New approach to the theory of coupled πNN-NN systems

NuckarPhysics A338 (1980) 377-412 Q North-Holland Publishing Co ., Amstcrrlam

Not to be reprodnoed by photoprint or microfilm without written penniaion from the publisher

NER' APPROACH TO THE THEORY OF COUPLED zrNN-NN SYSTEMS ()(n . Farther elaborations and relativistic extensions Y. AVISHAI* CENSaclay, DPh-N/HE, BP2, 91190 Gif-sur-Ytxtte, France and T . MIZUTANI

Physique Nucléaire Thforiquc, Inst. de Physique, Unitxrsité dt Liège, Sart ?îlman, B-4000 Liège 1, Belgium Received 2 July 1979 (Final version received 3 December 1979)

A6afra~i: We continue our investigation into the theory of wupled zrNN-NN systems itLSUOOession of an

earlier work, in which practical equations and unitarity have been established for all the pertinent amplitudes . First, the equivalence of our theory with that of Mizutani and Koltun is proved. Then, the question of whether theories which adopt the bound-state picture are appropriate for the description of systems of pions and nucleons is carefully discussed . We also show how the equations should be modified when we include the non-pole contribution to the ~rN Pt t channel . Fuuilly, we extend our equations to include relativity and show how to arrive at numerically solvable equations, with relativistic invariance guaranteed. In the NN sector, these equations contain the two time orderings of the OPEP in a correct manner.

1. Introdadion In this work we shall continue our investigation into the theory of the coupled vrNN-NN systems. This effort is motivated by the apparent need for a correct and practical theory both from experimental and theoretical points of view. In an earlier paper') (referred to as I hereafter) we have derived sets of integral equations which couple all the amplitudes of interest for the arNN-NN systems. In their antisymmetrized form, eqs. (L4.24) (with obvious notation for equations quoted from n are amenable for numerical solution which is now under way. This second part is devoted to the study of several subjects which have not been discussed in I but prove to have both theoretical and practical significance . Specifically, the subject matter contained in our present work covers (i) the demonstration of the equivalence of our set of transition operators to that of IVIizutani and Koltun 2) (IvIIC), (ü) comparison with the results based upon the so-called bound-state * On leave of absence from Dept. of Physic, Ben Gurion University, Beer Sheva, Israel .

377

378

Y. AVISHAI AND T. NIIZUTANI

picture (BSP) of e.g. Afnan and Thomas3) (AT), (iii) the inclusion of terms other than the nucleon pole contribution to the orN Pll partial wave in our scattering equations and (iv) the relativistic extension of our theory. These four subjects will be discussed respectively in sects. 2, 3 and 4. In this work again, we try to be as pedagogical as possible within a limited space. Before goinginto the main subject it may be appropriate to give a brief summary of the results obtained in I. There we have considered a system of two (initially distinguishable) nucleons N,, Nz and a pion ~r. The three-body states ds(~r, N,, Nz) and the two-body states ~(Ni, Nz) were coupled according to eq. (I .2.4) namely id! (E - HO - VNtNs - vaNt - VrNz)~ -(R1 +IZz)~ ~RN~ (A) (a) a ~=(Ri+Rz)fi=ltNChu~ : (b) (1.1) (Eho - vo)~~

Here Va (ß =N 1Nz, ~rN,, orNz) are three potentials describing the respective two-body interactions in the three-body channel and vo is a two-nucleon interaction which contains heavy-boson exchange (HBE) forces and non-interative multipion exchange forces, but lacks, roughly speaking, a one-pion exchange potential (OPEP). In ourtheory, the OPEP is generated by the irreducible pion absorption and emission operators R, and R; (pertaining to N,, i =1,2). The three-body kinetic energy operator Ho contains also mass renormalization of the nucleons, whereas the two-nucleon kinetic energy operator ho contains only that part of the mass renormalization which does not arise from repeated emission and absorption of a pion. Finally, the superscript A indicates the initial state of the system by giving the number of pious (0 or 1) before the collision. In (L2.6-9) we have listed all the amplitudes of interest in this problem, with the relevant processes explained in parénthesis namely ; T~ (pair ß +spectator-> pair a +spectator) TNa (pair ß + spectator~Nl +Nz) , TaN (N I +Nz ->pair a+spectator) TNN(Ni+Nz -i N,Nz) , Toa (pair ß + spectator ~ N, + Nz + ~r) ,

(a) (b) (c) (d) (e)

Too (NI+Nz+~r->N1+Nz+~r) .

(g)

(1 .2)

These amplitudes describe transitions amongthree kindsof asymptotic states, that is, two-nucleon plane wave Xz, three-particle (ar N, Nz) plane wave Xs and correlated pair plus free spectator states d~a. Transition operators UQß, UNa, etc. are then defined in such a way that the amplitudes introduced above can be written as matrix elements ofthe transition operators between the asymptotic states mentioned above. Specifically Ta~=(~alU~l~~>,

(a)

TaN = (~~Iv~Nlxz>,

TNN = CYi~ UNN~Xz) " TNS = ~Xz~ UNS ~ ~s ) , (b)

(d)

COUPLED ~rNN-NN SYSTEMS (II)

379

The main results of I include: (i) A definition ofthe physical transition operators in terms of the basic interactions of eqs . (1.1). Thus, UQß, Uxß, Uax and U,nv are respectively defined in eqs. (L3 .6), (L3.9), (L3 .14) and (L3.20) . (ü) Derivation of coupled integral equations among these operators [see eqs. (L3.2) and (L3 .3)] . The inhomogeneous terms and the kernels of these equations are completely determined by the operators which appear in eq . (1.1). Thus, we have derived the following equations Uaß = Gô 1 (1 - ô~ß)+ ~ tyGoUaß+RNTr,UNß y~a

and

Uxß =Rx +RNGo ~ tyGoUyß +(vo+Zxx)TxL1xs . y UQx =RN+RNTNUxx+ ~ tyGoUyIV, y~a Urnv =(vo+Zrmr) +(vo+ZNN)TNUNN+~RNGotyGoUyIV, (b) y

where ty [eq. (L2.1 lf)] is the scattering operator associated with Vy, Go ~ (E -Ho)-1 is Green's function for three free particles and Zrr,"r~RiGoRz+RZGoRi ~ZlZ+Z21

(1 .6)

[see eq. (L2 .13.b)] is the sum of the two OPE terms. The two-nucleon propagator [see eq. (L2.15 .a)] Tx ~ (E -ho -R 1GoRi-RZGoRZ)-1

(1 .7)

includes mass renormalization (see the appendix of I) and off-shell nucleon self energy through the bubbles b; ~ R;GoR;(i =1, 2). (iü) Eqs. (1 .4) and (1 .5) have been used in I in conjunction with the separable approximation (SA) to derive coupled integral equations amongthe physical amplitudes . Denoting these amplitudes by X's (rather than by T's) once the SA is applied we obtained eqs. (L3 .2~ and (L3.37 namely Xaß = Z~ß + ~ ZQyTyZyß +ZaNTNXxß , y~a

(a)

XNß = ZNß +~ ZIVyTyXyß +(vO+ZNN)~%~NXNß ~ y

(b)

(1.8)

and Xax=Zax+ ~ ZayTyXyN+ZaxTxXxx ~ yta

a)

Xxx =(vo+Zxx)+~ZNyTyXyx+(vo+Zxx)TxXxx,

(b)

y

(1.9)

38 0

Y. AVISHAI AND T. MIZUTANI

where, in the SA, the following quantities are defined Vy = ~ YS~Y tY

=

.

t

SYTyBy ~

TY = (~ rl

- 8YGo8Y)-1

Zaß = gaGo8~1-s~a) . ZaN - ôaGORN~ ZNß = RNGpgß "

(a)

() b (c)

(1 .10)

(d) e

(iv) We have also proved exact two- and three-body unitarity in our theory . (v) We have demonstrated how to carry out the antisymmetrization (AS) among the two nucleons, and obtained in eqs. (L4.24) a set of equations which involve properly antisymmetrized amplitudes and driving terms. This set can be obtained from eqs. (1 .8) and (1 .9) by first replacing the Z's by the antisymmetrized driving terms Z~'s [see eqs. (L4.23)]and dividing n,, by 2. (vi) Finally, we have also derived in I a general expression for the two-nucleon potential which is valid also above the single-pion

production threshold [see eqs. (L2.14) and (L4.37)]. Having thus briefly reviewed the main results of our previous work, we shall now move on to discuss the remaining subjects which have not been covered in I.

2. Relation to other models 2.1 . EQUIVALENCE OF THE PRESENT THEORY TO THAT OF IuIIZUTANI-KOLTUNZ) As we have mentioned in I, the starting point, namely, the coupled set of Schrödinger equations (1 .1) is the same for both our theory and that of MK . The novel feature here is the emphasis on the explicit coupling of the NN amplitude to all other amplitudes . In this subsection we show that in fact the physical transition operators UaßUNßUaN and UNN [see eqs. (1 .3)] are equivalent to those obtained by MK. In order to do so, we shall need a number of operators (interactions, Green functions, etc.) which are constructed from the operators appearing in eqs. (1 .1). First, we recall from I the following operators So ~ (E - ho)-1 .

(L2 V=~ V,., V4~RNSRN ~

w~v+v4 ,

(L2. l0a)

.lla) (L2.21a) (L2.23a)

COUPLED ANN-NN SYSTEMS (II)

38 1

Go ~ (E-Ho)- 1 ,

(L2.1 lb)

G,. ~ (E -Ho - V) -' ~ Go + GoTvGo ,

(L2.1 lc)

Tv ~ V + VGoTv ,

(L2.1 ld)

ta ~ Vß + VaGß V,~ = Va + VßGotß ,

(L2.1 lf)

G = (E-Ho- ~-1 .

(L2.23c)

In addition, we extend somewhat the notation used in I, so that channel indices ~, ß, y etc. include not only pair plus spectator asymptotic states, but also those in which there are three free particles, characterized by the three-body plane wave X3 . We use the subscript "0" for such channel, so that the physical amplitudes defined in eqs. (1 .2 .e-g) can be written as Toa = (X3 ~ Uop ~ ~a ) ~ # 0) (2 .1)

TON ~ ~ X3I UONIX2) ~

Too ~ (X3I UOOIX3) We therefore introduce a "potential" (2.2)

Vo~O,

and henceforth, Greek indices will include this channel as well. Note that this does not change the structure of the equations satisfied by the U's and that Uoo, Uoa etc. can be obtained readily from the solution of eqs. (1.4) and (1.5). We shall also need the operator Tt ~ (E - ho- vo-RNGvRN)-' , (2.3a)

which is the full propagator for the NN system including the coupling to the three-body ~rNN channel. ):fone uses two potential scattering formulas [eqs. (L2.12), (L2.15) and (L3.18-20)] and the statement following eq . (L3.18), one can express T~ in terms of the dressed NN propagator TN (eq. (1.7)) and the NN transition operator [eq. (1.S .b)] UNN viz T~=

TN+TNLÎNNTN .

(2.3b)

We shall now express the operators appearing in MK using our notation. It should be noted that in MK, the off-shell nucleon self-energy is implicitly included in the nucleon kinetic energy operator [cf. eqs. (2.15-16) of MK]. Thus Uo~o = Tv+,flvRrrTtRN,fl `'

(for Tr+N+N-~ ~+N+N),

(a)

UôK =,(a~,RN,(aN U~ =~NRN~ V

(for N+N->N+N+~rr),

(b)

(for yr+N+N-~N+N),

(c)

(2.4)

382

Y. AVISHAI AND T . I~IIZUTANI

where T~ is defined in eq . (L2.1 l.d), Rx and RN are given by eq . (L2 .3), z, has been introduced in eq . (2.3) and the Miller operators are defined as ,(i,. ~ (1 + TvGo), ,(l v ~ (1 + Go T,.) , (a) (2.5) ~1x5(1+UxxTx)~ SZx ~(1+~Uvx)~ (b) The role which the production or absorption operators play in the expression for the elastic operator is evident if one employs eqs. (2.3-5) and writes Uo~ o = Tv + U~zrrRx,(l v = Tv +,f1vR,tn,,, U~` . (a')

(2 .4)

We can see easily that in eqs. (2.4 a, a') the first term is ~r-absorption free, since the potential V has no absorption effect built in, (recall its definition in I or in MK) whereas the second term is responsible for this effect . Eqs. (2.4) are identical to eqs . (L1 .1-2) but the special form of T~ [eq. (2.3a)] facilitates the comparison with the original expression of MK [see eqs. (2 .25-28) in MK]. , Our goal is to show that U`oô` = Uoo, Uô` = Uox and Uxnc o = Uxo, where Uoo, etc. are the operators satisfying eqs. (L3.2-3) with an additional channel "0" to be explicitly considered as described in connection with eq . (2.1). Since both MK and our present work start from the same equations (L2.2) and (L2.4) [which is eq . (1.1)], the resulting transition operators in both formalisms must be identical, though satisfying seemingly different equations. However, the substantiation of this statement requires some work which utilizes straight forward operator algebra, and eq. (2.4) will be derived for our U opérators. For simplicity, we shall not consider three-body forces here since their presence does not affect the derivation. (a) Proof of the equality UoAÔ = Uoo. For a, ß ~ 4 [where Va ~ RNBRx, see eq . (L2.21a)] we have from the definition of the Alt-Grassberger~andhasa) (AGS) type operators [cf. eqs. (L2.24) and (L3 .6)] We need also the absorption free transition operators Uâß defined through Thus G - Gv = Ga(U~a - U~ä)Gs ~

(2.8)

On the other hand, G - Gv may also read where

G-G~ = GvV4G = G~T4G~ = GvRrrTeRxGv

(2.9)

Ta = Va + V4GVa = Va + V4GvTa = R ~eRx

(2.10)

COUPLED ~rNN-NN SYSTEMS (II)

38 3

Combining eqs. (2 .7) and (2.9) and comparing the two expressions for G - Gv we easily get (2.11) U~=Uâ~ +( ô°.+Uâ°,~.G,,)RNT~Rx(S,a+G U~.;~), where y, ~ ~ 4 but arbitrary otherwise. Eq. (2.11) is rather useful in the present study. First, one can put ~ = ß = y = ~ = 0 and recall that Uôô~ = Tv. In this case, the r.h.s. of eq. (2.11) coincides with that of eq . (2.4.a), so that we have proved the equality Uo~o = Uoo. Furthermore, one can use the AGS equations for U~~ and rewrite eq. (2.11) as [by setting y = ~ = 0 in eq. (2.11)] U,~ = UQ~ +(1+~ U~°,~.Got,.Go)RNr~lIx(1+~ GotGoU~.;~). n

(2 .11')

Both eqs. (2 .11) and (2.11') demonstrate that the total transition operator can be expressed as the sum of the absorption free part and the absorption correction term . The reason for introducing eq . (2.11') is that in the separable approximation (SA) it leads to a concrete form for the physical amplitudes X~a, XQa - Xaa + (Zax + ~ X~°yTrZ,x) (Tx+ Tx Uxx~r)(Zxa +~ ZxnT~.rXns) Y

TI

(2 .12)

where Xâ~ _ (4sa ~ U~ß ~d~a) are the solutions of the pure (absorption free) three-body problem. [e.g. eqs. (L3 .2') in which Zax , Zx~ and (Zx~,+v °) are switched off]. Eq. (2.12) proves our statement in I [see eq . (L1.1')]. (b) Proofofthe equality U~ = Uo1v. Eq. (L3.3 .a) for a = 0 reads (recall that to = 0) Uox = RN(1 + Tx Ux,,r) +~ t,.GoU,.x . Y

(2 .13)

Recallingthe defjnition of Uy x [eq. (L3 .14)] in terms of theAGS type operators 0,,4 [see eq. (L2.24)] and using the AGS type equation [e.g. (L2.25)] ~ t,,GoOrs = Oas , Y

(2.14)

we find where the Miller operator ~ ° ~ 1 + (E - ho - RNG°R N)-1 (vo + Zxx) has been defined in eq. (L2.17c). Consider now the two AGS type operators Ova and O~ which are related to a three body problem with total interaction W= V + V4 considered as two independent potentials V and V4 [see eq . (L2.23a)]. These two operators are coupled by AGS type equations, Ova = Go 1 + taGoO~ , Osa = TvGoDvs

(2 .16)

384

Y. AVISHAI AND T. MIZUTANI

from which we get a single integral equation for Oaa, namely Oaa = Tv + TvGotaGoOaa .

(2 .17)

Substitution of the r.h.s. of eq . (2.17) for O in eq . (2.15) leads to the following expression U°N =R,tv(1+~r,,,U,rN)+Tv(1+G°t4G°O~)G°RNw` ;

(2.15')

we now use the definition of UNN [eq. (L3.20)], together with the relations to = RN TRN [eq. (1.2.22 .b)], z = TNw ° (L2 .17 .a), w ° =1 +TNrI (L3.28), etc, and arrive at the expression (2.18) U°N = (1 + T~Go)RN(1 + TN UNN) , which is identical to UôN` as is evident from eq . (2.4.b). Likewise, we can show UQN =(1+~ Uâ°,~,G°tYGo)RN(1+TNUNN),

(2.18')

Uô = UNO .

(2 .19)

UN~ _ (1 + UNN-rN)R N(1 +~ GotYG°U~,°,~~ ) ,

(2.19~

Y

Y

thus we have shown that our transition operators are identical to those found in MK. In showing the above mentioned equivalence, we have assumed neither the vanishing of non-pole-type interaction V,~ for the two-body arN Pt t channel, nor the separable ansatz for V~,. 2.2 COMPARISON WITH THE BOUND-STATE PICTURE (BSP)

In our previous work, a brief comparison was made at various spots between our formulation and the one based upon the BSP [especially that of Afnan andThomas3) (AT). Here in this subsection we shall elaborate this comparison to some extent . A characteristic feature of the BSP (independent of whether it is based upon a relativistic picture) is that only the isobar N' which simulates the nucleon (as a orN P tt boundstate) can emit the pion whereas only N (the elementary nucleon) can absorb it (to become N') . As a consequence it differs from any description using the standard field theoretic TrNN coupling in various points . We shall mention some of them here in connection with the coupled vrNN-NN systems of our interest: (i) As for the one pion exchange (OPE) NN interaction the BSP gives only one of the two possible time orderings (fig. 1).

Fig, 1. OPE interaction in the BSP. Only one time ordering is possible [see point (i) in subsect 2.2 for further discussion].

COUPLED aNN-NN SYSTEMS (II)

385

(ü) No possibility exists for describing e.g.

an intermediate process where the pion is absorbed by one nucleon and is reemitted by the other as shown in fig. 2. (iü) During the off shell propagation of two nucleons resulting e.g. from the absorption of the pion, only the N' can have off-shell self energy effects due to virtual single-pion productions (fig. 3).

Fig. 2. An intermediate processin NN scattering in which the pion is absorbed by one nucleon and is then reemitted by the second one. This process is missing in the BSP [see point (ü) in subsect. 2.2].

(iv) The inclusion of the heavy-meson exchange NN interaction vo in the twobody sector does not come out naturally in the BSP. One might add it as an ad hoc three body force, which however will not lead to any constraint being imposed on va (see the discussion pertaining to eq . (L2.14)]. The asymmetric appearance of N and N' [especially manifested in points (i}-(iii) above] certainly shows the drawback of the BSP approach to the problem of coupled systems of pions and nucleons in general. (In addition, non-relativistic BSP requires the kinetic mass of N' to be mN+m and not mN). Nevertheless there appears to be some feeling that with the antisymmetrization (AS) of N and N' the BSP should give almost correct physical amplitudes compatible with results obtained from theories with field theoretical orNN coupling, notably presented in I. We shall see below whether this is really the case . Our description in the following may become somewhat lengthy, which however seems necessary for a clear understanding of the problem. First we follow the procedure of ATwithin the context of the Faddeev theory plus separable two body input. When separating the N' ( _ ~rN P tt isobar) explicitly from all other isobars (for ~rN and NN pairs), the AT equations (not antisymmetrized) read (with factors like (1-5,1) implied in the Z's) X~,~ = ZQ ~ +~ ZaYTYXYa +~ Zarr,Trr,Xrra ~

(a)

XNi9 - ZNiP +~ ZNjyTyxyd +ZNiNjTN'1XNjß

(b)

i

Y

Y

XaNj = ZaN(+~ ZayiyX yNj +~ ZaNfrNfX N,TIj, Y 1

a

XNfNf -ZNrNf +ZNiNfTNfXNfNf +~ ZNnTy XYNf . ~) Y

N~ N

N

N.

N Fiji . 3. Seff-energy effects in the BSP allowed only for N'. [See point (üi) in subsect. 2.2].

(2.2~)

(2.21)

38 6

Y . AVISHAI AND T. IrIIZUTANI

For definiteness, recall that a, ß, y stand for zrN and NN isobars other N' and N; signifies N' formed by the pion and the ith nucleon (i =1, 2). Eqs. (2.20) and (2.21) look, at first sight, rather similar to our equations (1 .8) and (1 .9), but in detail theyare quite different. However, we shall not elaborate on this point now. From eqs. (2.20) and (2.21) AT obtained sets of equations in which two nucleons (NN in the three-body ~rNN states but not NN') are antisymmetrized,ewploying the numerical identites of various Z's and T's. These equations then read Xâß) =ZQ~) +~Z~B)TYX~) +ZA~)TN. X ~), Y

(a) (2 .20')

XNâ) =ZN~) +~ZN'y )TYX Y9 ) +Z N~)TN'X NB) ,

(b)

s) =Z Ars) +~ ZâcH) TYXYN) +Zy(NS,)~,X

(a)

Y

Xâ

Y

(2 .21')

+~N.cH)TYX ~) XA NZB V~ =Zx~r +ZNT,~TrrXANTr . Y

(b)

where, with d and d/ signifying various NN and TrN/ isobars respectively AS (antisymmetrized) driving terms and propagators are defined as Z~acs) = 2'/z Zdld = -2i/z Zn~ - Za,ai = - Za za4 ,

- { - ZNtn, _ -ZNfa~ ,

(a -_ d~ ß =d) (a =d, ß =d ) _d)

(a) (b)

(2.22)

(similarly for ZQ~) )

TY=Td(~i=d)~

TI,1.= TNi =TNf ~

TY

- Tdt = Tda

(~i = d),

(e)

where two-nucleon states (marked by "d") are assumed to be already AS. Likewise AS amplitudes in BSP; XAB)'s, are defined, for example XANN

= 2-1 LXNiNi +X N§Ni -XNiNi -XN~N~ ~ = XN4Nf - XNiNi ~

(2.23)

where numerical identities XNfNi =X N~N~,and XN~)vf =XN~Ni (stewing from the identity of the nucleons) have been utilized .

COUPLED ANN-NN SYSTEMS (II)

38 9

One might think that in order to simulate the (true) physical NN state, it is enough to first solve eqs . (2 .20') and (2 .21') and then perform AS of NN' states in the resulting amplitudes as has been suggested in AT (the details of such procedure are not stated in detail in AT). Now it is not difficult to see that this idea implies that the following sets of equations should give the correct physical amplitudes : A

A

A

yA

A(B)

A(H)

Y (2 .20")

XNß = ZNß +~ ZNYTyX yß +ZNN (~TN')XNß ~ Y

(b)

Xx -Za +~ ZYTrXYN +Zy (z7IV')XNN

(a)

XrArN = ZrAnv +ZNN (iTN')XNAN +~ ZN,.TYX,A.N Y

(b)

Y

where

ZNß

~ 2 -i~z(ZN,~ ) -

ZrArN ~Ziwv(ZBV') -

ZN~)(ex) )

ZN,cNB,xex)

(ig = d, d) ,

Z~ ~Zââ),

(2.21")

(a) (b)

(2.24)

(c)

and similarly for ZN, XNß, XNN etc. [Note the equality Xâ = X~~) in eq. (2 .20 .] The amplitude X~) is not coupled to the others but is obtained from eqs . (2 .20'). The superscript (ex) in eqs. (2.24) means that the quantum numbers (spin, isospin, momentum) of N and N' are exchanged (for Z,v.,,, . the (ex) applies either to the initial or the final NN' pair). ïn order to obtain the above set of equations from eqs (2.207 and (2 .21') we have used the equalities XN'N'(P,q)=XNN'(-P. -9) etc,

(b)

(2 .25)

[q(p) is the initial (final) spectator momentum in the three-body c.m . frame] which are consequence of the invariance under the parity operation. Eqs. (2.25) also imply the numerical equality of the two different time ordering Z's. As it turns out, eqs. (2.21 are self contained and implement the Pauli principle appropriately but eqs. (2.20 do not share this property. As we have already noted, the third term on the r .h.s . of eq . (2.20"a) cannot be expressed in terms of ZNN and XN~. Thus intermediate NN' states cannot be made AS and this algorithm violates the Pauli principle [see the discussion associated with eqs. (L4.33}-(L4.36)]. Therefore, one must impose a forced AS on this term through the replacement ZA,,i(B.) Trr'X N'â) ~ Zâ(B)(~TN, )XNß . (2 .26)

38 8

Y. AVISHAI AND T . MIZUTANI

Unfortunately, this modification does not yet guarantee that the BSP indeed gives the correct description of the 'rrNN-NN systems. For example, we know that the OPEP Z~ simulates only one half of the full OPE interaction, reflecting a characteristic feature of the BSP. A careful observation [recalling eqs. (L3 .39), (L4.17) and (L4.33)] reveals that in order to account for the correct OPEP terms, the driving terms in this modified physical BSP have to be 21/2 ZrAr~) ~ ZrAra Z~ÎP) ~ 21/2 Z N

~ = d. d)~

(a)

(~ = 4r d)~

(b)

Z~) ~ 2 ZIA,rN, A(P) A ZaB ~ZaB

(c)

(2 .24')

(d)

and similar definitions apply for the corresponding amplitudes (which we write as XA~)'s). The equations coupling these physical amplitudes now read [as easily obtained from eqs . (2 .20 plus eq. (2 .26)] A(P)

Xaa

A(P) A(P) A(P) A(P) j A(P) =Zaß +~ZQY 7yX.s +ZaN (anv')XNß Y

) +~ ) ) XNB = ZN~ ZNYP)TY" YB Y

+Z,,A (4TN')X Nß (,,P)

),

A(P) _ A(P) A(P) yA(P) A(P) j A(P) XaN - ZaN +~ ZaY TYXyN +ZaN (4TN')XNN Y

A(P) - A(P) A(P) yA(P) A(P) j A(P) XNN -ZNN +~ ZNY TY" Y N +ZNr (a?N)ZNN Y

(a)

(2 .27) (b) (a) (b)

(2 .28)

where the occurence of â7N. [as compared with i'r,,r in our eqs. (L4.24) and (L4.25)] still reflects the remnant of the BSP; for example, a process like the one shown in fig. 2 is not yet taken properly into account in the above equations. In order to include such missingprocesses and at the same time to be consistent with two and three body unitarity, the following replacement must be made : TN' -~ ZTN

(but not TN. -> 27N-) ,

where 'rT.r is the NN propagator : eq . (1 .7), which has self-energy bubbles for both nucleons [see eqs. (L2.15), (L2.13 .c) and the appendix of I]. At this point we find that apart from the inclusion of heavy-meson exchange forces vo [see point (iv) above], the BSP has been improved to become almost identical to our set of equations (L4.24) and (L4.25) . However, its original characteristics are lost in the course of the improvements, which reflects the fact that the BSP cannot really describe correctly the physic of our interest. Beside AT, several other authors have studied the NN scatterin& problem using the BSP. Varnas) used Faddeev-Lovelace three-body theory, whereas Dodd6)

COUPLED arNN-NN SYSTEMS (In

38 9

obtained the Yukawa type NN interaction from minimal requirements on three body equations (such as analyticity and unitarily) . It has been noticed by Fuda') that in the BSP, the NN interaction has unphysical matrix elements, which connect singet and triplet states of the two-nucleon systems. Finally, we mention here the work of Kloet et a1. 8 ) on the inelastic NN scattering based upon the BSP and the relativistic three-body theory of Aaron et al.~ (AAY). The equations used there are actually a relativistic extension of (2.21 with vo included. In order to implement the full strength for OPE they have chosen vo in such a way that its combination with Z~ produces the full OPEP in the static limit. Since v o has no contribution to three body unitarity, its addition does not destroy the unitarity structure within the BSP. However, the other time ordering [not included in ZN~ ]also should contribute to three body unitarity. Therefore, the addition of interactions (e.g. vo), which do not destroy the unitarily structure of OPE driving terms does not lead exactly to the complete description of the inelastic NN scattering. The central importance of non-static effects in NN scattering encourage numerous investigations, some of them do not rely on three-body formalism in the sense described above [see e.g. Green et tzl. l ~] . On the other hand, there have been several works aimed at constructing a theory of the ~rNN system which is not based on the BSP, notably by Thomasll). Recently, Thomas and Rinatl2) have derived a set of equations exactly like our eqs. (1.9), using graphical summation techniques and the non-relativistic reduction method. It seems, however, that the separable assumption for the two-body input is essential for their derivation [thus it is hard to see how equations like ours (1.5) may be derived thereby] . Furthermore, their two-nucleon propagator (denoted by Go in ref. 12) but corresponds to our Trr) has no self-energy contribution due to the virtual single-pion production, which does not meet exact three-body unitarity to be satisfied by the solution . We also point out that with the methods used in ref. lz) it is rather difücult to incorporate the non-pole term contribution to the two-body input forthe aN P l1 partial wave. We addressourselves to this (somewhat unusual) feature in the next section. 3. Indosion of non-pole term rnntrlbution to the two-body ~rN Pll du~nnel (NP Pll) 3 .1 . PRELIA~IIIINARIES

The theory of coupled ~rNN-NN systems is plagued by unusual disconnectedness problems. Firat, there are those terms where thepion is absorbed byone nucleon and then remitted by the second one . This type of disconnectedness has been studied in great detail by Stingl and Stelbovics'3). There is another type of disconnectedness which has not been discussed so far. In this subsection we study it and show how to remove the obstacle which it raises.

390

Y. AVISI3AI AND T. MIZUTANI

We have mentioned in I that if an interaction V (a = ~rNl or aN2) contains a contribution to the arN Ptt channel, it generates the non-nucleon pole term which in what follows will be denoted as NP Pl,. When these interactions are assumed to be separable, the integral equations for the physical amplitudes Xa~, Xxa, Xax and Xxx [viz ., eqs. (L3.2'-3')] contain disconnected driving terms which prevent us from obtaining a unique solution as the kernels of integral equations become noncompact. More precisely, consider the driving term Za,x for the process N,+NZ -> at +NZ where a l is a Ptt - ~rN state for the ~ and Nl, whose form factor is ga, " Then clearly [see eq . (L3 .39c)] (3 .1) Za,x = Ba,GoRrr = BQ,GoRi+i~lGoRi=Zâ,x +Zâlx ~ where the second term Zâlx =ga,GoRz is connected but now we have a nonvanishing disconnected term Z~,x =galGoRi (see fig. 4). Similar disconnected terms occur in Zx~ where ß is a ~rN Ptl isobar. In order to avoid this problem we have considered in I only the case where the Va do not have NP Pt1. In the following we shall show how to solve the problem when this restriction on Va is removed.

Fig . 4 . (a) . The disconnected driving term Zd,N and (b) Zâ,N the connected part of Z Q,N [see eq . (3 .1) for discussion].

Before going into the main discussion one remark seems due; it is important to keep in mind that whenever a represents a correlated ~rN state in the Ptl partial wave due to the interaction Va, this is not physically observable . The physical ~rN P,1 state has contributions also from the dressed nucleon pole term generated by the repeated emission and absorption of the pion by nucleon i through R;and R, (i =1, 2). Hence, an amplitude Taa (or Xaa ) (where a or ß denotes the above mentioned arN state) is not directly connected with observed quantities . We can however use such amplitudes as a scaffolding upon which the complete physical amplitudes are constructed. To understand this (somewhat subtle) argument more clearly, consider the ~rN physical amplitude in the ~rN Pll channel. We shall translate the discussion of this subject from appendix A of MK into our nomenclature . For the purpose of minimizing the introduction of new notations we shall consider it in the three-body aN,NZ space with NZ as a spectator. Let ta, be the arNt Pl , amplitude resulting from Va, (a1= ~rNl in Pt1 state) . Thus

COUPLED ~rNN-NN SYSTEMS (II)

39 1

We can now combine tQ, and R, (R i )' to define one-particle irreducible (pion) absorption and emission vertices (in the presence of the spectator nucleon NZ!) Thus (see fig. 5) we consider Yt ~R i +R 1 Got,~ 1 , z1~Ri+tQ ,GoRi

(b)

(3 .3)

ri

Fig. 5 . Irreducible pion absorption (a) and emission (b) vertices in the case where the rrN P 1 ~ channel gets a contribution from both pole and non-pole tenors [see eq . (3 .3) for discussion]. For convenience, the spectator nucleon is absent in figs. 5-ß .

(they correspond respectively to Aol and .1 to in appendix A of MK). It isworth noting here that R, (R;) is a two-particle irreducible vertex (see appendix A for further details) . The introduction of Va, also modifies the self-energy operator for nucleon N, (in the presence of the spectator nucleon NZ!) as (see fig. 6) : vl=RiGoRi+RlGotQ ,GoRi=R l Gozl = yt GoRi.

(3.4)

ota~ t GpR~

RtG l

Fig . 6 . Nucleon self-energy term for the ~rrN P 11 interaction as in fig. 5 [ace eq. (3 .4) for definition and discussion].

Recall that in the absence of the interaction VQ, we have v1= bt = Rt GoR i [see eq. (L2.13c)] . The interpretation of vt as a self-energy will become clear later on. Note that in eq. (3.4), v1=R1Goz1 ~ YiGozi so that yl and zl enter in an asymmetricway. This construction ensures us against overoounting some part of the self-energy contributions. A similar situation arises in standard quantum field theory, where one avoids the overcounting of so called overlapping divergence contributions [see e.g. Lone r`)].

392

Y. AVISHAI AND T. MIZUTANI

We can now easily find the corresponding dressed propagator Tl for nucleon Nt (see fig. 7) zi = (B - ho - vi)-1 = Bo+SoviTt" Z~t

(3 .5)

90

Fig. 7. Graphical representation of eq . (3 .5) for the dressed zrN P tt propagator .

Then, the totâl ~rN, Ptt amplitude is shown to be given by (see fig. 8) shot) _ tat+zlTlYi .

(3 .6)

Fig. 8. Graphicalform of the totalaN t-matrix in thePtt channel, as discussedin corinedion with eq . (3 .6).

frotg which it is clear that Val creates only the non-nucleon pole part of the physical scattering state in this partial wave. In order to see how the self-energy operators v; (t =1, 2) come in let us now recall the Schrôdinger equation for NN scattering [eq. (L2.4b')] and inspect the term RN G~RN=Rtv(Go+GoTv Go )RN which enters the equation due to the coupling to the three-body -rrNN channel. It has been assumed in I that Va,= Va==0. Thus, dN =RNGoT~GoRN is completely connected there and the only disconnected terms in RNG,.R N are then bl + b 2 = aN [see eq . (L2.13)]. Here, on the other hand, dN will also have disconnected parts, thus it is useful to decompose dN into its connected (dN~) and disconnected (dN~) parts, and to put disconnected terms of RN G~RN together. We then write dN~ = RtGota,GoRi +R2GotasGoRi ~

(b)

(3 .7)

dN~ and dN~ are schematically depicted in fig. 9. Thus AN should obviously be

identified as the self energy of the two nucleons, with v, being the self energy for N;. (The propagator corresponding to AN will be introduced in the next subsection.) As a result of eqs. (3.7) it is clear that eq. (L2.14) for the total two-nucleon interaction should now be modified as v~Nr l =vo +Z~+dNi .

(3.8)

For the purpose of practical calculations, it is useful to give the form of operators y,, z; and v, when ta, takes on the separable form (I.lO.b) ta, = Sa1Ta~Sa~

(i =1, 2) ~

(3.9)

COUPLED ~rNN-NN SYSTEMS 2

393

No a ahso~ption

t

(d)

(ln

1

dN =

2

2

Füa 9 . (a) Connected parts of dN ~RN Go T~GoR,'r including: (1) Pion emission by N=, followed by ~ - ~Y fra P~Bation Ga Tr absorption free scattering processes Tom, three-body free propagation Go and pion absorption by N r . (2) Pion emission by N l , Propagation Ga scattering processes that invotve neither ~r absorption nor ~N l non-pole contribution to the P l i channel, propagation Go and pion absorption by N l . [See eq . (3 .7)]. (b) The disconnected part of dN, as defined in eq. (3 .7b) .

Then (with Zâ,N defined in eq . (3.1)) we have (with i =1, 2), yt -

R, +ZNQ,Taeôap

la)

z,= Ri +Ôa~+a,N,

(b)

Yi = RiGORi+ZNa~Ta~Za~N "

lC)

(3 .10)

In order that ti`°`) of eq. (3 .6) (restricted this time to the two-body rrN space) will reproduce the experimental phase shift in the P, t channel, R, and ga, should be chosen appropriately. The position of the mucleon pole, the corresponding residue (proportional to the square of the ~rNN coupling constant) and the scattering volume should also be reproduced reasonably well . Incidentally it should be pointed out that if one is only interested in reproducing low energy TrN Pl, data, then neglecting V~ and keeping the nucleon pole contribution alone (as has been assumed in I) may be sufficient. At low to medium energies the Ptl phase shift stays small and changes sign at an energy somewhat below T,~(lab) = 200 MeV. For T > 100 MeV one term separable interaction seems insufficient and one certainly needs the combination of R, and VQ,. 3 .2. TRANSITTON AMPLITUDES IN THE PRESENCE OF NP Pii

In this subsection we shall show how to incorporate the contribution coming from Va whichis a part of the two-body ~rN (Prr) interaction that produces the NP Ptr. In order for the presentation to be transparent we will make the following simplifications : (i) Amplitudes are not yet antisymmetrized with respect to the nucleons involved. The physical (antisymmetrized) amplitudes can readily be obtained by a standard prescription as we have carried out in I. (ü) We use compact

394

Y . AVISHAI AND T. MIZUTANI

matrix notation which is exemplified in the case where we consider one isobar for the two nucleons in the arNN space (denoted by "d") while for the TrN, system we consider two isobars, the Pll isobar a, and the second one d, ~ Pll. Certainly, generalization should be straightforward. Now we define __ [ S x S matrix with elements X11 Xaa,a,ß=d,di,dz,ai,az '

(a)

_ [ S x 1 matrix with elements Xlz~ Xax~a=d~dl~dz~ al .a z '

(b)

T Xz1~X,2,

( ) C

Zll , Zlz, Zzl -defined similarly ,

(d)

Xzz =XNx ,

(e)

Zzz s vo+Z~ . dlz =

(f)

I S x 1 matrix with elements d,.Ny d, dl, dz, al, az where d,.N

(3.11)

Zâ~N ~a

y at y~a,

dz1= d~ ,

(a)

(3 .12)

(b)

[recall eq . (3.1)] Tl

_ [ S x S diagonal matrix with elements ' Ta~a=d~dhdz~ al .a z

zz ~ Ty l

[see eq . (1.7)] .

) (b)

(3 .13)

Eqs. (1.8) and (1.9) (or eqs. (L4.2)) then involve the 6 x 6 matrices %Clzl X = (Xll Xzl Xzz '

(a)

Z = (Zu Zlzl Zzl Zzz '

(b) (3 .14)

In the present discussion it is important to define disconnected and connected matrices of driving terms as

Thus, in the presence of NP Pll, eqs. (1.8) and (1 .9) have the matrix form X =Z+ZBX,

(3.16)

COUPLED rrNN-NN SYSTEMS (I>)

395

whereas in the absence of NP P t1 the corresponding amplitudes Xco~ can be directly computed from the equations (3 .16') xco) = Z`+Z °BXco> as found in I. We now concentrate on eq . (3.16) and decompose the amplitude into connected and disconnected parts, namely

The formal definitions of X` and X° will be given below, and it should be kept in mind that X` is the physical quantity connected with observables (except for the elements with a, legs). A formal application of two potential formula to eq . (3.16) with the distortion Zd and the main transition induced by Z`, leads to the following results X`=(1+XdB)Y`(1+BX°)--=,fldl'`,fld,

(b)

(3.18)

where ,(ld=1+XdB(,(l°=1+BXd) is disconnected whereas Y` is connected and satisfies the following equation : with the propagator matrix ~r 3

9 + 9Xd B = 9 + BZ dvr = 9,fld = ,(l a 9 .

(3.20)

Eq. (3.20) admits an explicit solution which is expressed in terms of Tt, TZ, dtZ, du and propagator Trzz corresponding to the two-nucleon self-energy term AN (see eq . (3 .7)) . The solution is (see Fig. 10) 7lt t = Tt + Ttdt2?122d2tT1 ,

(a)

~tz = Ttdtz~zz ~

(b)

Tr z i = arzzdztTi ,

~rrzz=(g - ho - AN) -t =TN+TN dzt Tt d tz ~zz ,

(3 .21)

(d)

L >±
Y. AVISHAI AND T. I~ZUTANI

39 6

where the dimensions of matrices X11, ?rlz and ~rzl are identical to those of X11, Xlz and Xzl [see eqs. (3.11)], wheres ~zz is a single element. In eqs. (3.21), matrix algebra rules should be applied such that in (3.21a) for example, dlz ~zzdzl is a 5 x 5 matrix since one multiplies the rowmatrix dzl by the column matrix dlz~zz from the left . On the other hand, in (3.21d), dzl Tl dlz is asingle element. Thus, unlike the propagator B encountered in I, the propagator matrix ~ in the basic equation (3.19) is not diagonal. To proceed further, we utilize eq . (3.20) and solve eq. (3 .18a) forX° to find Xil = dlz~zzdzl ,

(a)

Xiz =dlz+dlz~zzdzlTldlz,

(b)

=dzl+dzlTldlz~zzdzl,

(c)

Xi1

Xiz =dzlTldlz+dz1T1d1z~zzdzlTldlz .

(3 .22)

(d)

With the matrix ~ as given in eqs. (3.21), eq . (3 .19) should be solved for Y`. This step is straightforward and essentially no extra effort beyond solving for e.g. X~a~ [eq. (3.16')] is needed. Actually, most of the non-diagonal elements in the matrix Tr do vanish : the 5 x 1 column matrix ~rlz has only two non-vanishing elements namely Ta~a,p7lzz (i =1, 2), whereas the 5 x 5 matrix -rr,l has only two non-diagonal elements which differ from zero (fig. lOb) namely TQ~~,N~rzzZxa,T~, (i ~j). The solution Y` of eq. (3 .19) must then be used in eq . (3.18b) to give the connected (physical) amplitudes X`=,(ldY`,fld . From the previous discussion these extra operations appear close to "unity" and the resulting physical amplitude X` should not be very different (numerically speaking) from Y` or even Xco~, at least at low energies . Before closing this section we would like to show that we can actually obtain an integral equation for X` itself. Let us recall eq . (3 .20) and write Recalling the relation between X` and Y` (eq. (3 .18.b) we know that the desired equation for X` must take the form where B`---,f1aZ`,(la

(3.25)

and the propagator matrix B is identified as e=(nd)_le,

(3 .26)

which can be evaluated explicitly to give B

T1

- ~ - T2d21 T1

- T1d12T2 Tz

(3.27)

COUPLED aNN-NN SYSTEMS (II)

39 7

Thus, we have found a single matrix integral equation for the physical amplitude X` whose kernel is connected. Comparing now eq . (3.24) with eq . (3.16') [or eq . (L4.2)], the following points should be observed : (i) The propagator matrix H is non-diagonal due to the non-vanishing transitions NN H (~N)P + N (which lead to disconnected terms) where (~N)r is an isobar induced by the non-pole part of the ~rrN interaction. (ü) B` now contains non-vanishing diagonal elements between two (-rrN) P +N type of states . Finally, we note that although 9 [eq. (3 .24)] contains Tz( _ -rr,) which is not the fully dressed NN propagator lrsz (eq. (3 .21d)), the solution X` contains only azz but not rz as may be proved easily. The renormalization of the two-nucleon propagator ar2z is worked out in appendix B. In the presence of NP P it is this modified propagator which should replace 7N in the integral equations for the transition operators [see eqs. (1 .4) and (1 .5)] . Lastly one can show that X` does satisfy two- and three-body unitarity as shown in I. 4. RelativlsHc exte~fon The final section is devoted to an extension of our results so far to incorporate relativity. The necessity to consider that may be clear but for definiteness we restate why it is needed : (i) due to its small rest mass compared with that of the nucleon, the pion becomes relativistic already at energies as low as TÂb = 50 MeV. Therefore in order to describe e.g. the pion scattering around the d (1236) resonance, a relativistic consideration is certainly called for, (ü) as is well-known, any non-relativistic field theory with a coupling ôf the type aHb+c suffersfrom the violation of the Galilean invariance when the conservation of kinetic masses m,=mb+m~ is not guaranteed at the coupling vertices (kinetic mass is what appears in the denominator of the ordinary non relativistic kinetic energy operator). This is often referred to as the violationof Bargman's super selection rule' s). Obviously, a = b = N and c = ~r in our problem and thus the super selection rule is violated . This violation brings in discrepancies of O(m/mN) where ~ is the pion energy (with the rest mass) among energy and momentum variables in intermediate states, which is bearable only at low energies. (It may be worth emphasizing that the so called Galilean invariant ~rNN couplings' 6) resulted from non-relativistic reduction of the ys or y,~ys theory are not Galilean invariant!) The only way to maintain the Galilean invariance in a non-relativistic theory is to adopt the BSP. However, compared with real (physical) situations, the BSP also brings in discrepancies of O(w/mN) in addition to various other diflïculties discussed already in subsection 2.2. In the following we shall try to develop a compact algorithm leading to a set of relativistic equations which is an extension of ournon-relativistic results. It would be

39 8

Y . AVISHAI AND T. MIZUTANI

possible to start out from e.g. Taylor's non-perturbative approach to Green function equations") and find the relativistic counterpart of eqs. (1.4) and (1 .5), and then proceed to adopt isobar assumptions etc. However, in order to keep our description contained, we shall at the moment short-cut this procedure and start with the isobar (separable) assumption for the two-body amplitudes . With this we write down côupled sets of linear equations exactly like eqs. (1 .8) and (1 .9) for various transition amplitudes X~~, XN~, etc. Then utilizing the fact that the amplitudes are assumed to be analytic functions of energy variables and that these equations must satisfy two and three body unitarity, the driving terms: (Z,~a etc.) and the propagators for various isobars (plus spectators): (T~ etc.) are determined . This method was first developed and applied by Aaron et a1.9) (AAY). It eventually coincided with approaches to relativistic three-body scattering problems with the BlankenbeclerSugar propagator'8) plus separable two-body amplitudes which satisfy cluster properties. As worked out by Kloet eta1.8) and by Klcet and Silbara), the AAY theory can be applied directly to the problemof inelastic NN scattering once the BSP is adopted for one ofthe nucleons in the two-nucleon channel. However, as we shall see shortly, the neat trick of AAY is not so powerful once we abandon the BSP for the nucleons. We will need some minor assumptions in order to arrive at practical results in this more general case . (Recall that several inconveniences of BSP in the theory of coupled NN-~rNN systems have just been discussed in subsection 2.2.] To be definite, we start byassuming thatour relativisiticequations take the form as presented in eqs. (1.8) and (1.9) where just as in the non-relativistic case Z's represent exchange driving terms and T's are two-particle propagators. This choice implicitly assumes that NP P, 1 contribution is not considered in what follows. The inclusion of its effect can be carried out along the lines discussed in the last section with minor modifications. We let a, ß etc represent channels with various isobars and spectators (d, d, etc.). As far as Z's and T's for isobars a, ß etc. are concerned, a direct application of the AAY technique is sufficient to determine their concrete forms once the isobar form factors are specified. We shall not get into the problem of their construction here, but refer the reader to various works on relativistic rr-d scattering'. Our main concern here is to determine Z,,,r, and -ry,, now without BSP [otherwise they can be found in ref. e )] . For this purpose we may neglect all the channels a, ß etc. except N since the existence of those other channels does not affect the construction of Znrr or T,,r. We therefore take the reduced form of eq . (1.9b) : X~_ (vo+Zxw)+(vo+Zxiv)TYvXNN .

(4 .1)

In order to avoid unnecessary complication we tentatively regard nucleons as distinguishable andspinless. The inclusion ofspin-isospin degrees of freedom may be followed along with refs .9''9). Also since vo is considered to contribute neither two (NN) nor three (~rNN) body unitarity, it can be thrown out of eq. (4.1) as long as the

COUPLED ANN-NN SYSTEMS (In

399

equation is used to determine the above mentioned driving term and propagator . Thus we start with Xxx =Z,,m,+Z,, ,,T,,,X,,r,a .

(4 .1')

We shall work in the c.m . of (i) the two-nucleon system for the NN channel, (ü) the ,rNN system for the three-body channel. Thus for the NN channel we specify the moments concerned as follows: k h kz four-moments for two nucleons , P = kl + kz = (W, O), (total momentum),

s = Wz,

k=2(k l -kz)

(square of the total c.m . energy?, (relative four-momentum).

(a) (b) (c)

(4.2)

Eq. (4.2) clearly implies

Now we can write eq. (4.1') as

kl =2P+k,

(a)

k z = 2P - k.

(b)

(4 .2')

(k'IXnnv(s)Ik)aXrrrr(k', k, s)

(2 d4k"Zxx(k', k", s)Trr(k ", s)XrrN(k", k, s) =Zrrrr(k', k, s)+ ~ J

(4.3)

giving the t-matrix for Ni(ki)+Nz(kz) -i Nl(ki)+Nz(kz)

where the delta function ensuring the overall energy-momentum conservation has been factored out. In order to exploit two and three-body unitarity resulting from eq . (4.3) to determine Z,a,,, and ~,, we need the single-pion production (N+N-" ~r+N+N) amplitude XoN and the amplitude for its time-reversed process (ar+N+NON+N) . In an isobar model of particle production, a production process like 2 ~ 3 is always described as (i) an isobar producing 2 -> 2 process whereone of the final state particles is the isobar, (ü) the propagation of the isobar-particle state and (iü) the isobar decay to give the final three-particle state. Since we have disregarded isobars d, d etc., the only pion production mechanism is the one in which the pion is produced by either one of the two nucleons after their two-body collision, and the process goes just like the steps (i) (iü) mentioned above when an isobar is replaced by nucleon therein. [We need not worry about the possible final state interactions among the ~rNN since they are taken care of by those isobars neglected before we started out eq . (4.1).] Now the t-matrix for the production process (see fig. 11) reads (4i9243~Xorr~ki, kz) = Rt(4i~ 43)S{(4i +93)z}XNx(iP - 4z~ k, s) (4 +Rt(4i, 4s)S{(4z +4s)z}Xrrrr(Ri -iP, k, s) ,

.4)

400

Y. AVISHAI AND T. MIZUTANI

k k~

_

Q

a,

k~ k=

Fig. 11 . The production amplitude XuN(gi4i9s ~ k~, k~ eq. (4 .4). The dota indicate the propagator S.

where S is the propagator ofthe nucleonwhichemitsthe pion. The production vertex Rt is assumed here to be of S-wave type for simplicity (generalization to non S-wave coupling should be easy : see, for example, AAY~ . We note that R t is numerically equal to R :the absorption vertex, thus from now on we drop the "dagger" . With eqs. (4.3) and (4 .4) at hand we shall study two and three-body unitarity (discontinuity) relations in the following. Writing fâf(stiE),

df~,f~-f,

the two and three-body unitarity (discontinuity) relations for Xrrr,(k', k, s) should read dXNN(k', k, s) _ ~~

J

d,flZXN N (k', k", s)Xr,i,, (k", k, s)

+Jd~3(k1k2IXN0IQ1Q2Q3)(Q1Q2Q3IXONIklk2)J .

(4 .5)

with + d,(1Z = (2~r)-4d4 k"S (ki s -m 2 )S+(k2 2 -m 2) ,

(a)

d,(l3=(2~r)-Sd4 Qid4QiS+(Qil- m2)S+(Qi2- ~+ 2)S + (QsZ- N- Z)~ (b) z z S+(a - mz) ~ 9(ao)S(a - m2) ~

(4 .5')

(c)

where m and ~ are the nucleon and the pion masses, respectively. Note that in eq . (4.5)~the overall four momentum conserving delta function has been factored out . On the other hand by directly taking the discontinuity of XNN(k', k, s) in eq . (4.3) we find dXN,(k', k, s) _ (2ar)-4

J

d4k"Xi,N (k', k", s)dn r(k ", s)XNN (k", k, s)+(2-rr)-a

x ~ d4 k"d 4k~Xir, (k', k", x Tr,(k~, s)XrrN (k ~, k, s) .

s)TN (k", s)dZ,.,N(k", k~, s) (4 .6)

COUPLED ~rNN-NN SYSTEMS (II)

401

Now putting eq . (4.4) into eq. (4:5) and comparing with eq . (4.6) we get [with S}(kZ) ~ S(k Z t iE)]

J

d4k"X

, ,a (k , k", s)dTx(k ", s)Xxty (k", k, s)

+ m Z )S +(kzZ- m Z ) = d4k"Xinv(k',k",s)[i(2a)Z S (ki ZJ +i(2zr)3{S +(kz )S + (kiZ - m Z)F(kz)S (kiZ )

J

d4k"d4kmX ,,,(k', k", s)T;,(k ", s)dZ,,,n,( k", k~, s) (4.7)

xTrr(k~, s)Xrra(k M , s)8 4(ki+ki - ki - kz) =

J

d4k"d4k~XNrr(k'k"s)[~(2~r)3{S+(kz

)8+(kiZ-mZ)

xR(k2 , kZ -k2 ) S+( t-Fc Z )R(ki , ki - ki) (kz Z xS + -m Z)S- (kiZ ) +(1H2)}~tvw(k~,k,s)S 4(ki+kz - ki - ki),~

where t =(k"- k^~Z and (1 H2) signifies the term whose form is identical to the first term in the curly bracket except that the indices 1 and 2 are interchanged . The function F in eq . (4 .7a) is defined as F(k ; )

d4 x{R(x, ki-x)}ZS+(xZ-mZ)S + [(k~-x)Z-WZ]

(i =1, 2)

(4.8)

which eventually is a function of invariant k;Z. Note that The integrands of eqs. (4.7a, b) are schematically shown in figs . (12a, b) respectively. O

O

ZN 0

ZN 0

2 Fig. 12 . (a) The integranda on the left (1) and on the right (2) hand aide of eq. (4 .7a). (b) The integranda on the left (1) and on the right (2) hand aide of eq. (4 .7b).

402

Y. AVISHAI AND T . I~IIZUTANI

Any set of Z,anr, T, and S will give us a set of solutions as long as they satisfy eqs. (4.6) and (4.7). In the AAY type of problems (including BSP of NN scattering there is only one term in the curly brackets of eqs. (4.7a) and (4.7b), respectively : the term represented by (1 H 2) is absent there. In such cases it is possible to find the solution by (i) identifying dn,, and dZir,,, with the quantities in the large brackets ofeqs. (4.7a) and (4.7b), respectively, (ü) setting T(k, s) = 2~rS +( k i - mz) S(k2) and (iü) solving for S(kZ) and Z~,v(k', k~, s) through dispersion techniques . In our non-BSP approach the above procedures do not seem so effective. Especially procedure (ü) cannot work because in the curly brackets of eq . (4.7a) appear two different delta functions. However, we can see easily that procedure (i) is sufficient but not necessary: for example, a condition of the type Jb f(x)dx= J 6 8(x)dx a a does not necessarily imply f(x)=g(x). In our present quest we shall adopt one assumption which, together with a somewhat relaxed condition corresponding to procedure (i) mentioned above will enable us to follow closely the rest of the procedures (ü) and (üi) of the AAY type to arrive at a solution for Zr,,, -r,,r and S. When one uses or solves relativistic two-body equations of the Bethe~alpeter type, one quite often adopts a procedure typified by the one due to Blankenbecler and Sugarl8 ). This essentially amounts to eliminating from the t-matrix the explicit dependence on the fourth components of the two-particle relative (four) moments: The result is that the integration in the equation becomes only three dimensional: a practical advantage. However, there is a deeper meaning to this procedure. (The reader is referred to a comprehensive discussion by Freedman et a1.2~.) In our present work we shall make an ansaz that Xrmr depends explicitly only on the spatial part of the two-nucleon relative moments also when it couples to the inelastic rrNN channel: (4 âCivrr(k', k. s) = Xivx(k', kô, k, ko~ s) ~ Xxrv(k~+ k, S) .

.10)

Once this ansatz is adopted, we can change the integration variables : kô-~ - kä, kô -~ -kô ,for the second terms in the curly brackets of eqs. (4.7a) and (4.7b) [terms represented by (1 H2)]. Note that this makes the following change in the variables for these terms ki->kz~(kio~ - ki)~ kz ->ki~(kio. - ki)

and similarly for ki and k2. This change preserved the following relations ki+kz=ki+ki=P (also fork) . (ki)2=(ki)2~ (ki)2=(ki)Z,

COUPLED ~rNN-NN SYSTEMS (In

403

Recalling that F(k~ eventually is a function of invariant k;? we find that eqs. (4.7) can be reduced to the following relations . J

d4

k"Xrrx (k', ~~ s)~{i(2~r)zS+(kiz -mz)S+(kZZ -mz) + i2(2~r) -3S+(ki z )S+(k iz - m z)F(k z)S -(kZ Z )}- d-r,,r(k", s)] x(a) Xrr,,, (~, k, s) = 0 ,

J

d4k"d4kmXrrx(k',k",s)[i(2~r)3S+(kiz)S+(kiz-mz) x{R(ki , ki - kz )R(ki, ki - ki)+R(ki , ki - kz )R(ki~ ki - ki)} x S+(t-p,z)S+(k? z -mz)S-(k ; z ) -dZ,n,(k", k~, s)]X r,a(kM, k, s)

with We notice that in eq. (4 .1 la) our ansatz has made two discontinuities coming from the self energies of two nucleons identical: a consequence which is quite reasonable . Now we put the quantities in the larger bracket ofeqs (4.1 la) and (4.l lb) equal to zero, which like the original AAY paper gives a sufficient (but not necessary) condition for d~r,,, and dZ,,,,. From this point on we can follow almost the same arguments as those in AAY. First we set r,,r (k,s)=2TrS+(ki-m z)S(ki)

( =2~rS+(kZ-mz)S(ki)) .

(4.12)

This reduces the relation for d-r,,, and dZ,,r,,, to dS(k;)=2~riS+(k ;-mz)+2i(2~r)-zS+(k ;)S (k;)F(k,)

with

z k "z i =m ,

(i=1,2),

(a)

(4.12')

k~z z =mz .

These on mass shell conditions together with another one from S+(t - ~, z) of the above equation make Z,,,,(k", k~, s) depend explicitly on spatial vcctors h" and k~ alone .

40 4

Y. AVISHAI AND T . MIZUTANI

Eq. (4.12'b) may be solved by assuming no cut contribution in s from R's and using the following relations (plus the on shell conditions mentioned above) . _ _ 8+(t Wz)

W 1 2~-+t- S(W -Er-E~-wr+r)-~r+~ - s+(s-wz)~

E~~(mz+kz)'/z~

mt~(fLZ+lkz)1/z~

W~Er+Et-, +mr~- .

(4.13) (b)

The result is ZrrN(k ", k~,

s) ~ Zna(k", k ~, s)

In the above expression the appearance of two terms in the curly brackets ensures that Zrnv contains two distinct time orderings, which is missing in the BSP approach to Z,vn,. Kloet et al.e) tried to include the missing time ordering by incorporating its effect in vo which, however, has no contribution to three-body (-rrNl~ unitarity as mentioned in subsect. 2.2. So far we have not specified any concrete form for the ~rrNN vertex R except that we have assumed it to be of the S-wave type of coupling, to simplify our description. We now wish to elaborate on this point a little . For this purpose let us consider the one-pion-exchange interaction of two on mass shell nucleons written in FeynmanDyson picture. This can be decomposed into two contributions coming from two distinct time orderings for the exchanged pion. In the case of constant TrNN vertices one can easily find that those two contributions become identical. In eq . (4.14), do the two contributions (corresponding to différent time orderings) coincide in general even when not all the nucleon legs are on shell? For the case where at least one nucleon (momentum x) and one pion (momentum y) are on the mass shell the vertex R(x, y) may be shown to depend only on the square of the relative momentum : â(x -y)Z . Therefore we can write Then we find immediately that indeed two OPE contributions coming from distinct time orderings in eq . (4.14) are equal. For practical purposes it is more convenient to consider R as a function of the square of the so-called (three-dimensional) special (or magic) vector9) as we shall do below (see the original paper by AAY for details) . We shall only sketch its main features : consider a two-body process â(kl)+ b(k z ) ~ c(ki)+d(k2)

,

COUPLED aNN-NN SYSTEMS (II)

40 5

where the particles a, b, c, d could be off-shell. Then with the total and relative momenta,

we form

K=k,+k z =ki+k2, 4 ~ i(ki - kz) ~

(a) (b)

Q~~i(ki - kz) ~

(c)

w~~,

(d)

P~ 4 - (4'K)KlK z ,

(a)

(4.15)

(4.16)

and (4.17) Then Q(Q') reduces to q(q') and Q ~ Q' to q " q' in the two-particle c.m .s. Note also that Q " Q' _ -P " P' and thus is Lorentz invariant As has been mentioned, we assume then we can see that R(kz , kz - kz )R(ki. ki - ki)+(k~ -> ki . ki" ~ki~)

=2R(ki , ki - kz )R(ki, ki - ki)=2R[{Q(ki ~ kz" -k2 )}zl

due to the relation Q(ki ~ kz -~c2 ) _ -Q(ki . ki-k2 ) Q(ki. ~i - ki ) _ - Q(ki~ ki - ki )

(4.20)

Therefore the vertex R considered as a function of the norm of the special vector also ensuresthe equivalence of twoOPE contributions due to distinct time orderings. Once specifying the momentum dependence of R we can determine S(kz), through eq.-(4.12a). Fast we go back to eq. (4.8) to determine F(k;). Because of the two on-shell delta functions, k; must be time like. Thuswe can choose to evaluate the Lorentz-invariant function F(k;) in the c.m. frame of x and k; -x where k; _

406

Y. AVISHAI AND T. MIZUTANI

(k;o, O). Then the special vector Q(x, k; -x) reduces to x. Thus F(ki)=

J

daxR(xz)ZS +(x 2 -m 2)8 +{(kio - xo)2- z z- WZ} (4.21)

where Ê_~~+E_.

Then combining this with eq . (4.12a) and requiring that S(k2) has a pole at kz = m z with a unit residue we find (cf. AAA S(~) ~ - h(~) -1 ,

(a)

d3xR(z2)2Ê= ~ ~ (b) h(Q)-(~-m2)~ 1- 2(Q-m2) ~ (2Tr) 2E=w=(Q -Q~(~=-m )

(4.22)

where Thus we have found S, Zrrx (and ~r,,r) in terms of the ~rNN vertex function R andwe can now write eq . (4.3) as

1

_d3q 2(Eq+Et+~,,~+,,)R ~ R

Xuv(q, k', s)

(4.23)

with Qq ~(~-E~ 2 -Eq2 +mZ and the arguments in R's have been specified (cf. eq. (4.19)). Note that from the structure of eq. (4.23) we now know that our ansatz (4 .10) has been consistently implemented therein. In realistic situations we should replace the vertex function R(k2, k2 - kz) by the appropriate zrNN vertex which takes the form (4 .24) R -' GX~Q(k"z~ ki - kz)Zl~(k"~ ~Ysu~~~ ~~ rT ' ~a~I M ~

where u(k"', r~ is the nucleon state with momentum 1~ and spin r"', ~~, ~ are nucleon and pion isopin states respectively and G is the ~rNN renormalized strong coupling constant. The vertex function X is normalized to unity when all particles are on their mass shell and should be adjusted toreproduce the low energy ~rN P data (when itis used to calculate the dressed nucleon pole contribution to the P partial wave).

COUPLED ANN-NN SYSTEMS (In

40 7

The nucleon spinor is normalized as This implies that no other modification in eqs. (4.22) and (4 .23) is required. We note that the relativistic vertex in eq . (4.24) also ensures the equivalence of the two distinct time orderings in the OPE driving terms Zrrta. As has been mentioned earlier, contributions from other isobars a, ß, etc. may be incorporated to form the equations ofthe form (1.8) and (1.9). The driving terms and propagators for those isobars can be determined just like we have done so far but we shall notrepeat the procedure here : see ref.' forconcrete forms for various Z's and T's . Also inclusion of the spin-isospin structure should be found there. One remark seems appropriate here before completng this section: when one antisymmetrizes the amplitudes and various driving terms as has been done in sub sect . 4 .2 of I, one may come across with the antisymmetrized driving terms like ZaN(p, q~ s) -ZéN

(4 .24)

(p~ 9~ s)~

where (ex) denotes that the quantum numbers of the two nucleons (including the moments) are to be interchanged . With a straightforward but somewhat involved algebraone can show thatwith our choice ofthe special vector dependence of various vertices, the angular momentum isospin reduced driving terms e.g. Z;, ors (p, q, s) obtain the correct Pauli factor

s

where l, S, I are orbital, spin and iospin angular moments ofthe two-nucleon channel considered . In conclusion, we have arrived at a Lorentz invariant set of equations satisfying exact two- and three-body unitarity which can be solved, just as in the nonrelativistic situation of I, by anumerical procedure for ordinary three bodyproblems. We thank M. Anastasio for his critical reading of the manuscript . One of us (T .M) appreciates the financial support of LLS.N., Belgium during the course of this work. The other, (Y.A .) would like to thank J. Miller, P. Catillon and the DPh-N/HE at Saclay for their hospitality and support. Appendu A PARTICLE IRREDUCIBILITIES

Here we shall explain briefly the concept of particle irreducibilities which may be useful in understanding the structure of scattering amplitudes . For an extensive discussion on this subject see, e.g. Taylor l').

408

Y. AVISHAI AND T. MIZUTANI

Let us consider an amplitude (or t-matrix) with the following specifications: (i) it is totally connected; (ü) the initial and final channels are defined with a definite number of particles. Diagrammatically we write it as a bubble with a definite number of particle legs specified in the initial and final channels as in fig. 13. Then the irreducibüities are defined as follows: an amplitude (t-matrix) is called k-particle irreducible if there exists no cut, separating the initial and final channels in all possible ways, which intersects less than or equal to k-particle lines in intermediate states. A k-particle irreducible amplitude is represented by a vertical line indexed by k intersecting a bubble (see fig. 13). k

Fig. 13 . k-particle irreducible n-m amplitude .

To understand the above definition clearly let us take a 2 -> 2 amplitude "A" resulted from the Lippmann~chwinger (or Bethe-Salpeter) equation with a oneparticle exchange potential. Obviously, this potential is two-particle irreducible (fig, 14). Then the amplitude is easily seen to be one-particle irreducible due to the two-particle propagation between the potential interactions (fig. (15)). Now we are going to study the irreducibility of R1, y, etc., stated in sect . 3 [in between eqs. (3.2) and (3 .4)]. Let us recall eqs. (L2.1-3). First we note that two-body interactions in the original Hamiltonian [eq. (L2.1)], viz. W,~(a =N,N 2, arNl, ~rNz) may be regarded as two-particle irreducible 2 -> 2 amplitudes as they are supposed to arise from e.g. heavy-boson exchanges. After eliminating multiparticle states to arrive at eqs. (L2.2a,b) (eliminating n > 3 particle states) the resulting effective interaction D(E) consists of (i) totally connected three particle irreducible 3-~3 amplitude (regarded as the ~rNN three-body force which we neglected in I), (ü) 2 -> 2 amplitude of two-particle irreducibility plus a spectator for N1NZ, arNl, TrNZ pairs, respectively (L, of eq . (L2.3a) correspond to those for ~rN, (i =1, 2)) and (iü) renormalization of (totally disconnected) free ~rNN propagation. Therefore we may observé that (a) regarding elementary ~rNN vertices A,(i =1, 2) as infinitely irreducible (as they should be in this irreducibility criteria, see Taylorl'), the vertices

Fig. 14 . Two-body potential by one-particle exchange .

COUPLED aNN-NN SYSTEMS (I)7

409

Fig. 15 . Example of one-particle irreducible 2 -~ 2 amplitude.

R;(i =1, 2) through eq . (L2 .3a) may be identified as two-particle irreducible and (b) 2-~ 2 potentials VQ (a = zrNl, ~rrNZ, N,NZ) [eq. (L2.4)] which resulted from a further deeoupling of eq . (L2.3) are two-particle irreducible and therefore the resulting tQ are all one-particle irreducible. Consequently, y; and z, (i =1, 2) of eq. (3 .3) are (disregarding the spectator particle) one-particle irreducible ~rNN vertices (see fig. t (16) . Incidentally, the total arN Pll amplitude ;a°`~(i =1, 2) can be easily seen to be one-particle reducible due to the existence of the nucleon pole (eq. (3 .6)) .

Yi

z

z

Ri

Ri

~

z

~ _ z ,:

ti

Ri

Ri

~ai

Fig. 16 . Equation for y, [eq. (3 .3a) in an explicit irreducibility decomposition.

With this concept of particle irreducibility and what is termed as complete unitarity, Taylor t') was able to derive e.g. Bethe-Salpeter equations for 2- and 3-particle systems with the possibility that one of the particles can be absorbed . This method was foundZ') to be quite useful in constructing ~rNN scattering amplitudes [like our Uaß, UN etc. a s in eqs. (1 .4 .5)] without overcounting pion exchange effects. A method adopted by Thomas and Rinatt2) (a graph summation method) to derive coupled equations for inelastic NN scattering is close in spirit to Taylor's approach but is less exacting in that without reooursing to complete unitarity it seems impossible to derive self reproducing equations by irreducibility arguments alone . Appendix B THE TWO-NUCLEON PROPAGATOR IN THE PRESENCE CONTRIBUTION TO THE zrN Pl1 CHANNEL.

OF

THE NON-POLE

In this appendix we shall discuss the physical (renormalized) two nucleon propagator in the presence of the aN NP Pt, interaction. The case in which this interaction is absent has been discussed in the appendix of I. In our present study we shall follow closely the method adopted there with some minor changes. We assume that (1) Hamiltonians ho and Ho contain rest mass contributions, (2) non-relativistic

41 0

Y. AVISHAI AND T. MIZUTANI

kinematics is adopted throughout . Then with eqs. (3.21d), (3.4) and (3.3) we have (in the two-nucleon c.m .s.) 2 rd3gR(q)Y(9+E - E - ma) (2~r) E-E-q /2~ -m q~R(9')ta (9', 9, E) ~ 1 s ~ d3 Y(9~ E) =R(9)+ (2~r) E -4 /2W

(c)

where p = relative momentum of the (off-shell) two nucleons E ~ 2mx +-,

mx

x

Ê---2mx+ 2mx

z

+ 2(mN ma)'

t,~ _ ~rN NP-P t-matrix (eq. (3.2))

m = pion mass , E,i. -1 -mNi +m41 .

mx = nucleon mass,

Smx = nucleon mass counter term,

In eq. (B. lb) we have explicitly included the nucleon mass counter term . Then the first step in the renormalization procedure is that this mass counter term be fixed so that for on-shell nucleons: E = Êthe self-energy contribution Ax vanishes . As in I (statement following eq . (LA.6) when setting Ê = É to correct for the violation of Galilean invariance we easily observe that Smx is expressed independent ofp and E, which when inserted back in eq . (B.2.b) gives a "once subtracted" form for Ax. Yet, in ourpresent study we want to have a neater representation for Ax using its analytic properties. Let us take the discontinuity of Ax: ~x(P~ E) °Ax(P+ E + ) - Ax(P+ E )

d34S(E-E-4z/2E,,-ms)~Y(9+E-E-m,.)~Z~ = -~ ( J where we have utilized eq. (3 .3), the reality condition : and the result of the hermitian analyticity: (see eqs. (3 .3)) . Recall that we have set Ê = B. From the condition

R=R',

y=z

Ax(p, E)=~

(B.2)

COUPLED ANN-NN SYSTEMS (II)

.411

mentioned above and eq. (B .2), we can see that Ax(p, E)

is an analytic function in E with a cut along the positive real axis starting at E _ É + m.,. So we can disperse hx(p, E) [eq. (B .3)] to get Ax(P~ E) =2(~

J

dZgly(q~ 4Z/2li)~Z

(B .4)

Notice that the spectral function p ~ ~y(q, q Z/2N,)~ Z has no dependence on E. As in the appendix of I the next step is to have a vertex renormalization to ensure a unitresidue of azz at E = É. The procedure to be taken here is identical to the one in I (appendix), thus we shall not repeat it here again except that we should note that here z X~xz~ (i) R Z (q) in I (appendix) has been replaced by ~y(q, q / 2l~)~Z , (ü) quantities renormalized twice in I (appendix) now read X~`~ . We then write the final result : arzz'(P, E) -1 = (E-E)-AN~ (P. E) , 2(B - E)z ~ d3 9~Y~~(q, 42/2~)~Z cR~ Ax (P~E) _ s (E-É-q (2 ,rr) l2E,,-m~)(4 l2N-+ir+  ) (AN'(P, E) _ ~) ,

1+AN~Trzï >

(a)

(b)

(c) wN~ = (wiv~(P. E) =1) ~ y cR~(q, (d) E) = f y (q, E) (similarly for z) , 2 r= 1 + -~ ~ d 3 gly(q~ 9 2/2~)~~ -1 (2Tr) (4 /2~+m ) (2or)

(4 /2W +m4 )

(e)

(B .5)

Furthermore the following quantities appearing in sects. 1, 2.2 and 3 are to be understood as renormalized [carrying implicitly the superfix (R)] (i) Rx -~ RN~ = fR x and similarly for RN ,

(u)

IX2) ~ IÀ~2 ~ )' `~~X2)

(iü) vo-i vô) $ rüo . The present results obviously reduce to the ones in the appendix of I when tQ (or va) 13 Set equ81 t0 ZerO. References 1) Y. Aviahai and T. Mizutani, Nucl. Phys . A352 (1979) 326 2) T. Mizutani and D. S. Koltun, Ann. of Phys. 109 (1977) 1

41 2 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)

17) 18) 19) 20) 21)

Y. AVISHAI AND T. MIZUTANI I. R. Afnan and A. W. Thomas, Phys . Rev. C10 (1974) 109 E. O. Alt, P. Grassberger and W. Sandhas, Nucl. Phys . B2 (1967) 167 V. S. Varna, Phys. Rev. 163 (1967) 1682 L. R. Dodd, Phys . Rev. C18 (1978) 2796 M. G. Fuda, Phys. Rev. C12 (1975) 2097 W. M. Kloet, R. R Silbar, R Aaron and R. D. Amado, Phys. Rev. Lett . 39 (1977) 1643 ; W. M. Iücet and R. R. Silbar, Los Alamos, preprint (1979) R. Aaron, R. D. Amado and-J. E. Young, Phys. Rev. 174 (1968) 2022 ; R. Aaron, in Modern three-hadron physics, ed. A. W. Thomas, (Springer, Berlin, 1975 .) A. M. Green et al ., J. of Phys . G4 (1978) 1055 A. W. Thomas, in Theoretical methods in medium energy and heavy ion physics (Plenum, NY,1978) A. W. Thomas and A. S. Rinat, Phys . Rev. C20 (1979) 216 M. Stingl and A. Stelbovica, Nud. Phys . A294, (1978) 391 D. Lurie, Particle and fields (Wiley, NY, 1968) p. 303 J. M. Levy-Leblond, Comet. Math. Phys . 4 (1967) 157 D. S. Koltun, Adv. in Nucl . Phys., vol. 3 (1969) 71 ; J. M. Eisenberg, in Proc . LAMPF Summer School on theTheory of pion-nucleus scattering, ed . W. R. Gibba, Los Alamos Conf. Pros . LA-5443-C (1973) ; M. V. Barnhill III, Nud. Phys. A131 (1969) 106 J. G. Taylor, Nuovo Cim. Suppl. 1(1963) 857; Phys. Rev. 150 (l96ä) 1321 R. Blankenbeder and R. Sugar, Phys . Rev. 142 (1966) 1051 R. M. Woloshyn, E. J. Moniz and R. Aaron, Phys. Rev. C12 (1975) 909; A. S. Rinat and A. W. Thomas, Nud. Phys . A282 (1977) 965; V. B. MandeLsweig, H. Gardhuo and J. M. Eisenberg, Nucl. Phys. A2S6 (1976) 163 D. Freedman, C. Lovelace and J. M. Namyslowski, Nuovo Cim. 43A (1968) 258 T. Mizutani, Ph .D. Thesis, University of Rochester 1976, u~ublished