Volume 2’7B, number 8 PHYSICS LETTERS 16 September 1968

THE PION CONSPIRATOR: A FURTHER EVIDENCE FOR ITS EXISTENCE AND A DISCUSSION ON ITS IMPLICATIONS IN MESON SPECTROSCOPY

P. DI VECCHIA, F. DRAG0 * Laboratori Nazionali di Frascati de1 CNEN, Frascati, Italy

C. FERRO FONTAN ** International Centre for Theoretical Physics, Trieste, Italy

and

R. ODORICO *** Istituto di Fisica Teorica dell’lJniversitil, Trieste, Italy

Received 19 July 1968

In previous papers [1,2] strong evidence for the existence of a class III pion conspiracy in n+ photo- production has been provided. This has been achieved by studying with the continuous moment sum rules [3] the amplitudes FL-) and Fh-) 1. In this note new evidence is presented for a Regge pole pion conspirator nc. In fact, analysing by the same technique the amplitude FIT) (to which A2 and nc can con- tribute) we have found, besides the A2, a Regge pole with the same trajectory previously determined for nc from the conspiring amplitude F$-). This result confirms the existence of the pion conspirator Regge pole.

We also find the Regge parameters of the A2, which provide the dominant contribution at small f; it results that the A2 chooses the Chew or the non-compensating mechanism when oA2 = 0 (t N - 0.60) $1.

Moreover, we discuss the connection between the existence of the 1~~ and the doublet structure of the A2 resonance [6], and give some arguments to classify the Al resonance as a recurrence of the first pion daughter. The possibility of considering the A3 [7] as a further recurrence of this trajectory is also discussed.

The sum rule used has the form:

G(Y) = (Vmax )-Y 1 jg fmax “0

[v2-i&‘YIm[exp(-+inY)F$-)(v,I)]dv-+cf [l+~p-~n][v~-v~]+Yj =

(1)

sin [&J(IY #) + y)] 21Mvmax a k(t)

ok(t) + Y (

SO >

where, at high energy

Fi-’ = c a (t)Pk(t) 1 + exp (-inor k)

k k sin 7ro k m4~/so) ok-1

and S o = 1 GeV’ .

As previously, the low-energy fit done by Walker [8] has been used for the evaluation of the integral. The range of t considered is -0.7 s t s 0.3.

* Sponsored in part by the Air Force Office of Scientific Research through the European Office of Aerospace Re- search, OAR. United States Air Force, under contract F 61052 67C 0084.

** On leave of absence from Departamento de Fisica. Facultad de Ciencias Exactas y Naturales. Universidad de Buenos Aires. Argentina.

*** Supported in part by Istituto Nazionale di Fisica Nucleare, $ We use the same notations as ref. 1 and 2. See also ref. 4.

Sottosezione di Trieste, Italy.

$1 This is in agreement with the indications obtained using integer moment sum rules [ 51.

521

Volume 27B, number 8 P H Y S I C S L E T T E R S 16 September 1968

F o r t > - 0 . 1 we f ind a c l e a r d o m i n a n c e of the A 2 po le ( s ee , f o r i n s t a n c e , in f ig . l a , ~)(y) a t t = 0.10). H o w e v e r , t he c u r v e s a r e no t p u r e o n e - p o l e and the i n t r o d u c t i o n of a s e c o n d po le d o e s not exp la in the d e v i a t i o n s f r o m the p u r e o n e - p o l e b e h a v i o u r ; we a r e o r i e n t e d to b e l i e v e tha t t h e s e d e v i a t i o n s a r e e s s e n - t i a l ly due to e r r o r s . We have then f i t t ed the c u r v e s in t h i s r e g i o n of t wi th t he A 2 po le only. We have

I

found OtA2(0 ) = 0.6 and OtA2(0) = 0.95. H o w e v e r t he e r r o r s a r e l a r g e . F r o m the i r r e g u l a r i t i e s of the ~ ( y ) ' s we m a y e s t i m a t e an e r r o r ~ 0.2 on (~A2(0), wh i l e c~2(0 ) m a y r e a l l y r a n g e b e t w e e n 0.35 and 1. T h e r e f o r e one has a r o u g h c o m p a t i b i l i t y wi th p r e v i o u s f i t s to ~ p r o d u c t i o n [9] $.

F o r t ~< - 0 . 4 5 a n o t h e r c o n t r i b u t i o n b e c o m e s d o m i n a n t . F r o m fig. lb , wh ich r e p r e s e n t s qS(y) a t t = = - 0 . 4 5 and t = - 0 . 7 0 , we can s e e $:~ tha t i t i s e s s e n t i a l l y a o n e - p o l e c o n t r i b u t i o n and tha t i t s t r a j e c t o r y i s s t r i k i n g l y s i m i l a r to tha t of the ~c, a s d e t e r m i n e d f r o m the c o n s p i r i n g F~ -) a m p l i t u d e $$$. In fig. l b t h e r e a r e a l s o r e p o r t e d the t w o - p o l e f i t s to the c u r v e s , f ix ing one of the p o l e s to have the ~c t r a j e c t o r y ; we f ind tha t the c o n t r i b u t i o n of the o t h e r po le , to be i n t e r p r e t e d a s t he A2, i s r a t h e r s m a l l and c h a n g e s s ign b e t w e e n t = - 0 . 4 5 and t = - 0 . 7 0 .

(~/sm ~ In f ig. 2a we r e p o r t ~(Oth2)fiA 2 and ~(C~c)fi~c (w h e re ~(o~) -= . z )(2M~max/So)°t) as a r e s u l t of

:~ We would also observe that all these fits assume that only A 2 is relevant in?7 production. This is justified by the small polarization observed in this process:<P> = +2 =L 7% at 11.2 GeV/c, averaged over the range 0.02 ~< - t --< ~< 0.35 (GeV/c) 2 [10]; however, a non-negligible weight of the ?r c cannot be excluded. Obviously this last poss ib i l ity could contemplate higher values for the A 2 t ra jec tory .

:~$ Disregarding the deviations for points at small y. which receive relevant contributions f rom the ra ther uncer - tain region near threshold.

$:~$ C~c(t) = -0.013 + 1.56 t, [2].

0.4

0.2

(lO

v

0.1

0.0

~e \ \ \

"\ \

I

t=O.lO

(a)

\ . .~ j . . ;~"~" I I I I

(b) •

I l I I I 0 1 2 3 4 5

Y

Fig. 1. a) The d isc re te points r ep re sen t qS(~) (see eq. (1)) at t = 0.10 and the continuous curve is its one-pole fit. b) The d isc re te points r ep resen t (b(~) at t - -0.45 and t - - 0.70. Continuous lines are two-pole fits to d)(~/) with one of the two poles (the Yc) constrained to have

ot = -0.013 + 1.56 t.

522

40O

~ 300

'3" :> 200

IO0

30O

[ ~ 200

100 (M

0

~ + + A 2 \ ' ~ o CONSPIRATOR

++~ (a) ++

+\ ~ o - O ~ O I - O - ~o'~'~~_

! I I I I l ! I I I -"

%~**~ (b)

+

i ' o : o ' ' ' ' ' " ' 0 2 -(~2 -0.4 -Q6

t ( 6 e V / c ) 2

Fig. 2. a) ~(aA2)~A2 and ~(ayc)~c determined from fits to ~b(~) (see the text). b) ~A2(t ) resulting from the assumption

0~A2 = 0.3 + 0.65 t.

Volume 27B. number 8 P H Y S I C S L E T T E R S 16 September 1968

t h e a b o v e - m e n t i o n e d t w o - p o l e f i t f o r 0 .15 --< - t < 0.70. T h e r e p o r t e d v a l u e s of ~ (OtA~)~A 2 f o r - t < 0 .15 h a v e b e e n d e t e r m i n e d f r o m a o n e - p o l e f i t to t he c u r v e s . In f ig . 2b we r e p o r t ~A2(t ) ffs r e s u l t s a s s u m i n g aA2(t ) = 0 .40 + 0 .65 t, a s g i v e n in r e f . 9. T h e o b t a i n e d f o r m of ~A2(t) i s , h o w e v e r , l a r g e l y i n d e p e n d e n t of t h i s a s s u m p t i o n , e s p e c i a l l y in t h e r e g i o n w h e r e OtA2 v a n i s h e s . F r o m f ig . 2b one can c l e a r l y r e a l i z e t h a t flA2 v a n i s h e s s o m e w h e r e b e t w e e n t = - 0 . 4 0 and t = - 0 . 6 . A s s o c i a t i n g t h i s v a n i s h i n g w i th t h e p r e s - e n c e of an (~ f a c t o r in t h e r e s i d u e , one can c o n c l u d e t h a t t he A 2 c h o o s e s t h e Chew o r t h e n o n - c o m p e n - s a t i n g m e c h a n i s m . On t h e o t h e r hand , t h e a b s e n c e of a d ip in 7/ p r o d u c t i o n s t r o n g l y f a v o u r s t he Chew m e c h a n i s m o v e r t h e n o n - c o m p e n s a t i n g one.

T h e v a n i s h i n g of t h e A 2 c o n t r i b u t i o n a l l o w s to s e e c l e a r l y t h e ~c, a n d t h i s new e v i d e n c e found f o r i t s e x i s t e n c e i s p e r h a p s t h e m o s t i n t e r e s t i n g r e s u l t of t he F t - ) a n a l y s i s .

W e would b r i e f l y d i s c u s s h e r e now t he p r e s e n c e of t h i s R e g g e po l e m a t c h e s t h e known m e s o n - r e s o - n a n c e s p e c t r o s c o p y .

T h e d o u b l e t s t r u c t u r e of t he A 2 r e s o n a n c e h a s b e e n c o n f i r m e d in t he r e c e n t e x p e r i m e n t by C r e n n e l l e t a l . [6]. If b o t h p e a k s h a v e t h e s a m e q u a n t u m n u m b e r s (which s e e m s to b e c o m p a t i b l e w i th e x p e r i - m e n t s ) , t h e s y s t e m of t h e A 2 and nc t r a j e c t o r i e s would p r o v i d e a n a t u r a l e x p l a n a t i o n f o r t he doub le t . H o w e v e r , one c a n n o t , a s u s u a l , e x t r a p o l a t e by s t r a i g h t l i n e s t h e two t r a j e c t o r i e s to t he t > 1 ( G e V / c ) 2 r e g i o n in o r d e r to p r e d i c t t h e m a s s e s . In f a c t . a s one m a y s e e f r o m f ig . 3a, t h e r e i s a d e l i c a t e p r o b l e m of trajectory crossing a r o u n d t ~ 0.5 ( G e V / c ) 2, w h i c h c a u s e s d e v i a t i o n s f r o m l i n e a r b e h a v i o u r f o r t he two t r a j e c t o r i e s . On t h e o t h e r h a n d , t he p r e s e n c e of a c u r v a t u r e in t h e A 2 t r a j e c t o r y , n e c e s s a r y to m a t c h t h e R e g g e f i t to p r o d u c t i o n w i th t he A 2 m a s s , i s a w e l l - k n o w n f a c t ( s e e , f o r i n s t a n c e , r e f . 9). O b v i o u s l y , t he w h o l e m a t t e r n e e d s f u r t h e r c l a r i f i c a t i o n .

T h e ~c c h a r a c t e r i s t i c s a l s o p r o v i d e a s u g g e s t i v e c l a s s i f i c a t i o n f o r t h e A 1. W e s h a l l i n d e e d g ive h e r e s o m e a r g u m e n t s to i d e n t i f y t he A 1 w i th t h e p ion d a u g h t e r . W h e n ot = 0 (t o ~ 0 .008) , t h e ~c c h o o s e s n o n - s e n s e [1,2]. T h e r e f o r e i t n e e d s a c o m p e n s a t i n g t r a j e c t o r y wi th ~ = - 1 a t t = t o. B e c a u s e of t he c o n - s p i r a c y of t h e p ion a n d t he nc a t t = 0 and the s m a l l n e s s of to, t he p ion d a u g h t e r h a s j u s t a n ~ a r o u n d - 1 f o r t = t o a n d t h e r i g h t q u a n t u m n u m b e r s to be t he n e c e s s a r y c o m p e n s a t i n g t r a j e c t o r y . W e a s s u m e t h a t t h i s i s j u s t t he c a s e . T h e n f r o m ~ d a u g h (0) = ~ ( 0 ) - 1 = ~ c ( 0 ) - 1 a n d C~daug h (to) - 1 i t f o l l o w s t h a t the p ion d a u g h t e r and t he c o n s p i r a t o r ~ [ r a j e c t o r i e s a r e p a r a l l e l in the r e g i o n ~ e a r t = 0 ( s ee f ig . 3b) [11]. Now, s i n c e t he p ion d a u g h t e r h a s no p r o b l e m of t r a j e c t o r y c r o s s i n g , i t s e e m s r e a s o n a b l e to a s s u m e f o r h e r a l i n e a r b e h a v i o u r up to t he r e s o n a n c e r e g i o n . If one d o e s so , one c r o s s e s ot = 1 f o r / t = 1130 MeV $ :~ , w h i l e MA1 = 1070 MeV wi th a wid th of 80 MeV.

W e would a l s o n o t e t h a t one can a l s o a c c o m m o d a t e on t h e s a m e t r a j e c t o r y t he A3, s u p p o s i n g t h a t i t h a s JP = 3 +. In f a c t , t h i s t r a j e c t o r y c r o s s e s a = 3 a t ~zt = 1600 MeV w h i l e MA3 = 1650 MeV wi th FA3 = = 110 MeV. T h i s i s an a l t e r n a t i v e to t he s u g g e s t i o n of L u b a t t i [12] to c o n s i d e r t h e A 3 a s the JP = 2- r e - c u r r e n c e of t he p ion t r a j e c t o r y : t a k i n g a 'n(0) = 0.65 (as d e t e r m i n e d in r e f . 2) one g e t s ot~ = 2 fo r / t = = 1760 MeV, w h i c h i s r e a s o n a b l e .

Conclztsion. M o r e d i r e c t e v i d e n c e f o r the e x i s t e n c e of t he p ion c o n s p i r a t o r h a s b e e n p r o v i d e d . M o r e - o v e r , i t h a s b e e n s h o w n t h a t i t i s no t a t r o u b l e s o m e o b j e c t bu t i s r a t h e r we l l i n t e g r a t e d in t h e e x i s t i n g Regge "panorama". It can provide a natural ex- planation for the doublet structure of the A 2 re- sonance and i t s c h a r a c t e r i s t i c s s e e m to s u g g e s t a t he A 1 a s a r e c u r r e n c e of t he f i r s t p ion d a u g h t e r . A l s o t h e A 3 can b e a c c o m m o d a t e d a s a f u r t h e r 2 r e c u r r e n c e of t h i s s a m e t r a j e c t o r y ; t h e r e a r e , h o w e v e r , o t h e r p o s s i b i l i t i e s and , f o r a d e f i n i t e c h o i c e , a d e t e r m i n a t i o n of i t s a n g u l a r m o m e n - t u m i s n e c e s s a r y .

As a b y - p r o d u c t we h a v e p r o v i d e d s o m e e v i - d e n c e fo r A 2 c h o o s i n g t he Chew m e c h a n i s m a t ( ~ = 0 .

I t i s a p l e a s u r e to t h a n k P r o f . L. B e r t o c c h i f o r h i s i n t e r e s t in t h i s w o r k a n d f o r c o n t i n u o u s

(x

A 2 doublet 1 s t X X •

i S

i • ss~ J~ t s s J tl 0 - -

t I $1S

• 1 ss ~ t~ J

Ao / I i t

1. 2. ( a )

i I t 1. 2.

(b)

$$$ See footnote on previous page.

Fig. 3. a) Cross ing of A 2 and ?T c t r a j ec to r i e s . b) See the text (for the sake of c lar i ty the t r a j e c to r i e s

are only schemat ic) .

523

Volume27B, number 8 P H Y S I C S L E T T E R S 16 September 1968

a s s i s t a n c e . We would a l s o thank P r o f . M. T o l l e r f o r f r u i t f u l d i s c u s s i o n s . One of u s (C. F. F . ) i s i n d e b t e d to P r o f e s s o r s Abdus S a l a m and P. Budini and the I. A. E . A . f o r k ind

h o s p i t a l i t y a t the I n t e r n a t i o n a l C e n t r e f o r T h e o r e t i c a l P h y s i c s , T r i e s t e .

References 1. A. Bietti. P. Di Vecchia, F. Drago and M. L. Paciel lo, Phys. Le t te rs 26B (1968) 457. 2. P. Di Vecchia. F. Drago, C. Fe r ro Fontgn, R. Odorico and M. L. Paciello, Phys. Le t te rs 27B (1968) 296. 3. M.G.Olsson, Phys. Le t t e r s26B (1968) 310;

A. Della Selva, L. Masper i and R.Odorico, Nuovo Cimento 54A (1968) 979. 4. J. S. Ball, W.R. F r a z e r and M. Jacob, Phys. Rev. Le t te rs 20 (1968) 518. 5. Shu-Yuan Chu and D.P .Roy . Phys. Rev. Le t te r s 20 (1968) 958;

K.Igi and S. Matsuda, in Proc . Topical Conf. on High energy coll isions of hadrons, CERN, January 1968; K. Igi, in Proc. of Meeting on Regge poles, Eugene (Oregon), March 1968.

6. G. Chikovani. M. N. Focacci , W. Kienzle, C. Lechanoine, B. Levrat , B. Maglic, M. Martin, P. SchUbelin, L. Dubal, M . F i s c h e r . P . G r i e d e r , H.A.Neal and C.Nef, Phys. Le t te rs 25H (1967) 44; D . J .Crenne l l , U.Karshon, Kwan Wu Lai, J . M . S c a r r and I .O.Ski l l icorn, Phys. Rev. Let ters 20 (1968) 1318.

7. lr A (1640) in Data on par t ic les and resonant s tates , UCRL-8030, printed at LRL and CERN. 8. R. L. Walker. A partial wave analysis of single pion photoproduction, Caltech prepr in t 68/158. 9. R . J .N . Phil l ips and W. Rarita, Phys. Rev. Le t te rs 15 (1965) 807; Phys. Le t te rs 19 (1966) 598.

10. Report of the Sac lay-Orsay-P i sa group to the Heidelberg Conference (1967). 11. This argument has been suggested to us by Prof. M. Toller . 12. H.J . Lubatti, Heidelberg Conference (1967), discussion following the repor t of I. Butterworth.

524