Bound states from the Bethe-Salpeter equation in QED3

N U CLEAR P H VS I C S B

Nuclear Physics B 387 (1992) 419—446 North-Holland

Bound states from the Bethe—Salpeter equation in QED3 C.J. Burden Department of Theoretical Physics, Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 2601, Australia Received 23 August 1991 (Revised 30 January 1992) Accepted for publication 28 August 1992 The bound-state spectrum of QED3 is studied using a quenched, ladder approximation to the homogeneous Bethe—Salpeter equation. Bound e+ —e states are classified in terms of their properties under space inversion and charge parity transformations. In addition to scalars and pseudoscalars, which form degenerate doublets, there exist doublet states, termed axi-scalars, whose transformation properties involve an arbitrary phase. Numerical solutions to the approximate Bethe—Salpeter equation for the lightest scalar and axi-scalar states are presented.

1. Introduction Quantum electrodynamics in three space-time dimensions (QED3) is, with good reason, becoming a popular area of study [1—81.It has important properties in common with quantum chromodynamics (QCD) without the complication of being a non-abelian gauge theory. QED3 is therefore an ideal place to study the non-perturbative methods of quantum field theory. In the quenched approximation at least, the bare photon propagator gives a confining, albeit logarithmic, e~—e potential. To one-loop order in the photon propagator, the confining potential is weakened to a 1 /r potential, but only at large distances [1], while more detailed analyses of the photon self-energy [6,7] indicate that full dressing of the photon propagator will restore the logarithmic nature of the potential. The low-energy spectrum of the theory can therefore reasonably be thought of as that of a strongly coupled system. Furthermore, the massless, four-component fermion version of QED3 which we study here, has a global, chiral-like U (2) symmetry. This symmetry is broken to a remnant U (1) x U (1), so generating a doublet of massless Goldstone bosons analogous to the pion in QCD. The global symmetry breaking has been well established by both lattice Monte Carlo calculations [1] and by solutions to approximate Schwinger—Dyson (SD) equations [2—8]. 0550-3213/92/S 05.00

© l992—Elsevier Science Publishers B.V. All rights reserved

421)

~.J. Burden / QED3 bound states

In this paper we take the next step beyond the SD equation, and solve an approximate Bethe—Salpeter (BS) equation for the lowest mass scalar bound states. The full SD and BS equations are, of course, intractable and we employ here a quenched, ladder approximation which we consider is the simplest reasonable approximation consistent with a qualitative determination of the bound-state spectrum. We note however that any simple truncation of the full equations made by replacing vertices with a finite set of diagrams is likely to break gauge covariance, as our approximation certainly does. Several attempts [6,8,9] at improved approximations to the SD equation designed to address the gauge covariance problem have been made, and these probably represent the most promising avenue for realistically improving the current bound-state calculations. A discussion of the transformation properties required of the propagators and vertex to maintain gauge covariance of the SD equations is given in ref. [7]. For the time being, however, our simple model suffices for the purposes of establishing techniques and developing a physical insight into the problem in hand. The methods we use here and the general form of the scalar bound-state vertices given in sect. 2 will no doubt be applicable to future, more rigorous treatments of QED3. The BS equation has often been used to model the binding of quarks. In fact attempts to obtain the meson spectrum from BS equations for tightly bound quarks were made well before QCD became entrenched as the accepted theory of strong interactions. Narayanaswamy and Pagnamenta [10] obtained solutions in terms of hyperspherical harmonics to the BS equation for a fermion and anti-fermion bound by the exchange of a pseudoscalar boson, while Böhm et a!. [11] considered a four-dimensional harmonic oscillator potential. More recently, Praschifka et al. [12] have looked for 0(4) symmetric solutions to a QCD inspired BS equation using a mass functional technique. Most other treatments of the BS equation applied to tightly bound systems have used the “instantaneous approximation” [13]. Our results will indicate that such an approximation is certainly not applicable to the low lying spectrum of QED3, since our solution functions are strongly dependent on the timelike momentum coordinate. In sect. 2 we give the transformation properties of four-component spinors under the action of space inversion and charge conjugation, which are necessary for classifying the bound e~—e states of QED3. In addition to the usual scalar and vector states and their parity partners we find it necessary to introduce new states which we refer to as axi-scalars and axi-vectors. General forms are given for the boson—fermion—antifermion vertices for the scalar and axi-scalar states. In sect. 3 we set out our model SD and BS equations. Our ladder BS equation is then reduced to sets of coupled integral equations, each set corresponding to one of the scalar or axi-scalar vertices given in sect. 2. The massless Gold-

C.J. Burden / QED3 bound states

421

stone boson doublet is explicitly obtained as a solution to one of these sets. Numerical solutions to the remaining sets are given in sect. 5 for the lightest bound state in each sector. A brief discussion of the analytic properties of the fermion propagator is given in sect. 6, and our conclusions are drawn in sect. 7. Appendix A is devoted to our conventions for relating quantities defined in Minkowski and euclidean space and in appendix B our model SD and BS equations are derived as a stationary-phase approximation to the path integral by first recasting the path integral in terms of bilocal fields. 2. Transformation properties of QED3 Consider the Minkowski-space action for massless, four-component QED3 [2], S[A,17.7,yi]

=

fd3x[—~F,vF’~~+Wyp(w’~+eA11)w]. (2.1)

The 4 x 4 matrices y,~satisfy {y~,y~,} = ~ ~ = diag(1, —1, —1) and the index ~s takes the values 0, 1 and 2. We introduce the complete set of 16 matrices

{ YA}

=

(ci

Yo —

4 =

—

(0I”\ ~i o) Yp4

0

3

I\ 0

=

=

{I, Y4,

)‘5, Y45,

\

~7)

Yl,2

,

1Yp~4,

=

Yp5

=

)

=

ZY/2Y5,

=

./a12 0

—l ~

(0 _iI\ ~ii 0

~ ~

‘

Y~L,Yp4, Y~t5,Yp45},

0

\

~l,2)

45 ‘

~

)‘p45

=

=

1YpY4Y5,

‘2 2

~t4,~5orp45 — Y

—

?7

Yv4,,i5orv45,

satisfying ~tr(yAyB) = ô~.The action (2.1) enjoys a global U(2) symmetry generated by {I, y4, y5, y45} which is broken in the full quantum theory [1,4,5] by the generation of a dynamical fermion mass to a U (1) x U (1) symmetry generated by {I, t45}. The action is also invariant with respect to discrete parity and charge conjugation symmetries, which for the fermion fields are given by ~(x)

-~

yj’(x’)

=

H~(x),

127(x)

C~7(x)T,

~7(x)~’(x)

—~

ii.7’(x’)

=

~7(x)H~,

(2.3)

_yJ(x)TCt,

(2.4)

and ~(x)—~ yi’(x)

=

=

C’.J. Burden / QED3 bound states

422

TABLE 1 1A under parity and charge conjugation. The Transformation properties of the bilinear currents matrices Rp, Rc and A are given by eqs. (2.7) and (2.8) JA

J=~,iii

PJAP’

CJAC’

J

J

(J4\\

(J

I .15 I

Rp~/54’\

J45

(J4 .15

—J45

J45

A;Jl,

(~,44~

R~A(~~4~

J,~5

R~(~/~4

—AJ~4s

1,x2). The matrices H and C are each determined up to where x’ = (x°,—x an arbitrary phase: H

=

y 14e~°’~’~~, C = y2e’~’~~~,

(2.5)

(0 ~ ç~p,q5-~ < 2ir) by the conditions {H,y1} = [H,y2]= 0, C~y,~C = -y~’, CT = ~C, C~ = Ct = 5p and q~chave previously been neglected [3,141, but are imporThe q tant phases for classifying the bound e~—e states. In order to classify these states, consider the transformation properties of the bilinear currents JA = WYA w under the action of parity and charge conjugation, listed in table 1. (To obtain the correct sign for the charge conjugation results we have taken into account that the fermion fields are anticommuting, Grassmann numbers.) For brevity we refer to the bound e+_e states as “mesons”. We see that, in addition to scalar, pseudoscalar, vector and pseudovector mesons, with the same transformation properties as J, J 45, J,~ and J,~45,there exist states which we shall refer to as axi-scalar, axi-pseudoscalar, axi-vector doublets ‘I~= ~ ~ )T, 1 T for which, and underaxi-pseudovector parity transformations /1 = (~4Lç~ ~AS(x) ~ ~AS~(x/) = Rp’I~~(x), [H,yo]

=

~APS(x)

~APsI(xF)

~AV(x)

,S

~1Av~(xl)

~APV(x)

.~

~,PAPv~l)

=

=RpA(X), =

—RpA~1~’1(X),

(2.6)

C.J. Burden / QED3 bound states

423

where (—cos2q5p —sin2q5p’\ srn 2çbp cos 2c1:’p I diag(1,—l,1).

Rp~

.

—

=

(2.7)

States which transform under charge conjugation according to the rules for JA given in table 1 we refer to as having positive charge parity, and states whose transformation properties differ from these by a minus sign we refer to as having negative charge parity. The matrix R~in table 1 is given by Rc

=

cos 2q~csin 2cbc ~ sin 2q~c cos 2q~c

2

In the remaining sections of this paper we will solve an approximate BS equation for scalar, pseudoscalar, axi-scalar and axi-pseudoscalar meson states of QED3. The BS equation is solved for the meson—fermion—antifermionvertex F, the invariance properties of which, under parity and charge conjugation, identify the meson. The invariance properties and consequently the general forms of the vertices in four space-time dimensions are given in ref. [15]. Here we give the analogous forms for three space-time dimensions. For a scalar or pseudoscalar state of momentum P with parity ~rand charge parity C, ~‘(P)—~-*,t~(AP), ~(P)—~-~C~(P),

m,C=±1,

the corresponding vertex satisfies the invariance property [151 F(q,P)

=

7rHF(Aq,AP)H~ = CCF(—q,P)TC’.

(2.9)

Similarly, for an axi-scalar or axi-pseudoscalar doublet transforming according to eq. (2.6) under parity and (P) CRc~(F) under charge conjugation, the corresponding axi-doublet of vertices must satisfy ~

~F(~)(q, P))

—+

=

(HF14) (Aq,AP)H~ irRp ~HF(~)(Aq,AP)H’

=

(CF~4~ (q, p)TC1’\ CR~~CF(5)(_q,P)TC~)

.

(2.10)

With these restrictions we see that the most general form of the scalar and pseudoscalar vertices is FS(q,P) = If+ ~g +~h+ ~ (2.11) F’~(q,P) = y~f+ qPy~ 1qr’yI~lk, (2.12) 5g+ PILy~45h+ ~0pP/ where f, g, h and k are functions only of q2, P2 and q•P. Vertices corresponding to the components ~ and of axi-scalars and axi-pseudoscalars take the ~

C.J. Burden / QED3 bound states

424

TABLE 2 Parity of the functions f,g,h and k in eqs. (2.ll)—(2.14).

C

Even functions of q . P

Scalar and Pseudoscalar

+1 —1

f, g, k h

f, g, k

Axial Scalar

+1 —l

f,h,k g

g f,h,k

Axial

+1

Pseudoscalar

—l

g f,h,k

f,h,k g

general forms r’(4~( F) AS (F(5)~P))

4)( ~

=

(~)f

+

Odd functions of q . P h

(~)

q~g+P~h~

(Y 4)k (2.13)

APS

r(

(F5~:P))

=

(~

4)f

+ (~~) (q~g+ P~h + f~0~~~qV (Y~5)k~ 2, P2 and q P. Furthermore, the (2.14) where, once again h the and bound k depend on isq determined by the parity of the charge parity C = f, ±1g,of states functions f, g, h and k under the transformation q P ~ —q F, as set out in table 2. Solutions to the approximate BS equation introduced in sect. 3 of this paper will be easily identified with the euclidean analogues of the vertices (2.ll)—(2.l4). .

.

.

3. Solving the ladder Bethe—Salpeter equation The main purpose of this paper is to find solutions to the ladder approximation homogeneous BS equation illustrated in fig. 1. Clearly it is not feasible to attempt a treatment of the full Bethe—Salpeter kernel, and the one-photon exchange, or ladder approximation we use here is the most obvious and commonly used starting point [131. It is convenient from this point on to work with a euclidean metric. Rules for transforming from Minkowski-space quantities to the equivalent euclidean quantities are given in appendix A. We also choose to work in the Feynman gauge. Reading from fig. 1 we have F(p,P)

=

_e2fd3qD(p_q)y~SF(~P+ q)F(q,P)SF(—~P+ q)y,~, (3.1)

where F (p, P) is the one-fermion irreducible meson—fermion—antifermion ver-

C.J. Burden / QED3 bound states

425

~P÷p Fig. 1. Diagrammatic representation of eq. (3.1).

tex [151*, D(k) = 1/k2 arises from the Feynman gauge propagator, and d3q = d3q/(2,r)3. It is important that the fermion propagator SF in eq. (3.1) be consistent with the result [1—8]that the global U(2) “chiral” symmetry of QED3 is broken. Any calculation of the bound-state spectrum which fails to allow for this symmetry breaking can not hope to generate the appropriate massless Goldstone bosons. For this reason we take for the fermion propagator the solution to a truncated SD equation, 1(p)

=

e2fd3qD(p_q)y/ 4SF(q)yp,

(3.2)

where 1(p) is the fermion self-energy: SF = (1 ~ + 1Y’. Our numerical solution to this quenched, bare vertex approximation is plotted in fig. 2 in terms of functions A and B defined by 1(p)= i~A(p2)+B(p2). (3.3) S~ The generation of a dynamical fermion mass and the breaking of chiral symmetry is signalled by the fact that B (p2) > 0. This solution is qualitatively similar to those obtained in more sophisticated studies [6,81 of the QED3 SD equation which have paid serious attention to the effects of fermion loops and to maintaining gauge invariance. Furthermore, as we shall see later, the combination of eqs. (3.1) and (3.2) ensures the existence of the required massless Goldstone bosons [16,171. In appendix B we derive the above model from a bilocal field theory formalism developed by Cahill and coworkers [12,18—211.This derivation is equivalent to lowest order in i~ of the meson-loop expansion of ref. [22]. In this formalism the chiral symmetry of QED3 is manifestly broken in a stationary-phase approximation to the path integral. It also has the advantage *

The vertex

x in ref. [15] is related to our vertex F via X(~P,p)= S~(~P + p)F(p,P)

XSF(—~P+p).

C.J. Burden / QED3 bound states

426

(a)

0.12 (b) 0.10

0.08

~0.06

-

-

0.04

-

0.02

-

0

0.2~

0.4’

‘0.6

0.8

p Fig. 2. The functions A and B in the fermion propagator SF from the SD equation (3.2) (solid curves) and analytic fits to these curves using the forms (4.3) (dashed curves).

that some indication is given as to how this model may be extended to include more diagrams. In the remaining part of this section we will reduce eq. (3.1) to sets

C.J. Burden / QED3 bound states

427

of coupled integral equations corresponding to the scalar, pseudoscalar, axiscalar and axi-pseudoscalar mesons described in sect. 2 at their mass poles = (0,0, iM). These sets can then be solved numerically. We begin by expanding F in terms of the 16 matrices of eq. (2.2): F(p,F)

=

~FA(P,P)YA.

(3.4)

Eq. (3.1) can then be written as the 16 coupled equations FA(P,P)

=

TAB =

_fa3q(pe2q)2TABFB(q,P),

~tr[yAy,~S~’(q + ~F)yBS~(q ~F)y,~]. —

(3.5)

Evaluating the traces we find this set partitions neatly into two disjoint sets of eight coupled equations, one in terms of 81 = {F1, F45, I~,f~45} and the other in terms of 82 = {F4,F5,I~4,I~5}. The set 81 contains the equations Fj(p,F)

=

3e~ j(—3 d ~(pq)2 x

{

[(~p2

—

1 (Q?A? +B?)(Q?A~+B?) 2)A q 1A2 + B1B2] Fj(q,F)

+i[(A1B2 +A2B1)q~+ ~(AiB2_A2Bl)Pp]Fp(q,F)

}

F~(p,F) =

+ :Ec~flpqc,FpAlA2F~45(q, F) (3.6) 3q ~p ~q)2 (Q?A? + B?)(Q?A 2 + B?) fd 2 x {_e [(A 1B2+A2B1)q~+ ~(A1B2_A2Bi)Pp]Fi(q,F) ,

+ie~ppqaFpA1A2F4s(q,P) 2)ôpa 2qpqa + ~FpPcT] A + ([(q2 ~F 1A2 + öp~B1B2)Fg(q,F) —

—

—

f~pa[~(A1B2 +

A2B1)Pc, + (A1B2 _A2B1)q~]1,45(q,P)} (3.7)

plus equations obtained from these by making the replacements F1~—~F45,

(3.8)

Fp+4J~45.

Here we have used the shorthand notation A1,2

=

A(Q?2),

B1,2

=

2 B(Q1 2),

Q1,2

=

q±

where the functions A and B are those defined in eq. (3.3). The set 82 contains

C.J. Burden / QED3 bound states

428

the equations F4(p,P)

2)(Q?A 2 + B?) 2

f~3q~2 (Q?A? + B1

=

+ q2 )A

{[(P2

1A2 + B1B2] F4(q,P) + [(A1B2_A2B1)qp + ~(A1B+A2Bi)Pp]Fp4(q,P) + ifc~flpqcxPflAlA2Fp5(q,F)}

~

F)

=

(3.9)

,

2 (Q~A~ + BI2)(Q?A? + B?) f~3q(~q) x {[(A 1B2_A2B1)qp + ~(A1B2+ A2B1)P~]F4(q,P) ifappqcx1’11A1A21’s (q,F)

2)öpa 2q~q~ + ~PPP~I] A —

+ —

([(q2

—

1A2

~P

—

o~~B1B2) F~,4(q,P)

i~ [~(A~B~—A2B1)PQ + (A1B2 + A2Bi)q~]I’as(q,P)}, (3.10)

plus equations obtained by making the replacements T~,,s—~—T,4.

~

(3.11)

With the help of eq. (B.12) in appendix equivalent (3.2) 2 =B 0,(which the setis 82 is solvedtobyeqs. either of and (3.3)), we find that, on setting P the ansätze F

2), 4(p,0)

=

B(p

or

2), F5(p,0)=B(p

F 5

=

=

F~= 0,

(3.12)

F 4=1~4=1~5=0.

These solutions, which are combinations of the forms (2.13) and (2.14), correspond to a massless doublet containing an axi-scalar state and an axi-pseudoscalar state. They are Goldstone bosons [16,171 arising from the breakdown of the initial U(2) symmetry of the action (2.1) to a residual U(l)j xU(1)45 symmetry. Evidence for these massless states has also been found in quenched lattic~simulations [231. The residual U (1) x U (1) symmetry is manifest in the relations (3.8) and (3.11). To determine the remaining meson states, we project F,, onto parts F~,F~ and 1, parallel and perpendicular to the vector P~~: F,,

=

u,,(q)F~(q,P)+ v~(q)1T~(q,P) + FpFp(q,P),

(3.13)

C.J. Burden / QED3 bound states

429

p

Fig. 3. The vectors u (q) and v (q) occurring in the expansion (3.13) of the vector part of the vertex function.

where the vectors F2q—(P.q)P u(q) — P2q2—(F.q)2’ —

v(q)

=

Pxq P2q2(P.q)2

(3.14)

are illustrated in fig. 3. f~4, F,,

5 and T~45are projected in a similar fashion. After some straightforward but lengthy algebra we find that the angle between u (q) and u (p) can be integrated out and that each of the sets S~and 82 further partitions into two sets of four coupled equations: Si = ss U Sps, 82 = SAS U SAPS, where 8s

=

{Fi,F~,F~ 45,Fp},

SAS

=

{F4,F~4,F~5,Fp4},

Sps

=

{F45,F~45,F~,Fp4s},

SAPs

=

{F5,F~5,F~4,Fp5}.

(3.15)

From the expansions (3.4) and (3.13) we see that they can be identified with the four generic vertices (2.ll)—(2.14) in sect. 2. The residual symmetries (3.8) and (3.11) mean that S~and Sr’s are equivalent sets of equations and that SAS and SAPS are also equivalent. From now on we will concentrate only on the sets Ss and SAS describing scalar and axi-scalar mesons, noting that each of these sectors actually corresponds to a degenerate parity doublet, the components of which transform into each other under U (1)45. Before displaying the integral equations explicitly, we set the bound-state momentum to F,, = (0,0, iM) so that the vertex functions F(q, F) depend only on = (q? + q?) 1/2, q3 and (parametrically) on M. We also rescale

C.J. Burden / QED3 bound states

430

the functions F1, F~,F~and F~to new functions f, U, V and W via

f(IqI,q3)

=

U(~qI,q3) =

i

V(~q~,q3) = ~~1~Fv4s(~qj,q3), W(jq~,q3) = MFp(jqj,q3), for the set Ss and a similar rescaling for the set SAS. With this choice of functions the integral equations take on a particularly symmetric form, namely 3e2 ~‘° 1 I~I dq3 dIqI 2_fl2)h/2IQ2A2+B2~2 = j0 (a x(Tfff(q) + T 1uU(q) + T1vV(q) + TjwW(q))

(2~)2j~I 2

U(p)

f

(~)2

=

dq3

I

2fl2)’/2a I~I d~qj(~ fl(a2—fl2)’/2 1Q2A2 + B2~2

x(T~ 1f(q) + TuuU(q) + TuvV(q) + TuwW(q))

V(p)

=

W(p)

=

2fl2)’/2a

~oc

2

/

I~I

d~q~ (~ fl(~2 fl2) 1/2 1Q2A2 + B2~2 x(Tvff(q) + TvuU(q) + TvvV(q) + TvwW(q)) 2 ~ 1 I~I ( 2 )2f dq3 d~q~ 2_fl2)l/2 Q2A2 + B2~2 Jo x(Twjf(q) + TwuU(q)(~+ TwvV(q) + TwwW(q)), (3.16)

dq3

(~)2

—

[

where

fl=—2~pIIqj,

~ =

~

(3.17)

A=A(Q2), and Tff, Tj ~

B=B(Q2),

are analytic functions of q

3, q I and M. These are the required sets of integral equations which will be dealt with numerically in sect. 4. The diagonal T’s are given by 2+ q~+ q12)1A12 i IBI2, Tff = (~M = (-~M2+ q~ q~2)~A~2 ±1B12, —

~

=

Tw~ =

1B12, —(~M2+ q~ q12)1A12 ±IBI2, (~M2+ q~+

—

q~2)~Aj2 ±

(3.18)

C.J. Burden / QED3 bound states

431

where the upper sign applies for the scalar meson equations 8~and the lower sign for the axi-scalar meson equations SAS. The off-diagonal T’s are, for the scalar mesons, Tj~Tui Tfw

=

Tvf

=

(A*B~B*A)~q~, —M~q~~A~2,

=

Twj

=

_(A*B + B*A)q

=

3 + ~i(A*B B*A)M, -

TuvTvu

=_[~(~*B~B*A)M~jq3(A*B_B*A)],

2,

Tuw = ~ = 2q3~q~IAI Tvw = Tw~= _j(A*B_B*A)~q~,

(3.19)

and for the axi-scalar mesons: Tiu Tfw

Tuv

=

Tuj

=

—Tv

=

=

2, 1

=

1

=

-Tw

M~q~~AI

=

~

= =

-T~v = (A*B + B*A)~q~.

_[~i(A*B_B*A)M_q3(A*B--B*A)], 2, —T~~ = —2q3IqIIA~ =

(3.20)

Before closing this section we note that the functions U, V and W are related to the functions occurring in the generic form of the euclidean space vertex (A.3) given in the appendix A by U(q

3,Iq~) = —IqIg(q3,IqI),

V(q3,IqI) = —iM~q~k(q3,IqI), W(q3,~q~) = iMh(q3,IqI)—q3g(q3,IqI). With the help of table 2, we see that, for a scalar state with positive (negative) charge parity, the functions f, U and V are even (odd) functions of q3, while the function W is an odd (even) function of q3. For an axi-scalar meson with positive (negative) charge parity, f, V and W are even (odd) and U an odd (even) function of q3. 4. Numerical results We have derived sets of coupled integral equations for the scalar and axiscalar meson vertices which can now be dealt with numerically. These sets, given explicitly by eq. (3.16), are of the form

f(IpI,p3M) =fdq3fdI~IK(IPI~P3IqI,q3M)f(IqJ~q3M)~ (4.1)

Cf. Burden / QED3 bound states

432

where f

=

(f, U, V, W )T~They are solved as eigenvalue problems of the form

f

dqK(p,q;M)f(q)

=

A(M)f(p),

(4.2)

the bound-state mass being obtained by adjusting M so that A(M) = 1. The kernel contains the functions A and B defining the dressed euclidean photon propagator, and obtained from a numerical solution to the approximate SD equation (3.2). The two coupled integral equations involved in eq. (3.2) were solved using simple iterative substitution on a non-uniform grid of 51 points with a momentum cutoff (p/e2)m~= 1000 and a 0.1% tolerance in the integration routine. Our results are plotted in fig. 2 in units for which e2 = 1*. The numerical solutions of eq. (4.1) entail the evaluation of the fermion propagator in the complex plane. In fact, the kernel K (p, q;M) contains the functions A and B evaluated at the complex arguments Q2, given by eq. (3.17), and its complex conjugate. To facilitate the numerical evaluation of the kernel we replaced A and B with the analytic fits Afit(p2)

=

(a? +p2)112 + a

2+ 1, 3e”~”

Bfit(p2)

2 +b 2, 2+p 3e~~~ which are also plotted in fig. 2. A least squares fit gives =

b

(4.3)

a 1

=

6.298x 10—2,

a2

=

9.8l2x 102,

a3

=

—2.379 x 102, a4

=

3.222,

=

1.102 x l0~,

b2

=

1.295 x 102,

=

2.694 x 102,

b4

=

1.216 x 102.

b3

We have checked numerically with these the fermion propagator has 2 axis,that as required forfits a confining theory. However, no poles on the p singularities in the propagator in the complex p2 the fit does givenegative conjugate plane at p2 = Sp = —0.00400 ±0.00666i, that is, spAfit (Sp)2 + Bfit (SF)2 = 0. Indeed, any singularities off the real p2 axis must come in conjugate pairs if the euclidean propagator is to be real. Such singularities have been observed previously in fermion propagators obtained by numerically solving model SD equations for QED4 [24]and QCD4 [25]. In these cases the singularities are conjugate branch points, and we suspect a similar situation may arise for the solution to eq. (3.2). If these singularities are to be avoided by the integral Our SD and BS equations are scale invariant, as can be seen by recasting the equations in terms of the dimensionless momentum p/e2.The functions A and B for an arbitrary value of e can be obtained from the e2 = 1 solution by the scale transformation: A (p2 e) = A (p2 /e4 1), B(p2e) = e2B(p2/e4 1). *

C.J. Burden / QED3 bound states

433

TABLE 3

Lowest-mass meson bound states from eq. (3.1). The axi-scalar, C Goldstone boson State

(a) (b) (d)

Scalar Scalar Axi-scalar

=

+ 1 meson (C) is a massless Mass

C C C

= = =

+1 —l —1

0.080±.OOl 0.123 ±.002 0.111 ±.002

over real q3 and I q I in eq. (4.1) we are restricted in our numerical calculations to values of M satisfying M<0.l528.

(4.4)

The question of the validity of a naive Wick rotation in these circumstances is very much open [26]. This point will be considered further in the next section. In the meantime we will assume that working with the euclidean metric is valid for the bound-state mass calculations [25] and suggest that the restriction (4.4) might be circumvented, if necessary, by an appropriate deformation of the contour of integration in eq. (4.1) in the q3 plane [27]. The eigenvalue problem (4.2) was solved by iterative substitution for the functions f, U, V and W defined on a 25 x 25 grid in the P3—IPI plane. The integrals were evaluated on each tile in the plane using a Simpson-rule quadrature, the p-dependent terms in the integrand being estimated by linear interpolation. The terms in the integrands in eq. (3.16) containing a and /3 have an integrable singularity at q = p. Contributions to the integrals from a small rectangle about this singularity were estimated by using the value of the p-independent terms in the integrand at q = p and integrating the remaining, singular terms analytically. We obtained solutions to eq. (3.16) for the lightest meson in each of the four sectors: (a) scalar mesons, C = + 1, (b) scalar mesons, C = —1, (c) axi-scalar mesons, C = + 1 and (d) axi~scalarmesons, C = —1, where C is the charge conjugation parity. The masses obtained for the lightest meson in each of the sectors (a), (b) and (d) are set out in table 3. The errors are fairly conservative estimates based on the effect of varying the momentum cutoff, the fineness of the integration grid in the I q I—q3 plane and the size of the rectangle over which the p = q singularity is estimated. Note that each of these solutions satisfies the restriction (4.4). For the axi-scalar, C = + 1 meson (c) we obtained for the eigenvalue A(0) in eq. (4.2) the value 1.012, comparing well with the expected exact value of 1 for the Goldstone boson. In fact the solution (3.12) is recovered to within 1% out to p = 0.4, providing a useful check on the accuracy of our integration routine. In fig. 4 we plot the functions f, U, V and W contributing to the vertex F of the scalar, C = + 1 meson [(a)of table 3]. We see that the principal

C.J. Burden / QED3 bound states

434

JØ~

(a)

,i

0.YQ

~.06 0.o~ ~

O.°~

Qo

5

0.00

0.101

(b)

0. 1~

0.06

0.05

0

o.03

0.03 ~~o0

0

o_.

a 0

~0

0

N..

0

Fig. 4. The parts f, U, V and W of the vertex function F for the scalar, C = + 1 meson [(a) of table 3]. The vertical scale is arbitrary, but the same for each plot.

C.J. Burden / QED3 bound states

435

(c)

fO.io

/

O.08\

0.0S 0.o o.0~

003

0 0 0

.7

0

‘,~

o 0

(d)

f0.i~

0.10

/o.oe

~

t0.03

0~~’~*’

Fig. 4. Continued.

Cf. Burden / QED3 bound states

436

0.18

0.18

o.14

0.74

o.09

00

0.0~

0.0~

~

~NN 0 LI

0

q

0,~.

0

Fig. 5. The part

f of the vertex function F for the scalar,

C

=

—1 meson [(b) of table 3].

~.25

o.19 o•~

07

o.06

0•~

0.00 07,~

a

0 •‘O

0

..~

q0

a Fig. 6. The part

f of the vertex function F for the axi-scalar, C

=

—1 meson [(d) of table 3].

Cf. Burden / QED3 bound states

437

contribution to the vertex comes from the function f. This also occurs for the scalar and axi-scalar mesons [(b) and (d) of table 3, respectively], for which we have plotted the function f in figs. 5 and 6. As required, f is an odd function of p3 for each of the C = —1 states. Immediately noticeable from these plots is that the function peaks at a positive value of the spatial relative momentum I~ I between the fermions for the scalar states, but at I~I= 0 for the axi-scalar states. If we interpret this as a greater degree of radial excitation in the scalar mesons, it is not surprising that their masses in table 3 are higher than those of the corresponding axi-scalarbymesons. Vertex functions peaking 2 were also obtained Praschifka et a!. [12] who used at a positive values of p mass functional method to obtain approximate solutions to the homogeneous BS equation for mesons and diquarks in QCD4. They assumed the vertex to be 0(4) symmetric (i.e. a function of p2 only) and ignored those parts of the vertex analogous to the functions we denote by g, h and k. If our QED3 solutions are to be taken as a guide, this seems reasonable at least for the scalar and pseudoscalar charge parity positive mesons for which the function analogous to our function f is an even function of p 4 [15]. Praschifka et al. interpret the peaks as a signal of the generation of constituent fermion masses within the mesons.

5. The analytic properties of SF We now return briefly to the question of the analytic properties of the fermion propagator. Our numerical solutions will to some extent be influenced by the conjugate singularities Sp in the euclidean momentum-space fermion propagator. It is therefore important to know how genuine these singularities actually are. Given the numerically determined branch point singularities obtained in refs. [24,25], and the closeness of the fits in fig. 2, it is unlikely they are purely an artifact of the particular analytic fits employed here. On the other hand, it is impossible to tell at this point if they are symptomatic of the particular approximation (3.2) to the full set of exact SD equations. If complex singularities in the fermion propagator are a genuine phenomenon, it is also important to understand their physical meaning. The problem of performing the Wick rotation in the presence of such singularities is considered in ref. [26]. There it is pointed out that Minkowski momentumspace propagators cannot in general be obtained by a simple analytic continuation of euclidean momentum-space propagators. Instead one should first Fourier transform to euclidean coordinate space and then analytically continue to the real time axis. Since n-point functions obtained from euclidean path integrals will be time ordered along the euclidean time axis, the function obtained from Fourier transforming the euclidean momentum-space propagator

438

C.J. Burden / QED3 bound states

will, when restricted to the positive euclidean time axis, be the analytic continuation of the two-point Wightman function*. The inclusion of step functions in real Minkowski time to give the appropriate time ordering and Fourier transforming back to momentum space then gives the Minkowski momentumspace propagator. In ref. [26] Roberts et al. consider this procedure for a model euclidean scalar propagator, based on a quark propagator proposed by Pagels and Stokar [28], which has complex conjugate poles. They find that not only does this procedure fail to produce the naive analytic continuation of the propagator in momentum space, but surprisingly a Minkowski-space propagator cannot be produced in this way at all: the Minkowski coordinate space function obtained has no Fourier transform. Of relevance to the above discussion is the work of Lee and Wick [29] dealing with the quantisation of theories with indefinite classical energy. In a path-integral (P1) treatment of Lee and Wick’s work, Boulware and Gross [30] consider in detail a one-dimensional field theory with conjugate poles, not dissimilar to the above modified Pagels—Stokar model. For this case they explicitly illustrate two quantisation procedures, a euclidean P1 quantisation which leads to positive energies but an indefinite-metric state space, and a Minkowski P1 quantisation which leads to a definite-metric state space, but indefinite energies. We suspect that the singular euclidean momentum-space propagator of Pagels and Stokar, and those singular propagators which may arise from euclidean SD equations, lie within the realm of the first of these quantisation procedures. One disturbing aspect of indefinite-metric theories is that they tend to produce acausal behavior. For instance, the vacuum expectation value of the field in the one-dimensional model of ref. [30], in the presence of a source, depends on the source at later times. It is not clear whether, in the context of QCD or QED3, such acausality will manifest itself in the physical observables of the theory. While it would seem possible that the inclusion of a quark source in QCD may have an acausal effect on the quark field expectation value, it is debatable whether the inclusion of a colour singlet source would have the same effect. This suggestion is based on the assumption that a full “hadronisation” of QCD [21] along the lines of the method employed in appendix B is possible. Propagators of the resultant colour singlet hadronic fields would not be expected to have complex conjugate singularities.

The Wightman function is defined as the Minkowski-space n-point function without time M~)~(x(M>)~vac) where x(M) = (x°,x). The ordering: W~(x~M) x~M)) (vacI~(x~ Minkowski-space propagator is =obtained from the Wightman function via G *

2 (xi

O(x?_x~)W2(xi,xi)-i-9(x~—x?)Wi(x2,xi).

—

x2)

=

Cf. Burden / QED3 bound states

439

6. Conclusions We have generated numerical solutions to an approximate homogeneous BS equation for positive and negative charge parity “mesons” in three-dimensional massless quenched QED. This is a particularly useful field theory to study, being a confining gauge theory which is much simpler to work with than QCD. The mesons we obtain are classified as being degenerate doublets of either scalars or axi-scalars (defined in sect. 2) according to their transformation

properties under space inversion. The axi-scalar, positive charge parity doublet is a massless Goldstone boson analogous to the pion of QCD. The masses

obtained for the lightest scalar meson in each remaining sector are given in table 3. The quenched, ladder approximation used in this paper is chosen to take into account the global symmetry breaking property of QED3. The model automatically generates the expected Goldstone boson doublet as a solution to the BS equation. Furthermore it can be derived as a stationary-phase approximation to the bilocal form of the path integral. For an abelian gauge theory, this formalism provides an exact rearrangement of the original path integral in terms of bosonic fields. The stationary-phase expansion should, in principle, allow the approximation to be systematically extended to include more diagrams. Alternatively, improved vertices specifically designed to address the problem of restoring gauge covariance, such as those in refs. [6,8], could be included within the BS equation. The bound-state spectrum must ultimately, of course, be insensitive to the choice of gauge. An important point to emerge from our calculations is the question of the analytic structure of the fermion propagator. The analytic fit to our approximate SD equation solution suffers from conjugate singularities in the complex euclidean energy plane. These singularities may well be a genuine property of the full fermion propagator, and may even be endemic of confining propagators in general. Evidence from earlier studies of QCD4 [25] and strong coupling QED4 [24] propagators points in this direction, and suggests that the singularities may in fact be complex conjugate branch points. Ifthe singularities are genuine, the interpretation and validity of euclidean momentum-space propagators for confining theories in general will need further investigation. We also stress that the vast majority of lattice gauge theory calculations have been carried out on euclidean lattices, and any questions concerning the validity of our above procedures in this respect must also be levelled at lattice calculations. Our numerical solutions to the homogeneous BS equation require knowledge of the fermion propagator away from the real euclidean energy axis and are almost certainly influenced by the singularities. It is likely that lattice calculations will for some time remain our best hope for accurate quantitative analyses of confining theories. However, we believe

C.f. Burden / QED3 bound states

440

a qualitative understanding of the confinement mechanism and of the boundstate spectrum is far more likely to come from improved analyses of the SD and BS equations. We have found QED3 to be an ideal avenue for exploring in this direction. Furthermore we feel that improved techniques for handling these equations may eventually provide a useful and independent cross check on the results of lattice gauge theory.

Note added Since this work was completed, Munczek and Jam [31] have published a fully covariant study of the ladder Bethe—Salpeter equation of QCD in which they obtain a good fit for the pseudoscalar q~mass spectrum.

Appendix A Euclidean-space conventions Below are given our conventions for relating quantities defined in Minkowski and euclidean space, denoted by the superscripts (M) and (E) respectively. Space-time coordinates are related by x 8~ = xl,2(M) x31~= ixO(M) 1,2~ while momenta, coordinate derivatives 9,, and the vector potential A,, are related by ,

p 3(E)

=

jp0(M)

p12(E)

=

p12(M)

euclidean-space Dirac matrices are defined in terms of the Minkowski-space matrices in eq. (2.2) by p3(E)

=

~0(M) ,

p12(E)

=

~~12(M)

y4, y5 and y45 are unchanged, and y~4(E) = ~

~,,5(E)

=

~~,,(E)~5

~,,45(E)

=

They satisfy {y~(E),y0(E)}= 2ô,,0, ~u,v = 1,2,3 and ~tr(yA~~yB~)= öAB, A, B = 1,... 16. Minkowski and euclidean actions are related by S~ = ,

_jS(M).

In appendix B we make use of the Fierz identity (y,,(E))rs(y,,(E))tu

=

~(KA)ru(KA)ts,

(A.1)

Cf. Burden / QED3 bound states

441

where KA

(no sum over A),

= KAYA

KjK

K4=K5=—~—,

45=—~--,

K4

K445

=

i =

~,

K44

=

K45

1 =

(A.2)

~.

Finally, note that the terms containing the functions g and h in the scalar meson vertices eqs. (2.ll)—(2.14) differ in euclidean-space from their Minkowskispace counterparts by a factor i. For example, eq. (2.11) becomes 5(q,F) = If + jq,,(E)y,,(E)g + jp,,(E)y,,(E)h + ,zvpP,,q~yp F 4s~k. (A.3) Appendix B Bilocal field theory In this appendix we derive approximate Schwinger—Dyson equation (3.2) and Bethe—Salpeter equation (3.1) introduced in sect. 3 through the bilocal field theory formalism of Cahill and coworkers [12,18—21]. We begin with the euclidean massless QED3 path integral

Z[J,~7]

=

fVAVY/DWeXP (_SQED[A, yJ,~]

+Jd3x[i~w +

~

+

J4A4])~

with SQED

=

f

3xE~F#~F,,~ + ~(O,,A,,)2

+ ~7y,,(a,,+

ieA,,)~].

(B.1) (B.2)

d

The action is quadratic in the gauge fields which can be integrated out, and the fermions rearranged using the Fierz identity (A. 1) to give [18,19] (in the Feynman, ~

Z[0,~]

=

=

1 gauge) fvwviii

xexP(_Jd3xd3y [~(x)o3(x—y)~w(y) —~e2l~7(x)K~u(y)D(xyYlii(y)K”w(x)]) —

(B.3)

xexpfd3x(~w + ~ where the matrices KA are given in appendix A, and (—3

D(x_Y)=jdk

~

k2

(B.4)

C.f. Burden / QED3 bound states

442

—3 is the free photon propagator where d k = d3k/(2ir)3. Introducing a set of bilocal fields B(x,y) = !3(y, x)* = 8A (x, y)KA [18] then enables the fermion field to be integrated out, leaving, up to a multiplicative constant

Z[0,~,ifl

=

f V5Ae ~ exp (f ~(x)G(x,y;[L3])1i(Y)d3xd3Y)~ (B.5)

where 5 S[t3]

=

G~(x,y;[B])

=

(.‘~x)—Trln(G’(x,y;[8]))

fd3xd3y ô3(x—y)

0

(B.6) (B.7)

+B(x,y).

The inverse G’ is defined by

J G~(x,w)G(w,y)d3w

=

while the trace in eq. (B.6) involves an integral over space-time coordinates as well as a sum over spinor indices. We stress that, for QED, eq. (B.5) is an exact rearrangement of the original path integral (B.!). This is not true for the QCD analogue of eq. (B.5) in which contributions from n-point gluon functions for n > 3 should also be included. In practice, however, one approximates these effects with an effective two-point function designed to exhibit asymptotic freedom and infrared slavery [18]. On the other hand, the bilocal formalism, when applied to QCD in this way, has the advantage that by minimising S [5] one gets a chirally asymmetric lowest-order approximation to the vacuum. This is clearly an improved starting point for a stationary-phase approximation to QCD. The equivalent phenomenon in massless QED3 is the breaking of global “chiral” U (2) symmetry present in the bare action to U (1) x U (1) and the consequent generation of a dynamical fermion mass and chirally asymmetric vacuum. To see that the action S[B] achieves this in the stationary-phase approximation, consider the fermion two-point function SF(x—y)

=

(0~Tyi(x)~7(y)~0) = fV5~~(~~,y;[131)exp(—S[5]) (B.8)

To zeroth order in the stationary-phase approximation,

SF(x—y)~S~(x—y)=G(x,y;[5~i]),

(B.9)

where 5cl (x, y) is the classical solution to the Euler—Lagrange equation !3~(x,y)=e2D(x_y)tr[G(y,x;[Bci])KAJ

.

(B.l0)

Cf. Burden / QED3 bound states

443

The ansatz 2)

!3~1(x,y)=

—1) + B(p2)],

(B.ll)

~(A(p

f~3pe1~’)[i

gives

i~(A(p2) —1) + B(p2)

=

e2f~3qD(p

—

q)y,,[i~A(q2) + B(q2)]~y,,.

(B.12) Making use of eqs. (B.9) and (B.7) we see that eq. (B.12) is just the truncated Feynman-gauge SD equation (3.2) with S~playing the part of the fermion propagator*. Our numerical solution plotted in fig. 2 has B (p2) > 0, indicating the existence of a chirally asymmetric unperturbed vacuum. Perturbations on top of the classical field configuration are interpreted as bound e+_e “meson” states [20,21]. We write 5(x,y.)

=

L3~ 1(x—y)+ b(x,y),

(B.13)

S~(x—y) + b(x,y).

(B.14)

so that eq. (B.7) becomes G(x,y;

[5])l

=

Redefining the action by Snew [b] = SOld [5] trace log in the action (B.6) then gives S[b]

=

—

S01d [ccl]

and expanding the

fd3xd3Y bA(x,y)bA(y,x) + ~Tr(S~’bS~b) + ~fd3Xd3Yd3Ud3VbA*(X,Y)A1(X,YU,V)~bB(U,V)

+...,

(B.15) where we have defined A(x,y;u,v)~

= ô(v_Yô(U_X)oABt(S~)KAS~)KB)

(B.16) Here we have retained only the quadratic, free field part of the action. The dots indicate higher order, interaction terms, while terms linear in b cancel with the help of eq. (B.l0). It is convenient to define Fourier-transformed fields b (p, F) and inverse propagator 4’(p,q;F) by

b(p,P)

=

fd3xd3yexp{_i

[~.(x

±~) + p. (x_Y)]}b(x~Y)~

(B.17)

*

The parallel derivation of an approximate SD equation in QED4 for an arbitrary gauge

parameter ~ is given in ref. [19].

C.J. Burden / QED3 bound states

444

fd3xd3yd3ud3v

A~(p,q;F)(2,r)3ô3(P—F’) =

xexp{_i

[~.(x

±Y) +p. (x—y)

(~‘~) -q. (u_v)]}

_p’.

(B.18)

x4~(x,y;u,v),

in terms of which the quadratic part of the action (B. 15) becomes S[b]

=

~

=

~

(B.19)

with A~(pq;p)AB

=

(2ir)3ô3(p f

‘d3

I A~(p,q;P) =

—

q)tr [S~(—~P + p)KAS~(~P + p)KB]

~AB~_iz.(p_q)

(B20)

e2D(z)

(B.21)

~(KA)~1A(p,q;F)~(KB)~1.

We now assume the existence of a unique expansion of b in terms of a complete, orthonormal set of eigenfunctions Fk, .fj~,... of A~

b(p,P)

=

~Fk(p,P)~k(P) k

+ ~I~4(p,F)~(F)

+...,

(B.22)

k

satisfying f~3qAr

2Fkrsp,n, 5,tu(p,q;P)Fkf~(q,P) =

f~3qArs,ju(p, q; P)T~(q, F)

=

B.23

kF

~2~”(F2)F~’rs(p, P),

(B.24)

and ~fa3ptr[1~t(p,F)Fi(p,Fn

=

ökl,

(B.25)

~f~3ptr[1~4t(p,F)I~’(p,P)]

=

~klô~’.

(B.26)

The F’s are identified with vertices for scalar, vector, etc., mesons, as will become apparent shortly for the scalar mesons. We further assume that the scalar meson eigenvalues tk (F2) and the transverse part of the vector meson eigenvalues ~ (F2) have zeros at P2 =

Cf. Burden / QED3 bound states

445

and —M,~”2respectively. In general they can be written in terms of constants and functions /-tk’ and tk, as = fk(F2 + ILk(P2)2), (B.27) V(p2) = fk~”[(~2 + i4’(P2)2)o,,~ tk(P2)P,,F~], (B.28) where /2k (—M,~~) = Mk, /4’ (—M,~”2)= M,~”and tk (—M,~)= 1. The quadratic part of the action then becomes

/4’

fk, fkV

—

S[~]

=

~fkf

( 21~)3~k(fl(P+

22o,,~ tk(P2)P,,Pv] ~F) + ~fkVf (2)3~k~ [(p2 + ~p + higher-spin meson contributions. (B.29) The first two terms are easily recognised as a sum of Klein—Gordon and Proca—de Broglie contributions. The functions ~uare running masses for the mesons. Now we concentrate on the scalar meson sector. Suppressing the index k, we write 1 5(p,P) = FA(p,P)K~. (Note that these FAGs differ slightly from those introduced in sect. 3.) Eq. (B.23) then becomes, on multiplying through by K~ 2)I,c~I2FA(p,P) (no sum over A), -

f~3q4_l(p,q;P)~4BFB(q,P) = ~(F

(B.30)

where the KA are defined in appendix A. Substituting for A and Fourier transforming gives, after some rearrangement, =

_fd3qe~tr[s~1(_~F~q)KA5~(~F

—

from eq. (B.20)

~q)KB]FB(q,F)

(B.3l) Multiplying through by e2D(z), Fourier transforming back to momentum space and multiplying through by KA gives finally F(p,P)

=

—e2

f ~3qD(p

—

q)y,,S~J(~P + q)F(q,P)S~(—~P + q)y,,

+e22(P2) ~IKAI2KJI

f ~3qD(p

—

q)FA (q,P).

(B.32)

At a pole in the bound-state propagator we have ).(—M2) = 0 and the last term in eq. (B.32) drops out. We are left with the approximate homogeneous BS equation illustrated in fig. 1. I would like to thank J. Praschifka and C. D. Roberts for helpful and stimulating discussions.

446

Cf. Burden / QED3 bound states

References [1] E. Dagotto, J.B. Kogut and A. Kocic, Phys. Rev. Lett. 62 (1989) 1083; NucI. Phys. B334 (1990) 279 [2] R.D. Pisarski, Phys. Rev. D29 (1984) 2423. [3] T.W. Appelquist, M.J. Bowick, D. Karabali and L.C.R. Wijewardhana, Phys. Rev. D33 (1986) 3704 [4] T.W. Appelquist, M.J. Bowick, E. Cohler and L.C.R. Wijewardhana, Phys. Rev. Lett. 55 (1985) 1715 [5]S. Rao and R. Yahalom, Phys. Rev. D34 (1986) 1194 [6] M.R. Pennington and D. Walsh, Phys. Lett. B253 (1991) 246 [7] C.J. Burden, J. Praschifka and C.D. Roberts, Photon polarisation tensor and gauge dependence in three-dimensional quantum electrodynamics, ANU/Argonne preprint, 1991 [8] C.J. Burden and CD. Roberts, Phys. Rev. D44 (1991) 540 [9] J.S. Ball and T.W. Chiu, Phys. Rev. D22 (1980) 2542 [10] P. Narayanaswamy and A. Pagnamenta, Nuovo Cimento 53A (1968) 635 [11] M. Böhm, H. Joos and M. Krammer, Nucl. Phys. B5l (1973) 397; B69 (1974) 349 [12] J. Praschifka, R.T. Cahill and C.D. Roberts, Mod. Phys. Lett. A3 (1988) 1595; Int. J. Mod. Phys. A4 (1989) 4929 [13] D. Flamm and F. Schöberl, Introduction to the quark model of elementary particles, vol 1, Section 3.4 (Gordon and Breach, New York, 1982) [14] R. Jackiw and S. Templeton, Phys. Rev. D23 (1981) 2291 [15] C.H. Llewellyn Smith, Ann. Phys. (NY) 53 (1969) 521 [16] R. Delbourgo and M.D. Scadron, 1. Phys. G5 (1979) 1621 [17] R.T. Cahill, C.D. Roberts and J. Praschifka, Phys. Rev. D36 (1987) 2804 [18] R.T. Cahill and C.D. Roberts, Phys. Rev. D32 (1985) 2419 [19] C.D. Roberts and R.T. Cahill, Phys. Rev. D33 (1986) 1755 [20] J. Praschifka, PhD thesis, November, 1988 [21] R.T. Cahill, Austr. J. Phys. 42 (1989) 171 [22] M. Lutz and J. Praschifka, Meson-diquark loop expansion, Regensburgpreprint TPR-90-47, 1991 [23] K. Farukos and G. Koutsoumbas, Phys. Lett. Bl78 (1986) 260 [24] D. Atkinson and D.W.E. Blatt, Nucl. Phys. Bl51 (1979) 342 [25] S.J. Stainsby and R.T. Cahill, Phys. Lett. Al46 (1990) 467 [26] C.D. Roberts, AG. Williams and G. Krein, mt. J. MOd. Phys. A7 (1992) 5607 [27] C.J. Burden, R.T. Cahill and J. Praschifka, Austr. J. Phys. 42 (1989) 147 [28] H. Pagels and S. Stokar, Phys. Rev. D20 (1979) 2947; D22 (1980) 2876 [29] T.D. Lee and G.C. Wick, Nucl. Phys. B9 (1969) 209; Phys. Rev. D3 (1979) 1046 [30] D.G. Boulware and D.J. Gross, Nucl. Phys. B233 (1984) 1 [31] H.J. Munczek and P. Jam, Phys. Rev. D46 (1992) 438