IL NUOVO CIMENT0 VOL. XLVII I A, N. 1 1 o Matzo 1967

Some Comments on Phenomenological Aspects of Regge Trajectories.

J. DI~F~ITLEIN (*) and K. ~{AHANTHAI'I'~

Department o/ I'hysies, University o] Colorado - Bo~lder, Colo.

(ricevuto il 19 Gennaio 1967)

The recent~ resurgence of inberest in Regge trajeetorios can be t:raccd lo a number of successful fits of the experi- mental data (:) with amplitudes whose form is suggested by the Chcw-Frau- tschi (2) hypothesis on the relativistic versioa of the Regge analysis. In this note we confine ourselves to those cases

in which the fits :~re achieved by including direct-channel resommce poles in addition to the crossed-channel Regge trajec- tories (3.4) and try to see whether such

(*) Al f red P. Sloan Fel low; s uppo r t e d in part by t he Office of ~N'aval l~esearch u n d e r c o n t r a c t No. I~14-002-()5-01, 1-2.1.

(z) ~ e f e r e n c e s to v a r i o n s fits of experi- m e n t a l d a t a c an be found in R. J . N. ])I~ILL]I,S: Lectures given at the Eriee Sure,nor School, 1966 (to bo publ ished) and H a r w c l l ]~,eport .No. I I L 66]3798.

('~) G. F . C~fEW a n d S. C. FRAUTSCIII : Phys. 1)~ev. LeH., 7 ,394 ([961). ,See also V. iN. GR]BOV: Sm,. Phys. J E T P , 14, 47~ (1962); S .C . FRAU- TSCIII, :'r GELL-~IANN a n d F. ZA('IIAR]ASE~N~: Phys. l~ev., 126, 220,t (1962).

(a) V. BARGEB, ~,nd I). CLINE: Phys. Rev. Lett., 16, 913 (1966).

(4) G. ]I(5~LER, J. JJAACKF~, H. SOHLAI:L~,, and R. SO.N'DEREGGER: Phys. Loll., 20, 79 (1966); A. S. CAI~ROLL, I. F. COIqBETT, C. J. S. ])A)IEI~ELL, x~-. ~]IDDLEMAS, D. NEV~'TO~', ~k. l~. CLEGG and

0 9 6 6 ) ; R. ] : . LOGAN a n d L. ~ERTOR]O: P]~ys. Rev. Lett., 17, 83,1 (1966); V. BAI:GER a nd M. OLSSON: U n i v e r s i t y of Wisconsin R e p o r t (to be publ i shed) ; R. J . N. PRn.LIrS: Nuovo Ci~nento, 4 5 A , '245 (1966).

fits could be consistently interpreted as success of the Rcgge-trajectory hypothe- sis or they imply something more than that. For the sake of simplicity of argu- ment, the discussion is restricted to the Bt~rgcr-Cline parametrization(a) although ~he argument applies equally to similar attempts to fit the experimental data.

Firs t we note tha t all the particles both in the direct and crossed channels t~re Reggcized; hence no elementary par- ticles are encountered in u dynamical calculation. As an example, a simpli- fied version of v.~N" scattering given by ChEw (5) (the reciprocal bootstrap mech- a.nism) in fact 5:ields ~,'V*aa predomi- nant ly due to forces given by ,N" exchange and ,~V predominantly due to forces giv- en by ~*ss exchange.

The gist of our argument is this: the coherently added Regge amplitudes, /r~,~, and the direct-channel resonance amTlitu- des,/~', are not dynamically independent and so cannot be independently specified i/ the physical interpretation o/each is to be retained. In particu]ur, the explanaiion of any phenomena as due to inter- ference between the crossed-channel Reg- ge and d i rec tchannel resonance ampli- tudes should be viewed suspiciously.

(~) G. F. CHEW: Phys. trey. Lelt., 9, 233 (1962).

276 j. DREITLE[N and K. MA.I[ANTH&~'PA

In more detai], if the resonances are treated as bound states lying on Regge trajectories, then a resonance amplitude is described Thenomcnologically by a Feynman graph with a single propagator associated with the resonance. Dynam- ically resonance amplitude arises via lmi tar i ty from crossed-channel forces (in some models as a sum of ladder graphs). The part of the amp]itude described by the Rcgge t ra jectory, /nr is dynamical and ~wt phenomelmlogical. Here the main point emers: the inehmion of the Rcgge amplitude means that already some information about the direct- channel resonance amplitude is incorpo- rated. A part of the Regge amplitude goes into building resonances which, hence, cannot be specified completely independently. In fact, i t is exactly in the region of interference used by BAR(r and CLI_~): (a) tha t the dynamical inde- pendence of / a ~ and /re, is most ques- t ionable (G).

The argument becomes clearer in the high-energy limit where, according to the Chew-Frautschi conjecture, the am- pli tude is completely dominated by Regge exchange. But if the Regge-cxchange amplitude is assumed impor tant at any

(6) T h i s c r i t i c i s m is a k i n to one t h a t e x i s t s i n n u c l e a r p h y s i c s c o n c e r n i n g t h e a n a l y s i s of e x p e r i m e n t a l d a t a in d e u t e r o n s t r i p p i n g reac - t i ons . H e r e if t h e d i s t o r t e d - w a v e B o r n a p p r o x i - m a t i o n ( D W B A ) is u sed , t h e n t h e d i r e c t - c h a n n e l r e s o n a n c e s h a v e a l r e a d y b e e n p a r t i a l l y t a k e n i n t o a c c o u n t v i a a n op t i ca l - r rmde l p o t e n t i a l a n d one c a n n o t a d d c o h e r e n t l y t h e D W B A a n d r e s o n a n t a m p l i t u d e s w i t h o u t r e s e r v a t i o n s .

particular energy it should specify at least partially the direct-channel reso- nance amplitudes at this energy. Pre- sumably high-energy resonances in the direct channel can be buil t up by ex- change of Rcgge trajectories in u- and t-channels.

Because Barger-Cline parametriza- t ion is so good, one would like to look for a solution to this dilemma. There are some possibilities: I) The resonances arise from Regge trajectories not explic- i t ly treated in the crossed channel. I t seems hard t o reconcile this picture with the Chew reciprocal bootstrap model. II) There are elementary particles which have not been included explicitly in the description but amount to gener- ating direct-channel resonances. Alter- natively, some of the resonances may be CDD poles, a rather unpleasant prospect. III) The resonances arise from the background integral in the Watson- Sommerfeld transformation. This seems to be a very interesting possibility to us. If this explanation is true, the Barger- Cline parametrization asserts an inter- esting dynamical si tuation over and above the usual hypothesis and needs further investigation.

In any event, the conclusions to be drawn from the simple parametrizat ion attempted (.%4) are not as straightforward as they appear at first glance.

* * *

We like to acknowledge interesting discussions with A. BAa~T, C. IDDINGS, D. KU~z and E. ROST.