How to Obtain a Covariant Breit Type Equation from Relativistic Constraint Theory

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How to obtain a covariant Breit typ e equation

from relativistic Constraint Theory

J. Mourad

LaboratoiredeModeles de Physique Mathematique,

UniversitedeTours, ParcdeGrandmont,

F-37200 Tours, France

H. Sazdjian

Division de Physique Theorique , Institut de Physique Nucleaire,

Universite Paris XI,

F-91406 Orsay Cedex, France

Octob er 1994

Unite de Recherche des Universites Paris 11 et Paris 6 asso ciee au CNRS. 1

Abstract

It is shown that, by an appropriate mo di cation of the structure of the interaction p o-

tential, the Breit equation can b e incorp orated into a set of two compatible manifestly

covariantwave equations, derived from the general rules of Constraint Theory. The com-

plementary equation to the covariant Breit typ e equation determines the evolution lawin

the relative time variable. The interaction p otential can b e systematically calculated in

p erturbation theory from Feynman diagrams. The normalization condition of the Breit

wave function is determined. The wave equation is reduced, for general classes of p oten-

tial, to a single Pauli-Schrodinger typ e equation. As an application of the covariant Breit

typ e equation, we exhibit massless pseudoscalar b ound state solutions, corresp onding to

a particular class of con ning p otentials.

PACS numb ers : 03.65.Pm, 11.10.St, 12.39.Ki. 2

1 Intro duction

Historically, The Breit equation [1] represents the rst attempt to describ e the relativistic

dynamics of twointeracting fermion systems. It consists in summing the free Dirac

hamiltonians of the two fermions and adding mutual and, eventually, external p otentials.

This equation, when applied to QED, with one-photon exchange diagram considered in

the Coulomb gauge, and solved in the linearized approximation, provides the correct

sp ectra to order [1, 2 ] for various b ound state problems.

However, attempts to improve the predictivity of the equation, by developing from it a

systematic p erturbation theory or by solving it exactly,have failed, due to its inabilityto

incorp orate the whole e ects of the interaction hamiltonian of QED [2]. Also, this equation

do es not satisfy global charge conjugation symmetry [3]. Finally, the Breit equation, as

it stands, is not relativistically invariant, although one might consider it valid in the c.m.

frame.

Nevertheless, despite these drawbacks, the Breit equation has remained p opular. The

main reason for this is due to the fact that it is a di erential equation in x-space and thus

it p ermits the study of e ective lo cal p otentials with standard techniques of quantum

mechanics. Improvements of this equation usually transform it into integral equations in

momentum space, b ecause of the presence of pro jection op erators [3, 4].

The purp ose of this article is to derive, from relativistic Constraint Theory [5], a

covariant Breit typ e equation, where the free part is the sum of individual Dirac hamilto-

nians. The latter framework ensures the relativistic invariance of the equations describ-

ing two-particle systems with mutual interactions [6, 7] and establishes the connection

with Quantum Field Theory and the Bethe-Salp eter equation by means of a Lippmann-

Schwinger-Quasip otential typ e equation relating the p otential to the o -mass shell scat-

tering amplitude [8].

The fact that a covariant Breit typ e equation can b e obtained from Constraint Theory

was already shown by Crater and Van Alstine [9]. In the present pap er we show that,

for general classes of interaction, the covariant Breit typ e equation is equivalent to the

Constarint Theory wave equations, provided it is supplemented with a second equation

which explicitly eliminates the relative energy variable and at the same time ensures

Poincareinvariance of the theory. The p otential that app ears in the main equation has a

c.m. energy dep endence that also ensures the global charge conjugation symmetry of the 3

system.

The pap er is organized as follows. In Sec. 2, we derive the covariant Breit typ e equa-

tion from the relativistic Constraint Theory wave equations. In Sec. 3, we determine the

normalization condition of the wave function. In Sec. 4, we reduce the wave equation, rel-

ative to a sixteen-comp onentwave function, to a Pauli-Schrodinger typ e equation, relative

to a four-comp onentwave function. As an application of the covariant Breit equation, we

exhibit, in Sec. 5, massless pseudoscalar b ound state solutions, corresp onding to a par-

ticular class of con ning p otentials, involving mainly pseudoscalar and spacelikevector

interactions. Conclusion follows in Sec. 6.

2 The covariant Breit equation

We b egin with the Constraint Theory wave equations describing a system of two spin-1=2

particles comp osed of a fermion of mass m and an antifermion of mass m ,inmutual

1 2

interaction [7] :

e e e

:p m = :p + m V ; (2.1a)

1 1 1 2 2 2

e e e

:p m = :p + m V : (2.1b)

2 2 2 1 1 1

Here, is a sixteen-comp onent spinor wave function of rank two and is represented as a

4 4 matrix :

e e

= (x;x ) ( ; =1;:::;4) ; (2.2)

1 2 1 2

where ( ) refers to the spinor index of particle 1(2). is the Dirac matrix acting

1 2 1

in the subspace of the spinor of particle 1 (index ); it acts on from the left. is the

1 2

Dirac matrix acting in the subspace of the spinor of particle 2 (index ); it acts on

from the right; this is also the case of pro ducts of matrices, which act on from the

right in the reverse order :

e e e e

( ) ; ( ) ;

1 2

1 1 1 2 1 2 2 2

e e

( ) ; = ; (a =1;2) : (2.3)

2 2 a a a

1 2 2 2

In Eqs. (2.1) p and p represent the momentum op erators of particles 1 and 2, resp ec-

1 2

tively. V is a Poincareinvariant p otential. 4

The compatibility (integrability) condition of the two equations (2.1) imp oses con-

ditions on the wave function and the p otential. For the wave function, one nds the

constraint:

2 2 2 2

(p p )(m m ) =0; (2.4)

1 2 1 2

which allows one to eliminate the relative energy variable in a covariant form. For eigen-

functions of the total momentum op erator P , the solution of Eq. (2.4) is :

2 2 2

iP :X i(m m )P :x=(2P ) T

1 2

=e e (x ) ; (2.5)

where wehave used notations from the following de nitions :

P = p + p ; p = (p p ) ; M = m + m ;

1 2 1 2 1 2

X = (x + x ) ; x = x x : (2.6)

1 2 1 2

We also de ne transverse and longitudinal comp onents of four-vectors with resp ect to the

total momentum P :

(q:P )

L T

^ ^ ^

P ; q =(q:P )P ; P = P = P ; q = q

q = q:P; P = P : (2.7)

L L

This decomp osition is manifestly covariant. In the c.m. frame the transverse comp onents

reduce to the three spacelike comp onents, while the longitudinal comp onent reduces to

T 2 2

the timelike comp onent of the corresp onding four-vector. (Note that x = x in the

c.m. frame.) Also notice that, with the de nition of the longitudinal comp onents, P ,

which is the p ositive square ro ot of P , do es not change sign for negative energy states

(under the change P !P); in this case, it is the longitudinal comp onents, q , of those

four-vectors which are indep endentofP that change sign, since these are linear functions

of P .

For the p otential, one nds the constraint:

2 2

e e

p p ;V =0; (2.8)

1 2

which means that V is indep endent of the relative longitudinal co ordinate x :

T T

e e

V = V (x ;P ;p ; ; ): (2.9)

L 1 2 5

Equations (2.5) and (2.9) show that the internal dynamics of the system is three-

dimensional, b esides the spin degrees of freedom, describ ed by the three-dimensional

transverse co ordinate x .

The relationship b etween the p otential V and Feynman diagrams is summarizedby the

following Lippmann-Schwinger-Quasip otential typ e [10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 8, 18 ]

equation :

e e e e

V T VG T =0; (2.10)

2 3

6 7

T 0T 0

T (P; p ;p ) T(P ; p; p ) ;

4 5

C (p);C (p )

where :

i) T is the o -mass shell fermion-antifermion scattering amplitude;

ii) C is the constraint (2.4) :

2 2 2 2 2 2

C (p) (p p ) (m m )=2Pp (mm)0; (2.11)

1 2 1 2 1 2

in Eq. (2.10) the external momenta of the amplitude T are submitted to the constraint

C ;

iii) G is de ned as :

e e

G (p ;p )=S(p)S(p)H ; (2.12)

0 1 2 1 1 2 2 0

e e

where S and S are the propagators of the two fermions, resp ectively, in the presence of

1 2

the constraint (2.11), and H is the Klein-Gordon op erator, also in the presence of the

constraint (2.11) :

2 2 2 2

P 1 (m m )

1 2

2 2 2 2 2 2 T2

H =(p m) =(p m) = (m+m)+ +p : (2.13)

1 1 2 2 1 2

4 2 4P

C C

In order to obtain the covariant Breit equation, we de ne the covariant Dirac \hamil-

tonians" :

T T

H = m :p ; (2.14a)

1 1 1L 1L

1 1

T T

: (2.14b) :p H = m

2 2 2L 2L

2 2

We then multiply Eq. (2.1a) by and Eq. (2.1b) by , resp ectively. After subtracting

1L 2L

the two equations from each other, we obtain the equation :

e e e

(P (H + H )) =(P +(H + H )) ( V ) ; (2.15)

L 1 2 L 1 2 1L 2L 6

which can b e rewritten as :

h i

e e e

P (1 + V ) (H + H )(1 V ) =0: (2.16)

L 1L 2L 1 2 1L 2L

Addition of the two equations to each other leads to the equation :

e e e

(2p (H H )) =(2p +(H H )) ( V ) ; (2.17)

L 1 2 L 1 2 1L 2L

which can b e rewritten as :

h i

e e e

2p (1 V ) (H H )(1 + V ) =0: (2.18)

L 1L 2L 1 2 1L 2L

2 2 2 2

Up on multiplying this equation by(H +H ) and noticing that H H = m m ,

1 2

1 2 1 2

it b ecomes, after using Eq. (2.16) :

h i

2 2

e e

V ) =0: (2.19) 2p P (m (1 + m )

L L 1L 2L

1 2

Wenow de ne the Breit wave function by:

e e

=(1 V) : (2.20)

B 1L2L

Then, Eq. (2.16) takes the form :

h i

e e

P (1 + V )(1 V ) (H + H ) =0; (2.21)

L 1L 2L 1L 2L 1 2 B

e e

while Eq. (2.19) yields, after factorizing the term (1 + V )(1 V ) :

1L 2L 1L 2L

h i

2 2

2p P (m m ) =0: (2.22)

L L B

1 2

Equations (2.21) and (2.22) are the twowave equations satis ed by the Breit wave

function . As far as the wave function transformation (2.20) is nonsingular, they are

equivalent to the initial twowave equations (2.1) of Constraint Theory.

Equation (2.21) is the obvious generalization of the Breit equation. Its interaction

dep endent part has an explicit c.m. energy (P ) dep endence which restores the global

charge conjugation symmetry that was lacking in the Breit equation. For each solution

of Eq. (2.21) with total momentum P , there will corresp ond, for charge conjugation

invariantinteractions, a charge conjugated solution with momentum P .

Equation (2.22) determines the relative time evolution law of the wave function, as in

Eq. (2.5), and ensures the relativistic invariance of the theory. While Eq. (2.21) might 7

b e considered alone in the c.m. frame, Eq. (2.22) indicates the way of passing to other

reference frames.

In the c.m. frame, with the standard de nitions = and = , Eq. (2.21)

0 0

b ecomes :

h i

e e

P (1 + V )(1 V ) (m + :p m :p) =0: (2.23)

0 1 2 1 2 1 1 1 2 2 2 B

In p erturbation theory, V has, in lowest order, according to Eq. (2.10), the structure

[18] :

V = U (x ; ; ) ; (2.24)

1 2

2 P

where U is the three-dimensionally reduced form of the propagator of the exchanged

particle, including the couplings at the vertices. To this order, Eq. (2.23) takes the form:

[ P (P )U (m + :p m :p)] =0: (2.25)

0 1 2 0 1 1 1 2 2 2 B

We notice here, in distinction from the Breit equation, the presence of the energy sign fac-

tor in front of the p otential U ; it is this factor which ensures the global charge conjugation

symmetry of the equation.

Finally, in the limit when m tends to in nity, Eq. (2.25) yields the Dirac equation of

particle 1, with the p otential U ( is replaced by 1 for the antifermion and (P )by

1 2 0

+1 in this limit).

Equations (2.1), or equivalently (2.21) and (2.22), were analyzed, in Ref. [18 ], in the

nonrelativistic limit, to order 1=c , in particular for the electromagnetic interaction case.

For an arbitrary covariant gauge of the photon propagator, the corresp onding hamiltonian

receives contributions (among others) from quadratic terms generated by the one-photon

exchange diagram as well as from the two-photon exchange diagrams. However, it turns

out that in the Coulomb gauge (and also in the Landau gauge to that order) the two-

photon contribution cancels the quadratic terms arising from the one-photon exchange

diagram and one then is left with the Breit hamiltonian [2, 19]. This explains why the Breit

equation in its linearized approximation provides a correct result to order .However,

in other gauges than the Coulomb and Landau gauges, it is necessary to takeinto account

the quadratic terms as well as the two-photon exchange contribution to obtain a correct

result. 8

3 Normalization condition

The normalization condition of the wave function can b e determined either from the

construction of tensor currents of rank two, satisfying two indep endent conservation laws,

with resp ect to x and x [7], or from the integral equation of the corresp onding Green's

1 2

function [14, 17 ]. One nds for the norm of [Eq. (2.5)] the formula (in the c.m. frame

and for lo cal p otentials in x ):

8 9

> >

< e =

@ V

3 y y 2

e e e e

d xTr 1 V V +4 P =2P ; (3.1)

10 20 0

> >

: ;

where V satis es the hermiticity condition :

e e

V = V : (3.2)

10 20 10 20

For energy indep endent p otentials (in the c.m. frame) the norm of is not p ositive

e e

de nite for arbitrary V . In order to ensure p ositivity, it is sucient that the p otential V

satisfy the inequality

e e

V ) < 1 : (3.3) Tr(V

In this case one is allowed to make the wave function transformation

e e e

V = 1V (3.4)

and to reach a representation where the norm for c.m. energy indep endent p otentials is

the free norm.

In this resp ect, the parametrization suggested by Crater and Van Alstine [20], for

e e

p otentials that commute with (and hence V = V ),

1L 2L

V = tanh V; (3.5)

satis es condition (3.3) and allows one to bring the equations satis ed by [Eq. (3.4)]

into forms analogous to the Dirac equation, where each particle app ears as placed in

the external p otential created by the other particle, the latter p otential having the same

tensor nature as p otential V of Eq. (3.5).

We shall henceforth adopt the ab ove parametrization (3.5). For more general p oten-

tials that do not commute with , the natural extension of parametrization (3.5)

1L 2L

is:

V = tanh( V ) : (3.6)

1L 2L 1L 2L 9

According to Eqs. (3.2) and (3.4), we shall intro duce the wave function transformation :

= cosh( V ) : (3.7)

1L 2L

The norm of the new wave function then b ecomes (in the c.m. frame) :

8 9

> >

< =

@ @

3 y 2 V V V V

10 20 10 20 10 20 10 20

d x Tr 1+2P e e e e =2P :

2 2

> >

@P @P

: ;

(3.8)

(The relationship b etween and is the same as in Eq. (2.5).)

Equations (2.1) then take the form :

( :p m ) cosh ( V ) =( :p + m ) sinh( V ) ;(3.9a)

1 1 1 1L 2L 2 2 2 1L 2L 1L 2L

( :p m ) cosh ( V ) =( :p + m ) sinh( V ) :(3.9b)

2 2 2 1L 2L 1 1 1 1L 2L 1L 2L

In order to determine the normalization condition of the Breit wave function ,we

rst de ne from V a p otential V as :

V = V: (3.10)

B 1L 2L

(Notice that b ecause of Eq. (3.2) V = V in the c.m. frame.) With this p otential, the

relationship (2.20) takes the form :

; (3.11) =

cosh V

while the relationship b etween and [Eq. (3.7)] is :

= e : (3.12)

The Breit typ e equation (2.21) b ecomes :

P e (H + H ) =0: (3.13)

L 1 2 B

The normalization condition of (de ned from as in Eq. (2.5)) is, in the c.m.

B B

frame :

8 9

> >

< =

3 2V 2 2V

B B

d x Tr e +2P e =2P : (3.14)

B 0

> >

: ;

We therefore end up with three di erent representations for the two-particle wave

function. The rst one, [Eqs. (2.1)], corresp onds to the framework where Constraint 10

Theory conditions, as well as connection with Quantum Field Theory and the Bethe-

Salp eter equation, are most easily established. The second one, [Eqs. (3.7) and (3.9)],

corresp onds to the \canonical" representation, for which the norm, for c.m. energy inde-

p endent p otentials, is the free one. This representation is also the one used by Crater and

Van Alstine [20]. The third one, [Eqs. (3.11), (3.12), (3.13) and (2.22)], corresp onds

to the Breit representation.

4 Resolution of the Breit typ e equation

By decomp osing the wave function along 2 2 matrix comp onents, Eqs. (3.9) can b e

solved with resp ect to one of these comp onents and transformed, for the case of p otentials

commuting with ,into a second order di erential equation of the Pauli-Schrodinger

1L 2L

typ e [18]. A similar reduction is also p ossible starting from the Breit typ e equation (3.13).

The relative time dep endence of the wave function b eing determined by Eq. (2.22),

with a solution of the form (2.5), one decomp oses the internal 4 4 matrix wave function

on the basis of the matrices 1; ; and by de ning 2 2 matrix comp onents :

B L 5 L 5

= + + + : (4.1)

B B 1 L B 2 5 B 3 L 5 B 4 i Bi

i=1

We consider the case of p otentials that are lo cal in x (but having eventually a c.m.

energy dep endence) and that are functions of pro ducts of and matrices in equal

1 2

numb er (general vertex corrections do not satisfy the latter prop erty); then V commutes

with :

1L 2L

V = V : (4.2)

1L 2L 1L 2L

Weintro duce pro jection matrices for the ab ove22 comp onent subspaces :

1 1

P = (1 + )(1+ ) ; P = (1 + )(1 ) ;

1 1L 2L 15 25 2 1L 2L 15 25

4 4

1 1

(1 )(1+ ) ; P = (1 )(1 ) : P =

1L 2L 15 25 4 1L 2L 15 25 3

4 4

(4.3)

They satisfy the relations :

P P = P ; P = (i; j =1;:::;4) : (4.4)

i j ij j i j ij j 11

(The 's are de ned in Eq. (4.1).)

Then, the most general (parity and time reversal invariant) p otential V [Eqs. (3.5)

and (3.10)] wemay consider has the decomp osition on the basis (4.3) :

V = a P : (4.5)

i i

i=1

The p otentials a themselves may still have spin dep endences. The spin op erators, which

act in the 2 2 comp onent subspaces, are de ned by means of the Pauli-Lubanski op er-

ators:

h h

W = P ; W = P ( = +1) ;

1S 2S 0123

1 2

4 4

2 2 2

h P ; W = W + W : (4.6) W = W =

S 1S 2S

1S 2S

They also satisfy the relations :

hP hP

L L

T T

W = ; W = : (4.7)

1L 1S 15 2L 2S 25

1 2

2 2

Weintro duce the op erators :

T T

2 2 W :x W :x

1S 2S

2 2

w =( ; (4.8) ) W :W ; w =( )

1S 2S 12

T 2

hP hP x

L L

then, the p otentials a [Eq. (4.5)] can b e decomp osed as :

a = A + wB + w C (i =1;:::;4) ; (4.9)

i i i 12 i

T 2 2

where the p otentials A ;B;C are functions of x and eventually of P .

i i i

The pro jectors (4.3)-(4.5) satisfy the simple prop erty:

4 4

X X

exp a P = P e : (4.10)

i i i

i=1 i=1

The Breit p otential V [Eq. (3.10)] has also a decomp osition like (4.5) :

V = a P ; (4.11)

B Bi i

i=1

with the following relations with the a 's :

a = a ; a = a ; a = a ; a = a : (4.12)

B 1 1 B 2 2 B 3 3 B 4 4 12

The relationship (3.12) b etween and can b e rewritten for their 2 2 comp onents

as well :

= e (i =1;:::;4) : (4.13)

Bi i

[ is de ned from a decomp osition of as in Eq. (4.1).]

The Breit typ e equation (3.13) is now easily decomp osed into four equations for the

four comp onents (i =1;:::;4) :

P e (m m ) + (W W ):p =0; (4.14a)

L B 1 1 2 B 2 1S 2 B 3

P e (m m ) W :p =0; (4.14b)

L B 2 1 2 B 1 S B 4

P e M + (W W ):p =0; (4.14c)

L B 3 B4 1S 2S B 1

P e M W :p =0: (4.14d)

L B 4 B3 S B 2

These equations allow one to eliminate the comp onents ; and in terms of

B 1 B2 B 4

, which is a surviving comp onent in the nonrelativistic limit. Up on de ning

B 3

2 22

(m m)

1 2

2(a + a ) 2h

1 2

e ; (4.15) e =1

2 2

one nds for and the relations :

B 1 B 2

2(a + a + h) 2a

1 2 2

P = e e (W W ):p

L B 1 1S 2S

2 2

(m m )

1 2

+ W :p e ; (4.16)

S B 3

2(a + a + h) 2a 2a

1 2 1 3

M = e +e W :p e

B2 S

2 2

(m m )

1 2

(W W ):p : (4.17)

1S 2S B 3

One then obtains two indep endent equations for and :

B 3 B 4

2a 2 2(a + a + h)

4 1 2

MP e = M + W :p e

L B4 B3 S

2a 2a

1 3

+ e W :p e

2 2

(m m )

1 2

(W W ):p ; (4.18a)

1S 2S B 3

P 13

2(a + a + h) 2 2a

1 2 3

(W W ):p e MP = P e +

1S 2S L B4 B3

e (W W ):p

1S 2S

2 2

(m m )

1 2

+ W :p e : (4.18b)

S B 3

Elimination of leads to the eigenvalue equation for :

B 4 B 3

2 2(a + a ) 2

3 4

P e M

B 3

2(a + a + h) 2a 2a

1 2 1 3

W :p e e W :p e

S S

2 2

(m m )

1 2

(W W ):p

1S 2S B 3

2a 2(a + a + h) 2a

4 1 2 2

e (W W ):p e e (W W ):p

1S 2S 1S 2S

2 2

(m m )

1 2

=0: (4.19) W :p e

B 3 S

Equation (4.19) is a second order di erential equation for the comp onent . Usu-

B 3

ally,bywave function transformations one can simplify the structure of the di erential

op erators in it. For the general p otential (4.5), the second order di erential op erator will

still exhibit a spin dep endence. However, for simpler typ es of p otential, the spin dep en-

dence of the second order di erential op erator also disapp ears. This is the case of the

p otential comp osed of general combinations of scalar, pseudoscalar and vector p otentials.

It has the following structure :

T T

x x

LL TT

V = V + V + g V + g U + T ; (4.20)

1 15 25 3 2 4 4

T 2

and the decomp osition of the p otentials a [Eq. (4.5)] along these p otentials is given by

the relations :

a = V + V + V + wU + w T ;

1 1 2 3 4 12 4

a = V + V V wU w T ;

2 1 2 3 4 12 4

a = V V + V wU w T ;

3 1 2 3 4 12 4

a = V V V + wU + w T : (4.21)

4 1 2 3 4 12 4

With p otentials of the typ e (4.20) and after using the wave function transformation

V V + V +2U +2T +h

1 2 3 4 4

= e

B 3 14

2V U T

2 4 4

(2 + ) e sinh(2U )

h P

1 1

2V + U + T 2V + U T

1 4 4 2 4 4

+ (1 w )e + (1 + w )e (4.22)

12 12 3

2 2

(w de ned in Eq. (4.8)), Eq. (4.19) reduces to a Pauli-Schrodinger typ e equation, where

the radial di erential op erators are those of the Laplace op erator. This equation, which

is also obtained from a wave function transformation in the equation satis ed by [Eq.

(4.13)], was presented in Ref. [18 ].

Equations (4.14) could also have b een solved with resp ect to instead of .

B 4 B 3

5 Zero mass solutions

As a straightforward application of the covariant Breit equation, with the class of p o-

tentials considered in Sec. 4, we shall exhibit, in this section, a class of solutions which

corresp ond to massless pseudoscalar b ound states in the limit when the masses of the

constituent particles tend to zero.

The key observation is that, b ecause of the presence of the kernel e in the nor-

malization condition (3.14), one is allowed to search for solutions in which some of the

comp onents are constants, provided the kernel e is rapidly decreasing at in nity.

The quantum numb ers of the state are determined by those of the comp onents

B 3

and , which are the surviving comp onents in the nonrelativistic limit. For the ground

B 4

state they have the quantum numb ers s = 0 (for the total spin op erator de ned in Eq.

(4.6)), ` = 0 (for the orbital angular momentum op erator) and j = 0 (for the total angular

momentum op erator); these quantum numb ers are those of a pseudoscalar state. We shall

restrict the searchby demanding that the comp onents and b e zero for the ground

B 1 B 2

state solution.

Insp ection of Eqs. (4.14a) and (4.14b) shows that must b e a constant:

B 3

= =const: : (5.1)

B3 0

(The vanishing of the comp onents and can then also b e checked directly in Eqs.

B 1 B 2

(4.16) and (4.17).)

One is left with the two equations (4.14c) and (4.14d), which b ecome simple algebraic

equations :

P e M =0 ; (5.2a)

L 0 B4 15

P e M =0 : (5.2b)

L B 4 0

These equations have a nontrivial solution only if a + a is a constant:

3 4

a +a = C = const: : (5.3)

3 4

Then :

P M

2a 2a

4 3

e = e ; (5.4) =

0 0 B 4

P M

P = Me : (5.5)

Wenowcheck the normalizability of the solution thus found. For simplicity,we shall

consider p otentials that are indep endentof P in the c.m. frame; the corresp onding

conclusions are not much a ected byaneventual smo oth P dep endence of the p otentials.

The normalization condition (3.14) b ecomes :

2 3

6 7

y y

3 2a 2a

3 4

4 d x e + e =2P ;

4 5

B 3 B 4 0

B 3 B 4

2 3 2a 2a 2C

3 4

=2P ; 4jj dx e +e e

0 0

2 3 2a

8jj dxe =2P ; (5.6)

0 0

where Eqs. (5.1) and (5.3)-(5.5) were used; furthermore, the spin 0 pro jection must b e

taken in the p otentials. Therefore, e must b e a rapidly decreasing function when

jxj!1 , or, equivalently, a must b e an increasing function of jxj at in nity, indicating

the con ning nature of the p otential.

Tohave a more explicit representation of the p otentials satisfying the ab ove conditions,

let us consider again the class of p otentials comp osed of general combinations of scalar,

pseudoscalar and vector p otentials [Eqs. (4.20) and (4.21)]. Condition (5.3) means that

2(V V )=C: (5.7)

1 2

Thus, the scalar and timelikevector p otentials cannot b e chosen indep endently from each

other.

The normalizability condition (5.6) implies :

lim (V 3U T )=1: (5.8)

3 4 4

jxj!1 16

(The spin 0 pro jection of the p otentials has b een taken.) This combination of the pseu-

doscalar and spacelikevector p otentials must therefore b e of the con ning typ e.

The comp onent [Eqs. (5.4) and (5.5)] of the wave function then b ecomes :

B 4

2(V 3U T )

3 4 4

= e : (5.9)

B 4 0

Equation (5.5) shows that when the masses of the constituent particles vanish, then

the mass of the b ound state also vanishes. [The fact that the right-hand side of Eq.

(5.6) vanishes in this limit should not lead one to the immediate conclusion that this

state disapp ears from the sp ectrum. It is its coupling to the axial vector current which

is imp ortantonphysical grounds, and this coupling involves the relationship of the wave

function to the Bethe-Salp eter wave function through nonlo cal op erators, where also P

is involved [8].]

As far as the p otentials do not have singularities at nite distances, the function

B 4

[Eqs. (5.4) and (5.9)] do es not vanish at nite distances and the corresp onding wave

function do es not have no des; it is then a candidate for the ground state of the

sp ectrum. To conclude that this is actually the case necessitates a detailed study of the

various p otentials in all sectors of quantum numb ers. Conditions (5.8) and (5.7) are not

sucient to guarantee con nement in general. There are cases of p otentials satisfying

these conditions, for which con nement do es not o ccur in a particular sector of quantum

numb ers or for which some solutions b ecome unnormalizable. However, there are also

cases for which the ab ove solution is the ground state of the sp ectrum; in particular,

when the con ning p otential is represented by the pseudoscalar p otential, Eq. (4.19)

can easily b e analyzed; in this case all solutions other than the one found ab ove remain

massive in the limit of vanishing constituent masses.

Equations (4.14) also have solutions for which = = 0 and and are

B 3 B 4 B 1 B 2

(a + a )

1 2

nonzero. In this case one nds the solution P = jm m je , with (a + a )

L 1 2 1 2

equal to a constant. This solution is, however, unphysical, since it b elongs to one of the

unphysical subspaces, where one of the longitudinal momenta, p or p , calculated from

1L 2L

Eqs. (2.6) and (2.22), may b ecome negative [7].

Finally, the solution found ab ove can also b e expressed in the \canonical" representa-

tion. Taking into account the relationship (3.12), one nds :

=(1+ ) e : (5.10)

L 5 0 17

The massless pseudoscalar b ound state solution found in this section do es not of

course exhaust all p ossibly existing solutions. Furthermore, several typ es of mechanism

may lead to the o ccurrence of massless pseudoscalar b ound states, in connection with the

sp ontaneous breakdown of chiral symmetry.Two such mechanisms are : i) the dynamical

fermion mass generation, due to radiative corrections in the fermion self-energy part [21];

ii) the fall to the center phenomenon, due to short distance singularities [22 ]. Our solution

di ers from the ab ovetwo in that it is a direct consequence of the particular con ning

nature of the interaction and therefore hinges on long distance forces, rather than on the

short distance ones or on the radiative corrections. The solution corresp onding to the

pure pseudoscalar interaction case was studied in detail in Ref. [23].

6 Conclusion

Wehave shown that, by an appropriate mo di cation of the structure of the interaction

p otential, the Breit equation can b e incorp orated into a set of two compatible manifestly

covariantwave equations, derived from the general rules of Constraint Theory. The

complementary equation to the covariant Breit typ e equation determines the evolution

law of the system in the relative time variable and also determines its relative energy with

resp ect to the other variables. Furthermore, in this covariantversion of the Breit equation,

the interaction p otential can b e systematically calculated in p erturbation theory from

Feynman diagrams by means of a Lippmann-Schwinger-Quasip otential typ e equation,

relating it to the o -mass shell scattering amplitude.

The normalization condition of the Breit wave function indicates the presence of an

interaction dep endentkernel in it, which should b e taken into account for consistent

evaluations of physical quantities, like coupling constants, or for the selection of accept-

able (normalizable) solutions to the wave equations. In this resp ect, we exhibited, as a

straightforward application of the covariant Breit equation, massless pseudoscalar b ound

state solutions, corresp onding to a class of con ning p otentials, essentially comp osed of

pseudoscalar and spacelikevector p otentials with eventually a particular combination of

scalar and timelikevector p otentials.

The covarianttwo-b o dy Breit equation suggests several p ossibilities for its generaliza-

tion to the N -b o dy case (N>2) or for the incorp oration of external p otentials. However, 18

one meets here the known diculty of the \continuum dissolution" problem [24, 25], which

prevents the existence of normalizable states. Usually, this diculty is circumvented by

the intro duction of pro jection op erators, either in the p otential [26, 27 ] or in the kinetic

terms [4]. It is not yet known whether some lo cal generalization of the Breit equation

mayavoid the ab ove diculty. 19

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