New meson configuration in the bag model (I). First order energy spectrum of qqg states

Nuclear Physics B185 (1981) 391--402 O North-Holland Publishing Company

NEW MESON CONFIGURATION IN THE BAG MODEL (I). First order energy spectrum of qqg states F. DE VIRON and J. WEYERS

D~partement de Physique Th~orique, Universit~ Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgique Received 20 October 1980 (Final version received 6 February 1981) Perturbative QCD in the bag is applied to unconventionalconfigurationof quarks and gluons. The meson spectrum is computed to second order in the strong coupling. Exotic states with jpc = 1-+, 0+-, 0-- are found in the energy range 1.57 ~
1. Introduction It is generally admitted in the framework of Q C D [1] that confinement is a large-distance effect which, among others, precludes the propagation of free quarks, but that it does not affect perturbative calculations of phenomena which occur at a scale Q2 7> 1 / R 2, where R is the dimension of a typical hadron. A phenomenological approach which incorporates these features is the (semiclassical) bag model [2]: quarks and gluons are confined to a cavity from which they cannot escape but where they obey the equations of motion of ordinary QCD. Lee [3] has given the form of the propagators for gluons and quarks confined to a spherical and rigid cavity of radius R, but otherwise free. In this paper, we use the formalism of Lee to develop a covariant perturbation theory for fields confined to a cavity. With the help of this formalism we are then able to calculate the energy shifts, due to one-gluon exchange, of the ordinary (i.e. qcl) mesons as well as for exotic mesons of the type qqg (quark-antiquark-gluon). For the ordinary mesons, we agree with previous calculations [4, 5]. Our results for exotic mesons, on the other hand, are new. In this paper we will restrict our discussion to the spectrum of qqg states [6, 16] and we postpone to a subsequent publication the study of their decay modes as well as their mixing with ordinary q~ as well as with glueballs or q?:tq~ states. It is worth insisting on the fact that in a "cavity approximation" to the hadrons, the excitation spectrum for mesons is a lot richer than in a naive non-relativistic quark model: in addition to the normal qq modes we have, among others, a whole spectrum of gluonium states gg, ggg as well as mesons with, e.g. a q~g configuration. There is no a priori mechanism to suppress this kind of state and it is certainly interesting to indicate what their spectrum might be like: this is the main purpose of our paper. 391

392

F. de Viron, Z Weyers / New meson configurations

In sect. 2, we develop the formalism for a perturbation theory of quark and gluon fields confined to a cavity. We give, in particular, the quark wave functions, the gluon modes, a s u m m a r y of the F e y n m a n rules and the formula for energy shifts. In sect. 3, we calculate the 0th order masses of qqg states; in sect. 4 we discuss energy shifts to order a~ and in sect. 5 we summarize our results.

2. Confined perturbation theory Before discussing the Feynman rules and energy shift calculations, let us first give to lowest order in as (the strong coupling constant) the quark wave functions and the gluon modes.

2.1. QUARK WAVE FUNCTIONS

The restriction to a rigid spherical cavity of radius R implies that our discussion will be valid only for the lowest propagation modes namely, in the standard spectroscopic notation, $1/2 and P1/2.

(

~.,s.~.~(x, t) =N(x.,s,~)

r)

+i]o X.,s.~-~ ~

e-'~°'~'~', (2.1)

dg.,e,/~.M(x, t) -- N (x.,p,/2 )

+i]l(x"P'/2 R) ° " nulVt]e_i,,.,.,/2¢,

4-dg

+Y0(x-.,',,~ R)u~' _J

where the eigenfrequencies are given by 8n,lr =Xn, cr/R ,

71" = 5 1 / 2 , P1/2,

(2.2)

and xn.~ is determined by the boundary conditions that no quark current crosses the surface of the bag. The lowest values are

xl.sl/2

= 2.04,

X2.s,/2=

5.40,

(2.3)

x1.el/2

= 3.81 ,

X2.p~/2=

7.00.

(2.4)

In eq. (2.1), uM is a t w o - c o m p o n e n t spinor with angular m o m e n t u m M along the quantization axis, and the normalisation factor (using the notation of [7]) is given by

N(x~.,,) - \2R3(x,.,, q: 1) sin 2 xn.,,

F. de Viron, J. Weyers / New meson configurations

393

2.2. GLUON MODES To lowest order in the coupling constant, the gluon fields are described by Maxwell's equations, just as eight independent abelian photons and hence the complete solution to the equations of motion can be written in terms of transverse electric (TE) and transverse magnetic (TM) radiation modes [8]. These radiation modes can be expressed in terms of solutions to the equations ~72~b+ k2cb = 0 which can be written as: q b ~ M ( r ) N j j ( k r ) Y j M ( O , ~p), where r, O, ¢p -~ spherical coordinates of r , YIM --=spherical h a r m o n i c s , jj -- spherical Bessel function, k - eigenfrequency. In the Coulomb gauge, the T E m o d e reads as follows: TE

Vk.j.M(r) = A

TE

(k, J)~r x (r~b~M),

(2.5)

with k determined by the boundary condition J j j ÷ l ( k R ) + (J + 1)]j_l(kR) = 0 ,

and with parity P = ( - 1 ) J÷l. The lowest eigenvalues for k are as follows: J P = 1 ÷,

k = 2.74/R, 6.12/R .....

j e = 2-,

k = 3.87/R, 7.44/R .....

(2.6) The T M m o d e on the other hand is described by TM Vk.ZM(r) = A r M ( k , j)~r x (V x ( r ~ J M ) ) ,

(2.7)

k is determined by ] j ( k R ) = 0 and its parity is P = ( - 1 ) J. W e recall the lowest values for k: je = 1-,

k = 4.49/R, 7.73/R .....

JP = 2+ ,

k = 5.76/R, 9.91/R .....

(2.8) ATE(TM)(k, J ) is a normalisation constant which is fixed by the condition Ib

] 17TE(TM) / \12 . 3 ag

--k.zo

~X)l o ~.= l .

2.3. F E Y N M A N R U L E S

Let us now recall the results of Lee [3] concerning the interaction hamiltonian and express the gluon propagator, in the Coulomb gauge. The hamiltonian is given by H = Ib

ag

d3x ~ i n t +

S " A,

394

F. de Viron, Z Weyers / New meson configurations

where s is the surface tension and A the area of the cavity, 1 a ~ain t = [~'r • 1r a ..[_gfa

.

1 a .a "b~gVo (1o

_~_ c a b c v b

1 a • ¢r c) + ~B "B a

Va + p + ~ b + ( _ i a " V+m)~b,

B a is the color magnetic field inside the cavity, ~ra is the conjugate momentum of V ~, ] "0 = a2 g! " h + ~a ' " ,~x,

ja=l~l+olAa~l

and p represents the pressure of the medium on the "bubble". In this gauge, the propagator of the gauge field consists of a longitudinal, ab D L (x, y), and a transverse part D~b(x, y) [3-9]: ab

ab {DL (X, y ) , D~,~(x, y) = D ~ b ( x , y ) ,

/Z=V=4, /z=i#4,v=]#4.

The longitudinal (Coulomb) part is instantaneous in time and is given by DLab (x, y) = 8(t-- t')8=bG(x, y ) , G ( x , y) = ~ l G o ( x

, y) + Gl(x, y) + ~ooG2(x, y) +" • •,

GI = - -1 r - - 1+ 4~[[x-yl

~ I + 1 ( x y ' ~ ' Pt (cos 0)1 • Y~ , = I - - ~ - \ R 2]

(2.9)

(2.10)

~ is the dielectric constant of the vacuum. To insure confinement we will let ~'~ -+ 0 at the end of all calculations. D'~b(x, y ) =

E

~I--~'5"b(v*MX(X)),(VuaMa(Y))ie~'k('-C)

k.JMh Z K

(2.11)

where V~aMa is the vector potential (a = T E or TM) and the exponential factor is e ik(t-t') , when t > t',

but

e +ik(t-t')

when t < t'.

Note that we have a sum rather than an integral over the gluon eigenfrequencies. This is, of course, due to confinement in a cavity. Similary one can write the quark propagator as a sum over spins and eigenfrequencies of the allowed modes in the cavity.

2.4. E N E R G Y

SHIFT FORMULA

For level shift calculations we use the S-matrix formalism as derived by Sucher [10]. It gives an exact formula for the level shift in terms of the adiabatic S-matrix constructed from the perturbation which produces the shift. In the interaction picture, let Ifio) be an eigenstate of the zero-order hamiltonian Ho, and [fi) that eigenstate of the perturbed hamiltonian H = H o + gHi, into which

F. de Viron, J. Weyers / New meson configurations

395

[/30) goes as g, the coupling constant, increases from zero to g. Thus

1-1ot3o) = Eo[/3o),

HI3) = E[3) .

The level shift A E = E - Eo is then given by [11]

1.

A E = lim ~tag

0g(30[ Ue, (00, -(x))[3o)

q olSolt o)

(2.12)

This formula provides a direct connection between the level shift and the adiabatic S-matrix; it allows the use of covariant techniques and considerably simplifies the computations. The adiabatic S-matrix, S ~, is defined by S ~ = U~(oo, -oo) with d i - ~ U~(t, t o ) = g e-~l'tHl(t)U~(t, to). The convergence factor a has been introduced by Dyson [12] in the interaction hamiltonian, in order to give a prescription on how to smooth out any transient behavior due to the assumption that at time to the state vector is assumed to be an eigenstate of Ho, yet its subsequent time dependence is determined by Hi(t). With formula (2.12) one can now compute the energy shift due to gluon and quark exchange to first order in as, the strong coupling constant. 3. Zeroth-order mass spectrum Because of our restriction to a spherical and static bag, we will only consider the lowest modes of quarks and gluons, namely quarks in an $1/2 o r P1/2 m o d e and constituent gluons with total angular m o m e n t u m J = 1. Furthermore, these constituents will be restricted to be in an S-wave (no relative angular m o m e n t u m ) in order to respect the spherical symmetry of the problem. The energy levels at zeroth order in the strong coupling constant can be computed as follows:

M o ( R ) = E v + Es + EK

(3.1)

where E v = volume term = 4~-R3p, Es = 4rrR2s, where s is the surface tension [13]. This contribution will be neglected in the following. Its numerical effect is very small anyhow. EK = kinetic energy of quarks and gluons as given by the formulae (2.2), (2.4), (2.6), (2.8). The hadronic radius R is determined by minimizing the total energy. The parity of a quark antiquark pair is given by P(qq) = 7rq • 7r~ because they are in a relative S-wave. The intrinsic quark parity in the bag, rrq is defined by the quantum numbers of the orbit occupied by the quark. Charge conjugation for a quark-antiquark pair in the same state is then given by C = ( - 1 ) s, with S the total spin. For states composed of a quark in an $1/2 state and an antiquark in a P1/2 state and vice versa, we define linear combinations of the

396

F. de Viron, J. Weyers / New meson configurations TABLE 1 First approximation to the energy spectrum of qclg states [16] Quark mode

Antiquark mode

Gluon mode

Sl/2

$1/2

1+1--

P1/2

P1/2

1+1--

t~+(Sl/2P1/2)

1+

1--

~-(81/2P1/2)

1 +-

1--

jeC(q~g)

M0(GeV)

R(GeV -1)

1-(0, 1, 2) -+ 1+(0, 1, 2) ++

1.27 1.27 1.51 1.51

7.15

1-(0, 1, 2) -+ 1+(0, 1, 2) ++

1.74 1.74 1.96 1.96

7.94

1+ (0, 1, 2) +1(0, 1, 2)--

1.52 1.52 1.74 1.74

7.58

1" (0, 1, 2) ++ 1(0, 1, 2)-*

1.52 1.52 1.74 1.74

7.58

7.57

8.26

7.94

7.94

The values of the parameters are taken as follows: mq = 0, ce~= 2.20, p 1/4 0.120 GeV, s = 0. e n e r g y e i g e n s t a t e s , which a r e c h a r g e c o n j u g a t i o n e i g e n s t a t e s w h e n t h e q u a r k a n d a n t i q u a r k a r e e a c h o t h e r ' s antiparticles. T h e s e states will b e d e n o t e d b y thc, with C=+I. T h e first a p p r o x i m a t i o n to the ?:tqg e n e r g y s p e c t r u m is given in t a b l e 1. In sect. 4 w e briefly c o m m e n t o n t h e f i r s t - o r d e r c o r r e c t i o n s to this s p e c t r u m .

4. Energy shift calculations W e a r e n o w t o o l e d to c o m p u t e the mass splittings in t h e e n e r g y s p e c t r u m of qqg states, d u e to o n e - g l u o n e x c h a n g e . F i r s t of all we h a v e verified t h a t t h e u n w a n t e d t e r m s v a r y i n g as ~ in t h e l o n g i t u d i n a l p a r t of the g l u o n p r o p a g a t o r go a w a y at t h e e n d of the c o m p u t a t i o n , as t h e y s h o u l d . This p a r t of t h e gluon p r o p a g a t o r is always d i v e r g e n t for c o l o r n o n - s i n g l e t states in o r d e r to m a k e sure t h a t no c o l o r n o n - s i n g l e t s such as free g a u g e p a r t i c l e s c o u l d e s c a p e f r o m the bag. S e c o n d l y , we h a v e to calculate all t h e d i a g r a m s which c o n t r i b u t e to first o r d e r in a~. T h e y a r e listed in fig. 1. W e will n o t c o n s i d e r in o u r r a t h e r n a i v e c o m p u t a t i o n , the s e l f - e n e r g y d i a g r a m s ( l a - c , 2a, b). T h e r e a s o n is t h e following. T h e exact f o r m a l i s m for a c o n f i n e d p e r t u r b a t i o n t h e o r y has to b e r e n o r m a l i s a b l e a n d g a u g e invariant. D i a g r a m s of t y p e l a - c , 2a, b t h e n e n t e r in t h e r e n o r m a l i s a t i o n of t h e gluon a n d q u a r k p r o p a g a t o r s a n d

397

F. de Viron, J. Weyers/ New meson configurations

la.

lb.

Ic

@@ 2a

2b.

3.

~.

5.

Fig. 1. First-order contributions to energy shifts. wave functions. In a complete and consistent approach to confined gluons and quarks, we should compute these diagrams, to determine the renormalisation constants and to define the renormalized propagators, in a general gauge, and finally we should verify that physical quantities are gauge invariant. Here, we adopt a phenomenological point of view. We consider that we have already taken into account the self-energy diagrams in our computation by putting the "phenomenological" masses in the end results. The quark mass values are fitted to reproduce the usual meson spectrum. The prescription followed here is different from the one followed by the MIT group [4]. To respect the boundary conditions on the gluon field, they have to include part of the self-energy diagrams in the computation of the interaction energy [14]. What is left of the self-energy diagrams is then absorbed in the quark mass renormalisation. In our case, however, since we use the propagator of a gluon restricted to a cavity, the boundary conditions on the gauge field are automatically satisfied. Let us briefly describe the contribution of diagrams 3, 4 and 5 respectively, in terms of formula (2.12). Diagram 3

---ig (~AaAb)

d3x ag

ab

day ag +

+

× [+2DL (x, y}{ + ~,~ (x)J,2(x)4,~ (Y)63(Y)- ~,1 (x)4~3(x}4,4 (Y)~2(Y)}

398

F. de Viron, J. Weyers / New meson configurations

2

V~jM(y))i + A,J,M,k Y k~_K~( V~l~(X))i( *~ X { Jr"Ill; (X)Olil[12(X)~) ; (y)ai4,3(y) - 6 ; (x)ad,3(x)4,~ (y)ai6dy)}] = AE~ 2 + AE~2.

(4.1)

H e r e 0(~b) is the quark (antiquark) wave function given by (2.1), (2.2), the number index specifies the quark state in the cavity, em corresponds to its e n e r g y m o d e and

o,=(o o) Ori

Vk.ZM is the gluon radiation field as defined in sect. 2. The exchanged transverse gluon is, of course, off-shell. In formula (4.1), there is a factor 1 / (k 2 _ K2) instead of 1 / 2 k as in the free (transverse) gluon propagator (2.11). This factor is, of course, never equal to zero since k is the gluon energy [equal to one of the values listed in (2.6) and (2.8)] while K is the difference between two quark eigenfrxquencies [given by (2.2)-(2.4)]. Let us point out one difference between the standard (quark and antiquark) mesons and the " n e w " ones (qqg). For the first type of states (qq) we have only one type of contribution,

which gives no mass splitting between the J(qft) = 0 and J(q£1) = 1 states. On the other hand, for (qCtg) states, there are two types of contribution:

The first one has a spin dependence varying as ~ and the second as o% and hence it does give a mass-splitting contribution. In the transverse part of the propagator, (4.1), there is a sum over the gluon spins and eigenffequencies. We restrict ourselves to an exchanged gluon with angular m o m e n t u m equal to one and two, since we only examine the lowest quark states. We also keep only the lowest eigenffequency in the sum over different energies: we have verified for each case that the sum converges rapidly enough to justify such an approximation. For instance, the energy shift due to the transverse electric gluon

F. de Viron, J. Weyers/ New meson configurations

399

propagator varies as follows with the values of k, for the (q $1/2, ~ $1/2, 1+-(1--)) state: AEI~ (k = 2.74) :AEI~ (k = 6.12) : AEp~ (k = 9.32) = 1 : 10 -3 : 1 0 - 4 . Mass splittings are proportional to as/R. With formula (4.1) we reproduce the numerical coefficients of a s / R obtained semiclassically by the MIT group for the usual qq and qqq hadrons.

Diagram 4 aE

e =

+

AE~[~. +~g2

,

d3y (cCtbc 1 a ab

ag

x [{- 83.4g,~ (x)g,2(x) + 8 1 . ~ (x)g,3(x)} X{(W, +w~f,*d, ~ . . , ~ y ) ., v . (y)}]. The contributions of quark and antiquark have opposite signs. Hence, to give a non-zero contribution, they have to be of different flavors:

4 AEI~ 2 (S1/2(m1)S1/E(m2) gluon) ~ 0,

if m x # mE,

=0,

if m l = m E ,

and the same conclusions hold for P1/2, Px/Eg, 4'+(P1/2, S1/z)g and ~-(P1/ES1/E)g states. Note that our results are in agreement with those obtained by Barnes [6] when we consider one type of flavor only, but that we completely disagree for states which contain a quark and an antiquark with different flavors. The last contribution of diagram 4 is given by

AE,~2 = + l g 2 f d3x f d3y (cabc ½Ad6ad) Jbag Obag X [{--~3,41//~- (X)Ogml~2(X)"[-~1,2~4- (x)otm~3(x)} X ~k ~2 1=1 ~ Eli'i{'4-OYi(J~Jm(X'Y)) x ( v * ~ (y ) V ; (y ) - v*b (y ) VT (y ) )

-/5i~ ((a, v* ~ (y)) v~ (y) - (a, v~. (y)) v* ~ (y) + D ~ ((a, v *~ (y)) v~ (y) - ( a , v ; (y)) v *b (r)] with

D,j(x, y) = Y ( V ~*~( x ) , ( V ~ ( y ) ) j . J,M,A Due to the quark-quark-gluon vertex structure, only the transverse electric propagator contributes to this correction for J(qcl) = 1 states.

400

F. de Viron, J. Weyers / New meson configurations Diagram 5 aE~,2 =

1. 21~ta . . . . a b -AbA o) +~tg

[ X

i,]-2

+1,5 ej+e+oJ +

+ i,/2 e j - - e +to -i,/2

+

,

-i,/2 EJ -- E --0.)

+ i ,/-2 ej --e

+

+~o

*

+

( -- 411 O~/)jMVi Xx)(~jMOld, 2 V/)(y)

(- ¢d-,~d,J~v/)(y)(¢,;~,~;¢,: V ~ ( x ) +

E j - - E --03

+

+

~. Z ( - - ~ 1 1 0 1 i ~ ) z M V i ) ( y ) ( ( ~ j M O l i ~ 1 2 ) V i (X) e r,J,M L E j 4- E -- 0,)

*

+

( -- 01 a i O m V i )(x)(Om~]~2

V,.)(y)

(+,/,2o~j~,~V,.)(y)(6;M,~d,~

V i*)( x )

+

*

+

( 4- (~40lil~)JMVi )(X)((~JMOI,~)3 V I ) ( Y )

+ i,/2

G, --e +ca - i,/~-

e 1 + e --~o

( 4- ~)40lil~jMVi.)(y)(~JffMOgi~)3 Vi@)(X)

( + e~,~i~v*

)(x)(~d,~ v,.)(y)], J

where we restrict ourselves to an exchanged quark with angular m o m e n t u m 21-.This correction enhances the splittings given by A E 41~2.

5. Energy spectrum of quark-antiquark-gluon states Including all corrections to order c~s, we obtain the energy spectrum for qCtg configurations as given in table 2. For simplicity we have only included in table 2 the results for massless quarks. Needless to say, the precise numbers listed in table 2 are only of limited interest: for each set of quantum numbers given in the table there are other configurations such as gluonium, qqqq or even qCt states (or all of these) with which the qQg states can and will mix. To give but one example, the lowest exotic qCtg state at 1.57 GeV with j e c = 1-+ will mix with a glueball state [15] which at zeroth order lies at roughly 1.69 GeV. We defer to a forthcoming publication the systematics of such mixing effects which turn out to be relatively small. The important point, however, is the qCtg states do exist in the bag approximation to hadrons. Some of them are exotic and lie very low but there is simply no way to get rid of them: to the extent that the bag is a viable approximation to Q C D and to the

F. de Viron, J. Weyers / New meson configurations

401

TABLE 2

Energy spectrum of qqg states Components

S1/2S1/2 I+-(TE) Sl/2Sl/2 1 (TM)

j P c (qqg)

M = Mo +/tEig2 (GeV)

1-0 -+ 1-+

1.32 1.87 1.57

1+-

1.56

0 ++

1.69

1++

1.60

P1/2P1/2 1 + -

1-0 -+ 1-+

1.76 2.09 1.92

P1/2P1/2 1--

1+ 0 ++ 1+÷

1.98 2.05 2.01

4J+(S1/2P1/2) 1+-

1+

1.52

0 +-

1.93

1+-

1.70

6+ 1

10-1--

1.74 1.32 1.76

~b_ 1+-

1+ 0 ++ 1++

1.57 2.04 1.82

~b 1 -

1

1.79

0 -+ 1-+

1.93 1.87

h a d r o n s p e c t r u m w e a r e l e d to e x p e c t t h e d i s c o v e r y of t h e s e s t a t e s a n d , m o r e g e n e r a l l y , t h e d i s c o v e r y of s e v e r a l f a m i l i e s of u n c o n v e n t i o n a l m e s o n s ! O u r p u r p o s e in this p a p e r has b e e n to g i v e an e d u c a t e d g u e s s as to w h e r e o n e s u c h f a m i l y , n a m e l y m e s o n s of t h e q q g t y p e m i g h t lie a n d o u r r e s u l t s , i g n o r i n g m i x i n g effects, a r e s u m m a r i z e d in t a b l e 2. H o w to l o o k f o r s u c h s t a t e s is, of c o u r s e , a n o t h e r m a t t e r . A s a final c o m m e n t let us s i m p l y p o i n t o u t t h a t p e r h a p s t h e b e s t w a y to d i s c o v e r a n e x o t i c 1 ÷ s t a t e is via p h o t o n t r a n s i t i o n s f r o m o r d i n a r y v e c t o r m e s o n s , s u c h as p ' ( 1 6 0 0 ) , p r o d u c e d in e ÷ e annihilation. It is a p l e a s u r e to a c k n o w l e d g e v e r y h e l p f u l d i s c u s s i o n s w i t h A . J . G . H e y a n d J. K u t i .

402

F. de Viron, J. Weyers / New meson configurations

References [1] H.D. Politzer, H. Fritzsch and H. Gell-Mann, Phys. Rev. Lett. 30 (73) 1346; H. Fritzsch, H. Gell-Mann and H. Leutwyler, Phys. Rev. B47 (73) 365; D.J. Gross and F. Wilczek, Phys. Rev. D8 (73) 3498; S. Weinberg, Phys. Rev. Lett. 31 (73) 494 [2] A. Chodos, R.L. Jatfe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D9 (1974) 3471; P. Hasenfratz and J. Kuti, Phys. Rev. 40 (1978) 75, and references therein; A.J.G. Hey, Lectures at Louvain-la-Neuve, October, 1977 [3] T.D. Lee, Phys. Rev. DI9 (1979) 1802 [4] T.A. DeGrand, R.L. Jatfe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060; T.A. DeGrand and R.L. Jatfe, Ann. of Phys. 100 (1976) 425 [5] F.E. Close and R.R. Horgan, Nucl. Phys. B164 (1980) 413 [6] D. Horn and J. Mandula, Phys. Rev. D17 (1978) 898; T. Barnes, Nucl. Phys. B158 (1979) 171; P. Hasenfratz, R.R. Horgan, J. Kuti and J.M. Richard, CERN Preprint TH. 2837 CERN (1980). [7] J. Kuti, Quark confinement and the bag model, Lecture at 1977 CERN-JINR School of Physics [8] J.D. Jackson, Classical electrodynamics (Wiley, New York, 1962); A. Akhiezer and V. Beretstetski, Quantum electrodynamics (Wiley, New York, 1963) [9] J. Breit and T.D. Lee, Columbia preprint CU-TP-145 [10] J. Sucher, Phys. Rev. 107 (1957) 1448 [11] S.S. Schweber, An introduction to relativistic quantum field theory (Harper and Row, New York, 1964) [12] F.J. Dyson, Phys. Rev. 82 (1951) 420; 83 (1951) 608 [13] P. Gnfidig, P. Hasenfratz, J. Kuti and A.S. Szalay, Proc. Neutrino 75 IUPAP Conf. (1975) vol. 2, p. 251; Phys. Lett. 64B (1976) 62 [14] K.C. Bowler and P.J. Waiters, Phys. Rev. D19 (1979) 3330 [15] R.L. Jatte and K. Johnson, Phys. Lett. 60B (1976) 201; D. Robson, Z. Phys. C3 (1980) 199; J.F. Donoghue, MIT preprint CTP 874 (August, 1980) [16] F. de Viron, Annales de la Soci~t6 Scientifique de Bruxelles T92, IV (1978) 271