Potential models of hadrons

Nuclear Physics B (Proc. Suppl.) 23B (1991) 315-327 North-Holland

315

P O T E N T I A L MODELS OF HADRONS Andr~ Martin CERN- TH, Geneva, Switzerland

I would like to thank our host, Stephan Narison, an expert of QCD sum rules, to have invited a representative of the opposition to give a talk at this conference. As you know there are several approaches to spectroscopy: - Lattice QCD, still very remote if one want to go beyond the quenched approximation, - QCD sum rules which give very interesting results, but cannot give an answer to every question in spite of what believes (or believed) my friend Hector Rubinstein, - the Bag model , - potential models. In fact the Bag model and potential models are not incompatible. It was shown long ago by Hasenfratz, I-Iorgan, Kuti, and Richard[l] that if one treats the bag model in the BornOppenheimer approximation, i.e. if one minimizes the energy of the Bag for a given configuration of the quarks, one ends up with a SchrSdinger equation and an effective potential. Some of my friends from the ITEP group, like M. Voloshin, are very skeptical on the potential approach. My position is that it is mostly justified by its successes, and I hope to convince you of that. I agree that there are many mysteries: why is the flavor independent of the potential working so well, why relativistic effects can be disregarded at least for energy levels etc. But as you will see, it works. During the sixties, it seemed that the SchrS-

dinger equation was becoming obsolete, except perhaps in calculating the energy levels of muonic or pionic atoms, or the medium energy nucleonnucleon scattering amplitude from a field theoretical potential[2]. Elementary particle masses were hoped to be obtained from the bootstrap mechanism[3], or with limited but spectacular success from symmetries[4]. At the beginning of the quark model only very few physicists considered quarks as particles, and tried to calculate the hadron spectrum from them. Among these I could quote Dalitz[5] almost "preaching in the desert" at the Oxford conference in 1965 and Gerassimov[6]. The situation changed drastically after the "October" 1974 revolution. As soon as the J/¢[7][8] and the ¢*[9] had been discovered the interpretation of these states as charm-anticharm bound states was universally accepted and potential models using the SchrSdinger equation were proposed[10][ll]. In fact the whole hadron spectroscopy was reconsidered in the framework of the quark model and QCD in the crucial paper of De Rujula Georgi Glashow [12] and the independent papers of Sakharov and Zeldovitch[13] and Sakharov[14]. Impressive fits of spectra of baryons (including those containing light quarks) were obtained in particular by Stanley and Robson[15][16], Karl and Isgur[17], Richard and Taxil[18], Ono and SchSberl[19], Basdevant and Boukraa[20] and

0920-5632/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved.

A. Martin/Potential models of hadrons

316

many others. I would like to return now to the case of quarkonium, i.e. mesons made of a heavy quarkantiquark pair. By heavy quark, I mean the c and the b quarks of effective masses ,-, 1.8 and 5 GeV, but also the strange quark, for which the effective mass turns out to be 0.5 GeV. The strange quark occupies a borderline position and can be considered either as heavy, as we do here, or as light as in SU3 flavour symmetry. In this list one would like to include the top quark which is certainly heavier than 90 GeV, from the CDF experiment[21] and which, from fits of experimental results by the standard model including for instance masses and widths of the W and Z ° particles and low energy neutrino experiments (with a standard Higgs), weights 122_+41 GeV[22]. My estimate is that if the mass of the top quark is heavier than 110 GeV the notion of toponium becomes doubtful, because the width due to single quark decay, t ~ b + w, exceeds the spacing between the 1S and the 2S state[23]. However Ugo Amaldi, claims that for a top mass of 130 GeV one can still distinguish the I S and 2S states. I hope he is right and I hope that the top quark is not too heavy. Fig 1 and 2 give a summary of the experimental situation for the c ~ ( J / ¢ etc) and bb(T etc.) bound states respectively. There are of course many potential models which have been used to describe the c~ and bb spectra. The first one was a potential V = - - a + br,

(1)

r

in which the first term represents one-gluon exchange, analogous to one photon exchange, and the second confinement, by a kind of string. I shall restrict myself to two extreme cases of fits, the first one, by Buchmiiller and collaborators[24], is a QCD inspired potential in which asymptotic freedom is taken into account in the

short distance part of the potential; the second is a purely phenomenological fit[25] that I did with the central potential V = A + Br ~.

(2)

Figure 3 represents the excitation energies of the c~ and bb systems.The full lines represent the experimental results (for the triplet P states we give only the spin-averaged energies). The semi continuous lines represent the Buchmfiller et al. result and the dotted lines my potential, to which a zero-range spin-spin interaction cm#l m ( r2) has been added, where ml and m2 are the quark masses, and C was adjusted to the J / ¢ - 7/e separation. My central potential is given by[25] V = - 8 . 0 6 4 + 6.870r '1 ,

(3)

where the units are powers of GeV, and quark masses mb = 5.174 me = 1.8, and as we shall see ms = 0.518. The smalless of the exponent, a 0.1, means that we are very close to a situation where the spacing of energy levels is independent of the mass of the quarks which is the case for a purely logarithmic potential. Table 1 represents the relative leptonic widths, i.e. the ratios of the leptonic width of a given 1 = 0 state to the leptonic width of the ground state. "Theory" uses the so-called Van-RoyenWeisskopf formula. We see that the fits are both excellent. The QCD inspired fit reproduces somewhat better the low energy states, in particular the separation between the l -- 0 and I -- 1 states for the bl~ system. This is presumably due to the fact that the QCD inspired potential has a correct shortrange behaviour while the phenomenological potential has not (there is a discrepancy of 40 MeV, which would have been considered negligible before 1974, but can no longer be disregarded with the new standards of accuracy in hadron spectroscopy). On the other hand the phenomeno-

A. Martin/Potential models

317

hadrons

of

yr (3770)

t N,,,,,,

MD)

N////N///////////////

w (36861

1

hadrons58 JPc=

--

1

()-+

0 Cc

++

++ 1

2

++

bound states Fig. 1. strange

Table 1

This

quark effective

, following

is because

Gell-Mann, logical

a suggestion

beyond

its limit

GeV agrees with experiment of A. De Rujula

masses of the cs states. logical

potential

citations, meson

gives a better

fit of higher

close to the dissociation pairs,

fact that

Do,

of confined

channels,

CC, bb, due to their coupling

to open channels.

In the list of parameters

of the phenomeno-

logical

potential,

we have already

indicated

Re-

but

M+I = 1.634

(1.650 GeV). At the I also calculated

the

One gets 2.01)

into

(Y= 0.1 takes into account

of the energies

of M.

predictions.

MD, = 1.99 (exp 1.97,previously MD; = 2.11

BB. This may be due to the

the optimal

the lowering

threshold

ex-

GeV.

of validity!

one gets a lot of successful

M+ = 1.020 GeV is an input request

= 0.518

I have tried to push the phenomeno-

model

markably

mass m,

and since, very recently, what

is presumably

(exp 2.11) [26] Argus

has observed

a C = 1 CS state

which

can

be JP = l+ or 2+ of mass 2.536 GeV I have calthe

culated without

the spin-averaged changing

any

mass of such a state, parameter

of my model,

A. Martin/Potential models of hadrons

318

m

Y (11019) 11.0

-

_

i-

y (10577)

2M(B)

10.5

BB

Y (10355)

3,13

(3 U~

~3

:~ 10.0

9.5

m

y + hadrons 1.6 jPC=

1--

0 ++

1++

2 ++

bb" bound states Fig. 2.

and obtained[27]

MD:. (£ =

1) = 2.532.

One can convince oneself that the state observed by Argus is not more than 30 MeV away from the centre of gravity. In conclusion it is impossible not to be impressed by the success of these potential models. But why is it so? The fact that various potentials work is understood: different potentials agree with each other in the relevant range of distances, from, say, 0.1 Fermi to 1 Fermi. However relativistic effects are not small : for the c~ system v2/c 2 is calculated a posteriori to be of

the order of 1/4. The only partial explanation I have to propose is that the potential is only an effective potential associated to an effective SchrSdinger equation. For instance one can expand ~ + m s, the relativistic kinetic + mass energy, around the average < p2 > instead of expanding around zero. For a purely logarithmic potential the average kinetic energy is independent of the excitation, and it happens that the potential is not far from being logarithmic. Anyway we must take the pragmatic attitude that potential models work and try to push the consequences as far as we can.

A. Martin/Potential models of hadrons

319

y~Z

m ~ u m ~

y,1000 .....

° ....

0=2)'

y,, A ¢1 m ~°°**

.....

t~

1=2

m ¢¢-

0

500

Tic . . . . .

y'

¥' .B. . . . . .

°°°°°

Zb

X,c (I,,1) ILl

i

J/~

""lib" Y

~C .....

L

J

L

J

c~ .....

bb

Expedrnent BuchmOIler et al. Martin

Fig. 3.

Concerning baryons, I shall be more brief. Baryons made of purely heavy quarks have not yet been seen, though they must exist, like bbb and ccc. Bjorken[28] advocates the study of ccc which possesses remarkable properties: it is stable against strong interactions and has a lifetime which is a fraction of 10 -13 seconds. Its lowest excitations are also stable or almost stable. If one accepts to believe that the quark-quark potential inside a baryon is given by[29]

= 1/2vqc

,

(4)

one can calculate all properties of ccc from a successful c~ potential. Bjorken thinks that such a state can be produced at a not too small rate. In the mean time we remember that the strange quark can be regarded as heavy. J. M. Richard, using the fit (3) of quarkonium and the rule (4) has obtained a mass for the ~ - baryon, which is sss, of 1.665 GeV[30] while experiment[31] gives 1.673.

A. Martin/Potential models of hadrons

320

Table 2

Table 3

~/~,p EXP

Theory QUARK

N

input

.939

MODEL

MODEL

A

input

1.232

Ae

1.111

1.115

rnu/ms = 1

m u / m s = 0.625

N

- 0.68

- 0.67

-0.67

A

-0.22

- 0.333

0.207

--

1

0.958

1.176

1.193

1.304

1.318

*

1.392

1.383

--=*

1.538

1.533

-

input

1.672

Ac

input

2.282

4- 0.09

Ec

2.443

2.450

-0.24

r.*

2.542

4- 0.01

.-.c = A

2.457

-0.45

-9.50

S

2.558

S*

2.663

-0.625

~c = T

2.664

T*

2.775

4- 0.02 E+

0.85 4- 0.01

E-

E=o

-0.50

-0.333 -0.333

-0.666

-0.37 -0.17

4- o.o5 1"2°

Experiment

QUARK

-0.72

-1

4-0.07

For baryons made of lighter quarks, after the pioneer work of Dalitz came the articles of De Rujula, Georgi, Glashow[12], Sakharov and Zeldovitch[13], and Sakharov alone[14]. In these works, the central potential is taken to be zero or constant i.e. incorporated in the quark masses and the dominant feature is given by the spinspin forces "derived" from QCD, which lead to remarquable results, in particular the first explanation of the E - A mass difference. In this approach which is 0 th order in the central potential the calculation of excited states is excluded. Another remarkable success, at this stage, is the prediction of magnetic moments of baryons presented in Table 2. You will notice the inclusion of the magnetic moment of the Q- particle recently measured by Luk and and collaborators[32] and which definitely found a ratio of the non strange to strange quark effective masses less than unity. The next step is to add a soft central potential

2.460

2.740 (3 events!)

and try to solve seriously the 3-body SchrSdinger equation. This has been done by many people. Stanley an Robson[16] were among the first Karl, Isgur, and Capstick[18][33] were among the most systematic. Here I would like to limit myself to the study of ground states, which has been down for instance by Ono and SchSberl[19] and by Richard and Taxil[18], and show for example in Table 3 the results of Richard and Taxil with a potential V = A + Br °'1 and a spin-spin Hamiltonian

c

mimj

(ri - rj)

(5)

Though the results are nice, it is not completely obvious that the rigorous treatment of the central potential does lead to a real improvement. To demonstrate this, we have considered some ratios, which in the DGG model[12] have simple values, and shown the values calculated and compared to experiment. This is the content of Table 4. Perhaps it is worth noticing that the equal

A. Martin/Potential

321

models of hadrons

Table 4 De R u j u l a Georgi Glashow

Richard

Sakharov

Taxil

Experiment

Zeldovitch M .~* -- M-~ ME* --ME

2M~* -{- M E

--3 M A

2(MA--MN)

3MA-{-MI~ 2MN-}-2M..z

1.08

1.12

1.07

1.05

1.005

1.005

1.10

1.007

1.09

1.08

G.M.O O C T E T M~, --MA MZ* --ME*

G.M.O M.* --Ml~* M~_ --M--* DECUPLET

spacing rule of the SUn flavour decuplet which was the triumph of Gell-Mann comes here in a rather funny way. Naturally if the spin-spin forces and the central potential are neglected the equal spacing rule is absolutely natural since the mass of the state is obtained by merely adding the quark masses: therefore the mass is a linear function of strangeness. However Richard and Taxil[34] discovered by numerical experiments that if one takes a "reasonable" flavor independent two-body central potential the masses of the decuplet are concave functions of in strangeness. In other words M(ddd) + M(dss) < 2M(dds)

(6)

This problem was submitted to Lieb who proved the following theorem[35] (in fact a bit more): M is a concave function of strangeness if the two-

body potential V(r) is such that VI>0,

V" < 0 , V m > 0 .

This is the case for instance for V = _ a + br, V = r °'1 etc. r

(However the property is not true for power potentials with large exponents, like for instance V = r5136]). Now let us return to the linearity in the decuplet: it happens that it comes from a cancellation between the effect of the central potential and the spin dependent potential. This is perhaps a good point to turn to the second part of my talk which concerns rigorous results on the SchrSdinger equation stimulated by potential models. What we have just seen is only one example: the calculation of the energy levels of the decuplet led to the discovery of this

322

A. Martin/Potential models of hadrons

property of concavity with respect to strangeness for a certain class of potentials. Now we shall concentrate on two-body systerns. I would like you to return to figures 1 and 2, when we see the c~ and bb spectra. There is one remarkable property which is that the average p state (t--l) mass is below the first (g=0) excitation. This property is satisfied by all existing models, in particular the initial model[10][ll] which made this statement before the discovery of the p state (at the Dijon congress of the "Societ~ Franqaise de Physique" K. Gottfried said that if the p states would fail to be at the right place their model would be finished. An experimentalist from SLAC, who knew that there was evidence for these p states, kept silent!). I was asked by M. A. B. B~g if this prediction was typical of the model and could be changed by modifying the potential. The question, therefore was to find a simple criterium to decide which is the order of levels in a given potential. Though some preliminary results were obtained in 1977 by Grosse and myself it was only in 1984 that the situation was completely clarified. We denote by E ( n , g ) the energy of the state with angular momentum g and radial wave function with n nodes. For a general potential we have only the restrictions

V(r) = 0

(9)

i.e. the Laplacian of the potential is zero outside the origin, while the harmonic oscillator potential satisfies d ldV -dr r dr

-

0

(10)

i.e. it is linear in r 2. Now in the case of quarkonium what can we say about the potential? First of all we have "asymptotic freedom". "Strong" asymptotic freedom is the fact that the force between with quarks is ~ d~(r) > 0

(11)

which is equivalent to r2A V(r) > 0

(12)

On the other hand, lattice QCD implies, according to Seiler, that V is increasing and concave, which implies d ldV d'-~r d-T < 0.

(13)

E(n + 1, g) > E(n, g)

(7)

Now come the two main theorems obtained by Baumgartner, Grosse and myself[37] which happen to be relevant to this situation (and as we shall see to other situations). - Theorem 1

E(n, ~ + 1) > E(n, g)

(8)

E ( n , e ) <>E(n - 1 , e + 1)

which follow respectively from Sturm-Liouville theory and from the positivity of the centrifugal term. W h a t we want is more than that! There are two potentials for which the solutions of the Schrbdinger equation are known for all n and ~ and exhibit "accidental" degeneracies which are the Coulomb potential and the harmonic oscillator potential. In fact one can go from one to the other by a change of variables. The Coulomb potential can be characterised by

if r2A V(r) <> 0 Vr. - Theorem 2 E ( n , g ) <> E ( n - 1,g-b 2) d l d V > 0 Vr, if dr r dr < i.e. if V is convex or concave in r 2. There are however other important results among which I choose.

A. Martin/Potential models of hadroas

323

- Theorem 3 (Common + A. M.[38]) E (0, g) is convex (concave) in g

N=3

if V is convex (concave) in r 2 (E(0, g)is what used to be called the "leading Regge trajectory" in the Sixties!) - Theorem 4 (Richard, Taxil, A. M.[39]) a) The spacing between g = 0 energy levels increases (decreases) with n if V=r 2+Av,

N=2

N=n+l+l r2AV(r)=0

A small

d 5d ldv and ~rr r ~r r-~-r > 0 Vr

.......

r2AV(r)~>0

.................

r2AV(r)~0

b) it decreases if V" < 0 (here the proof is not yet complete, so that is it really only a conjecture that we believe to be correct with 99.9% probability). - Theorem 5 (Stubbe and A. n.[40]) if r2A V > 0

N=I

::=:=:= Fig. 4.

d2E (0, e) 3 dE(O,g)> 0 de 2 + g +~ d ~ < which implies E(0, e + 1) - E ( 0 , g ) > Ec (0, g + 1) - Ec (0,t)

E(0, g ) - E ( 0 , e -

1) < Ec (0,g) - E¢ (0, g - 1)

where Ec is the Coulomb energy:

charge distribution, the potential it produces, by Gauss's law, has a positive Lalpacian. In the tables by Engfer et al.[41] one finds abundant data on p - 138 Ba atoms. In particular, in spectroscopic notations, one finds for the N = 2 levels

Ec = - c o n s t . ( g + 1) -2 2Sl/2 -- 2pl/2 = 405.41 Kev Figure 4 illustrates Theorem 1. The full lines represent the pure Coulomb case. The semi continuous lines the case r2A V > 0, and the dotted lines the case r2A V < 0. Now the application to quarkonium is obvious. It corresponds to r2A V > 0, and we see this very clearly on figures 1 and 2. There are, however other applications. First of all muonic atoms, in which the size of the nucleus cannot be neglected with respect to the Bohr orbit. Since the nucleus has a positive

which is positive and which would be zero for a point like nucleus (even for the Dirac equation) Similarly for N = 3 one has 3d3/2 - 2pl/2 = 1283.22 Kev while 3p3/2 -- 2p1/2 = 3p3/2 -- 2s1/2 -t- 2s112 -- 2p1/2 = 1291.91 key

A. Martin/Potential models of hadrons

324

2P~raa~2 P

2S1/2 5.37

2sv2 $

2P~z

2Plrz

Pl

p2

2 Ds,2 a~ 2 FTr~sa d

!

0

!l

- 5d

5f

-I-~,

6,-J

5000

4 -

10000 4

-~sp~ ,4-~ r 7-- s~

~_ / I ~; ~~'~

/ I g///I ~ 'e" ~ , : I I I"/' .• 1

sooo

- ~oooo

15000 "7

3-

"~

20000 E o

F

,

"

>

°

2-

i

!25000

_~

1.84

30000

/

,_

-

30000

35000

40000

40000

_3s

45000

45000 Fig. 6.

Fig. 5.

which means that 3p3/2 is above 3da/: For N = 4 a violation of 4 Kev is found but this is compatible with relativistic corrections.

Of course, the t~ - 2 state of the c~ system, called ¢ " satisfies this Theorem: Me,, = 3.77 GeV while Me, = 3.68 GeV. To finish we would like to illustrate Theorem 5, the last published Theorem[43], which concerns purely angular excitations: if r2A V > 0

Another application is to alcaline atoms. There, the outer electron is submitted to the potential produced by a point like nucleus and by the negatively charged electron cloud, so that, in the Hartree approximation we have r2A V < 0. Figures 5 and 6 illustrate this situation for Lithium and Sodium, respectively [42]. Now we turn to Theorem 2. Figure 7 shows the level diagram of the harmonic oscillator (full line), of a potential with dr d r1 dV ~1; > 0 (semi continuous line) and of a potential with ~7 d 7"~1d V < 0 (dotted line).

E3d -- E2p >0.185... < E2p El,

(14)

E4r - Zaa $0.35

(15)

Esg - E4f >0.463... E4f - Ead'<

(16)

-- E2p

Inequality (14) with > is of course satisfied by the c~ system 54¢,, - Mxe Mx¢ - M j / ¢

3.77 - 3.51 - 0.634 >> 0.185, 3.51 - 3.10

but this is not very exciting. It is, we believe, in

A. Martin/Potentied models of hax/rons

325

6 5 4 3 2 1

. . . . . . .

I=0

I=1

1=2

1=3

1=4

N = 2n+1+1

d dr d dr d dr

1 r 1 r 1 r

dV =0 dr dV>o dr dV
Fig. 7. muonic atoms[41] that we find the most spectacular illustration. On figure 8 we have represented the ratios (14), (15) and (16) as a function of the charge

tween experiment, phenomenology, matical physics. How long will the equation remain useful for particle do not know but we see that there

Z of the nucleus. Relativistic effects have been eliminated by a procedure that I shall not describe here. We see that the first two ratios deviate from the Coulomb value very clearly as Z increases, while the last one, insensitive to the non-zero size of the nucleus, remains constant. An illustration going in the opposite direction is obtained by looking at the spectrum of Lithium on figure 5. We find

settled questions like the ccc spectrum or the d states of the upsilon spectrum for which it plays a role. The d states of the upsilon system have been predicted with high accuracy by Kwong and Rosner [43] from a reconstructed potential obtained from the/~=0 states and their leptonic width by an "inverse" procedure and the observation of these states would be a crucial test of potential models. Anyway, even when the Schrhdinger equation will become uninteresting for particle physics, the exact results obtained will survive and be used in other areas of quantum physics. Note Added: since this talk was given some

4f - 3d - 0.326 < 0.35, 3d - 2p as we should. We believe that in this talk we have demonstrated how fruitful can be the interaction be-

and matheSchrhdinger physics? We are still un-

326

A. Martin/Potential models of hadrons

I

I

I

I

L=0 1--

SINGLETTRIPLET L=I L=I

TRIPLET L=2

10.6

0.5 ................................................

..................

, = t

!::

.....

10.5160

10.5007

10.4 -

10.3555(5)

0.4 0.350

10.2686(7)0++ 10.2557(8) 10.2305(23)

10.2

0.3 10.0

--

10.1599 10.1562 10.1501

10.0234(4) -+

~pO

0.2 -0.185

10.4443 10.4406 10.4949

-+

~n

g

/

. 1 _

ee

2::

9.9133(6) 9.8919(7) 9.8598(13)

9.8

0÷+

• E2pE3d- Els E2p t

0.1

O E4! " E 3 d

E3d- E~

corrected data

A E 5 ~ E,~ E4f -

I

0

20

bb level diagram according to Kwong and Rosner

9.6

9.4600(2)

E3d I 40

I 60

I 80

9.4

Z

351

Fig. 8.

of the results presented here have been generalized and adapted to the ease of the Klein Gordon equation [44].

References [1] P. Hasenfratz, R. Horgan, J. Kuti and J.M. Richard, Phys. Lett, 9513 (1980) 299 [2] R. Vinh Mau in "Mesons in Nuclei" vol. 1, M. Rho and D. Wilkinson eds., North Holland (1979) 151. [3] G. F. Chew, The Analytie S Matrix, a Basis for Nuclear Democracy, Benjamin, New York (1966). [4] M. Gell-Mann in Proceedings of the 1962 International Conference on High Energy Physics at CERN, J. Prentki Editor, CERN (1962) 805. [5] R. H. Dalitz, Oxford International Conference on Elementary Particles, Sept 1965, T. R. Walsh Editor (Rutherford High Energy Laboratory, 1966) 157.

3pj

_

3Dj

Fig. 9. [6] S. B. Gerasimov, Zh Eksper. Teor. Fiz. (USSR) 50 (1966) 1559, English translation Soviet Physics J E T P 23 (1966) 1040. [7] J. J. Aubert et al., Phys. Rev. Lett. 33 (1974) 1404. [8] J. E. Augnstin et al., Phys. Rev. Lett. 33 (1974) 1406. [9] G.S. Abrams et al., Phys. Rev. Lett. 33 (1974) 1453. [10] E. Eichten et al., Phys. Rev. Lett. 34 (1975) 369. [11] T. Applequist et al., Phys. Rev. Lett. 34 (1975) 365. [12] A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev. D12 (1980) 147. [13] Y. B. Zeldovitch and A. D. Sakharov, Acta Physica Hungarica 22 (1967) 153. [14] A. D. Sakharov Zh. Eksper. Teor. Fiz. (URSS) 78

(1980) 2112; English translation J E T P 51 (1980) 1059. [15] D. P. Stanley and D. Robson, Phys. Rev. D21 (1980) 3180. [16] D. P. Stanley and D. Robson, Phys. Rev. Lett. 45 (1980) 235.

A. Martin/Potential models of hadrons [17] N. lsgur, in Proceedings of High Energy Physics 1980, Madison, L. Durand and L. G. Pondrom Eds. (AIP Conference Proceedings 1981) 30; See also N. Isgur and P. Kociuk, Phys. Rev. D21 (1980) 1868. [18] J . M . Richard and P. Taxil, Phys. Lett. B123 (1983) 4531; Ann. Phys. NY 150 (1983) 267. [19] S. Ono and F. Sch~beri, Phys. Lett. B l 1 8 (1982) 419. [20] J. L. Basdevant and S. Boukraa, Ann. Phys. F. 10 (1985) 475 and Z. flit Phys.C30 (1986) 103. [21] See for instance G. P. Yeh, in "Les Rencontres de Physique, de la Vall6e d'Aoste", Results and Perspectives in Particle Physics La Thuile 1990, M. Greco Ed. (Editions Fronti~res 1990) 451. [22] J. Ellis and G. L. Fogli, Physics Letters 249 (1990) 543. [23] A. Martin, in "Heavy Flavours", Nucl. Phys. (Proc. Suppl.) I B (1988) 133; In this article, toponlum widths should be multiplied by 2 (the fact that toponium contains 2 quarks was overlooked!). [24] W. Buchmliller, G. Grunberg and S. H-H. Tye, Phys. Rev. Lett. 45 (1980) 103; W. Buchmliller and S. H-H. Tye, Phys. Rev. D24

(1981) 132. [25] A. Martin, Phys. Lett. B100 (1981) 511. [26] H. SchrSder, in "Les Rencontres de Physique de la Vall6e d'Aoste" La Thuile 1989, M. Greco Ed. (Editions Fronti~res 1989) 461. [27] A. Martin, Phys. Lett. B223 (1989) 103. [28] J . D . Bjorken, in "Hadron Spectroscopy" J. Oneda ed. (AIP, New York 1985) 390 see also: A. Martin, in "Heavy Flavours and High Energy Collisions in the 1-100 TeV Range", A. Ali and L. Cifarelli eds (Plenum Press, New York and London 1989) 141. [29] J. P. Ader, J. M. Richard and P. Taxil, Phys. Rev. D12 (1982) 2370; S. Nussinov, Phys. Rev. Lett. 51 (1983) 2081; J. M. Richard, Phys. Lett. B139 (1984) 408. [30] J. M. Richard, Phys. Lett. B 1 0 0 (1981) 515. [31] V . E . Barnes et al., Phys. Rev. Lett. 12 (1964) 204. [32] K. Johns et al. Proc. 8 th International Conference on, High Energy Spin Physics, K. Heller ed. AIP (1989) 374.

327

[33] S. Godfrey and N. Isgur, Phys. Rev. D32 (1985) 232; S. Capstick and N. Isgur, Phys. Rev. D 3 4 (1986) 2809 see also K.T. Chao, N. Isgur and G. Karl, Phys. Rev. D23(1981) 155. [34] J. M. Richard and P. Taxil, Phys. Rev. Lett. 54

(1985) 547. [35] E. H. Lieb, Phys. Rev. Lett. 54 (1985) 1987. [36] A. Martin, J. M. Richard and P. Taxil, Phys. Lett. B 1 7 6 (1986) 224. [37] B. Baumgartner, H. Grosse and A. Martin, Phys. Lett. B 1 4 6 (1984) 363 and Nucl. Phys. B 2 5 4 (1985) 528; Simplified proofs have been given by M. S. Ashbaugh and R. D. Benguria, Phys. Lett. A131 (1988) and finally by A. Martin, Phys. Lett. A 1 4 7 (1990) 1. [38] A. K. Common and A. Martin, Europhysics Lett. 4 (1987) 1349. [39] A. Martin, J. M. Richard mid P. Taxil, Nucl. Phys. B 3 2 9 (1990) 327. [40] A. Martin and J. Stubbe, CERN preprint TH 584390 (1990) to appear in Europhysics Letters. [41] R. Engfer et al, Atomic Data and Nuclear Data Tables 14 (1974) 509. [42] These figures are taken from E. Chpolsky, "Physique Atomique", Mir Editions, Moscow, French translation 1978. [43] W. Kwong and J. L. Rosner, Phys. Rev. D38 (1988) 279. [44] H. Grosse, A. Martin and J. Stubbe, preprint CERN TH 5914-90 (1990) and UW Th-Ph 1990-50, to appear in Phys. Lett. B.