The U (6,6) ⊗ O (3,1) peripheral absorption model and the production of 2+ and 1+ mesons

Nuclear Physics B43 (1972) 253-271. North-Holland Publishing Company

THE U (6,6)® O (3,1) PERIPHERAL MODEL AND THE PRODUCTION

ABSORPTION

OF 2 + AND 1+ MESONS

T.K. GUJADHUR Dept. o f Mathematics, Imperial College, London SW7 R. MIGNERON * and K.J.M. MOR1ARTY Dept. o f Theoretical Physics, Imperial College, London SW 7 Received 1 November 1971 Abstract: The peripheral model incorporating U (6,6)@O(3,1) couplings is applied to the production of 2+ and l + mesons in pion-nucleon and kaon-nucleon collisions at intermediate energies. The agreement with experiment is good where pion exchange is the dominant mechanism and fails for the other reactions.

1. INTRODUCTION It has been found that forward high energy two-body collisions can be explained by the exchange of a few poles in the t-channel [1,2]. However, there is usually considerable freedom of manoeuvre in these peripheral calculations, which arises from three main sources: (i) there is often a wide variety of particles to be considered as intermediary; (ii) there are often alternative couplings possible at the vertices; (iii) in many cases the values of the coupling constants may be chosen freely. Previous calculations on the production of high spin mesons have always been plagued by one or all o f these features, making comparison with experiment obscure. Rather than use this arbitrariness to introduce parameters which could then be varied to fit the experimental data, it seems preferable to use some higher symmetry scheme in which the Lagrangian and couplings are uniquely determined. We choose U(6,6) symmetry [3], imposing unitarity requirements through the absorption model. Then the relative contribution o f pseudoscalar and vector exchanges, the F/D ratio, the electric and magnetic contributions to the relevant vector currents are uniquely prescribed. Previous calculations [ 1] have only involved ( 3 5 - ) mesons and (56 +) baryons. In this paper we are interested in the production of 2 + and 1+ mesons. We are here faced with the choice o f either using U(6,6) [3] (i.e. Global Symmetry) or U(6,6)@ * Present address: Dept. of Applied Mathematics, University of Western Ontario, London, Ontario, Canada.

254

T.K. Gujadhur et aL, U(6,6)@ 0(3,1) model (I)

O(3,1) [4, 5] (i.e. the kinetic supermultiplet scheme) as our symmetry group *. The fact most well established resonances are naturally classified in octets, nonets and decuplets [5] and that exotic particles do not seem to exist makes the latter scheme a natural choice. It should be noted that the Lagrangians in the kinetic supermultiplet scheme are much simpler than those involving the higher representations of U(6,6). We consider the following reactions, where the exchange particles are indicated in brackets: 2 +production

rr-p Or) f°n , rr+n Or) fOp , rr+p (rr) f°N*44 , rr+p (r/,p) A2+P , K - p (n, r/, p, ¢, co) K*-(1420)p , K - p 0r,p) K*°(1420)n .

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

1 +production

7r+p (p) a~ p , (g) K - p (p,co) K*-(1320)p . (h) In sect. 2 the derivation of the production amplitudes in the U(6,6) @O(3,1) symmetry approach is presented. Absorption effects are considered in sect. 3. Comparison of the model with experiment is given in sect. 4.

2. THE U(6,6)@ O(3,1) INTERACTION We wish to consider the production of 2 + and 1+ mesons (see fig. 1). The U(6,6) @(3,1) interaction Lagrangian corresponding to the upper vertex of fig. 1 is given by

(2.1)

mA

/s

where G1, the universal U(6,6)@ O(3,1) (2+-meson-meson) coupling constant, is discussed in appendix A, m A is the average U(6,6)@ O(3,1) meson mass embedding the (O-), ( 1 - ) , (1 +) and (2 +) mesons and q is the momentum transfer. Eq. (2.1) may be written formally as (2.2)

L = G 1 (J5 ¢5 + J# (bin) ' A

For instance a 2+ meson is described in the kinetic supermultiplet scheme by q ~ with one constraint equation corresponding to the (35,.~1; L=I)_,~_SU(6)multiplet. In the glbUbalsymmetry scheme it can be described by either q ~ or 4~{~} corresponding to 4212 and 5940 multiplets of U(6,6) - the corresponding SU(6) representations are 189 and 405. Needless to say the algebra for the Yukawa type interactions is much simpler in the kinetic supermultiplet scheme.

T.K. Gujadhuret al., U(6,6)@0(3,1) model (I) I~, - - -

_9_o-_ - , _ _ _

255

2-~ ]*

..@

'- - )----

1~3

I i ) ) I

q,A

o;1-

I I I

142 •

[D2

) @

1" 3+ 2, 2 >

.~)

Fig. 1. Peripheral production of resonances. where q55 and q5u are the pseudoscalar and vector fields. Higher spin exchange other than the O - and 1 - will not be considered [7]. The U ( 6 , 6 ) @ O(3,1) predictions for those parts of the pseudoscalar and vector currents relevant to the interaction of the 2 + meson nonet with the O - and 1 mesons are (omitting SU(3) indices) 4G l [

m2

JS=~SAS[2+s-)PluPl7

(2.3)

(~,(P3) ~ (Pl)) D

4G 1

J-

euwflP3~Plp

(~-vo(P3) ~b ( p l ) )

m A$2

(2.4) F

F o r the p r o d u c t i o n of 1+ mesons the relevant current is G1

J~= mA A# (~-x(p3) q5 ( p l ) )

Plx

'

F where

4 ((Sml_SV_S 2 + 2 P l . p3)P3u Au- mlSV + ( S m l - V m l - m ~ + 2p 1 • p 3 ) P l u )

•

(2.5)

In eqs. (2.3) - (2.5), S is the pseudoscalar meson mass, V is the vector meson mass, m I is the (1 +) meson mass and m 2 is the (2+) meson mass. Their numerical values are obtained by taking the average mass of the respective nonets: S = 417 MeV, V = 850 MeV, m 1 = 1244 MeV, m 2 -- 1375 MeV, and rn A -- 975 MeV. The corresponding expressions for the b a r y o n - b a r y o n - m e s o n vertex of fig. 1 are

256

T.K. Gujadhur et al., U(6,6)@0(3,1) model (I)

given by 2mo\ p2 J5= G 2 ( ( 1 + ~ - ) 4 m 2 ( N i 7 5 A V ) D + Z 3 F _ S O

1 ( m °-+ m oS]/ - " - ") +-- 1 + -IP2xDx(P4)N(P2)

(2.6)

m o

3S +

+

(1 2rn° \ [ ru +-~V N ) D + ~sF-S . V ) [ 4m 2

'tm°+mo

2morn D 1 + ~

O

1

euvpopvqoDaN

(2.7)

,

where P = P2 + P4" N is the (½+) baryon of mass m o, D is the (3/2 +) baryon of mass m D. As for the case of the mesons, m o and rnD represent the average of (½+) and (3/2 +) and are: rno= 1100 MeV, and roD= 1385 MeV. P u and r u are the conventional forms for "electric" and "magnetic" interactions and the coefficients ofpu/2m o and rJ4mo2 are the conventional Sachs [8] form factors F c and F M respectively. For each of these calculations we write the nucleon-nucleon vector current as P~/,

--

ju= G2 (M 1 (q2)~m__m(NN) + M2(q2 ) (N v/a N)) ,

(2.8)

O

where the relations between M1,M 2 and FC, F M are given by

Ml(q2 ) = F c F M,

M2(q2 ) = ( 1 \

q2 ) F M . 4m 2

(2.9)

O

The matrix element for the diagram of fig. 1 with exchange of pseudoscalar and vector particles is given by

T(s,t) = ~ J51 t 2 JsII+ 23 J u i t-la i j

+ t-u/2

v'

(2.10)

the summation over i and / represents the number of allowed exchanges of 0- and

T.K. Gujadhuret al., U{6,6)@0(3,1) model (I)

257

1 - particles respectively. The masses appearing in the propagators are the physical masses of the exchanged particles; s and t are the usual Mandelstam variables. The relevant 3-meson coupling for each case is determined from the A 2 ~ On decay (see appendix A).

3. ABSORPTION EFFECTS As in the previous work the effect of competing inelastic channels is represented by elastic scattering in the initial and final states. We adopt the original Sopkovich [2] formula

where S ~ a and sJfl~ are the elastic S-matrix elements in the initial and final states respectively. This formula is different from the one derived by Watson [9] in the relativistics case and by Durand and Chiu [10] in potential scattering. Their prescription is given by

TJ~ = ½ (SI~ TI~ + T ~ S ~ )

.

(3.2)

The two formulae become identical when the initial state and final state elastic scattering parameters are set equal; we shall make this assumption in the present work as it is impossible to determine final state elastic scattering parameters. However, for the sake of completeness, we shall take, in some of the reactions, v final to be both less and greater than v initial to show how this affects the results. The unitarity correcting factor is taken directly from experiments on ~N and KN scattering, fitted to a Gaussian form (purely real). S / a = 1-Ca(s ) exp

( -j (j + 1)/uaZ (s) p2 ) .

(3.3)

The radius u(s) and the opacity factor Ca(s) are determined from the observed exponential slope of (do/dt)acc As Ca(s) ~ 1 experimentally we shall make the not unreasonable assumption of taking it to be equal to unity throughout. Then for small / this will have the effect of eliminating from the Born term the large contributions coming from the low partial waves, which certainly do not represent the physical situation correctly for a small impact parameter. The amplitudes qSi(s, cos 0) are decomposed in the helicity representation of Jacob and Wick [ 11 ]:

4~i(s, cos 0) = ~ J

(2/" + 1)

T/(s) d / u (0) ,

(3.4)

T.K. Gujadhur et al., U{6,6)@0(3,1) model (I)

258

where X = )k1 -~t2,/1 = ~k3 -~-4 and i labels the various helicity combinations. Making use of the orthogonality of the d/xu(O ) functions 1 •,

,t

2

f dJxu(0)d~(0)d(cos0)-2j+l -1

8/]' '

(3.5)

we obtain the partial wave helicity amplitudes 1

f

c~i(s, cos O) d ~ (0) d (cos O) .

(3.6)

Once this partial wave amplitude has been modified according to the prescription (3.2), the unitarised scattering amplitudes can be written in the form gS~(s,t) = ~ (2j + 1) 7"ii' (s) d/xn (0) . Y

(3.7)

The differential cross section is given by do

d.-q

7r

{b,i

i

12

,

(3.8)

where k and q are the initial and final state centre of mass momenta; the summation is over all the helicity amplitudes. We give in appendix B a brief summary of the calculations.

4. COMPARISON WITH EXPERIMENT AND CONCLUSION Some ambiguity arises in the choice of the masses in U(6,6)@ O(3,1). This is resolved by taking ad hoc the masses to correspond to the average of the respective multiplets. The exchanged particles have been assigned at a position corresponding to their physical mass. We have further assumed the parametrization of the final state elastic scattering to be identical to that of the initial state. However, for the sake of completeness we have taken u final to be less and greater than u initial in reactions (a) and (c), to see its effects on the results. We would like to emphasize that we are not treating this as a parameter but only trying to determine the sensitivity of the results to this number. The results of the calculations for the differential cross sections and the density matrices are shown in Figs. 2 - 1 6 .

"..}

......

oz. - t [(GeVlc)21

WITHOUT ABSORBTION WITH ABSORPTION

o 8

Fig. 2. D i f f e r e n t i a ] cross section f o r n - p ~ ton at 4.0 G e V / c . D a t a f r o m ref. [ 1 7 ] .

oo

I0"t

E

101

oo

•

"n'+'n~ ......

J

J

I

o4 -t ~oev/c) 2]

I °~ AT 6Ù GeW¢ Wl rHOUT ABSORPTION WITH ABSORPTION

J

i

L

1

O8

1

Fig. 3. Differential cross section for 7r+n ~ fOp at 6.0 GeV/c. Data from ref. [ 17 ].

~0~

10-I

~Ls

~6

10!

®

-

00

-

I

I

~-r.

I

- t[( GeV/c)']

04

[

~ r - J 3 ~ f°n AT 80 OeV/¢ -WITHOUT ABSORPTION "~,-1.'~(1 • 0.270 Gcv, ..... "~-l, O.400 Gev O. '100 Gev

l

I

[

08

/

J

Fig. 4. Differential cross section for 7r-p ~ f°n at 8.0 GeV/c. The radius of interaction in the final state is taken to be less, equal and greater than that of the initial one. Data from ref. [18].

~d

10"1]

T

101

J

\

~

I

[

0/, t [(GeVk)' 1

I

1

I

I

08

.I/

Fig. 5. Differential cross section for ~+p ~ fo N*-H- at 8.0 GeV/c. The radius of interaction in the final state is taken to be less, equal and greater than that of the initial state. Data from ref. [ 1 8].

10 .2

O

>~

lCf

X\

~

Gev

V;l=

......

0.100

~;'~= 0.400 Gev ~(' ='~;'= 0 270 6ev.

°NI**AT 8'0 OeV/c WITHOUT ABSORPTION

.....

101 - ~ b ~ --,--


®

4

t,~ C~

oo

,

,

,

_t[(GeV/~)2]

o,

(

,

DENSITY MATRIXELEMENTS FOR n ' ~ 1 * ~

,

,

o8

I

AT 80 OeVIc

Fig. 6. (a) Density matrix elements of the fo resonance in ~r-p --* tOn at 8.0 GeV/c.

-o :

-0.2

- 0.1

00

01

02

03

,

t,-I

,

I

ol,

- t [,G~V/c) 2]

,

,

,

,

L

08

Fig. 6. (b) Density matrix elements of the fo resonance in n - p -* ton at 8.0 GeV/c.

o0

-0/.

-0.3

-02

-01

0'I

02

03

04

tO

Cb

®

t

I

I

-t [(GeV/c )2]

0~

I

TT+p ~ A 2 P A T S 0 GeV/¢ ...... "q EXCHANGE ----D EXCHANGE x l 0 "~ .... q-*~ EXCHANGE xl0 -I

I

t

1

08

Fig. 7. Differential cross section for n+p --, A2P at 8.0 GeV/c. Data from ref. [18].

00

10-2

I0-~

E.

to

~a

io'

IO~:

00

- - - -

...... ......

~

,

t

04 - t [(GeV/¢) 2]

..~

.

I

>

~

t

l

---~

rr EXCHANGE 0 EXCHANGE x l 0 -I TOTAL WITHOUT ABSOPBTION xlO "2 TOTAL WITH ABSORPTIONxl0 -I

08

I

' ~ - - - ~~

Fig. 8. D i f f e r e n t i a ] cross section f o r K - p ~ K * ° ( 1 4 2 0 ) n at 4.1 OeV/¢. D a t a f r o m ref. [ 19].

l(J

10-~

,E

to

101

®

2

b~

.....

i

----9

. . . . . .

~

.

,

I

04 -t [(GeVIc) 2]

i

~,.~,

,

,

" It

"""---. t

r

$2.'7"-2~ -7

~

"n" EXCHANGE EXCHANGExl0 -1 n+ 9 WITHOUT ABSORPTIONxl0-~.~TI ÷ @ WITH

,

l

08

t

Fig. 9. Differential cross section for K p ~ K *° (1420)n at 5.5 GeV/c. Data from ref. [6].

o.o

10 .3

o

10"1

10~

0'0

I

J

I 04

-t [(GWIc) 2]

L

I

,

, 08

Fig. ]0. Differentia! cross section for K - p ~ K *° (1420)n at 10.0 OeV/c. Data from ret'. [19].

lC

10:

T

10"1

K-P ~ K1°(1420)1"~AT 100 GeV/c ...... •n" EXCHANGE --.--@ EXCHANGEx10 "1 - - ' n + 9 W THOUT ABSORPTION xl0 "2 --,~-~- --st + @ WITH ABSORPTION x 10 -1

®

o,

J

I

I o/-t [(c~zv/c)z]

L

I

I

n + ' q EXCHANGE t @ +~+,,.~ EXCHANGExl0TOTAL WITHOUT ABSORPTION x 10-2 TOTAL WITH ABSORPTION x 10-1

t

L o8

Fig. 11. Differential cross section for K - p ~ K * - (1420)p at 5.5 GeV/c. Data from ref. [19].

10-21 oo

10-I

E

.y

10 0

- -

.........

E

1 0 1F- . . . . . . . -

-

i

K-b ~ K ........

I

I

I

04 - t [(OeVlc)2]

I

= - (1420) P AT 100 GcV/c "n ~.1"1. EXCHANGE - - T O T A L xl0 -1 ~)÷W,,~ EXCHANGE x l 0 -1

J

I ,

I

08

Fig. 12. Differential cross section for K - p ~ K * - (1420)p at 10.0 GeV/c. Data from ref. [6].

oo

1~ 3

10-2

~E,

LD

r~

10-t

lo'

®

c~

1",9

T.K. Gu/adhur et al., U(6,6) ~ 0 { 3 , 1 ) model (I)

-

-

PSEUDOSCALAR

. . . . . .

VECTOR

CONTRIBUTIONS

CONTRIBUTIONS

~

02

~ 2.2 . . . . . . . . . . . .

/

0[

01 ,

oo

,

,

[

,

o4 -t [(GeV/c)2]

,

,

L

08

Fig. 13. The density matrix elements P 1,1, P2,2 and Re P2,1 for the K*- resonance in K-p K*-(1420)p at 10.0 GeV/c.

0.3

DENSITY MATRIX ELEMENTS FOR K-p ~ Kl°(1/,20)'n AT 10 0 OeV/c PSEUDOSCALAR CONTRIBUTION ..... VECTOR CONTRI B U ~

i . . _ 0,2

01

Z___J'---<-- ----- ...........

O0

04

0,8

t [(G~v/c )2] Fig. 14. The density matrix elements P29, Re O21 and Re o 2 l for K*° resonance in K-p --* K*° ~420)n ~t 10.0 GeVI~.

265

oo

/

,

/

/

/

,

/

I

/

/

04 - t ~(oeV/c )' l

[

11 /

}t

//I

/

' t " l " + p ~ A i p AT 8 0 GeV/c WITH ABSORPTION xl0 "2 -:" . . . . WITHOUT ABSORPTION x 104

O6

Fig. 15. Differential cross section for n+p ~ AlP at 8.0 GeV/c. Data from ref. [ 18 ].

16

10 "I

E

t~

I0 c

10

oo

I

,

,

I

04 -t[(GeV/¢) 2]

,

K-t(1320)p AT 10'0 GeV/c D EXCHANGE x10 .2 EXCHANGE x10 -2 9 + ~ EXCHANGE xl0 "2

t Fig. 16. D i f f e r e n t i a l cross section f o r K - p ~ K * G e V / c . D a t a f r o m ref. [6].

10 -2

I0"I

~o

>

lo°

...... ----~ - -

f011 K-p ~

]

08

( 1 3 2 0 ) p at 10.0

i

{

S

®

2

c~ 4.

T.K. Gujadhur et al., U(6,6) ~ 0(3,1) model (I)

267

For those reactions where the O - is the dominant exchange mechanism the agree. ment with experiment is good, as shown in Figs. 2 - 1 6 , both for the magnitude of the differential cross section and for small t values (O ~< -- t ~< 0.6 (GeV/c)2). We would like to emphasize that the model has no arbitrary normalization factors and no arbitrary parameters. For the other reactions where either O - and 1 - or just 1 - are the exchanged particles, the results are in disagreement with experiment. The contributions from the O - particle(s) alone fit the data reasonably well whereas those from the vector exchanges are one to two orders o f magnitude too large, although the t dependence improves with increasing m o m e n t u m . This is in contrast to the results of the earlier U(6,6) applications [1 ] where the vector contributions were smaller than the pseudoscalar ones (at intermediate energies), thus leading to a good agreement with experiment for a wide variety of processes. This particular feature may be due to the presence of multiple derivative couplings at the meson vertex. Were the vector contributions smaller than the pseudoscalar ones at intermediate energies, the model, although successful for O - exchange, would still be met with the breakdown of the vector exchange mechanism at high energies. We would like to thank Professor P.T. Matthews, Drs. S. Hochberg and R. Delbourgo for useful discussions.

APPENDIX A. COUPLING CONSTANTS. The universal (2 +) MM coupling constant G 1 is obtained from the observed decay width for A 2 -+prr [12]:

p=

G2 A2PTr 1 Q5 4rr 10 M 4 A2

42(X/2)2 G2 _ G 2A2Prr 4rr 4rr

(A.1)

(A.2) '

where Q is the three-momentum of the vector meson in the rest frame o f the (2 +) meson. MA2 is the physical mass o f the A 2 and P is taken to be equal to 88 MeV. Assuming a mixing angle of 10 °, 40 ° and 30 ° respectively for the O - , 1 - and 2 ÷, the various 3-meson couplings of interest are given in terms of the basic one as follows:

T.K. Gu]adhur et aL, U(6,6)(~0(3,1) model (I)

268 (a) 2 +Production."

g n - f on +

=~

--~] G 1 '

g

_+ nA27r

=0.955G 1 ,

gn+K,OK - = G 1

1

gp+K,OK - = G 1 , g

g

glroK*+K- = V ~ G1 '

-1

~K*+K - - x/2 a , ,

q~K* +K -

=G 1 ,

1

gooK,+K - = - - ~

G1 •

(b) 1 +Production." g OK,+K -

--X/~ G l,

g

oOK*+K -

=-- G 1

•

APPENDIX B In this appendix we give a brief account of the calculations. Following Salam et al., [3], the wave function ¢ABu(P3) may be decomposed in the following way:

~ A ( p ) = (S / (p) + ~ , %i , (p) + ½ % r ~i . (p)

i (p)+ 75 q~u5 i (p))~ ( tT'a ) b +(i'Yx~/5)Axu

(B.1)

which, by means o f the Bargmann-Wigner [13] equations, can be simplified to

The first and second term describe a 2 + and 1+ particle respectively. The parity P o f such a system is given by P = ( _ ) L × (parity of q~) where L is the angular m o m e n t u m and q (q) describes the quark (antiquark) [14]. (a) The pseudoscalar amplitude for the single resonance production of a 2 + particle is then given by

T.K. Gufadhur et al., U(6,6)@0(3,1) model (I)

269

m2 4) (s,t) - ~4G1G2 mA (2 +-~-)ph, phu (~vu (P3) O (Pl))D

X

2too\ p2 1 1 +-- - (IV(P4)i75 (p2)) t__l~2 S ) 4m 2 D + ~F-S

(B.3)

Similarly the vector exchange amplitude is -4G1G 2 (s, t) -

eux~oP3a Pl# Plo (~-xo (P3) ~b(pl))F ( --guv+tquqv/m2m2e')

- -

mA S 2 Pv

X(MI(q2) 2-m AT(P4) N (p2) + M2(q2 ) N(P4) 7 v N (p2))

(B.4)

O

where Ml(q2 ) and M2(q2 ) have been defined in eq. (2.9) (b) The pseudoscalar amplitude for the production of the double resonance (e.g. 7r+p Or) fo N*++)

~(s,t)=

4G1G~2 9 MA S (2+ ) plvPlu (~uv (p3) O (pl))D

2rook l 1 1 + - - ~ ) P 2 x /gh (P4) g (p2) t_m2 mo

(B.5)

(c) The amplitude for the l + production (e.g. 7r+p (p) Al+P) is given by ¢) (s, t) -

[ -guy + ququlm 2 \

G1G 2 mA

X(MI(q2) ~

P /)

N(P4) N (p2) + M2(q2 )/V(P4) 7v N (p2)) ,

O

where P = P2 + P4 and A u is defined by eq. (2.5).

(B.6)

T.K. Gujadhur et al., U(6,6) ~ 0(3,1) model (I)

270

The various amplitudes are d e c o m p o s e d in the helicity representation of Jacob and Wick [11] as

c~i(s,t) = ( ~.3~4 I T I ~1~2 ) where the X's refer to the helicity of the four particles and i is an helicity label. F o r simplicity the kinematics is chosen as P l = ( E l ' K sin 0, O, K cos 0) ,

P3 = ( E 3 ' O, O, q)

P2 = ( E 2 ' - K

P4 = ( E 4 ' O, O, - q )

s i n 0 , O, K

cos0)

,

The construction of high spin wave functions, required for the evaluation of the helicity amplitudes, has been widely described in the literature [ 15 ]. The partial wave projection follows the m e t h o d of Migeron and Watson [1 ] and the differential cross section is obtained from eqs. (3.7) - (3.8). Finally, the density m a t r i x elements for 2 + -+ 0 - 0 - and 2 + ~ 1 - 0 - are [16]

1 ~ < ~3X4 { T I X1X2 > < X3' X4i T I Xl~2 ) * Ph3h3' = N hlh2h4 where N = ~

IqSil2, ( s u m m a t i o n over all i's). i

REFERENCES [1] H.D.D. Watson and R. Migneron, Phys. Letters 19 (1965) 424; R. Migneron and H.D.D. Watson, Phys. Rev. 166 (1968) 1654; R. Migneron and K. Moriarty, Phys. Rev. Letters 18 (1967) 978; D. Fincham et al, Nuovo Cimento 57 (1968) 588; H.D.D. Watson et al, Nuovo Cimento 62 (1969) 127; D. Fincham et al, Nucl. Phys. B13 (1969) 161. [2] N.J. Sopkovich, Nuovo Cimento 39 (1962) 186; J.D. Jackson, Rev. Mod. Phys. 37 (1965) 484; J.D. Jackson et al., Phys. Rev. 139 (1965) 428; these two papers contain an exhaustive list of references to the absorptive model. [3] A. Salam, R. Delbourgo and J. Strathdee, Proc. Roy. Soc. 284A (1965) 146; 285A (1965) 312; M.A. Beg and A. Pais, Phys. Rev. Letters 14 (1965) 267; B. Sakita and K.C. Wali, Phys. Rev. Letters 14 (1965)404; Phys. Rev. 139 (1965)B1355; R. Delbourgo et al, Seminar on high energy physics and elementary particles, Trieste (International Atomic Energy Agency, Vienna, 1965), p. 455; Phys. Letters 15 (1965) 184.

TK. Gufadhur et aL, U{6,6) @ 0{3,1) model (I)

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