Acta Physica Academiae Scientiarum Hungaricae, Tomus 26 (1--2), pp. 35--46 (1969)

ANALYTICITY, FACTORIZATION, A N D LORENTZ SYMMETRY*

By

P. DI VECCHIA CALIFORNIA INSTITUTE OF TECHNOLOGY, PASADENA, CALIFORN1A, USA

AND LABORATORI NAZIONALI DEL CNEN, FRASCATI (ROMA), ITALY+

a n d

F. DRAGO** CALIFORNIA INSTITUTE OF TECRNOLOGY, PASADENA, CALIFORNIA, USA

The properties of the scattering amplitude near t ~ 0 have been studied on the basis of analyticity, factorization and Lorentz symmetry.

During the last few years, the propert ies of the scat ter ing ampl i tude ncar t = 0 have been extensively studied in the f ramework of the Regge pole model. The origin of the interest to this par t icu la r point , which at high energy in the general mass configurat ion, is ve ry close to the physical region, has to be found in the existence at t -~ 0 of two kinds of complications. In fact, the ana ly t ic i ty propert ies of the scat ter ing ampl i tude and the crossing s y m m e t r y require the existence of some kinematical constraints between different heli- c i ty ampli tudes. These constraints in the f ramework of the Regge pole model g i re rise to the concept of " consp i r acy" be tween different Regge trajectories .

Addit ional interest in the point t ~ 0 arises in connect ion with the Regge expansion of the scat ter ing ampl i tude, which is singular at t ---- 0 in the u n e q u a l - u n e q u a l (UU) and e q u a l - - u n e q u a l (EU) mass configurat ion. The concepts of daughte r t r a j ec to ry and of Regge pole family come into p lay in order to avoid these unwan ted singularities and to restore the pure Regge behavior .

The propert ies of the scat ter ing ampl i tude near t ---- 0 have been s tudied in the following two different ways: the group theoret ical approach and the analyt ic approach.

The first approach is based on the invariance of the scat ter ing ampli- tude under the Lorentz group 0(3, 1) at t = 0 in the pairwise equal mass configurat ion. In fact , TOLLER [1--3] Reggeized expansions of ampl i tudes

* Work supported in par t by the Air Force Office of Scientific Research th rough the European Office of Aerospace Research, OAR, Uni ted States Air Force, under con t rac t F61052 67 C 0084; and by the U.S. Atomic Energy Commission under cont rac t A T ( l l - - 1 ) - - 6 8 of the San Francisco Operat ions Office, U.S. Atomic Energy Commission.

Pe rmanen t address. ** "Angelo Della Riccia" Fcllow, on Icave of absencc from the Laborator i Nazionali

del CNEN, Frascati ([ loma), Italy.

3 " Acta Physica Academiae Scientiarum Hungaricae 26, 1969

36 P. DI VECCHIA and F. DRAGO

in terms of the representation of the group 0(3, 1); the simpler eompaet group 0(4) was used later by FaEEDMAN and WAr~c [4] and by DOMOKOS [5].

The results of this approach ate that a TOLLER pole leads to an infinite family of Regge poles with defined relations between the trajectories and the residue functions at t = 0. These families are characterized, apart from the internal quantum numbers, by the ToLLwa quantum number M which, for boson trajectories, can assume all the integer values.

Since these formalisms apply rigorously only at the point t = 0 and for equal mass scattering, more general formalisms have been developed recently in order to overcome these limitations [6, 7]. The most general work in this direetion has been done by Cosvr~ZA, SCIARRINO, and TOLI~En [8, 9] by a generalization of the Lorentz group formalista and the introduction of quite strong assumptions.

The analytic approach on the other hand is based on the usual assump- tions of the S-matrix theory, which adapted to the Regge pole theory permits us to clarify the properties of the scattering amplitude near t = 0 for any mass configuration [10--15]. The assumptions made in the analytie approach are the following: (a) analyticity; (b) simplicity; (c) crossing symmetry; and (d) factorization.

Assumption (a) is the fundamental one of the S-matrix theory and claims that the scattering amplitude once its kinematical singularities have been removed, should have only the singularities required by the Mandelstam representation. In the UU and EU mass configurations, the contributionof a single Regge pole is not ah analytic function at t ---- 0; assumption (b) requires that the analyticity of the scattering amplitude has to be realized by the introduction of the minimum number of daughter trajectories with residue sufficiently singular. Moreover, the requirements of analyticity and crossing symmetry impose some constraints on the helicity amplitudes that must be satisfied by the Regge pole families.

The assumption (d) is a consequence of uni tar i ty and is necessary in order to connect the various mass configurations. Following the above assump- tions, it is possible to define a quantum number M which has been found to be the TOLLER quantum number, and to classify the Regge poles in families with well-defined quantum numb'ers. This classifieation is, in some sense, equivalent to the group theoretical one; moreover, the t = 0 behavior of the factorized Regge pole residue functions, satisfying all the kinematical eonstraints, is the same as that derived in [8] and [9] by the group theore- tical methods. I t is possible then to reconstruct the scattering amplitude at t = 0 for the process N + N --, N + N due to the exchange of a Regge pole family with a defined value of M and to show the complete equivalence bet- ween the group-theoretical and the analytic approach at t = 0 in the equal mass configuration. This equivalence eliminates the possibility that an "analy-

.Acta Physica .,4cademiae Scientiarum Hungaricae 26, 1969

ANALYTICITY, FACTORIZATION AND LORENTZ SYMMETRY 37

t icity family" represents a string of integer spaeed TOLLER poles rather than a single TOLLE~ pole. As a consequence of the analytic approach, it is possible to go away from t = 0 and evaluate the "mass formulas" for the Regge trajee- tories. FSEED~Xr~ and WANG [10] showed that in the spinless case the intro- duction of daughter trajectories is necessary in order to ensure the analyticity of the full amplitude. In the scattering between particles with spin, the daughter trajectories have necessarily to come into play bu t the singularity structure of their residue functions is, however, somewhat different than in the spinless case [12]. Therefore, in any spin configuration, the analytieity requires, for every parent t rajectory exchanged, the exchange of ah infinite family of Regge poles with well-defined quantum numbers with respect to the parent pole. In order to classify the Regge poles into families with well-defined quan- tum numbers, we first s tudy the "minimal" solutions of the factorization conditions and the constraint equations. In this paper we will limit ourselves to the reactions S + N - ~ J + N and the others related to these through factorization. Here N is a nucleon particle with mj vi= ms :g: nueleon most interesting from the physical

We gire now the kinematical in our discussion: (1) EU case (i.e., S -4- N --~ J + N).

and J(S) is a spin J(S) and mass mj (ms) mass. Obviously these reactions are the point of view. constraints at t = 0 for the various cases

The eonstraints turn out to be [12, 14, 16]

i~~~/,-1/2 -f;~,1/2,~/~ = 0(t) (1)

for any c and d satisfying the inequality c ~= d, and

i~~-) -.f;~,~1~,11~ o(t) J cc;112-112

for any c. (2) UU case (i.e., S + S --~ J -t- J) . The constraints ate

I :~= +Ÿ = o(~.),

(2)

(s)

if I ~ - - p [ < l ~ + # ] ; and

Y+d,ab - - . ~ ; d , ~ = 0 ( t a ) , (4)

if ] 2 - - # [ > [ Ÿ [, where m = M i n i m u m ([Ÿ l p ] ) . (3) EE case. In the simplified t reatment given here, we have to eonsider only the nucleon--nucleon scattering. Inthis case the constraint is well-known

[171:

�9 J~1-)2,112|1]2,1/2 - - ~~Ÿ --,J~1-)2-1/2;1/2-1/2 = O( t ) . (5)

Acta Physica Aeademiae Scie~r Hun$ar/cae 26, 1969

3 8 P. DI VECCHIA and F. DRAGO

I t is well-known tha t in the Regge pole model these eonstraints can be satisfied in three different ways; by evasion, by conspiracy between differ- ent poles and by a daughter- l ike conspiracy. In order to discuss this problem, we first s t udy the minimal solutions of the factor izat ion conditions. The objects which are assumed to factorize are the residues of the individual poles in F~+ab; therefore, in our case, the residue is

Kcd;a £ (t) + qi qo ~'cd;ab (t) - - t S o l

(6)

where K�91 is the WANG kinemat ica l factor, (qiqo/So) ~+-(O-N is the usual threshold fac tor and ~c~;ab(t) in the reduced residue free f rom kinemat ical singularities at t = 0.

The factor izat ion condit ions in all the three channels considered ate consis tent and, in general, f rom a given solution, one can obtain other solu- tions b y increasing the n u m b e r of t powers of some of the Regge pole residues in the original solution. I f a given solution cannot be obta ined from ano the r in this way, we shall call it "min ima l " . Once the factor izat ion conditions are wr i t ten down, it is not too difficult to f ind the " m i n i m a l " solutions. For the reactions with unequal masses of the type S + S -* J -f- J o r S ~- S' --* J + J ' , one finds [12, 18, 19]:

a ~ t l l t z [ - M [ - n 7ca;ca , (7)

where/~ = c - - d and a is the Regge na tura l par i ty .

When an equal mass ve r t ex is involved, the selection rules due to pa r i ty and G-par i ty invariance must be t aken into account . This implies the ident ical vanishing of the residue if the following conditions are not satisfied:

a � 9 1 1)S+I+N __ 1,

S = I , if a - - - - + l . (8)

is the s ignature of the pa ren t Regge t ra jec tory , the integer n is the order of the daughter , and S is the to ta l spin of the N � 9 system. The quan t i t y is defined in terms of the in ternal quan tum numbers I (isospin) and G (G- par i ty) of the exchanged fami ly by

�91 = G(-- 1 ) r .

In the equal mass react ion N + N - * N + N for the residues which do not vanish identically, we have [12]

Acta Physica Academiae Scientiarum Hungaricae 26, 1969

ANALYTICITY, FACTORIZATION AI~D LORENTZ SYMMETRY 39

{ t M ir (__ 1)a+n = a,

t IM- l l if ( _ 1 ) a + n = _ . a ,

where 2 : a - - b . For the families wi th a =: + 1 and z : - -~ , whose p a r e n t (and even

daughter ) t ra jee tor ies do not couple to the N � 9 sys tem, we have for the odd daughte r s [12]:

+ [t M+I if ~ = 0 (10) ~a£ if q : 1

The solutions for the e q u a l - - u n e q u a l mass case can be ob ta ined th rough the fac tor iza t ion and can be found explici t ly in [18].

M i s a n u m b e r t h a t we int roduce in order to label the min ima l solu- t ions and can assume all the in teger va lues be tween zero and infini ty. An in teres t ing fea ture of our results is t ha t , for MI the values of the masses, a f ami ly wi th a given va lue of M cont r ibutes a symp to t i c a l l y only to the forward s-channel helicity ampl i tudes wi th helici ty f l ip equal to ~ :M. M i s therefore the TOLLER q u a n t u m n u m b e r in t roduced in the group theoret ica l approach .

The Regge pole families can be then classified [12,20] according to the values of M, a, �91 These families and the residue behav io r nea r t = 0 are coincident wi th those found in the general group theore t ica l approach .

Class I : M = 0, a = + 1 , �9 : ~. Only the t ra jee tor ies wi th n even can couple to the NN sys tem. Poles of this elass ver i fy the eons t ra in ts 1, 2, 3, 4, 5 b y evasion.

Class la: M : 0, a : + 1 , z = - -~ . The pa ren t and the even daughte r s of this elass do not eouple to the N � 9 sys tem. This explains w h y this elass is not eonta ined in the FnEEnMAN and WANG elassifieation.

Class II: M = 0, a = - -1 , �9 = - - ~. The poles of this elass sa t is fy the cons t ra in ts 2 and 5 b y a daughter- l ike conspiraey. All the o ther eons t ra in ts are satisfied by evasion.

Class IIa: M = 0, a : - -1 , z = ~. The pa ren t t r a j ee to ry of this elass, which is deeoupled f rom the N � 9 sys t em at t = 0, satisfies the cons t ra in ts b y evasion. Conspi racy be tween daughte rs is allowed.

Class I I I : M = 1, z : ~. In this elass we find the wel l -known pa r i t y doubl ing phenomenon , which not only allows us to sat isfy the cons t ra in ts 1, 3, 4, 5 b y conspi raey , bu t ir is als6 imposed b y other general ana ly t i e i ty requi rements . The m e m b e r s of the double t have the following q u a n t u m num- bers:

] n e v e n : P n : ~ : T ,

a - - - - - + l [ n o d d : P n : - - � 9 1

~4cta Physica Academiae Scientiarum Hungaricae 26, 1969

4 0 P. DI VECCHIA and F. DRAGO

[ n e v e n : - - P n : ~ : 3 - - - - l l n o d d : -- P~ : -- �91 : ' -- 3,

where P means pari ty. Class Ilht: M : 1, ~ ~ --~. The poles of this class have the following

quan tum numbers:

[ne ven : P n = - - ~ = 3 ,

( ~ : + [ n . o d d : P,, = �91 -- 3 ,

a : __ { ¡ - - P . = - - ~ = 3 , odd : -- Pn = �91 = -- 3.

The parent trajectories satisfy the constraints by evasion. Conspiracy between daughter trajectories is allowed. Poles with M > 1 are decoupled, at t = 0, from the N�9 system, in agreement with the group theoretical results.

Using the assumptions (a)--(d), we were able to classify the Regge fami- lies aceording to the values of M, a, ~ and we showed t h a t this classification is equivalent to tha t obtained in the group theoretical approach. However, the previous discussion does not eliminate the possibility t ha t an "ana ly t i c i ty fami ly" represents a string of integer spaced TOLLEa poles ra ther than a single TOLL~R pole.

In the following, we will reconstruct the scat ter ing ampli tude at t : 0 for the procesa N + N ~ N + N due to the exchange of a Regge pole fami ly with a definite value of M and we wfll show t h a t the scattering ampl i tude obtained in the various eases is the same as t ha t dedueed using the group theoret ical approach [11, 14, 15]. We will reeonstruet only the classes I, I I , I I I beeause the reconstruction of the other classes is quite similar to the pre- vious ones.

Class I : M : 0, a = -~1, z ---- ~. In order to s tudy the Class I fami ly in the UU case, we have to eonsider the ampli tude f0+;0. The con t ¡ of a family of Regge poles to this ampli tude is given by:

where

J~~0 (s, t ) - - . ~ 2a, + 1 rl=O s i n g O~ n

- - [ l + 3, e-i"=']~=.(--cosOt)fl,(t), (11)

,..0= (Z) = t a n =Of - - Q-=-I (z) = 2= T'0t + 1/2) z=" F(~ + 1) ~tz/2

- - - - , - - - - - - ; - - - - a r =

2 2 2 2

= F(= + 1/2) 2 = z= . ~ ak (a) z -2k ~1/2F(= + 1) k=0

A~= -Pkyt.ka Aca&miat Sc/~~iatu.-a H~ngarieae 26, 1969

ANALYTICITY, FACTORIZATION AND LORENTZ SYMMETRY 41

$

z = c o s o t - 2q iq~

t t - - m~ + ( m E - m . ) ( r n ~ - m ~ - - +

4tqi qo

(12)

,,~ (~) = . F ' ( - - ~ - + k ) F ( - - - ~ ---4- 1 + 2 k)F ( ~ - - - z t )

rl-~-)F{--~+l/r(-2 - - - - ~ + k) k!

and qi and q0 are the initial and final m o m e n t s in the c.m. f rame of the t channel .

E x p a n d i n g the r igh t -hand side of Eq. (11) in a power series, af ter some rear rangements , we get

N(m) m - 2 k J~+o(s,t) = ~," [B(t)/t] m (s/s~ 2 " . ~ dmk (t) (s/s~ ~'(')+n '

m = 0 k=o n=O

(13)

where s o is a seale fac tor

14j t t - - . ~ m ~ + ( m E - m 2 ) ( m ~ - m i ) i = 1

B(t) = 2s ~ ,

N(m) =

m

2

m m l

if ( - - 1)'n = 1,

if (--l)m:-- i, 2 (14)

,n t 2~, + 1 F(~~ -4- 1/2) 2% d~,~ ( ) - - - - [1 + ~~ ~-'"~ y~ (t) a~ (~,,).

sin yt ~ n :ta/2F(~n -~- 1)

�9 [D(t)/B(t)] 2k [B(t)/to]-~ F ( % -- 2k + 1)

( m - - n - - 2 k ) ! F ( ~ n + l - - m + n )

D(,) = [ [ ' - - ( m i -4- m3)21 I r - - ( m , - nrj) 21 [ t --(m2 -+- m4)~l [ t - (m~ - m4) 2] ]1/2.

4s2o

Since the functionJ~o;o has to be analyt ic at t = O, for any s, we m u s t require t h a t :

N(m) m- -2k ~ ' .~ ' dmk(t) (S/So)"(O+n= O(t m) for m ~ 1. (15) k=o n--O

Acre Physica Academias Sr Hunsarir 26, 1969

42 P. DI VECCHIA and F. DRAGO

These are the fundamenta l relations of our approach; f rom these equa- tions in fac t not only the quant i t ies ?n(0) can be expressed in terms of ?0(0) bu t the behavior of the fami ly for t =~0 can be studied. I f we evaluate the expression (15) at t = 0, we get a sys tem for the residue funct ions of the daugh- ter t ra jec tor ies at t = 0 in func t ion of ?0(0). The solut ion of this system has been shown to be [11, 14]:

?n( 0 ) - (_l)n F(n--l--2~) ?0(0). (16) n! /"(-- i - - 2~)

Once these relations are known we can deduce the mass formula. In fac t , f rom the system (15), d i f ferent ia t ing with respect to t, we obtain for m >~ 2:

N(m) m--2k

.a~ . ~ d ~ k ( 0 ) ~ ~ ( 0 ) = 0 . (17) k = 0 t imO

The solution of this sys tem is the DOMOKOS--SURJ, NYI [21] mass fo rmula

~~ (0) = al + a2 (a - n) (~ -- n + 1) . (18)

We then s tudy the Class I in the EU mass configurat ion; this case is somewhat simpler t han the UU one, due to the presence of weaker singula- rities. At the equal mass ver tex , the selection rules due to pa r i ty and G-par i ty invar iance must be taken into account . In the present case one finds tha t the paren t and the even daughter t ra jector ies couple to the ampl i tude f+;1/2,1/2~f0+;0 while the odd daughters are decoupled from the N�9 system.

Using then essentially the same methods described for the UU case,

[ 1 ) / ' n :r

?2n (0) - - ( - 1)n 2 ?o (0) (19)

we get

and ~2,(0) = a 1 + a 2 ( a - 2n) ( ~ - - 2n + 1) (20)

which, as expected, is again of the form of Eq. (18). The factor izat ion theorem provides now the bridge necessary to s tudy the cont r ibu t ion of the ana ly t i c i ty family wi th M = 0 a n d a ~ + 1 to the n u c l e o n - n u c l e o n scattering.

Using therefore the factor izat ion theorem, we get:

n(+_)2n (2n)! F ( a -4- 1 - - n) (+)n=0 yl/2,1/2;1/2,1/2- fll/2,112;112,1/2. (21)

2 2n(n!) 2 F ( ~ ~ - I ) F a + 2 n

Acta Physiea Academiae Scientiarum Hungarieae 26, 1969

.ANALYTICITY, FACTORIZATION AND LORENTZ SYMMETRY 43

The same expression can be found using the group theoretical approach. This proves tha t the Class I "ana ly t i c i ty families" ate indeed the same as the Class I "group theoret ical families".

Class II: M----0, ~ = --1, ~ = --~. In order to s tudy the Class I I family in the U U case, we have to consider the amplitudeJ~o:o, whose discus- sion goes on exact ly the same way as tha t for the Class I UU mass case. The results, therefore, are exact ly the same as repor ted above.

In the E U case, however, there is a new complication due to the spin. Due to the selection rules at the nucleon vertex, one finds tha t the parent and the even daughters contr ibute to the ampl i tude j~r ~J~o;1, while the odd daughters contr ibute to the ampl i tude J~cc:1/2,12 ~J~0;0. Moreover, there is the constraint 2 which is satisfied b y conspiracy. I f we impose tha t the ampli tudes f0;0 and f0;~ are analytic at t = 0 and satisfy b y conspiracy the constraint ecluation, we get the following relations for the residue functions:

n! [ (~-- 2n)(~-- 2n-~ l) ]U2 ~l n 21 ~)

] ' ( n + 1 } - - - - (~

},o2~ '+1 (0) -- (- 1)n 2 ~£ x (0), (22) n! a/ F ( 1

2

2~-~- 1 t i - 1 Y0;0 (0)--~i V~(~_~]) y~;=~

Using the factorizat ion theorem, we get for the complete residue func- tions in nuc leon- -nuc leon scattering:

zn (2n)! (~ -- 2n)(~ -- 2n + 1) F (-- ~) ~1/,,._~/~,1/,,._ ,~ (o) - x

22n (n!) 2 ~ (r162 ~- 1) F (n -- ~)

F{n 21 ~) ~n =o [0 ~ X I J 1 1 2 - 1 / 2 ; 1 / 2 - 1 / 2 ~ ]

1 a) (23) - P I 2

2n+1 fll/2,1/2;1/2,1/~(0 ) = (2n + 1 ) ! F ( - - ~) 2 ~ (n!) 2 I '(n -- ~)

n = l fli/',,i/~:u2,1/2(0),

Acta Phys ica Academiae Sc ien t iarum Hungar icae 26, ]969

4 4 P. DI VECCHIA and F. DRAGO

n = l ~112,1/2;1[2,1/2( 0 ) - - _ _

2 ~ + 1

~ ( ~ + 1 )

These results are coincident wi th those obtained using the group theoretical approacl~. I t is easily seen t h a t in this case the analyt ic and the group theore- tical families ate the same. The mass formula for poles belonging to the Class I I is the same t h a t has been obtained for the Class I.

Class III: M ---- 1, z = ~. The discussion of this family is much more involved, essentially due to the fact t ha t a is no longer diagonal with M. Here, for the first time, the pa r i ty doubling phenomenon appears. In order to s tudy the Class I I I family in the UU case, we mus t consider the ampli- tudes J~~l(S, t). In fact, because of the existence of the par i ty doubling pheno- menon, we cannot restrict consideration of only one ampli tude like in the first two classes. I f we require t ha t the contr ibut ion to the amplitudes f~ l of the two parent 1Regge trajectories aud of their daughters is analytic at t ---- 0 and satisfies the constraint 3 by conspiracy, we get the following rela- tions for the residue functions:

yx(;~ )n (0) -- ( - 1)n P(n -- 1 --2~) r~;l"'•176 (0),

n! ]"( - - 1 - - 2~)

r~;~~ . -o (o) = - ~~,~-o (o) , (24)

and the mass formula for M = 1 Regge families:

~n ~ ( 0 ) = c 1 + [ c 24- % 1 ( ~ - - n ) ( ~ - - n + l ) - (25)

Owing to the par i ty and G-parity selection rules, in the EU case the parent and even daughters of the sub-family wi th a- - - - - -1 couple to the ampli tude F~;~ and the odd daughters to the ampl i tude F~;~. The parent and even daughters of the sub-family with a ---- + 1 couple to the ampli tude F~;~ while the a = + 1 odd daughters ate decoupled from the N�9 system.

In this case we have to impose the analy t ic i ty a t t ---- 0 to the ampli tudes … and f~ l and the constraint 1. In this way, we get

{ 1 ) F n ct

~~~~~ (0) = ( - 1)" 2 ; 1 7~~ )"-~ ( 0 ) ,

nX r { 2 ~J

y~~).-1 (0) = 2~ +____! ~,~+>.=o / 1 ; 1 (26)

Aves Physica Academias 8~i~mi~um Hungaricae 26, 1969

ANALYTICITY, FACTORIZATION AND LORENTZ SYMMETRY 4 5

~(~)~.+1 ( o ) - - -

+ + ! = ) ( _ 1)n 2

71~-1> ,-1 (0), 21 a) ni F [+

(

7(70 )n-~ (0) = i V ~ +a 17~;~)n=~ (0),

a n d a mass formula which is consistent with Eq. (25). The factorization theo- rem permits one to get the following resuhs for the nucleon--nucleon scattering:

( * ) 2 n ~1/2-1 /2 ;1 /2-1 /2 - -

F {. 1 ~) (2n)! F ( - :r 2

2 2" (n!) 2 F(n -- a) .F { 2 (+)n=0

~1/2-112;1/2-1]2

t 1 I ff n o~ R(-.)2. (2n)! F(-- ' ct) , 2 /~1/2,1 [2;1/2,1]2 -- X

22n(n!) 2 F ( n - - ~ ) / ' ( . 21 ~1

X (~ - - 2n) (~r - - 2 n + 1) fl(~)Ÿ ( 0 ) , ~(~ + 1)

fl(-)2n+: 112-1[2;1/2-1/2 - - "

r(n§ } (2n q- 1)l F(-- ~) 2

,•(-)n-1 112-1[2;1]2-1/2 "~

( - ) n - 0 - - fll/2-1/2;1/2-1/2 (0), ~1/2,1/2;1]2,1[2 ( 0 ) ( + )n =0

(-).=i 2zr -4- 1 - - ~1]2-1{2;1{2-i]2 "J fll/2-112;1/2-1\2 ( + ) n - 0 (zr -k 1)2

which ate coincident with those derived from the group theoretical approach. The proof that the M--~ 1 Regge pole families ate the same as the

Regge pole families derived from M---- 1 TorT.~R poles in EE mass scatter- ing is therefore complete&

Acta Physica .4cademiae Scientiarum Hungaricae 26, 1969

4~ P. DI VECCHIA and F, DRAGO

REFERENCES

1. M. TOLLER, Nuovo Cimento, 37, 631, 1965; University of Rome Reports No. 76, 1965 and No. 84, 1966 (unpublished).

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4/1968.

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Acta Physica Academiae Scientiarum Hungaricae 26, 1969