Practical relativistic three-particle theory with arbitrary particle interactions

[ ~

Nuclear Physics B44 (1972) 573-593. North-ltolland Publishing Company

PRACTICAL

RELATIVISTIC

WITH ARBITRARY

THREE-PARTICLE

PARTICLE

THEORY

INTERACTIONS

U. WEISS Institut fgtr Theoretische Physik der Universitgtt Stuttgart

Received 12 January 1972 (Revised 22 March 1972) Abstract: Relativistic three-particle equations are established by the rules of Blankenbecler and Sugar, which put the intermediate particles on the mass shell. By use of the WightmanGfirding momenta we preserve the correct cluster properties, time-reversal invariance and avoid a spurious s = 0 singularity. A reduction to effective multichannel two-particle equations which is not restricted to separable potentials is discussed. Our approach admits the systematical application of perturbation theory to relativistic composite particle scattering problems.

1. INTRODUCTION Since the famous work of Faddeev [1 ], collision processes of three non-relativistic particles have been treated extensively. In between, many advances of a formal and a technical nature have led to considerable improvements in our ability to do explicit calculations. It has been shown by many authors [2] that in the separable potential model three-particle equations exactly reduce to effective two-particle equations, independently whether one starts from Lippmann-Schwinger or from Faddeev equations. This reduction was generalized by Grassberger and Sandhas to arbitrary (e.g. local) potentials [3]. In this paper we apply these techniques to suitable relativistic three-particle equations with arbitrary potentials. We shall especially take care of Lorentz-invariance, unitarity at all energies and of the correct cluster decomposition of the appearing three-particle quantities. As in the non-relativistic case the final equations are only one dimensional after partial wave decomposition and amenable to practical calculations. In relativistic linear off-the-energy-shell theory the Bethe-Salpeter equation often is used as a convenient scaffolding. This equation, however, in its usual truncated form (i.e. ladder approximation) does not satisfy unitarity at all energies, since it always contains some multiparticle contributions but not all. This is one reason why we shall replace the two- and three-particle Bethe-Salpeter equations by equations which have some good properties of potential scattering. The new equations involve

574

U, Weiss, Three-particle theory

free two- and three-particle propagators which are constructed by the corresponding two- and three-particle unitarity conditions, respectively [4, 5]. It is wellknown that this replacement will also reduce the dimensionality of our equations and will remove the strong singularities of the Bethe-Salpeter equation without violating Lorentz invariance. By use of the Wightman-G~rding relative momenta [6, 7] we preserve the correct cluster properties and avoid a spurious s = 0 singularity which appeared specifically in relativistic Faddeev-type calculations in a direct [8] or indirect [9] mannet. Both, the two-particle and the three-particle equations assume the operator form of the Lippmann-Schwinger equations. Like these they are only three- and sixdimensional, respectively, and do not have the strong singularities of the Bethe-Salpeter equation. Following the methods of Grassberger and Sandhas [3], our equations will exactly be reduced to effective multichannel two-particle equations, which satisfy threeparticle and bound-state two-particle unitarity at all energies. The generalized potentials occuring in them are calculable via an iteration scheme, which should rapidly converge in most cases of physical interest. A relativistic three-particle problem is dealt with by the separable potential with remainder method by Riordan [10]. Different from our approach, it is based on a model [11 ] where the internal lines represent 6-functions rather than propagators. In sect. 2 fundamental unitarity relations and appropriate free propagators, constructed out of these, are given for the two- and the three-particle case. In sect. 3 Lippmann-Schwinger-type equations are established and examined. In sect. 4 effective two-particle equations for the scattering and bound state problem of three particles are derived. Some conclusions are discussed in sect. 5.

2. UNITARITY RELATIONS AND FREE PROPAGATORS Since unitarity will be of fundamental importance in our treatment we first review our notation for it. The transition matrix, which is defined in terms of the Smatrix according to Sfi=6fi+ 2i~4(/~f

/£i)rfi,

(2.1)

fulfills the unitarity relation H

T n - Tt~ = 2i ~ f d r 8 4 ( / ~ f n

- ~ £i)TfnT,~i , i=1

(2.2)

where tl

dr, = I-I d4£i~+(/~? - m2), i=1 is the n-body phase-space element.

(2.3)

U. Weiss, Three-particle theory

575

This equation reads in the real two-particle case* T(/~ a, b~; s+) - T~(/9 a , / ~ ; s~) = 2 i f d4p~' T ( / ~ , / ~ ' ; s+) Ac~(/~,/~a) T(b~',/~¢~;s2) , (2.4) where A3(/~ 3 ;/~3)= 6 +(/~2 _ m~)6 +(/¢~ - m 2 ) ,

(2.5)

and in the real three-particle case T(/~c~, qc~;/~a~,q~; s+) - T(/~a, qc~;/~2,q2;s-)

= 2ifd4~"fd4~'

' y(/~, da;/~2'' qa;'"s+) A°(p~,4";I~lY(~",qa';p~,d2;s-), . . . . . . . .

(2.6)

where A0(/~ , q~,/£) = 6+(/¢~ - m~)6+(Ic~ - m~) 6+(/¢~ - m ~ ) ,

(2.7)

sa(s) is the square of the total energy in the c.m. system of the two-particle (threeparticle) system. As to the choice of the relative momenta/3 and q,~ we refer to appendix A. We now consider linear off-shell integral equations for the transition amplitude. In the two-particle case we write them in the form T(/~ , / ~ ; s ~ ) = I (/~ ,/~) +

d4/~ Iu(pa,p'~)E (p";s;)T(fi'a',p~,s-a) ,

(2.8)

where I (/~ ,/~' ) is the interaction potential and E0(/~ ; s~) is the free two-particle ^ Pa) ^" is real and symmetric, we~ obtain ~ ~ from eq ( 2. 8 ) propagator.a If aIc~(pc,, "

P.; a) - T ( / ~ , / ~ ; sd) =fd4/~2'T~(/~,~,P, 'A'''s+a) [EO(fi2';s+)-eO(¢'~';s;)] T(/~2',/~2;s;) .

(2.9)

From direct comparison of eq. (2.4) and eq. (2.9) we conclude that eq. (2.8) will be an integral equation that satisfies two-particle unitarity for all energies, provided that * For later convenience we introduce the index a indicating the various two-particle subsystems in the three-particle case.

576

U. Weiss, Three-particle theory

(2.1o)

EO(s +) - EO(s2) : 2i A (s).

Similar to Blankenbecler and Sugar [4] and Aaron, Amado and Young [5] we now construct a covariant two-particle Green's function from the unJtarity condition (2. l 0) according to 0 + 1 ~ A3(S;) EBts?):Tf 2as3 (mr+m2) s'3 - s 3 ¥ ie •

o

(2.11)

.

Using the Wlghtman-Gardmg momenta the integration can be done in a covariant way and the result is 6{/)3' /(3~ - 0 =2 ± \ ~3 ]E3(P3;S3)'

E0(/7~'/~3 "/(3;s;)=

(2.12)

where

uO(p~)

- 0 =2 +E3(P3;S3)=

1 g-3(p~)_s3 ¥ i e '

(2.13)

with 1 [ K3(P~) ] 21 vO(P~):2~ (m 2 _P3Xm2~2 2 _ ~ ) "

(2.14)

Using these methods in the three-particle case we are led to the covariant Green's function EO(s ~) =

17

+

~-(m 1 m2

a°(s ' )

ds' +

m3)

s'

2

(2.15)

sV-ie

which in terms of the Wightman-G]rding momenta reads ^ ^ . + .

_

^

A

[q3"K\

A

[P3"K3\-

0

=2

P3;S-),

.

\

~¢'5

/

\

V33

(2.16)

/

where ~0(q~, =2

+

1

(2.17)

U. Weiss,Three.particletheory

577

with

pO(q2, p ~ ) =

~- =2, =2 ~. ~2

I[ ~ _

.(q3 P3) 3(P3 ) m2 - ~ ~72~(s3(P3)-q3)(m3-q3)(ml-~Xm~

1

i]~

(2.18)

ff~

With eq. (A. 14a) R 0 (s+-) assumes the form *-2. +- ey(qy, ±2 p2) , -E (py,sq,)

3' = 1,2,3 ,

~ 0 , ±tqq,, 2 P,y ±2 ;s-) + --0

(2.19)

where -

=2

±2

l

2 L - =2 =2 2 -:2 ] s (q~,, p,r) ( m r - q~,)

½

(2.20)

"

Eq. (2.19) plays a fundamental role in our treatise. It directly exhibits the correct cluster properties. Here e~, is the phase space factor of the third particle for the situation that the subsystem 3' is in a scattering state. In close analogy to eq. (2.20) we introduce the phase space factor

~vr(#2)=l[r(q2, r)(m2 _ ~ ) j

,

(2.21)

which is appropriate to the situation that the subsystem is in a bound state of mass

M~r" 3. THE EQUATIONS

3.1. Two-particle case As the starting point of further considerations, we use eq. (2.8) where EO(s~) is defined in eq. (2.11). Just as in the non-relativistic case we consider this integral equation as an operator equation

T(s )=I (s )+ I(sa)E2(s )T(sa).

(3.1)

The operators are acting between states

v 7 2 , aoL, with the normalization condition

I¢ 2 >--

v72

,

,

U. Weiss, Three-particle theory

578

2

*

'

~oz

X/r~~ ](5(p~*2 fi22)5(a ~ - a'ce),

Pc~" c*~

(3.2)

and the closure property

(15 I.

(3.3)

This notation involves the energy and radial variables m a manifest covariant way, whereas we use, as a notational simplification, instead of the covariant variables fic~'fi~. and/~ •/)2' the angles a s = (Oce ~oc~) of the three-vector [12] p . The 6-function in eq. (2.12) effects that eq. (3.1) actually is a three-dimensional integral equation. Thus it is apparent to replace eq. (3.1) by the equation T (s)

=~(s

)+ I (sc~)E2(s)L(s),

(3.4)

where the operators with a bar over them are only acting between states (/Sa I and Ip2 ) with the normalization condition

= 2

~(a~2 p~=)a(a _~'2,

(3.s)

and the closure property o

r= 5 dp:: ll&l ffda I~ ><& r.

(3.6)

_oo

Sometimes, it will be suitable to consider the Green's function Ec~(sc~), defined according to

fo(,o) = ~O(so) + ~o(,o) L%)Eo(,o),

(3.7)

which fulfills the equation E ( s ) = E2(s ) + E2(s ) I E(sc~ ) .

(3.8)

The two-particle bound-states will occur as poles in ~ ( s )

~P%)= ~ I q~r>< %r j , r

M ~2r - s o~

(3.9)

579

U. Weiss, Three.particle theory A

where Mar is the bound state mass and (pc~ I ffar ) is its wave function. Assuming that ~ does not depend on s a, we obtain from eq. (3.8) by some operator algebra [13]

aE(s a) a~

aF°-~(Sa ) G(~a) as Ea (sa)

a

(3.1 o)

a

Inserting eq. (3.9) and taking out the residues of tile double pole at s a = M2an' one gets the normalization condition

(~kan I -

as

ItPar)=6nr'

(3.11)

which reads with eq. (2.13) (3.12)

~a,I v° - j I @~r > = 6nr • Using the obvious relations lim - ieEa(M2an + ie) vO -' I @an ) = I @an), e.--~O

lim - i e E ° ( M 2 +ie)uOa-~l@c~n)=O

(3.13)

e -* O

we obtain from eq. (3.8) the equation I ~,,

) = ;~°(M2 ) F I ~,, C~ a?l

).

(3.14)

Introducing k IO~n)= v0-2 I ~an ) '

(3.15)

this equation is easily transformed into the hermitian Schr6dinger-type equation

[ - ±2 so:(Pa)

2

^

IOcm )

0 + f d~'a2½1p2 I ffd~'a (~a I U l f i ~ )(fi'a [Oa,, ) = 0 ,

with the symmetric interaction

(3.16)

580

U,. Wciss,.Three-particle theoo' 1

(<[U l;~): u02(:%(
I

p02 : ' 2

(3.17)

where I¢cm ) is normalized as usual. 3.2. Tkree-particle case We now set up three-particle equations, which have tire free three-particle propagator E0(s) as an ingredient. From our discussion in the preceding subsection we conclude that, due to the (5-functions in eq. (2.16), the various three-particle operators are only significant in the reduced space of vectors IPc~) Iqc~ ). In the following context we have to distinguish between a two-particle operator and the corresponding operator acting in the three-particle Hilbert space. The general rule, which connects their matrix elements, is

- ---

2

--2 -

6(q~

-

A')

<

lO(s)

Ip2)

,

(3.18)

where s = 7 2 ~- - - - -~

s.

(3.19)

Eq. (3.19)contrasts with the corresponding eq. ( 6 . 9 ) o f ref. [8]. It directly exhibits that no spurious s = 0 singularity can occur. Defining the Green's function i

i

E ~(s) = c Ec, (s) e~

" (_,.20)

one obtains from eq. (3.8) with eq. (2.19) tire equation ~'~(s) = L~°(s) + ~7 °(s) v E ~(s),

(3.2l)

with the symmetric interaction operator

< %1.

(3.22)

Obviously, eq. (3.21) is a three-particle equation of the Lippmann-Schwinger operator form describing three particles where only two of them interact. Its manifest generalization is E(s):/~0(s)+E0(s)

~

VE(s),

(3.23)

U. Weiss, Three-particle theory

5 81

where E(s) is the Green's function of three interacting partiCles. Like the non-relativistic equation this is a six-dimensional integral equation and suffers from the same disconnectedness [1,14].

4. THE MULTICHANNEL QUASIPARTICLE METHOD

4.1. Two-particle subsystems The relativistic eqs. (3.4) and (3.8) have rather good convergence properties. Their kernels are completely continuous and even belong to the Hilbert-Schmidt class for square integrable potentials. Therefore, the quasiparticle method of Weinberg [15] can be applied. We now state this method in a manner suited for the following treatment. To simplify our notation, we make the realistic assumption that there is not more than one bound state or resonance in each partial wave. This assumption, however, could easily be removed by considering a potential matrix in each partial wave. Splitting ]-~ in a finite sum of separable terms and a small non-separable remainder term

lc,= ~a [Xc~r)~,c~r(Xo~rI+I£,

(4.1)

r

eq. (3.8) may be replaced by the equations (4.2) r

E~(s ) = ~ ' 2 ( s ) + P2(so~)-I~'E'a(se,)=E2(so~)+Ea(s)72EO(se,).

(4.3)

By use of a sufficient large number of separable terms in eq. (4.1), the Schmidt norm of/~°.(sc~)Y~ becomes arbitrary small, so that eq. (4.3) can be solved by iteration, i.e. by the Born series shown in fig. 1. On the other hand, eq. (4.2) can be solved algebraically. If there is not more than one separable term in each partial wave, the solution is Ec~(sa ) = E2(sc~ ) + ~

R2(sc~) I Xc~r) t~r(Sa) ( Xc~r I E2(sc~),

(4.4)

r

where

1 t~cn(Sa) - )k-lore-- ( Xom [E'o~(Su) l Xom }' which is shown in fig. 2.

(4.5)

U. Weiss, Three-particle theory

582

LD-

= - -

+

~

+

[

~

+ . .

Fig, 1. The Born series or' k:2, The wavy line represents the rest potential I-~.

1

2 -~F'ig. 2. Structure of the quasiparticle propagator toaz defined in eq. (4.5).

If we require

(4.6) the equation (4.7) r

which results from eqs. (4.1), (4.3) and (3.14), is reduced to

kY~(M2 n) I X~ n ) = I ~ n }"

(4.8)

Then, by substituting eqs. (4,6), (4.8) and the equation

i : ( M L ) _ ~2(so)=~2(ML){po '(, ) _ f.0-'~M~ . ~,,),• ~2(so) ,

(49)

into eq. (4.5), and using eq. (2.13), we get

1 t~n(Sc~)-- [M2n - s ] (×~nIE;(M2n)uO-'E~(s

(4.1o)

)Ixu n )

It follows from eqs. (4.4) and (4.8) that the sum in eq, (4.4) directly exhibits the pole contributions (3,9), provided that t n ( s ) has a pole with residuum one for s =M 2 tf.%-

2

_

J

(4.11)

M~,, ) - M 2~. - s

This is confirmed by the normalization condition (3.12). Substituting eqs. (4.9), (4.8) and (3.12), we finally get the complete expression

_

I oLn

+

0-'-'

0

i

(4.12)

u, Weiss, Three-particletheory

583

This equation exhibits that ten(S~) may be called the propagator of a composite (quasi) particle. The second term is the contribution of the two-particle continuum states. 4.2. Rearrangement reactions The above method is now applied to eq. (3.23) to derive effective two-particle equations with a connected kernel [3]. Starting from the decomposition (4.1) of / , V assumes the form

~= ~

IX~r) 2t r (Y(~rl+ V£,

(4.13)

r

where !

12~ r > = e~ I X~r >, _p

I

_ ,

(4.14)

1

V = eel--=1 e~-~ .

(4.15)

Now eq. (3.23) is superseded by if(S) = i '(S) + ~ ~ i '(S) [ )~3`r ) ~'3'r ( )(3'r [ i ( s ) , 3' r

(4.16)

where E (s) fulfills the equation

ff'(s)=iO(s) + ~ iO(s) V£ E'(s),

(4.17)

3, or equivalently i ' ( s ) = it3'(s) + ~

i S ' ( s ) V; i ' ( s ) ,

ie'(s) = i°(s) + i°(s) Fe'ie'(s),

(4.18)

(4.19)

which can be solved by iteration for sufficient small values of the two-particle Schmidt norms 11F'.°Y£ 1]. Multiplying eq. (4.16) from the left with ( Xt3m [, we get expressions of the form ( 5@m [ i ' ( s ) [ )~.),r), which for 7 ~ are already connected. The only disconnected term appears in the case y = 13 from ira'(s), occuring in eq. (4.18). Using eqs. (4.19), (4.15), (4.3)and (2.19), it reads

U. Weiss, Three-particle theory

584

=

>

(4.20)

Transferring tiffs term to the left-hand side o f e q , (4.16), we get with eq. (4.5)

"y

r

or after little algebra the more evident form

R(s) -- R°(s) + RO(s) W(s)R(s) ,

(4.22)

where tile elements of R, R 0 and 1¢ are

R~m, an(s) - ! 7r {~.~ ~m t¢~ ~rn ()~rn[/~(s)I)~c~n)t¢~an ?tom + 6~a6mn~an K~n} ,(4.23) 1

(4.24)

ROrn, an (s) -=6 ~a6mn "Can(S) = 6~a 6 rnn ~ Kc~n tan(S)' 1

1

-~- " W~,n, an(S)=TT ~mY ( )(fire I [/~ '(s) --6t3aff/3'(s)] I Xan)~an

(4.25)

These matrix elements are still operators acting on states (qB I and I q' ). Eq. (4.22) has the structure of a multichannel two-particle Lippmann-Schwinger equation. Like the genuine two-particle equation (3.8), it is a three-dimensional equation with a connected kernel. It is appropriate to the scattering of three-particles two of them being bound in the initial and in the final states. The generalized potentials W~rn an(s) are energy dependent and become complex above the break-up threshold. It sho'uld be noticed that the phase-space factor Kan, which is defined in eq. (2.2 l) is introduced in eq. (4.22) with the purpose of interpreting RO(s) as the free propagator of a bound two-particle system and a third free particle. Now it is convenient to pass over to the transition amplitude and its equations. As in the single-channel case we have

R(s) = R°(s) + RO(s) T(s) RO(s) ,

(4.26)

r(s) = W(s) + W(s) R(s) W(s) ,

(4.27)

with

U. Weiss, Three-particle theory ,,, a

~

B

~

585 a*

g-~r- a

o~

-t-

n

Fig. 3. The effective two-particle equation (4.28) in graphical form. oC

oC

= a

~

X o~

x~t):~=+

,,

/3

+ ~ Z Y,c,

~o

I "(

×

d'

× a. '~

o~

q>~=+

9""

Fig. 4. Perturbation expansion for the generalized potential Wwith the conventions used in the --1/2 preceding figures. The signs 0 and × correspond to the factors ey and ~y , respectively. --1/2

-

where T(s) fulfills the operator Lippmann-Schwinger equation

T(s) = W(s) + W(s) R°(s) r(s) = If(s) + T(s) R°(s) If(s).

(4.28)

It is proved in appendix B that T(s) is related to the S-matrix as follows

=6#a6mn 2~/m2+k 2 2 MV~ffm+K~ 63(k~-k'o)63(K~

K~)

(4.29)

+ 2 i ~ 4 ( K - / ~ ' ) (qa [ T~m, an(S)lq' a ), The graphical significance of the composite particle scattering equation (4.28) is shown in fig. 3. The generalized potential W are demonstrated for/3 4: c~ in fig. 4. 4.3. Break-up reactions We now give equations for the break-up amplitude. Instead of the matrices R, If and T w e now have column vectors R, W and Twith elements R 0 apt, W0 an and T0,an , which are still operators acting on states ( q#l (p#l and [ qa). Defining Ro,~n by

Ro,~n(S) =

fv

E(s) I;~an) K~n ~ Xan ,

(4.30)

which is suggested by eq. (4.23), we get, using eqs. (4.16) and (4.23) (4.31)

R(s) : ~ ° ( s ) W(s) R(s) , where t

Wo,an(S) = ~ f f

!

0 -' (s) £ (s) l y(oen ) u.om~ "

(4.32)

U. Weiss, Three-particle theory

586

Inserting eq. (4.31) into the equation T(s) = ~ o - ' (s) R(s)R °-' (s) ,

(4.33)

which is easily obtained from the analogue of eq. (4.26), i.e. (s) = £ o (s) T(s) R °(s) ,

(4.34)

we finally obtain with eq. (4.26) (4.35)

T(s) = W(s) + OJ(s) ~,°(s) T(s) ,

which is illustrated in figs. 5 and 6. This equation combines the bound-state disintegration amplitude T(s) with the bound-state scattering and rearrangement amplitudes T(s). It is proved in appendix B that the S-matrix for the bound-state disintegration process is given by the following relation (;11(k2

I(k3

IS0,c~n[K~)l£;>=2i64(/~

/(')(q~l(fl/31T0,c~n(s)lq; )" (4.36)

4.4. Three particle bound states In quite analogy to the two-particle case the three-particle bound states will occur as poles in E(s) EP(s) = ~ i

I~PMi)(q'Mil-, M.2

(4.37)

s

l

where MiKm 1 + m 2 + m 3 is the bound state mass and (%L( t,, tion with normalization

= -d@_ + >--- - @

is its wave func-

@_

Fig. 5. Eq. (4.35) in graphical form. +

g"

*

,~ oc

]

Fig. 6. Perturbation expansion for the break-up potential Wdefined in eq. (4.32).

U. Weiss. Three-particle theory l pO-' l *Mj) =

%,

587

(4.38)

where p0 is defined in eq. (2.18). The wave function obeys the equation I~M/) = E0(M/2) ~ ~ IqJM/), (4.39) 3' which has the same undesired properties mentioned above. Applying, however, to eq. (4.21 ) the relations lim - ieE(M.2) p0 1 [ qJg/) = I q'M/') ' e~+0 lira - ieE'(/14/.2) pO '1 q~M/) = O,

(4,40)

e~O

we get, using eqs. (4.24) and (4.25),

II % , > = Ro(M )

*Mj

(4.41 )

where 11q'Mj >> is a row vector with elements I

11qXM/>~13m = )kl3mK•gm ( )(~3m [ qtMj ) '

(4.42)

^

still acting on states (0~ 1. The integral equation (4.41) which is illustrated in fig. 7 is very similar to the corresponding equation of the genuine two-particle case. Like this it has a connected kernel and is only one-dimensional after partial wave decomposition. Finally we note that the full state vector [XPM/) is obtained from the equation (fig. 8) 1

]%)=

~ ~F'(M/2)[XTr}Kyrg[[ %>2>7r, 3" r

which results after applying eqs. (4.40) to eq. (4.16).

Fig. 7. Eq. (4.41) in graphical form.

(4.43}

588

U. Weiss, Three-particle theory

+ g'r"

cl;,g,r-

o"

Fig. 8. Eq. (4.43) in graphical form with the Born series for E '.

5. CONCLUSIONS We have proposed a systematical treatment of relativistic three-particle boundstate scattering including break-up reactions, which is exact within the scope of a mere three-particle theory. The relevant amplitudes obey one-dimensional integral equations. The generalized potentials, occurring in them, can be calculated by iteration, which should rapidly converge in most cases of physical interest. The present paper only gives the equations when the elementary particles are spinless. The spin. however, can be incorporated into our equations in a manner described in ref. [7]. Besides this, three-body forces can be considered by the methods of ref. [8]. A special virtue of our approach is that we are not restricted to the separable pole approximation. As Basdevant and Oran,s [16] have emphasized, this approximation is unreliable in many cases of physical interest, e.g. when calculating three-body bound states, as the poles of the subsystems are outside the region of integration. In between, practical calculations which involve non-separable rest potentials turned out to be successful both in the non-relativistic Coulomb [17] and Yukawa [18] case. Non-relativistic multichannel resonant reactions have also been treated by this method [19]. Of course, the considerations quoted there can easily be applied to our equations to derive relativistic nmltichannel Breit-Wigner relations, where level shift and level width are expressed by accessible quantities. I wish to thank Prof. Dr. W. Weidlich for valuable discussions.

APPENDIX A Kinematics We have three particles of masses m l, m 2 and m 3 and four-momenta/~1, £2, £3" We use a metric, where the fourth component is positive. If the four-momenta are restricted to the mass-shell, a bar is placed over them. In the (1.2) subsystem we introduce the total momentum

/~3 = k l + £ 2 ' and the relative nlomentum of Wightman and G~.rding

(A.I)

U. Weiss,Three-partieletheory

A

--

I

^

P3 - 7(kl

/¢2)

589

rn~ -- m 2 2/£~ K3"

(A.2)

Using the inverse formulae /¢, =p3 +1(1 4 m ~ - - m ~ ) / ~ 3

K3

) R3

,

(A.3)

(A.4) ,

the mass-shell conditions k"2s -_ m2 (u = 1,2) can be expressed by the equations P3"/(3 = 0 '

(A.5)

s3 =K~ = [v/mm~ p~ +~m~2 - ff~] 2 ,

(A.6)

P3 is such P3 that P3"/£3 = 0. The total momentum of the system is k =/~3 +/~3 "

(A.7)

In exact analogy to eq. (A.2) we define the relative momentum of the {(1,2): 3} system according to q3 -2(k3 -/~3 )

^ _l(k^ q3 - g--3

/:3)

rn~ - s 3 /~, 2s

(a.8a)

m~

(A.8b)

M~n /~,

2g where we have discriminated whether the a b o u n d - s t a t e o f mass

(1,2)

subsystem is in a scattering state or in

M3n.

With the inverse formulae ]~3=q3 +½ (1 +m~ s s3)/~,

(A.9)

U. Weiss, Three-particle theory

590

/~3 = - q3 +1

1

-

£~,

(A.10)

and corresponding relations for the bound-state case, the mass shell conditions

£2 = m23' K23 = s3" (I~ = M~n), read as follows q3" /~ = 0 '

(A.11)

(A. 12a)

(A.12b) where q3 is such q3 that q3 ' /~ = 0. By cyclic permutation of the indices we get other sets of the momenta*/)1, ql and/~2, c12, which are appropriate to the {(2,3); 1} and to the {(3,1); 2} system, respectively. They immediately prove that the mass-shell restrictions /~3" /~3 =tt3"/~= 0 ,

(A.I 3a)

imply the conditions /)1" ]t~l = ql " /~ = 0 ,

]92" ]~2 = 42" /~ = 0 .

(A.13b)

In the subspace defined by eq. (A.13) we have s3- s3 -

s 3 -- M~n

~_

-

q3

" (s

F),

(s

-

(A.14a)

~),

(A.14b)

in the two-particle scattering state and bound-state case, repectively. Sometimes it will be necessary to use q~c,' @(~4=/3) as independent momenta. The formulae expressingp 3, s 3 and ~ in terms of~-l, q3 are = 2(ml2+m2

, - ~(m , 2I , q ) - 4s3

l

~3 '

* Note that only two of the six vectors Pw qa(c~= 1, 2, 3) are linear independent.

(A.1 5a)

591

U. Weiss, Three-particle theory

s-3 = [v/m~ - q~ + v/m~ -- (q3 + ql )2]2 + q i '

(A.15b)

s-= [%//m-m~- q ~ +~m~

(A.15c)

q~ + J m ~ - ( q ,

+~3)2] 2 .

It is appropriate to do practical calculations in the c.m. system ~ = (W, 0, 0, 0), where -0_ qc~-

0

,

=2 = qc~ - q a2'

(a = 1, 2, 3)

APPENDIX B Unitarity proof for rearrangement and break-up amplitudes If we allow two-body and three-body intermediate states, the unitarity expression (2,2) reads

{q~l T [3m,an(S +)[q'o)-<41Tt3m,.n(S-)l~t' a) :2i ~ ~'fd4~(dt3l 3"

* 2i

r

~m, Tr(S + )lqy')A3.r(d;)(q7"lrr.,(s)14~>_,

~

(B.1)

r

fd44~fd4~
with A3,r = 5+

- M r)5+(k - m

Ao = <(~ 12_ ml~)~ +(i 22 m~)<(i~ - m~),

,

(S.2) which in terms of the Wightman-G~lrding momenta are written ATr=8[Cl3,. I(~. A')r'(qr';S)= ' =2

v7 I

kvTl

-

(02~

a~r-" s

P7'

= K~(qy) =2 6(s7 M27r) ) '

(B.3)

' (B.4)

U. Weiss. Three-particle theory

592

where KW and p0 are defined in eqs. (2.21) and (2.18), respectively. Restricting ourselves to the subspace defined in eq. (A.I 3), the unitarity expression (B.1)assumes the operator form

+),.%,

2; E E

,

~/ r

+ 2 i T rn, 0 (S+) ~0 7o,~,,(s- ),

(B.S)

-

where T~m a,, and T~m 0 are acting on states I qa )and I pa )1 qc~ )' respectively. We now' prove that {he transition amplitudes T~m,om and T0,an introduced in sect. 4 fulfill the unitarity expression (B.5). As a first step tile following expression is obtained by straightforward manipulations of eq. (4.28) ref. [81

T(s +) T(s-)= T(s+)[RO(s +)

RO(s-)] T(s )

+ I+T(s+)RO(s +) [W(s+)-W(s )] I+RO(s-)T(s

) .

(B.6)

Let us now evaluate the discontinuities of R°(s) and W(s). Using eqs. (4.24), (4.5), (4.11) and (B.3) one obtains

R ~JJYl, 0 a P l ~ts+~-R~m,c~n(S ) = 6~a6mn A 0~11 ~ +6~a6,nn~K;) ' Tc~,,(s+) [t~l(s2)

t],(s+)]"can(s )

(B.7)

and using the equation

E '(s+) - k '(s-):k'(s+)E o-' (s+1 [F0(s +) _ kO(s )1E°-' ( s ) k ' ( s +) = 2iE'(s+)E 0-1 (s+) YxOE O-I (s )Y-'(s-),

(B.8)

and eqs. (4.25), (4.32) and (4.5), which define W, ~) and t, one finds W

~m,~.(s + )-w~j,,,~,,(s )=

2i

W~m,o(S+)X 0 ~;O,~.(s )

Substituting eqs. (B.7) and (B.9) into (B.6), one finally arrives at the expression T#m.an(S +)

Tl3rn,¢n(S-) : 2i E "1"

•

+

+2tZ~m,0(s ),5

0

G

T&n,vr(S+) avr Lr, an ( s - )

r

%,an(S--)+Zmn, an(S,S ), +

(B.10)

u. Weiss, Three-particle theory

593

where

Z~m,~.(s +, ~ ) 5 ~ , ~ m : % ~ , [

-1 +

+ 7rKgI It t ~3m" -1 (s+'l ~3: -- tl31 (S~-)l 7gm (S-) Tt3m,an ( s - ) +).:o<.(s +)It -1 (~) + - t~1(s~)] ~-1 + T ~,.,o,.(s O~Y/

"

(B.11)

Eq. (B.I 0) directly agrees with the unitarity relation (B.6), provided that

Zorn, c,n ( s+, s- ) vanishes. Indeed this holds, if the initial and the final states are on the mass shell. There h a v e s = s ( q 2 2 ) = ~-(q2) and st3 = M2m , sc~ = M2n, which i m m e d i a t e l y yields that the expressions in the square bracket can not be singular so that Z~m,~n(S +, s - ) vanishes. we

REFERENCES [1] L.D. Faddeev, JETP (Sov. Phys.) l 2 (1961) 1014; Mathematical aspects of the three-body problem in quantum scattering theory (Israel program for scientific translations, Jerusalem, 1965). [2] Three-body problem in nuclear and particle physics, ed. J.S.C. Mc Kee and P.M. Rolph (North-ttolland, Amsterdam, 1970). [3] E.O. Alt, P. Grassberger and W. Sandhas, Nucl. Phys. B2 (1967) 167; P. Grassberger and W. Sandhas, Z. Phys. 217 (1968) 9. [4] R. Blankenbecler and R. Sugar, Phys. Rev. 142 (1966) 1051. [5] R. Aaron, R.D. Amado and J.E. Young, Phys. Rev. 174 (1968) 2022. [6] A.S. Wightman, L.invariance dans la mdcanique quantique relativiste, in the book dispersion relations and elementary particles, ed. C. de Witt and R. Omn~s (Hermann, Paris, 1960). [7] J.M. Namyslowski, Nuovo Cimento 57A (1968) 355. [8] D.Z. Freedman, C. Lovelace and J.M. Namyslowski, Nuovo Cimento 43A (I 966) 258. [9] V.A. Alessandrini and R.L. Omn~s, Phys. Rev. 139B (1965) 167. [10] F. Riordan, Nuovo Cimento 58A (1968) 649. [11 ] P.V. Landshoff and D.I. Olive, J. Math. Phys. 7 (1966) 1464. [ i 2] J.M. Namyslowski, Phys. Rev. 160 (1967) 1522. [13] N. Nakanishi, Progr. Theor. Phys. Suppl. 43 (1969) 1. [14] S. Weinberg, Phys. Rev. 133B (1964) 232. [15] S. Weinberg, Phys. Rev. 131 (1963).440. [16] J. Basdevant and R.L. Omnes, Phys. Rev. Letters 17 (1966) 775. [17] U. Weiss and W. Weidlich, Z. Phys. 228 (1969) 1. [18] E.O. Alt, P. Grassberger and W. Sandhas, Phys. Rev. D1 (1970) 2581. 119] U. Weiss, Nucl. Phys. A156 (1970) 53.