IL NUOVO CIMENTO VOL. LVI A, N. 4 21 Agos~o 1968

Minimal Solutions to the Conspiracy Problem

and Classification of Regge-Pole Families .

P. Dt VECCmA and F. DnAGO

]5(!boratori Naz iona l i del C . N . E . N . . l~rascati (Roma)

~[. l , . P A C I E L L 0

Is t i tuto di F i s ica dell'U'niversit& - R o m a

(rieevuto il 17 Luglio /968)

The most general appro~ch to the conspiracy problem is, up to now, a group- t.heoretieal one, essentially based on the expa.nsion of the scattering amplitude in terms of the irreducible representation of the Lorentz group Oaa or of the group 04. Since this formalism applies rigorously only at the point t = 0 and for equal-m~ss seat- tering, more genera.1 formalisms have heen deveh)pcd recently in order to overcome the.ae limitations. ]n particular very interesting results have been obtained recently b y (JOSI.;NZA, 8CIARRINO a n d TOI ,LER (1.2). Moreover the group-theorelieal~ approach provides a classification of the lteggc poles in families with well-deiincd quantum numbers (3).

In this note we will giw, some of the resulls of a nmlgroup-thcorctieal n,ethod, which will be described in full detail in a forthcoming paper.

We consider only the problems which arise in a neighbour]mod of the point t .... 0. Ore' fundamental assumptions arc: analiticit, y, crossing symmetry and the faetoriza- tion principle for the residue functions of the Regge trajectories, which are suppes,~d to be bosonic and with well-defined quantum numbers, hi this note we will limit our- selves to the reactions 8 + . \ ' - > J t-,,\' and the others related to these through t'ae- torization, lle, 'e , \ ' is a nlteleon and J (S) is a spin J (S) and mass u~j ('ms) particle with

(1) (~. COSI"NZA, ,~. ,~CIAI~RINO ~tnd ~T. T()I,IA';R: [II|(~l'II~l] [N'I)Ill'! NO. 15~, [stit lflO di I"isie~ r162 G, Mar- coni ,), R o m ~ (1!)[;8).

(a) G. COSEXZA, A. SCIAI~RIN'O ~nd M. TOLLER: TriesI( ' p r c p r i n t It'/{;S/21. I n t h i s p a p e r a comph: i e l is t of r e fe rences abouf t he g r o u p - t h e o r e t i c a l work c~n b,' fmt~d.

(3) F o r ~ s imple expos i t ion of l, he r e l e v a n t rcsu l l s of the g r o u p - t h e o r e t i c , d a p p r o a c h to the s c a t t e r i n g p rob l em see L. ]~ERTllCCItI: Itapporh'ur's talk at Ihc lIcir (/o~ffcre~wc o~ Elemetllary Parlieles (1967).

1186 P. DI VECCHIA, P. DRAGO and M. L. PACIELLO

mj # ms ~ nucleon mass. We suppose in the following tha t , I ~ S. Obviously these react ions arc the most in teres t ing from the physical po in t of view.

We will deduce the k inemat ica l constraints a t t = 0, wi thou t making use of trans- vers i ty ampli tudes (4), the (~ min ima l ~) behaviour of the residue funct ions for the paren t and daughter Regge t ra jector ies and a classification of Regge t ra jec tor ies in families equ iva len t to those deduced in the most general group- theore t ica l approach. In our approach the in t roduct ion of daughter t ra jector ies is necessary, as in the spinless case (5), in order to ensure the ana ly t ic i ty of the full ampl i tude . The s ingular i ty s t ruc ture of the daughter residue funct ions is however somewhat different than in the spinless case (6).

Le t us quickly in t roduce some notat ions . The asympto t ic cont r ibut ion of a single Regge pole to the usual par i ty-conserv ing hel ici ty ampl i tudes is g iven by

cd;ab s--~-co sill Jt~* '

where K ~ is W a n g ' s k inemat ica l factor (s), X = max( lA I, I~1), ~ = a - - b , t~ = c - - d , and Y~a,ab() is the reduced residue free f rom kinemat ica l singularit ies at t = 0. The objects which are assumed to factorize are the residue of the individual Regge poles: in our case the residue is

\ So /

We give now the k inemat ica l constra ints at t = 0 for the var ious cases in our dis- c u s s i o n :

1) E U case (i.e. S - [ - z V - + J § The constraints turn out to be

(1) i7(+) t 7 ( - ) t J~d,�89189189189 = O ( t ) ,

for any c and d sat isfying the inequa l i ty c # el, and

(2) -2 A~.�89 _�89 o(t)

for any c.

2) U U case (i.e. S § The const ra in ts are

7(+)$ ~(--)t m (3) J~d.~b + ]c~.~b = O(t )

(4) G. OOHEN-TANNOUD,II, A. MOREL and t t . NAVFLET: .']n?~. O/ Phys., 46, 239 (1968). (5) D. Z. FREEDMAN a nd J . hi. WANG: Phys. Rev., 153, 1596 (1967). (~) This difference is m a i n l y due to the fac t t h a t the so-called *, pa r i ty -conserv ing ,~ hel ici ty ampli -

tudes h a v e definite p a r i t y only a sympto t i ca l ly , so t h a t , when the pari ty-boundii~g phenomenon appears , a s o m e w h a t compl ica ted cancel la t ion be tween the m e m b e r s of the p a r i t y double t is requi red ('). We are g ra t e fu l to l ' rof . L. ]:~ERTOCCllI a l ia Dr. A. SCIARRINO for en l igh ten ing discussions about th is point .

(7) p. DI VECCHIA, F. ])RAGe and M. L. PACIELLO: to be published. (8) L. L. C. WAN(~: Phys. Rev., 142, 1187 (1966).

M I N I M A L S O L U T I O N S TO T H E C O N S P I R A C Y P R O B L E M E T C . 1187

if I ~ - , 1 < I ~ + , l ; and

(4) 7:;:>I 7:Za' = O<t m)

if SI, where min(I2 l,

3) E E case: in the simplified t reatment given here we have to consider only the nucleon-nucleon scattering. In this case the constraint is well known (9):

(5) l~+.t~ + f "~ l' = o ( t ) + - + a -+ - + t+.-+ - t - .~ -+ .+-+

The origin of the constraints (1), (2) and (5) are quite different: in fact while the latlcr follows from the forward conservation of the angular momentum in the direct s-channel, the former ones do not have so clear a physical origin. In fact they are a consequence of analyticity and crossing symmetry. They can be deduced following a method originally proposed by COHEN-TAN:NOUDJI, MOREL and NAVELET (4), writing down the crossing relations between the helicity amplitudes, ]ree ]rom both s and t kinematical singularities, in the s and t channels. The requirement that no spurious sing~flarities are introduced leads to the constraints (1) and (2). The constraints (3) and (4) have essentially the salne origin, although they can be deduced more simply without reference to the crossing matrix, following a method due to ]~RAUTSCIII and JONES (lO). The details of derivation of these constraints can be found in ref. (v.n).

I t is well known that, in the Regge-pole model, these constraints can bc satisfied in three different ways (1~): by evasion, by conspiracy between different poles and by a daughterlike conspiracy. In order to discuss this prohlem we first study the <( minimal ~> solutions of the factorization conditions. In general, from a given solution to all the factorization requirements, one can obtain other solutions by increasing the number of t powers of some of the Regge-pole residues in the original solution. If a given solu- tion cannot be obtained from another in this way, we shall call it (( minimal ~). I t is important to note that in order to decide if a solution is a minimal one, one must con- sider simoultaneously all the channels connected by the factorization.

In order to study the behaviour near t = o of the rcsidue functions one must write in compact form the factorization conditions, taking into account Wang's (s) kinematical factors. Once this is done, it is not too ditticult to find the solutions given below (7.n). For the reactions with unequal masses, of the type S + S - + J + J or S + S ' - + J + J ' , one finds

(6) ~,~ gb~l-~l-- + a b g t b ~ ~

When an equal-mass vertex is involved, the selection rules due to parity and G-parity invariance must be taken into account. This implies the identical vanishing of the residue if the following conditions are not satisfied: a~T(--l)S+l+n= l , S : l i f a = + 1.

U) ]). V. VOLKOV a n d V. N. G m n o v : ,b'ov. l 'hys. ,]ETI ' , 17, 720 (1963). (1o) ~. FRAUTSCHI a n d L. JONES: Phys. ]~ev., 167, 1335 (196S).

(tx) 1'. DI VECCHIA, F. ])RAGO a n d M. L. ])A(dlI']LLO: F r a s c a t i I n t e r n a l Hepor t LNF-68[5 , F r a s c a t i (196S) (unpubl i shed) .

(t~) E. LEADER: l 'hys, l~ev., 166, 1599 (1968).

118g r . D I V E C C t I I A , F . D R A G O and ~r. T,. P A C I E L L O

Here a is the:Rcgge-polc natural parity, v is the signature of the parent Regge trajec- tory, the integer n is the order of tlm daughter, and S is the total spin of the ,;\<'0V system. The quantity ~ is defined in terms of the internal quantum numbers I and G of the exchanged family by ~ = G(--1)C In the equal-mass reaction ~N'+,N'-+,N'§ ), for the residues which do not vanish we have

(7) [ t ~ if (--1) a+"---a ,

~ab, ab

t I~ -~ l i f ( - - 1) ~+" = - - a .

For the families with a = -4-1 and ~ = --$, whose parent (and even daughter) trajec- tories do not couple to the ,N)2T system (~a), we have for the odd daughters

(+) (8) 7.~.~b "-~

t M+I i f ~ = O,

t I~1-11+1 i f t,~] = 1 .

(In this case however there are two other cvasive solutions not present in the group- theoretical approach and that we do not discuss here.) Tho solutions for the equal- mmqual mass ease can be obtained through the factorization and can be found explic- itly in ref. (7). M is a nmnber that we introduce at this point in order to label the minimal solutions. I t can assume all the integer values between zero and infinity. Every family of Reggc poles is characterized by a value of M. An interesting feature of our results is that, for all the values of the masses, a family with a given M con- tributes asymptotically only to the forward s-channel helJcity amplitudes with helicity tlip equal to =[:M. M is then tho Tol]cr quantum number introduced in the group- theoretical approach. This conclusion will be enforced by the following considerations.

The Regge-pole families can be classified according to the values of M, a, ~. These families and tim residue behaviour a.re coincident with those found in the general group- theoretical approach (1,2)o

(:lass I : M = O, a = + l , z ~ - - ~ .

A parent trajectory requircs the existence of a family of trajectories with << angldar momentum )> a,~(0)= ~.(0)--n, where a(0) is the intercept of the parent trajectory.

1,'or n evcn we have 3 , , - - P n = ~ = r and for n odd r n = P ~ = - - ~ = - - v . Only the trajectory with ~ even can couple to the A'A ~ system. This is class I of Frcedmau and Wang (la). Poles of this class never conspire, as can bc sccn from the behaviour of the

residues listed above.

(;lass Ia: M = 0 , ~ = + 1 , 3 = - - ~ .

Poles with aevcn have 3n = P~ = - - $ = ~ and with ~oddT~ = Pn = ~ = - -v . Tile 1)arent and the even daughters of this class do not couple to the ~N'JW system. This

(,s) Wc t h a n k Prof. M. TOLLER for calling our ~tttention to these families. (t*) 1). Z, ]!'REFd)MA~ ~ltl([ 5. ,~[, ~VANG: I'liyS. Rev., 160, 1560 (1967).

M I N I M A L S O L U T I O N S T O T I l E C O N S P I R A C Y P R O B L E M E I ' C . 1 1 8 9

explains why this class is not conta ined in the Freedman and Wang classification. Also

these poles never conspire.

C l a s s 1 I : M = O . ~ - - 1 , r - - ~ .

The poles of this class satisfy the constra ints (2) and (5) by a daughter l ike conspiracy. All the o ther constra ints are satisfied by evasion. For ~r even we have r,~ = - - P . = = - - ~ = r and for n odd r ~ = P ~ = ~ = - - r .

Class I I a : M ~ O, ~ = - 1 , r = ~.

Poles with ~ even have r,, ] ) n - - ~ T and with n odd r,, = - - P ~ = - - ~ = - - r . The paren t t ra jector ies of this class, which are decoupled froin the ~N'd~" sys tem at t = O. satisfy the constraints by evasion. Conspiracy between daughters is allowed.

Class I I I : 31 1, r = ~.

In this class we find the well-known par i ty -doubl ing phenomenon. The par i ty double t s t ruc ture not only allows us to sa.tisfy by conspiracy the constra ints (1), (3), (4) and (5), but is also imposed by other general ana ly t ic i ty requirements . One has the following quan tum numbers :

+ 1 / n even : r n - - P~,~ ~ = r , ff

[ n odd: r ,~= P n ~ - - ~ - - - - r

(only tvaject~)rios with ~ even ca.n couple to the oX~3~ ~' sys tem);

~ even :

n odd :

Class I l i a : M = 1. r - - - - ~ .

The poles of this class have the following quan tum numbers :

+ l / ~ even: v n = t ' , ~ = - - # r .

/ n odd: v ~ P ~ = ~ - - - r

(only t rajector ies with a, odd can couple to the , \ L ' , s) 's tem);

~ e v e n : T~ - - ]O n ~ - - ~ ~ r .

n odd: r , , ~ P ~ = ~ r .

76 - I I .Vuoco C imcn to A .

1190 P. DI V:ECCIIIA, F. DRAGO and ~. L. t'ACIELLO

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 ~'~'~ sys- tem, in agreement with the group-theoretical results.

$ $ $

I t is a great pleasure to thank Prof. L. BERTOCCHI and Dr. A. ScIA~I~-o for very useful discussions, and Prof. M. TOLLER for helpful correspondence.