark QQ can exist in colour ~ or 6 and the antidiquark QQ in 3 or 6. Th,,s high spin hadron states in which the diquarks are separated by a high angular m o m e n t u m ~, ~

(QQ)X

(~)x

(19)

will have flux tubes carrying colour x = ~ or 6. According to (18) then, Regge tra'ectoriesj with x = 3 will have ~ - - t husual e slope a'~ ~ .9 ..GeV-2~ but those with x = 6 will have a different slope ¢@3/~6 x .9 GeV-L,i.e. ~(16/3)/(40/3) X .9 = .63 GeV -2. Such an effect if observed will be a triumph both for the MIT bag model and for chromodynamics. As a third example, consider the spin-dependent part of the interaction between quark constituents. It is suggested that spin-dependent forces, being shortranged, may by asymptotic freedom be approximated by perturbation theory. In particular, the one-gluon-exchange diagram between two quarks is quite similar in structure to the one-photon-exchange diagram between electrons. Through the Breit reduction familiar already in atomic physics, one obtains a spin-dependent interaction between quarks of the form (see e.g. de Rujula, Georgi and Glashow, 1975): 13

vspin lj

=-

C,S(~ %a ~ I 2v i ~ ) 3 a +--4r 3

1 m.m. l j

(P r - i x-

63(r) (o_i • o_j) --

" ~i)

- m. j

1

+ m m.. - i j + -

4r 3

m 1. m . j

( xr p j _

(r x p i • o_ j

-o_i

•

~.

--j

+

3

(20)

• .0) _ ;

- r x p j • o_i )

(~.

--1

•

r)

--

(0.

--j

•

r_)

where la (a = I, ..., 8) are 3 x 3 Gell-Mann matrices representing the colour charges of the quark constituents while o r (r = i, 2, 3) are 2 x 2 Pauli matrices representing their spins, and for antiquarks, the following replacements in the formula are understood: %a ~ _ la,, ur ÷ -o r* . Of the three terms in (20), the first spin-spin term proportional to o . • o . is the least dependent on the details --1 --j of the confinement mechanism. Being a 6-function potential it contributes only to relative s-wave states, and leads for example to the splitting between the triplet QQ meson state (p) and the singlet state (7), or between the J = 3/2 QQQ baryon state (A) and the J = 1/2 state (N). In QQ mesons, the quark-antiquark pair has colour I, giving a

a

16

Z Xl x2 =--i-

(21)

a

13

See e.g.

the lectures of Professors

Robson and C.W. Wang in this school.

Quark Interactions

311

and Colour Chemistry

while in baryons, any QQ pair must necessarily have colour 3, giving thus lust an overall factor X~ %~ = ~ 1 3 3

a

for all pairs. yields

(22)

An easy calculation of o . • o . for the various spin states then --i --j m

-

0

m

=

~

-

~

]A (- -¢~-) J s

(4)

K_

,

(23)

and

m A - m N : - es(~)

(- 6) K B .

(24)

where K M and K depend on the quark wave functions. Note that a comparison of .m . . (23) o ~ ( 2 4 ) individually to experiment affords no test of basic chromodynamics, but only of the confinement model used to calculate the wave functions. Further, since the values of K M and K B are in general unrelated in (23) and (24), only the sign of the splitting remains as an unambiguous prediction of (20), which is not much indeed for such an ambitious theory as QCDI Again, the situation is improved by extending the consideration to multiquark hadrons, since the existence of constituent pairs with different colours, such as (QQ)~ and (QQ)6 in ( Q Q ~ ) , will give much greater scope for testing the crucial colour dependence of the formula (20). Also the experience gained in the QQ and QQQ sector will have refined the phenomenology sufficiently to make consistency checks more meaningful. For these reasons, we believe that the spectroscopy of hadrons other than the simplest hydrogen-like QQ mesons and QQQ baryons are of fundamental interest. But will such hadrons exist? Now in electrodynamics very complex atoms and molecules can be obtained by combining many constituents. Can we do the same in colour chemistry? This is not obvious. In ordinary atoms, we have strong short-ranged nuclear forces which overcome the electrical repulsion between like-charged protons, concentrating them in the nucleus; this is what makes a complex atom stable. In a system of coloured quarks, however, we know as yet of no equivalent to the nuclear forces between nucleons in atoms, so that a complex system of many quarks may_readily fall apart. Take again the simple example of QQQQ. This can form two QQ pairs each in a colour singlet, namely mesons, and there being now no colour forces to hold these colour neutral mesons together, the system can simply dissociate thus (QQQQ) + (Q~)I + (Q~)I. One can see in general that the more constituents a quark system contains the more channels there are into which it may dissociate. Thus, it would appear that barring the possible discovery in the future of a new short-range interaction between quark constituents which can play the role of the nuclear forces in atoms, the construction of stable complex hadrons out of many quarks will require a certain amount of sophistication. Of course, it may happen that due to some enhanced interaction between its constituents a multi-quark system acquires a mass which is below the threshold of all the channels into which the system may dissociate. In such a case, the system will appear as a metastable object. For example, the colour magnetic spin-spin interaction between quarks in (20) is known empirically to be quite strong, and given the right conditions may in certain cases be sufficient to shift the mass of a multiquark system below all its dissociation thresholds. In order to explore this possibility, let us write first the spin-spin interaction terms in the Hamiltonian as:

312

Chan Hong-Mo H (I) =

[ •

l>j

where the sum runs

V.. .

(25)

lj

over all pairs of constituents with

V (I) ! I a I~ "' : - C. l] lj i ]

[ ori r

o.r , ]

(26)

and several ill-defined quantities such as ~ and factors dependent on the quark wave function having been absorbed into th~ single parameter C.. which we shall try to estimate from experiment. In order to calculate the mass1~f a system containing several quarks all in the s-wave state, we need to diagonalise the matrix H(1)above, and it may often happen in diagonalising a big matrix with elements of comparable size that one state gets a particularly large eigenvalue. Indeed, for the matrix (25), this tends to be the case for the state with maximal symmetry in colour and spin, or because of the Pauli principle, the state which is most antisymmetric in flavour. To see this, take as first approximation Cii to be the same for all pairs, independent of their flavours etc. The Hamiltonian ~hen becomes

H (I) =

- C

I i>j

which, being highly symmetric, (Jaffe, 1977): 14

H (I) = - C

~ %a a

i

la I or~ J r

-

can be readily diagonalised.

½ ~6(TOT) + ½ ~3(TOT)

o .r J

(27)

The answer is

+ ~6 (Q) + ~6(9) - ~3(q)

- ~3(Q)

(28)

4 sTOT (sTOT 8 8 + ~ + I) - -~ S O (SQ + i) - ~ S~(S~ + i) 8N

where ~6 represents the quadratic operator of colour - spin SU(6) a n d S 3 the same for colour SU(3). One notes that by far the biggest contribution in (28) comes from the first t e r m s w i t h ~ 6 . For a system of quarks only (no antiquarks) ~6(Q) gives attraction and takes a maximum value for the most syrmnetric representation in colour-spin. For example, for a system of three quarks QQQ, the greatest attraction attains for the 56-dimensional totally symmetric representation of colour-spin SU(6), giving a value of - 14C for (28). Because of the Pauli principle, such a state can exist only when all three quarks have different flavours, e.g. uds, which we may denote by (uds)~ with I = O, where the superscript denotes the total colour and the subscript the multiplicity of the total spin. On~ can estimate the colour-magnetic binding energy of (uds)~ by fitting the value of C in (24) to experiment giving C ~ 20 MeV, and a large attractive value of < H ~ - 280 MeV for (uds)~ . Now the state (uds)~, being coloured, is not a hadr~n, but it can exist e.g. Zas (uds)~ (uds)~ in a six quark hadron, which will then possess an enormous colour-magnetic binding. This may well bring its mass below the threshold of its natural dissociation channel into AA, and gives the famous

14

See also the lectures of Professor G.E. Brown in this school

Quark Interactions

and Colour Chemistry

313

"stable dihyperon" predicted first by Jaffe (1977b). Experimental searches for this state have so far been unsuccessful, but need not as yet imply that the state is nonexistent since the production cross section is expected to be very small in any case. With some ingenuity, one may hope to construct theoretically further metastable states with the colour magnetic spin-spin interaction. It is not easy, however, and I know of only one other likely example, suggested first by Gelmini (1980), and also by Isgur and Lipkin (1981). Take a QQQQ system all in the s-wave state. If all the constituents are u, d quarks, we can put all Cij in (26) approximately as C ~ 20 MeV and use the diagonal form (28). It can then be seen that a particular state with IG(j P) = O+(O +) acquires a huge colour-magnetic binding of the ordel of - 40 C ~ - 800 MeV. This was first noted by Jaffe (1977a) who identified the state with the broad structure in ~ scattering called o at around 600 MeV mass. In spite of the huge colour-magnetic binding however, this state is not stable against dissociation into ~ (hence the broad width), because it happens that the pions themselves have a large colour magnetic binding of the order - 16 C ~ 320 MeV for each ~. Now the spin-spin interaction on the other hand is known empirically to be much weaker for heavy quarks, decreasing roughly as Cij ~ i/mim i . Thus if one replaces a q~ pair in (qq~) by a heavy quark pair, say (qc~E) or (qbqb), the colour magnetic binding becomes negligible in the dissociation products (q~) and ~ b ) but remains still largely in force between the q~ pair in the four-quark state, which may thus become stable against dissociation. Such 'narrow' resonances if existing, may be observable as decay products of 4" and T" in e+e - experiments with small cross sections, but are best looked for in the formation channel of low energy p~ colliders with good resolution, e.g. the LEAR project at CERN. 15 At present, one has no experimental indication of their existence. Both the dihyperon of Jaffe and the possibly stable four-quark states just discussed may be regarded as examples of complex colour atoms. However, it is already clear from their construction that they are rare objects, whose existence in a multiquark system is the exception not the rule. A naive application of the bag model of course will give many states for a multiquark system whether they lie above or below the thresholds of their dissociation channels. In the former case, however, they are merely an artifice of the boundary conditions put into the model to simulate confinement, and should not be regarded as genuine hadron states. 16 Their significance then is similar to the zeros of the R-matrix in the WignerEisenbud formalism of a compound nucleus; they represent structures in the scattering continuum,.but are no real analogues of bound states or resonances (Jaffe and Low, 1979). Thus, it would appear that our attempt at developing a colour chemistry is so far not very successful. There are not going to be many complex atoms other than the hydrogen-like QQ mesons and QQQ baryons which we already know. Can one however construct complex molecules out of these simple atoms? In ordinary (electrical) chemistry, we have two common types of molecules: - (i) co-valent molecules such as H 2 in which two electrically neutral atoms combine with each other by sharing a pair of their valence electrons, and (ii) ionic (electrovalent) molecules such as Na+CI - in which two oppositely charged ions are held together by electrostatic forces. Can we have analogues of these in colour chemistry? Consider first covalent molecules. Take for example two nucleons, which being colour singlets are colour analogues of neutral atoms. We make them share a pair

15 16

See lectures of Professor B. Povh in this school. See also lectures of Professor C.W. Wang in this school.

314

Chan Hong-Mo

of their valence quarks, thus as in Fig. 3.

N

Will the resulting interaction between

N FIG 3

them lead to bound states? The diagram in Fig. 3 is familiar of course to both particle and nuclear physicists; it is just the quark diagram representing the exchange of mesons between the two nucleons and is widely believed to be the origin of nuclear forces at long to medium ranges, say > .7 fermi. For instance, the deuteron is a bound state between a proton and a neutron due to just these nuclear forces. It would appear therefore that the deuteron, and by the same token nuclei in general, ought to be regarded as the colour analogues of co-valent molecules formed by the combination of several hydrogen-like atoms, i.e. nucleons. One needs not of course restrict one's considerations only to nucleons as atoms. For example, in the days of S-matrix theory, attempts were made to construct the o-mesons as a bound state of two pions via the exchange of the 0-meson itself. The efforts were however not very successful. More recently, there are suggestions for bound states or resonances between a nucleon and an antinucleon via nuclear forces, which we shall have occasion to discuss again later. If such exist, they would in our language again be colour analogues of covalent molecules. What about ionic molecules? Are there analogues in colour chemistry? Suppose we take two conglomerates of opposite (i.e. conjugate) colour charges, will they stay together to form a molecule? By virtue of the confinement hypothesis, the potential of interaction between the two ions is believed actually to increase with increasing separation. There will thus be no possibility of ever ionising such a molecule. In other words, the formation of bound states from these ions would be a certainty. However, there is a threat to the stability of such a molecule from a different direction, namely that we know of no repulsive forces here analogous to that due to core electrons which keep ordinary ions apart, so that the colour ions will have a tendency to merge reducing the whole system to a large complex atom of the type considered above which is liable to instability against dissociation. Hence to guarantee the stability of such a colour ionic molecule, some mechanism has to be devised to keep the ions apart. One possibility is an arrangement of the type considered already in Fig. 2 where the ions are separated by an angular momentum barrier. Provided that the concept

Quark Interactions

and Colour Chemistry

315

of angular momentum barriers continues to make sense for quarks, the ions will be prevented from merging into one large atom. It may be wondered why an elongated ionic molecule like this could not simply split along its length, e.g. as that shown in Fig. 4, dissociating thus into two smaller molecules of the same form. If one believes in the bag model, however, this is never possible for the

x

Xl

x2 FIG 4

following reason (Chan and H~gaasen, 1977). For large angular momentum ~, the mass of the molecule is carried mostly by the flux tube whose linear mass density Px according to (18) is proportional to /~x, where ~ x is the Casimir operator for the colour x carried by the flux tube. Suppose the molecule splits into two others with flux tubes carrying colours x I and x 2 satisfying x I x x 2 ~ x then for the 'decay' to be kinematically possible,

Px > Px I + Px 2

,

or

/~x > /~x I + /~x 2

(29)

which violates however a certain triangular inequality satisfied by Casimir operators in general. We conclude therefore that these molecules are stable against such dissociations. They will have to decay by creating quark-antiquark pairs just like ordinary QQ and QQQ baryons. In other words, they are hadrons. If these colour ionic molecules exist, then they will be very useful for testing the basic tenets of chromodynamics. First, the colour degree of freedom as well as its non-Abelian nature are reflected in the colour representations available to the chromions. For example, for the simplest ions with two or three constituents, the possible colour charges are:

Secondly, occurred,

QQ

:

3x3=~+6

QQ

:

3 x 3 = i + 8

QQQ :

3 x 3 x 3 = I + 8 + 8' + I0

QQQ :

3 x 3 x ~ = 3 + 3' + 6 + 15

(30)

in contrast to the hydrogen-like atoms QQ and QQQ where only colour 3 different molecules can now be constructed with different internal

,

316

Chan Hong-Mo

colour configurations from the same quark constituents, e.g. (QQ)3 _ ( ~ ) 3

, (QQ)6 _ ( ~ ) 6

;

(Q~)8 _ (Q~)8 ;

(31)

(QQ)3 _ (QQ~)3,~ (QQ)6_ (QQ~)6, (Q~)8_ (Q~)8_ (QQQ)8, etc. The complexity of the observable hadronic spectrum is therefore greatly increased directly as a consequence of the colour degree of freedom. Thirdly, since differently coloured flux-tubes (chromionic bonds) occurred, e.g. in (31) x = 3, 6, 8, one can check the variations of the Regge trajectory slope as a function of the colour, thereby testing confinement mechanisms, such as the bag model through the formula (18). Fourthly, assuming that within each ion the quark constituents are in relative s-wave to each other, one can apply the formulae (20) or (28) to calculate their colour magnetic splitting, checking its validity also in this new situation where the total colour is not zero (i.e. nonsinglet). Finally, we have here a chance of studying unambiguously for the first time the effect of quark pair annihilation to leading order in spectroscopy (Chan and H~gaasen, 1980). As a quantum field theory, QCD must give diagrams of the form Fig. 5, the equivalent of which in quantum electrodynamics give clearly observable

FIG 5

effects in the spectroscopy of positronium. They give no contributions in ordinary QQ mesons since the Q~ pair there must be a colour sin$1et, but should be observabl~ in the ions (Q~)8 or in the QQ pairs of the ions (QQ~)O,6 as an interaction:

~s H2

m.m., i

1

~

32 (3 + o_i " o_i , ) (-- +

%a a i ~ ,)

(32)

i

of the same leading order as the colour magnetic spin-spin interaction (20) considered above. A naive estimate shows that it can lead to large mass shifts and mixing between quark pairs of different flavours, which should readily be measurable once such hadrons are identified. There is yet another property which makes these chromionic molecules interesting to study. Theoretically one has some reason to expect that colour mixing forces

Quark Interactions

and Colour Chemistry

317

between chromions will decrease with increasing separation. To illustrate this point, let V(r) represent the interaction potential between two quarks or antiquarks at dist-ance r. Consider now a chromion formed from quarks with colour charges (matrix) %i--at nearby positions x i. The potential they exert on another quark at the origin is then proportional to .~ %i V(-~i)" We now make an expansion about the centre-of-mass

of the ion -r = ~. --i x l ' thus

:

l

%i V(x--i) = ( 1 % i ) V(r) + ~ %i (xi i

i

The leading term is proportional

~V - r) ~ r + "'"

1

(33)

--

to ~ %. which is the total colour charge of the i

ion and cannot therefore change its ~olour. It follows then that colour mixing forces between two chromions are 'dipole' forces which must decrease faster than the confining potential V by some power of the ionic separation. This may well be sufficient to suppress the mixing between colour configurations when the clusters are spatially well-separated. If the above conjecture is true, then for chromionic molecules of large L we may neglect the mixing between colour configurations, e.g. between those in (31). We may then assign to each chromionic molecule a definite colour-bond x and to the chromions themselves also a definite colour. Further, since spin-dependent interactions between quarks become small also for large separation as we shall demonstrate later, we can assign to each ion a definite spin. In other words, ions and bonds now take on individual significance independently of the molecules in which they occur. One can thus meaningfully compile a list of ions and bonds each characterised by a number of parameters (such as colour, spin, isospin, and mass) just as one did in ordinary chemistry, from which one can even imagine constructing one's own molecules (in much the same way that Crick and Watson constructed the double helix of DNAI) and predict to some extent their properties (Chan et al., 1978a). Do these chromionic molecules exist? Two years ago I would have answered with an unqualified "yes". In the meantime, however, the experimental situation has changed. So, I am much less sure, but would still tend to answer in the affirmative. The main evidence came from the so-called baryoniam states which decay preferentially into baryon-antibaryon pairs, and from some heavy hyperon states which decay preferentially into several strange particles. Both types of states are hard to explain except as multiquark chromionic molecules. Baryoniums are mesonic states whose quantum numbers are such that no known slection rule forbids their decay into mesons. Given that they have masses around 2 to 3 GeV, they have ample phase space for doing so. Hence, the fact that they prefer to decay into baryon-antibaryon pairs must reflect some unusual internal structure. Now a chromionic molecule of the type: (QQ) - (QQ) is indeed expected to have such decay properties. The angular momentum barrier inhibits the quarks from combining with the antiquarks to form mesons. The molecule prefers therefore to break by creating a QQ pair, thus (QQ)

,~'

(QQ)

->

(QQQ)

+

(QQQ)

(34)

leading to BB pairs in the final states. Similar arguments apply to some heavy hyperon states recently discovered, e.g. R(3.17). This has the same quantum numbers as an ordinary hyperon resonance but decay mainly into final states containing three strange particles such as K~J~, KE etc., in preference to A ~ . . . , which should be much favoured by phase space. The natural explanation would be

318

Chan Hong-Mo

that it contains already of a chromionic molecule. firmed in their existence fidence in QCD in hadron a state of flux and I am is to be believed.

two s quarks and one ~ antiquark, e.g. qqss~, in the form If either baryoniums or these heavy hyperons were conand properties, it would be a great boost to our conspectroscopy. However, the experimental situation is in not competent to tell you how much of the data reported

So far I have kept to generalities, quite deliberately in order to develop better our chosen theme. However, so as not to leave you with the impression that nothing is done in this subject except qualitative conjectures, I wish to finish by a more detailed discussion of baryonium spectroscopy on which many papers have been written in the last few years. In view of their unusual decay properties, baryoniums are presumably not meson resonances of the ordinary QQ type. The two most popular suggestions are that they are: (a)

baryon-antibaryon

(BB) states held together by ordinary nuclear forces,

(b)

diquark-antidiquark

(QQ)(QQ)

states held together by quark confinement forces

By themselves, (a) and (b) cannot explain why mesonic decay modes of baryoniums arc apparently suppressed. For example, the BB pair in (a) may readily annihilate and lead to large mesonic widths. Indeed, the annihilation cross section of the NN pair at low energy is known empirically to be very large. To achieve the suppression of mesonic modes, it is usually assumed in both (a) and (b) that quarks and antiquarks are separated by an angular momentum barrier. In other words, (a) and (b) are what we would call in the present context (a) colour covalent molecules formed from the B and B atoms, and (b) colour molecules formed from the (QQ) and (QQ) ions. Nuclear forces, being short-ranged, are incapable of forming metastable states with very high orbital angular momentum or very high mass between B and ~ - also tightly bound states are very unstable because of annihilation. Hence, one expects the covalent type (a) of baryoniums only around the BB threshold. On the other hand, the confinement potential is not only long-ranged but is supposed even to increase, perhaps linearly, with the distance. It is capable therefore of forming baryonium states of the ionic typ~ (b) with essentially any high angular momentum and any high mass. Around the BB threshold, presumably both types (a) and (b) can exist and will be quite hard to distinguish except by detailed spectroscopy, es~ecially since the two types can mix via quark annihilation. High above the BB threshold, however, only chromionic molecules of type (b) can remain. Chromionic molecules at high ~ come in two varieties depending on their colour configurations x = 3 or 6. It was argued above that at sufficiently high ~, mixing will be small so that the colour configurations will be essentially pure. In the particular case of diquark-antidiquark molecules constructed out of only u, d quarks, it happens that one may go further and estimate the amount of mixing which may occur. Let us assume that the two quarks in each ion are in relative s-wave. The diquark ion can then exist in the following states:

(qq)3 i,I

~ ;

(qq)3,3

6 ;

(qq)l,3 ;

6 (qq)3,1

'

(35)

where superscripts denote the dimension of colour representations, the first subscripts the multiplicities in spin, and the second subscripts the multiplicities in isospin, the Pauli principle between quarks having been properly taken into account. We notice then that in order to change the colour representation of a

Quark Interactions

and Colour Chemistry

diquark ion by the exchange of flavourless also to change the spin.

319

gluons at large distances,

one needs

Now the effect of spin dependent interactions between quarks is small at large distances. This can be demonstrated phenomenologically as follows. ~ and P are both s-wave qq states differing only by the total quark spins; for ~, S = O (singlet), and for p, S = i (triplet). Therefore, the mass difference Am (~ = O) = m_ - m~ measures the effect of the spin-dependent interaction between q and q in relative s-wave, namely ~ = O. Similarly B and A 2 are p-wave qq states differing only by the quark spin S. Am (~ = i) = mA2 - m B measures the effect of the spin-dependent interaction b e t w e e n q and ~ for ~ = I. Hence, plotting the mass difference Am (£) between the triplet and singlet states along Regge trajectories gives the variation of spin-dependent effects as function of ~, or equivalently of the distance between q and ~. Fig. 6 shows such a plot for the q~ and q~ mesons. In both cases, Am (~) is seen to decrease rapidly with ~.

.7 .6

• p-~ A2-B g - A 3 .5

• K-K K - % K

-L

>.~ t,D

~

<3

:3

/

.2 .1

l

°t

-1

I

,I

0

1

!

3

FIG6

320

Chan Hong-Mo

Assume that this is true also for diquarks. The amount of colour configuration mixing at a given ~ can then be estimated as follows. The mixing angle e~ is given by perturbation theory as: tan 0 h


(36)

The numerator represents the mixing matrix element as measured by the effect of the spin-dependent interactions at angular moment ~, e.g. by Am (I) = mA2 - m B at ~ = I and by Am (2) = mg - mA3 at ~ = 2. The denominator E 1 - E 2 is the mass difference between the colour configurations before mixing. This is characterised by the mass difference between the colour ~ and 6 diquarks, which ar~ themselves split by one-gluon exchange between the two constituents. Now since the constituents are in relative s-wave by assumption, this mass splitting is of the same order as Am (0) = m - m . Hence we obtain the estimate via Fig. 6 p

Am (O tan 6

% A---~-o~ % .2 at ~ = I,

Thus one sees that the mixing is already quite small at decrease further as ~ increases.

.i at ~ = 2(37)

~ = I, and is expected to

Given the small mixing between the two colour configurations, we may regard them as corresponding to two difference varieties of chromionic 'baryoniums' (Chan and H~gaasen, 1978b) (Jaffe, 1978). The two varieties are expected to have quite distinctive properties both from ordinary QQ mesons and from each other. First, as already explained above, they both have suppressed decays into purely mesonic channels, which should readily distinguish them from ordinary QQ mesons. Second, the variety with x = 3 can decay by breaking the colour bond and creating a Q~ pair, resulting in a BB final state, as illustrated in (34). This m e c h a n i s m is similar to the decay of say f ÷ ~ , and is expected to yield similar widths, namely of order IO0 MeV except when inhibited by phase space. For the variety with x = 6, however, the breaking of the colour bond requires the creation of at least two QQ pairs, since a single Q carrying colour 3 cannot neutralise a colour 6 diquark ion, and the result is not BB but two 'baryoniums' of the same x = 6 variety: (QQ)6

/"

(~)6

÷

(QQ)6 _ ( ~ ) 6

+ (QQ)6 _ ( ~ ) 6

(38)

QQQQ Besides, such decay modes are expected to be suppressed relative to normal hadronic decays by a factor similar to that in other processes requiring double quark-pair creations, e.g ( ~ ÷ p p ) / ( ~ ÷ ~ ) . A more likely decay mode for the x = 6 variety is by emitting light mesons and cascading into a lower ~ member of the same variety, thus (QQ)6

~

(~)6

÷

(QQ)6

(~)6 + ~

(39)

The cascade chain will continue until the angular m o m e n t u m separating the ions is low enough for colour m i x i n g to occur, at which point the molecule will break up into a BE pair via the x = 3 configuration, leading to an overall final state of the form p ~ . . . Now, cascade decays of this type are known to be strongly governed by angular momentum and tend to be suppressed, suggesting therefore rather narrow widths for the x = 6 variety of 'baryoniums', although such widths cannot as yet be reliably estimated. In any case, the predicted decay properties

Quark Interactions

and Colour Chemistry

321

of the two varieties of diquark-antidiquark chromionic molecules are so different that there should be little difficulty in distinguishing them if they are ever found in experiment, and once identified, their properties should afford very valuable tests for the basic tenets of QCD. For a more detailed comparison of theory to experiment, one will need some calculation of the baryonium spectrum. This has been attempted by many authors for both the chromcovalent (a) and chromionic (b) types. In what follows I shall give brief indications on how these calculations were done. It is sufficient to illustrate with those molecules built only from the light quarks q = u, d. For NN chromcovalent molecules, the binding forces are supposed to be ordinary nuclear forces at long to medium ranges. Now, the nucleon-nucleon potential for r > .8 fermi is known to be well described by the exchange of mesons each of which carry definite G-parity. If this description is accepted then the NN potential can be obtained from the NN potential simply by charge conjugation. Thus the repulsive contribution from the m-exchange changes sign under charge conjugation but the attractive 2~-exchange does not. Since the 2~- and u-exchange terms are now both attractive, we have very strong attractions between N and N at intermediate ranges, leading thus possibly to many bound and resonant states. Starting with any meson exchange potential for NN interactions such as the Paris potential, for example, we can try to calculate the chromcovalent baryoniam spectrum. The outstanding ambiguity in such calculations is our ignorance of the annihilation interactions at short distances which are usually simulated only by some ad hoc assumptions. For example, in some calculations of the Paris group, it was assumed simply that the potential between N and N can be approximated by U(r) = I uPariS(r)

r > r '

uPariS(rc )

(4o)

c

~r < r '

c

where r ~ .8 fermi was treated as a parameter. It was believed that the level • c orderlng is insensitive to how the potential is cut-off, although the details of the spectrum can be quite strongly affected. An example of such a calculation can be found in the work of Vinh Mau (1978). For chromionic molecules of high ~, one can neglect the spin-dependent interaction~ between the ions. The mass differences between molecular states with the same are then just given by the 'hyperfine' splitting of the ions themselves due to one-gluon-exchange (20) between their constituents. For ions in the s-wave state, we have from (28):

(qq)~, i Am =

-8C

(qq)~, 3 8C

(qq)~,3 4C

(qq)63,1

(41)

4 -~c

Taking the previous estimate of C ~ 20 MeV obtained from ordinary QQQ baryon spectroscopy, one can calculate all these mass differences. Next, the mass differ. ences between molecular states with different ~ are assumed to be given by the linear formula for the Regge trajectory (17), with the values of ~' ~ .9 GeV -2 for x = 3 and ~' ~ .6 GeV-2 for x = 6 suggested by the bag model formula (18). One has then only one parameter ~ left for each variety which may be either estimated theoretically by means ~f a confinement model with bag or potential, or

322

Chan Hong-Mo

otherwise simply fitted to experiment. For ~ = i, 2, one may expect some deviation of the Regge trajectory from linearity, and some non-negligible effects from spindependent interactions between the ions, which have to be handled by injecting further phenomenological inputs. An example of such a calculation can be found in the work of Chan and H~gaasen (1978b). Up to about two years ago, the experimental situation was quite encouraging. For chromionic molecules, in particular, I have made an attempt at the 4th European Antiproton Symposium at Barr to compare in detail the then existing data to theoretical calculations and the outlook seemedoptimistic (Chan, 1979). Since then, however, the situation changed - many experiments done recently (often by the same groups) trying to confirm previous observations have given negative results, while other experiments obtained new and different effects. I am incompetent to say what is right and what is not in experiment. However, I should like to remark that most of the experimental groups involved were highly competent and experienced in spectroscopy, and the experiments themselves quite massive and painstaking. In view of this, it seems that the only reasonable conclusion one can draw in the present confused situation is that the experimental conditions are not yet sufficient to resolve the question of existence and nature of baryoniums. Let us hope, however, that in the near future, the greatly improved conditions to be provided by the LEAR project at CERN 17 will lead to a definitive answer. Baryoniums are the simplest examples of colour molecules. Although there are indeed also experimental candidates of other colour molecules some of which I have already mentioned, none of these has yet been subjected to the scrutiny as some baryonium states. For this reason, I think it will be in baryonium spectroscopy that the basic tenets of colour chemistry are going to be ultimately confirmed or disproved. From the theoretical view point, the ideas of colour chemistry as outlined above are, I think, a fairly logical development from the current concepts of quantum chromodynamics, although one cannot guarantee that the predictions will be correct in every detail. Personally, I shall be surprised if the many multiquark states predicted turn out not to exist. 18 If in the near future we do indeed find these colour atoms and molecules, we shall have discovered another verse of this great poem of Nature on chemistry in the ancient Chinese tradition (one needs not dispute here about priority), which will probably keep busy a new generation of colour chemists. If we do not, we shall have to go back to the drawing-board to see whether Nature has given us yet another twist. If so, the truth when we do find out will most likely be even more intriguing and beautiful than anything we can imagine now. In either case, life will be interesting.

17 18

As described by Professor B. Povh in his lectures in this school. Of course, it may happen that these states exist but have properties (e.g. high instability) which make them unsuitable or uninteresting for detail study. In this context, Professor R. Dalitz told me after my lectures of the parallel situation in the early days of QED of which I was unaware. At that time apparently, multielectron states were also predicted and discussed. As far as we know, the predictions are correct and presumably such states do exist, but as yet they have not made the slightest impact on our understanding of QED.

Quark Interactions and Colour Chemistry REFERENCES Anonymous (c. 500 B.C.). Shijing ( ~ ,~ ), ed. Kong Fuzi (de (publisher unknown, China). Chan, Hong-Mo, and H. H~gaasen (1977). Phys. Letters, 72B, 400. Chart, Hong-Mo, et al. (1978a). Phys. Letters, 76B, 634. Chan, Hong-Mo., and H. H~gaasen (1978b). Nucl. Phys., B136, 401. Chan, Hong-Mo (1979). Proc. 4th European Antiproton Symposium, Barr, ed. A. Fridman (CNRS, Paris) 477. Chan, Hong-Mo, and H. H~gaasen (1980). Z. Physik, C7, 25. de Rujula, A., H. Georgi and S.L. Glashow (1975). Phys. Rev., DI2, 147. Gelmini, G.(1980). Nucl. Phys., B174, 509. Isgur, N., and H. J. Lipkin (1981). Phys. Letters, 99B, 151. Jaffe, R. L. (1977a). Phys. Rev., DI5, 267, 281. Jaffe, R. L. (1977b). Phys. Rev. Letters, 38, 195, 617. Jaffe, R. L. (1978). Phys. Rev., DI7, 1444. Jaffe, R. L., and F.E. Low (1979). Phys. Rev., DI9, 2105. Johnson, K., and C. Thorn (1976). Phys. Rev., DI3, 1934. Matsuyama, S., and H. Miyazawa (1979). Progr. Theor. Phys., 61, 942. 't Hooft, G. (1979). Nucl. Phys. , B153, 141. Vinh Mau, R. (1978). Proc. 13th Rencontre de Moriond, ed. Tran Thanh Van (Editions Fronti~res, France). Wu, Tai Tsun, and Yang, Chen Ning (1976). Nucl. Phys., BIO7, 365. Yang, C. N., and R. L. Mills (1954). Phys. Rev., 96B, 191.

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