SPIN EFFECTS IN THE T FAMILY AS SEEN IN e+e- INTERACTIONS

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SPIN EFFECTS IN THE T FAMILY AS SEEN IN e+e- INTERACTIONS

A. Fridman

To cite this version:

A. Fridman. SPIN EFFECTS IN THE T FAMILY AS SEEN IN e+e- INTERACTIONS. Journal de Physique Colloques, 1985, 46 (C2), pp.C2-157-C2-174. �10.1051/jphyscol:1985215�. �jpa-00224526�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°2, Tome 46, février 1985 page C2-157

SPIN EFFECTS IN THE T FAMILY AS SEEN IN e+e ~ INTERACTIONS A. Fridman

DPKPB, Centre d'Etudes Nualeaires de Saelay, 91191 Gif-sur-Yvette, France and

University of Hamburg, I. Institut fttr ExperimentaVphysik, Hamburg, F.R.G.

Résumé - Quelques effets de spin liés aux transitions radiatives dans la famille du T sont discutés. Ceci couvre les résultats expérimentaux sur les états Xv. et xû ÇRi sont résumés et discutés dans le cadre des modèles â potentiel. En outre,quelques

commentaires sont présentés sur la structure hyperfine,domaine dans lequel n'existent pas de résultats expérimentaux actuellement.

Finalement l'utilité de la polarisation transverse des faisceaux pour la détermination des spins des états Xv, (Xv.^ e s t brièvement discutée.

Abstract - Some spin effects related to radiative transitions within the T family are discussed. This covers the experimental results obtained on the Xv. ari(3 X K states which are reviewed and discussed in the framework of potential models. In addition, some comments are presented about hyperfine splitting for which there exist no data as yet. Finally the usefulness of the transverse polarization of the beam particles for determining the spin of the xb (Xb) states is briefly discussed.

1 — Introduction

In this report we will essentially discuss the radiative

transitions observed in the T family built from bound bb quarks. In this context, we will investigate some of the spin effects related to these transitions. All the experimental information which will be used throughout was obtained from the CESR (Cornell) and DORIS (DESY) e e~

storage rings.

Fig. 1 presents the T family below the free beauty threshold in the form of an energy level diagram with some of the observed transitions

(full lines). We use the usual notation which labels the triplet (singlet) S-states by the T = T(1S), T' H T(2S), T" E T(3S) ... (nb, n', n" ...) symbols. In addition we will also use the spectroscopic

2 S+1

notation n LT where n is the radial quantum number associated with the bb system (L is the relative orbital momentum, S = 0,1 the total Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985215

C2-158 JOURNAL DE PHYSIQUE

spin of the quarks and J the total spin of the system).

Here it is in place to make some general comments about the relation between the energy level ordering in potential models when fine and hyperfine splittings are neglected. There is a new theorem (1) stating that, if for any r, one has a spherically symmetric potential V(r) such that

the energy levels obey the inequalities:

Here r is the modulus of the relative coordinate of the two quarks and N is the principal quantum number given by:

N = n.

nodes +l+L

with nnodes (n = n nodes +l) being the number of nodes of the radial wave function. As the general features of the charmonium and the T family are well described by a Cornell type of (static) potential(2) V(r) = -z/r + ^ar (X and

a

for any r. This implies for instance that

or equivalently

which appear to be verified by the levels seen in the charmonium and the T family

.

Thus the above theorem gives a simple way to remember some

BB threshold 7'- BE

F i g . 1

-

p r o p e r t i e s of t h e l e v e l o r d e r i n g i n quarkonium. The f i n e and h y p e r f i n e s p l i t t i n g s o b s e r v e d i n t h e d a t a a r e b e l i e v e d t o r e s u l t from r e l a t i - v i s t i c c o r r e c t i o n s t o t h e s t a t i c p o t e n t i a l , a s , f o r example, g i v e n above. T h i s w i l l be b r i e f l y d i s c u s s e d i n S e c t i o n 2 , whereas S e c t i o n 3 w i l l be d e v o t e d t o t h e f i n e s p l i t t i n g o b s e r v e d i n t h e T f a m i l y f o r t h e

1 3 p J and 2 3 ~ J s t a t e s .

We w i l l g i v e i n S e c t i o n 4 some b r i e f comments on t h e h y p e r f i n e s p l i t t i n g f o r which t h e r e i s no e x p e r i m e n t a l i n f o r m a t i o n y e t . We w i l l i n p a r t i c u l a r d i s c u s s t h e importance a t t a c h e d t o t h e d i s c o v e r y o f t h e ' P I s t a t e s . A t t h e T(2S) t h e beam p a r t i c l e s a r e t r a n s v e r s e l y

p o l a r i z e d a t t h e DORIS s t o r a g e r i n g s . T h i s w i l l g i v e u s t h e opportu- n i t y t o d i s c u s s i n S e c t i o n 5 t h e u s e f u l n e s s of t r a n s v e r s e p o l a r i z a - t i o n . f o r measuring t h e s p i n o f t h e

xb

xb ^{+ y T} ( I S ) and T ( I S ) + V + V - ( L E e , p )

.

2- Q u a r k o n i a and P o t e n t i a l Models

A s a g u i d e f o r t h e d i s c u s s i o n s o f t h e e x p e r i m e n t a l r e s u l t s , l e t u s c o n s i d e r a C o r n e l l t y p e of p o t e n t i a l w r i t t e n t h i s t i m e a s :

The f i r s t t e r m c o r r e s p o n d s t o t h e one gluon exchange (as i s t h e QCD r u n n i n g c o u p l i n g ) , whereas t h e c o n f i n i n g p a r t h a s two p i e c e s c o r r e s -

C2-160 JOURNAL DE PHYSIQUE

ponding t o a v e c t o r ( g v r ) o r a s c a l a r ( g s r ) exchange i n t h e t - c h a n n e l ( s e e F i g . 2 ) . R e l a t i v i s t i c c o r r e c t i o n s t o o r d e r v/c have been i n v e s t i - g a t e d by many a u t h o r s , some o f them b e i n g l i s t e d i n Ref. 3 t o 7 . Here we w i l l simply d i s c u s s t h e s p i n dependent c o r r e c t i o n (VSD) t o t h e p o t e n t i a l g i v e n above. T h i s s p i n dependent e x p r e s s i o n can be w r i t t e n

as :

one g l u o n exchange, s h o r t r a n g e

c o n f i n i n g p a r t , long r a n g e

Here, mb i s t h e b quark mass. The t e n s o r o p e r a t o r S 1 2 i s g i v e n by:

vector: l gluon, g, scalar : gs

F i g . 2

-

where

z1

-f +

The L O S and S I Z t e r m s a r e r e s p o n s i b l e f o r t h e f i n e s p l i t t i n g , w h i l e t h e terms c o n t a i n i n g ^{- 2 2} g e n e r a t e t h e h y p e r f i n e s p l i t t i n g - . The e x p e r i m e n t a l d a t a a r e u s u a l l y d e s c r i b e d u s i n g

qv = 0 (See f o r i n s t a n c e Ref. 4 , 8 and 9 ) o r

g v - g s - = 0 . 1 5 ~ e ~ ~ ^{( l1} " ^I# Ref. 6 and 10)

I f one chooses gv = 0 , t h e r e i s o n l y one term i n t h e l o n g r a n g e p a r t + +

o f VSDl which w i l l t e n d t o d e c r e a s e t h e L - S c o u p l i n g . Moreover, i n

t h i s c a s e , t h e h y p e r f i n e s p l i t t i n g w i l l o n l y a r i s e from t h e ( s o - c a l l e d

- + + -+

c o n t a c t ) term: 32nas/ (gm;) 6 ( r ) S1 ss2. T h i s y i e l d s a h y p e r f i n e s p l i t t i n g :

(11)

$(O) b e i n g t h e wave f u n c t i o n a t t h e o r i g i n

.

With t h e s p i n d e p e n d e n t c o r r e c t i o n g i v e n by VSD, t h e masses o f t h e b 5 bound s t a t e s c a n be e x p r e s s e d a s :

The a , b and c p a r a m e t e r s depend o n t h e chosen p o t e n t i a l , w h i l e K i s a s c a l e f a c t o r . With t h e above c h o i c e o f qv and gs o r , i n t h e frame- work o f u s u a l p o t e n t i a l models, one h a s a>1.2b > 0 . T h i s g i v e s t h e s p i n o r d e r i n g o f t h e

xb

For 3~ s t a t e s , t h e p a r t 0 3 2 ) +K i n VSD r e p r e s e n t s simply t h e c e n t e r o f g r a v i t y o f t h e ^3~ m u l t i p l e t which by d e f i n i t i o n i s g i v e n

J by:

The e v a l u a t i o n o f <L*s> ^{and < S 1 2 >} f o r t h e 3 ~ Js t a t e s l e a d s t o :

from which one c a n e x t r a c t :

and

By measuring t h e masses w i t h i n a ^{3 ~ J}m u l t i p l e t , one i s t h u s a b l e t o a p p r e c i a t e t h e c o n t r i b u t i o n o f t h e Z - g c o u p l i n g and t h e t e n s o r f o r c e s t o t h e f i n e s p l i t t i n g . Note t h a t t h e measurement o f a and b do n o t depend o n t h e s y s t e m a t i c u n c e r t a i n t y i n t h e mass s c a l e of t h e ^{3 ~ J} s t a t e s . The q u a n t i t y ^!

JOURNAL DE PHYSIQUE

i s a l s o v e r y s e n s i t i v e t o t h e p r e s e n c e o f t e n s o r f o r c e s ( R = 2 i f b = 0 whereas R < 2 i f b

*

f o r c e s . _only

_1.5 -.

-

_sI2

F i g . 3 - The f i n e s p l i t t i n g p a t t e r n w i t h o u t ( l e f t ) and w i t h ( r i g h t ) t e n s o r f o r c e s .

3 - F i n e S p l i t t i n g

B e f o r e summarizing t h e e x i s t i n g d a t a , l e t u s p r e s e n t i n some d e t a i l t h e r e s u l t s o b t a i n e d by t h e C r y s t a l B a l l C o l l a b o r a t i o n ( 1 2 f . F i g . 4 p r e s e n t s t h e i n c l u s i v e photon spectrum i n t h e T(2S) decay (from which t h e can be e x t r a c t e d ) . The t h r e e photon l i n e s o f lower e n e r g y r e s u l t from t h e El t r a n s i t i o n s T ( 2 S ) + y13pJ and a r e w e l l s e p a r a t e d from e a c h o t h e r . The f o u r t h photon peak r e p r e s e n t s t h e Doppler broadened enhancement d u e t o l i n e s 4 and 5 i n F i g . 4 . The photon e n e r g y spectrum h a s been f i t t e d w i t h a polynomial background ( f i f t h o r d e r ) , 3 g a u s s i a n s ( f o r t h e l i n e s 1 , 2, 3 i n F i g . 4 ) w i t h f i x e d w i d t h s g i v e n by t h e e n e r g y r e s o l u t i o n o f t h e C r y s t a l B a l l [ t h e d i s p e r s i o n o f o r a photon o f e n e r g y E i s o/E = O . O Z ~ / ( E ( G ~ V ) ) ' / ' ] and two a d d i t i o n a l g a u s s i a n s w i t h means c a l c u l a t e d assuming t h e c a s c a d e h y p o t h e s e s i n o r d e r t o d e s c r i b e t h e Doppler broadened peak ( l i n e 4 + 5 ) . The r e s u l t s o b t a i n e d f o r t h e e n e r g i e s o f t h e t h r e e photon l i n e s a r e g i v e n i n Table 1.

A measurement o f t h e photon l i n e s h a s a l s o been done by means o f t h e c a s c a d e r e a c t i o n s :

l e a d i n g t o t h e f i n a l s t a t e y y ~ f ~ - ( R = e , ~ ) . On t h e b a s i s o f 200 K (12) T ( 2 S ) decay e v e n t s accumulated by t h e C r y s t a l B a l l C o l l a b o r a t i o n ,

we p r e s e n t i n F i g . 5 t h e e n e r g y d i s t r i b u t i o n o f t h e l e s s e n e r g e t i c photon (Elow) o b t a i n e d from t h e c a s c a d e e v e n t s . The yyR + R- e v e n t s were s e l e c t e d a f t e r k i n e m a t i c a l f i t t i n g and r e q u i r i n g , i n a d d i t i o n , t h a t t h e mass (M,) r e c o i l i n g a g a i n s t t h e two photon system s h o u l d be c o m p a t i b l e w i t h t h e T (IS) mass. I n p r a c t i c e , t h e f o l l o w i n g c u t was

5000 PRELIMINARY 4000

D 3000

y

a

3

c 400 S

t

Q

0

-200

60 80 100 200 400 600 800

r

F i g . 4

-

a p p l i e d

500 < M[T(2S) 1

-

M[T (2 s ) 1 i s t h e mass o f t h e T ( 2 s ) r e s o n a n c e . D e t a i l s on t h e s e l e c t i o n p r o c e d u r e c a n b e found i n r e f e r e n c e ( 1 2 ) . One s e e s from F i g . 5 t h a t o n l y two photon l i n e s a r e p r e s e n t i n t h e c a s c a d e e v e n t s a s indeed e x p e c t e d from t h e o r e t i c a l c o n s i d e r a t i o n s ( 1 3 ) . The spectrum was f i t t e d w i t h a f l a t background and two g a u h s i a n s w i t h f i x e d w i d t h s g i v e n by t h e e n e r g y r e s o l u t i o n o f t h e C r y s t a l B a l l . A s a r e s u l t from t h i s f i t , o n e o b t a i n s t h e f o l l o w i n y v a l u e s for t h e l i n e s :

Elow l = 105.6+1 .4+2.0 MeV

~i~~

Here t h e f i r s t e r r o r dehotes always t h e s t a t i s t i c a l o n e and t h e second e r r o r t h a t due t o s y s t e m a t i c s .

C2-164 JOURNAL DE PHYSIQUE

These v a l u e s a r e i n agreement w i t h t h o s e o b t a i n e d from t h e a n a l y s i s o f t h e i n c l u s i v e photon spectrum ( l i n e 1 and 2 i n F i g . 4 , s e e a l s o

Table 1 )

.

15.0 ^{I l r l I ~ t ~ ~ ~ r v ~ r ~ v ~ +}

PRELIMINARY

-

F i g . 5

-

T a b l e 1 g i v e s a c o m p i l a t i o n o f t h e e n e r g i e s o f t h e photon l i n e s a s o b t a i n e d by t h e ARGUS(14), CLEO(15), C r y s t a l B a l l ( l 2 , and CUSB ( 1 6 ) c o l l a b o r a t i o n s . I n t h i s t a b l e a r e a l s o g i v e n t h e weighted a v e r a g e s o b t a i n e d from t h e s e e x p e r i m e n t s . Using t h e s e l a t t e r v a l u e s and assuming t h a t t h e s p i n o f xb s t a t e s correspond t o t h e t h e o r e t i c a l p r e d i c t i o n s ( s e e F i g . 1 ) , one o b t a i n s t h e v a r i o u s p a r a m e t e r s

c h a r a c t e r i z i n g t h e f i n e s p l i t t i n g o f t h e 1 3 p J s t a t e s (Table 2 ) . One n o t i c e s , i n p a r t i c u l a r , t h e i m p o r t a n t c o n t r i b u t i o n o f t e n s o r f o r c e s t o t h i s s p l i t t i n g . For t h e T(3S) ^-+ yz3pJ t r a n s i t i o n s , t h e r e a r e o n l y r e s u l t s from o n e experiment ( l 7 ,

.

.

I n F i g . 6 w e have c o m p a r e d , t h e measured mass d i f f e r e n c e s M 2 - M 1 and M,-MO w i t h v a r i o u s t h e o r e t i c a l p r e d i c t i o n s . The p r e d i c t i o n s a r e d i s t r i b u t e d o v e r a r e l a t i v e l y s m a l l r a n g e , b u t some o f them c l e a r l y d i s a g r e e w i t h t h e v a l u e s o b t a i n e d from t h e weighted a v e r a g e s o f t h e photon e n e r g i e s . The measured b r a n c h i n g r a t i o s o f t h e observed

t r a n s i t i o n s a r e g i v e n i n Table 3 t o g e t h e r w i t h t h e i r weighted averages.

C2-165

Table 1 - The energies (MeV) of the photon lines obtained from the T(2S) decays as observed by various experiments together with their weighted averages <EY>. The energies of the daughter lines (lines 4+5 in Fig. 4) are not given.

R e f e r e n c e C U S B< 1 6 )

C L E O( 1 5>

A R G U S ( 1 4 )

C R Y S T A L , . , , BALL1 " '

< E >

Y

A s s u m e d t r a n s i t i o n

2S - 1!P2

1 0 8 . 2 1 0 . 3 : 2 . 0 1 0 9 . 5 1 0 . 7 : 1 . 0 109.0+1 = 1 . 0 1 0 8 . 2 1 0 . 7 : 4 . 0

109.011.1

2 S •+ 1 3P i 1 2 8 . 1 1 0 . 4 + 3 . 0 1 2 9 . 0 t 0 . 8 ± 1 .0 1 2 9 . 8 1 0 . 8 1 1 . 0 1 2 7 . 1 1 0 . 8 1 4 . 0

1 2 9 . 1 1 1 . 2

2S -> 13P0

1 4 9 . 4 + 0 . 7 1 5 . 0 158 17 11 1 4 7 . 2 1 1 . 4 + 1 . 0 * 160 + 2 . 4 1 4 . 0

1 5 4 . 8 1 3 . 7

* This value has not been taken in the calculation of the weighted average because of the small statistical significance of the observed signal.

Table 2 - Quantities characterizing the fine splitting of the Xjj an d Xv. stated (see text) .

a (MeV)

b (MeV)

„ _ M 2 - M 1 M i - M o

T(2S)-+Yl 3P j 12.8 + 1 . 4

2.2 ± 0 . 9 0 . 7 8 + 0 . 1 3

T ( 3 S ) - + Y 23P j 9.4 + 2 . 5 1.4 + 1 . 5 0 . 8 5 1 0 . 4 0

Table 3 - Experimental branching ratios (percent) together with their weighted averages [<Br>] for the T(2S) -»- y1 3P j transitions. The data are compared with some theoretical predictions for which the errors are essentially due to the uncertainty in the total width of the T (2S) resonance.

R e f e r e n c e C U S B< 1 6 )' C L E O *1 5 )

A R G U S( 1 4' C R Y S T A L . ,

B A L L1 ^

< Br>

B u c h m u l l e r ' M o x h a y '5' R o s n e r M c C l a r y( 4 )

B y e r s

A s s u m e d t r a n s i t i o n 2S •+ 1 P2

6 . 1 + 1 . 4 1 0 . 8 1 1 . 8 1 2 . 1

8 . 9 ± 3 . 0 ± 1 . 2 6 . 0 1 0 . 7 1 0 . 9

6 . 5 1 1 . 0 7.4 + 1 .2 7 . 4 * 1 . 2 5 . 0 1 0 . 8

2S + 1 P, 5.9+1.4 8.011.7+1.6 8.8+2.211.0 6.610.811,0

6.6H .0 9.111.4 7.1+1.1 4.310.7

2S + 1 Po 3.5+1.4 4.412.310.9 4.011.811 .0 2.9l0.8'+0.9

3.4+1 .0 6.8±1.1 3.4+0.5 1.8+0.3

*)Phys. Lett. 112B, 479 (1982)

JOURNAL DE PHYSIQUE

Experiment

E i c h t e n , F e i n b e r g ,

Ref. 3 Buchmiiller ,

PL 112B, 479 ( 1 982) Gupta e t a l . Ref. 8

McClacy,Byers, Ref. 4

Moxhay, Rosner, Ref. 5

Bander e t a l . , PL 134B, 258

(1 984)

F i g . 6

-

From t h e t h e o r e t i c a l p r e d i c t i o n s g i v e n i n t h e same t a b l e , o n e n o t e s t h a t t h o s e g i v e n by Moxhay and R o ~ n e r ' ~ ) a g r e e t h e b e s t w i t h t h e weighted a v e r a g e s .

To c o n c l u d e t h i s b r i e f comparison, one can s a y t h a t t h e r e i s room f o r improvement i n b o t h e x p e r i m e n t s and t h e o r y . P r e c i s e measurements

o f P l o I 1 , z and b r a n c h i n g r a t i o s w i l l c e r t a i n l y h e l p t o improve t h e t h e o r e t i c a l d e s c r i p t i o n o f quarkonium f e a t u r e s . I n t h i s r e s p e c t , one should n o t e t h a t t h e s t u d y o f toponiurn

(tz

.

4

-

A s y e t t h e r e is no e x p e r i m e n t a l i n f o r m a t i o n a b o u t h y p e r f i n e s p l i t t i n g i n t h e T f a m i l y . T a b l e 4 , t a k e n from Ref.20 , g i v e s t h e o r e t i c a l e s t i m a t e s f o r t h e photon energy ( k ) f o r t h e allowed M1 t r a n s i t i o n T ( 1 S ) ⁺ yqb a s w e l l a s b r a n c h i n g r a t i o s . One s e e s from t h i s t a b l e t h a t t h e p r e d i c t e d photon e n e r g i e s a r e r a t h e r s m a l l ( l e a d i n g t o s m a l l b r a n c h i n g r a t i o s a s t h e t r a n s i t i o n p r o b a b i l i t y i s

p r o p o r t i o n a l t o k 3 ) . Thus t h e d i s c o v e r y o f t h e T(1S) ⁺ yqb t r a n s i t i o n would r e q u i r e l a r g e s t a t i s t i c s , a s i d e from t h e f a c t t h a t it i s

always d i f f i c u l t t o d e t e c t low energy photons w i t h g r e a t e f f i c i e n c y . The s i t u a t i o n may even be worse f o r t h e T(2S) ⁺ y q i and

T(3S) ⁺ yqg ( a l l o w e d M1) t r a n s i t i o n s . Indeed w i t h a s p i n dependent p o t e n t i a l a s g i v e n i n S e c t i o n 2, t h e t r i p l e t - s i n g l e t mass d i f f e r e n c e w i l l be o f t h e form

where F depends on t h e chosen p o t e n t i a l ( s e e t h e e x p r e s s i o n o f VsD i n S e c t i o n 2 ) . Here, $ n ( 0 ) d e n o t e s t h e wave f u n c t i o n a t t h e o r i g i n f o r b 5 i n t h e S - s t a t e ( i . e . t h e T(nS) r e s o n a n c e ) . I f t h e f i r s t t e r m i s t h e dominant one, t h e n AMH n w i l l d e c r e a s e i n t h e same way a s l $ n ( 0 ) 1 2 does when t h e r a d i a l quantum number n i s i n c r e a s i n g . The b e h a v i o r of

\ $ n ( 0 ) l 2 i s known from t h e o b s e r v a t i o n t h a t t h e l e p t o n i c w i d t h s

rn ( T (nS) ⁺ R + % - ) - which a r e p r o p o r t i o n a l t o l $ n ( 0 ) 1 v i a t h e Van Royen-Weisskopf formula - a r e d e c r e a s i n g when t h e r a d i a l e x c i t a t i o n i n c r e a s e s (21 ⁾

Another p o s s i b i l i t y o f o b s e r v i n g t h e v a r i o u s n'so s t a t e s would c o n s i s t o f d e t e c t i n g t h e h i n d e r e d M1 t r a n s i t i o n s ( F i g . 1 ) . The photon energy i s h i g h e r t h a n i n t h e a l l o w e d M1 t r a n s i t i o n s and c a n t h e r e f o r e be d e t e c t e d much more e a s i l y . A s t h e M1 t r a n s i t i o n s a r e p r o p o r t i o n a l

C2-168 JOURNAL DE PHYSIQUE

Table 4 - Photon energies and branching ratios estimates for the allowed M1 transition T(1S) -»• ynb (taken from Ref. 20).

Reference

Steiger, Phys. Lett.

1298, 335 (1983) Buchmuller et al.

Phys. Rev. D24, 182 (1981)

Leutwyler, Phys. Lett.

888, 447 (1981) Moxhay, Rosner, Phys. Rev. D28, 1132

(1983)

Martin, Phys. Lett.

100B, 511 (1983); and Reinders et al., QCD sum rules, NP B186, 109

(1981)

Eichten, Feinberg, Phys. Rev. D23, 2724,

(1981)

Voloshin, QCD sun rules (See Martin, Paris Conf.

1982)

Iwao, Yamawaki, QCD sum rules, Univ. of Rochester prepr. UR 771

McClary, Byers UCLA/821/TEP/1 2

Photon Energy(MeV)

22-28

46

50±15

57

60

85

90

100

101

BR(T(1S) - Ynb) (1.1-2.2)x10~5

(9.7±0.2)x10~s

l1-2-0.'8»*10""

(1 .9±0.4)x10-'

(2.2±0.4)x10~'-

(6.1±1 .2) xio"1"

(7.3±1 ^(xlO"1"

(1.0±0.2)x10"3

(1.0±0.2)x10~3

to k3, the hindered transition can hcive a sizeable branching ratio despite the fact, that its matrix element squared has a small value ( 2 2 ).

Let us now turn to the 1P1 states. From the expression giving V one sees that M will be proportional to a term containing |t|/(0) [2

and another one proportional to g . Here, iMO) represents the wave function at the origin when there is no L*S coupling or tensor forces.

But for P-wave, one has \j> (0) = 0. Therefore the hyperfine splitting between the n3PT center of gravity and the n*Pi state will be

proportional to g , i.e.

Mc o g( n3P j ) - M(nlPi) « gv

Thus the measurement of the masses of the n3P j and n'Pi states would allow a test of the long range part of the b-b potential.

The ' p 1 s t a t e s a r e , however, d i f f i c u l t t o d e t e c t a s t h e y have t h e same c h a r g e c o n j u g a t i o n a s t h e T ( n S ) r e s o n a n c e s . F i g . 7 p r e s e n t s some t r a n s i t i o n s which would a l l o w one t o r e a c h the l l p l s t a t e . Most o f them i n v o l v e 2 t r a n s i t i o n s from t h e T ( n S ) s t a t e s and l e a d t h e r e f o r e t o s m a l l b r a n c h i n g r a t i o s . The T(3S) + T T ~ ' P ~ t r a n s i t i o n , however, i s e x p e c t e d t o b e of t h e o r d e r o f a 1 % (Ref. 2 3 ) . Moreover, t h e ' P , s t a t e c a n decay v i a an E l t r a n s i t i o n t o t h e n b ( F i g . 7) w i t h a b r a n c h i n g

F i g . 7

-

r a t i o o f a 40% ( R e f . 2 4 ) . Thus t h e s u b s e q u e n t decays

T(3S) + T T I ' P ~ + n y q b may l e a d t o an e a s i l y i d e n t i f i a b l e f i n a l s t a t e which c o u l d a l l o w o b s e r v a t i o n a t t h e same t i m e o f t h e l l p l and t h e nb. Also h e r e , a l a r g e s t a t i s t i c s experiment would be needed (600 t o 1000 K e v e n t s ) , r e q u i r i n g a few y e a r s o f r u n n i n g w i t h t h e a v a i l a b l e l u m i n o s i t i e s o f a c t u a l s t o r a g e r i n g s .

C2-170 JOURNAL DE PHYSIQUE

5 - T r a n s v e r s e P o l a r i z a t i o n and Cascade E v e n t s

The measurement o f t h e s p i n of t h e xb ^{( o r}

x i )

T(2S) ⁺ yxb t r a n s i t i o n c a n , i n p r i n c i p l e , l e a d t o a s p i n d e t e r m i n a t i o n s i n c e :

Here, Q i s t h e a n g l e of t h e photon d e f i n e d w i t h r e s p e c t t o t h e e'e- d i r e c t i o n ( t h e z a x i s ) , Y cpy i s i t s a z i m u t a l a n g l e around t h e z a x i s and P i s t h e t r a n s v e r s e p o l a r i z a t i o n o f each beam. A t t h e T ( 2 S ) , t h e p o l a r i z a t i o n a t t h e D O R I S s t o r a g e r i n g i s l a r g e , namely P = 0.73?0.04.

T h i s v a l u e i s a n a v e r a g e o b t a i n e d by t h e C r y s t a l B a l l C o l l a b o r a t i o n from a s t u d y o f t h e a z i m u t a l d i s t r i b u t i o n o f muons i n t h e e+e- ⁺ p+v- r e a c t i o n .

One a l r e a d y s e e s from t h e above f o r m u l a t h a t t h e p o l a r i z a t i o n a d d s some r e d u n d a n t i n f o r m a t i o n a s t h e s p i n o f t h e e X b s t a t e s c o u l d b e d e t e r m i n e d from t h e c o s 0 d i s t r i b u t i o n a l o n e . T h i s r e d u n d a n t informa- t i o n i s , however, v e r y u s e f u l s i n c e d e t e c t o r s g e n e r a l l y have a l i m i t e d Y

( f u l l ) a c c e p t a n c e i n O (cp ) . U n f o r t u n a t e l y , i n t h e i n c l u s i v e photon

Y Y

spectrum t h e t h r e e photon l i n e s c o r r e s p o n d i n g t o t h e T ( 2 s ) -+ y1 3~

t r a n s i t i o n s s i t o n a l a r g e background ( s e e f o r i n s t a n c e F i g . 4 ) . I n t h i s c a s e , it i s t h e r e f o r e d i f f i c u l t t o s t u d y t h e a n g u l a r d i s t r i b u t i o n o f t h e photon. I n c o n t r a s t , t h e y y ~ + R - ( R E e , p ) f i n a l s t a t e s o b t a i n e d from t h e c a s c a d e r e a c t i o n s a r e p r a c t i c a l l y f r e e from background (see F i g . 5 ) .

The sample o f T ( 2 s ) -+ yy R+%- e v e n t s s u f f e r s , however, from low s t a t i s t i c s . T h e r e f o r e f o r a s p i n d e t e r m i n a t i o n o f t h e xb s t a t e s , one i s l e d t o u s e a l l t h e a n g u l a r i n f o r m a t i o n s o f t h e f i n a l s t a t e

p a r t i c l e s u s i n g a maximum l i k e l i h o o d t e c h n i q u e . Such a n a n a l y s i s i s now underway by t h e C r y s t a l B a l l C o l l a b o r a t i o n , u s i n g t h e f u l l a n g u l a r d i s t r i b u t i o n s W J ( R l , . . . , R ~ ) a s c a l c u l a t e d i n r e f e r e n c e ( 2 5 ) .

I n o r d e r t o i n v e s t i g a t e t h e u s e f u l h e s s o f p o l a r i z a t i o n i n t h i s c a s e l e t u s d i s c u s s i n some d e t a i l t h e p r o c e d u r e which can be u s e d f o r s p i n d e t e r r n i n a t i o n ~ ' ~ ~ )

.

J ' = JTEST i s t r u e . Using t h e d a t a one t h e n c a l c u l a t e s , t h e l o g a r i t h m

i i i

o f t h e l i k e l i h o o d f u n c t i o n L::~= f : W J , ( Q l , Q 2 , ...Q. ) , N b e i n g t h e i = l

t o t a l number o f a v a i l a b l e e v e n t s . T h i s v a l u e may t h e n be compared w i t h Monte C a r l o p r e d i c t i o n s o b t a i n e d w i t h v a r i o u s J v a l u e s , e a c h of

F i g . 8

-

.

which i s t e s t e d a g a i n s t t h e same h y p o t h e s i s , J' = ^J~~~~ The

p r e d i c t i o n s a r e o b t a i n e d by g e n e r a t i n g Monte C a r l o e x p e r i m e n t s , e a c h e x p e r i m e n t h a v i n g N e v e n t s ( t h e same number a s i n t h e d a t a ) a c c o r d i n g t o t h e t h e o r e t i c a l W J ( Q l ,

...,

C2-172 JOURNAL DE PHYSIQUE

a c c e p t a n c e p r o p e r t i e s o f t h e d e t e c t o r . Thus one o b t a i n s from t h e Monte C a r l o e x p e r i m e n t s from each J, a d i s t r i b u t i o n o f I n LJ v a l u e s which c a n be compared w i t h t h e d a t a p o i n t . A s an example, w e p r e s e n t i n F i g . 8 f o r v a r i o u s P v a l u e s t h e d i s t r i b u t i o n s o f t h e LJ v a l u e s t e s t e d f o r JTEST = 0. The c u r v e s a r e o b t a i n e d from a l a r g e number o f Monte C a r l o e x p e r i m e n t s ( a b o u t 1800) h a v i n g each N = 50 e v e n t s . One

s e e s from t h i s p l o t t h a t t h e s e p a r a t i o n between t h e < I n LJ> f o r

d i f f e r e n t J becomes l a r g e r when P i n c r e a s e s . Thus t h e s p i n a s s i g n m e n t s a p p e a r t o become e a s i e r w i t h l a r g e t r a n s v e r s e p o l a r i z a t i o n . T h i s c a n b e s e e n i n F i g . 9 i n a somewhat more q u a n t i t a t i v e manner. T h i s f i g u r e

BEAM POLARIZATION

F i g . 9

-

p r e s e n t s t h e a v e r a g e and t h e d i s p e r s i o n (aJ) o f t h e d i s t r i b u t i o n o f t h e I n LJ v a l u e s f o r v a r i o u s s p i n s and t e s t e d h y p o t h e s e s . One s e e s from t h e p l o t t h a t f o r JTEST = 0 , l t h e s e p a r a t i o n between t h e < I n LJ>

i n c r e a s e s w i t h P f a s t e r t h a n t h e c o r r e s p o n d i n g oJ do. I n t h i s r e s p e c t , a l a r g e r t r a n s v e r s e p o l a r i z a t i o n i s advantageous. F o r t h e c a s e

JTEST = 2 , t h e s i t u a t i o n i s n o t improved when P i s i n c r e a s e d .

6

-

We have d i s c u s s e d some s p i n e f f e c t s r e l a t e d t o r a d i a t i v e t r a n s i t i o n s w i t h i n t h e T f a m i l y . The a v a i l a b l e e x p e r i m e n t a l i n f o r m a t i o n a b o u t t h e f i n e s p l i t t i n g o f t h e

xb

r e s o l u t i o n and h i g h s t a t i s t i c s e x p e r i m e n t s . Concerning t h e s p i n d e t e r m i n a t i o n o f t h e xb ^{( X ' )} s t a t e s , we have b r i e f l y d i s c u s s e d t h e

b

u s e f u l n e s s o f t h e t r a n s v e r s e p o l a r i z a t i o n o f t h e beam p a r t i c l e s . A f u r t h e r i n c r e a s e o f s t a t i s t i c s a t t h e T ( n S ) , 1 2 n 5 3 a l s o o f f e r s t h e p o s s i b i l i t y o f s e a r c h i n g f o r t h e ^qb, qL and r$ states and t h u s t o o b t a i n i n f o r m a t i o n a b o u t t h e h y p e r f i n e s p l i t t i n g i n t h e T f a m i l y . I n t h i s c o n t e x t , we have a l s o n o t e d t h e i m p o r t a n c e of

d e t e c t i n g t h e ' p 1 s t a t e s . I n p o t e n t i a l l a n g u a g e s t h i s would a l l o w a t e s t o f t h e n a t u r e o f t h e l o n g r a n g e b-5 p o t e n t i a l .

W e have d i s c u s s e d f e a t u r e s o f t h e T f a m i l y i n t e r m s o f p o t e n t i a l models. We c o n s i d e r such an a p p r o a c h a s a c o n v e n i e n t g u i d e f o r c l a s s i f y i n g t h e e x p e r i m e n t a l r e s u l t s r a t h e r t h a n t h e u l t i m a t e model f o r d e s c r i b i n g quarkonium p r o p e r t i e s . There i s no d o u b t t h a t

a d d i t i o n a l l a r g e s t a t i s t i c s e x p e r i m e n t s would a l l o w a more complete d e s c r i p t i o n o f t h e T f a m i l y , b o t h e x p e r i m e n t a l l y and t h e o r e t i c a l l y . T h i s can be c o n s i d e r e d a s t h e c h a l l e n g e f o r t h e n e x t few y e a r s .

Acknowledgements

I t i s a p l e a s u r e t o t h a n k t h e "I. I n s t i t u t f i i r Expezimentalphysik"

and t h e DESY L a b o r a t o r y f o r t h e i r h o s p i t a l i t y and f o r t h e s t i m u l a t i n g s c i e n t i f i c atmosphere. I wish a l s o t o thank P r o f . E . Bloom,

D r . S. Cooper, D r . B. Klima, P r o f . J. Rosner, P r o f . U . S t r o h b u s c h , and D r . A. Schwarz f o r u s e f u l d i s c u s s i o n s .

R e f e r e n c e s

1 ) B. Baumgartner, H. G r o s s e , A . M a r t i n , CERN P r e p r i n t , CERN-TH. 3946/84 ( 1 984)

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2) E. E i c h t e n e t a l . , Phys. Rev. D21, 203 (1980)

3 ) E. E i c h t e n , F. F e i n b e r g , Phys. Rev. 023, 2724 (1981) 4 ) R. McClary, N . Byers, Phys. Rev. D28, 1692 (1983) 5 ) P. Moxhay, J. Rosner, Phys. Rev. D28, 1132 (1983) 6 ) D. B e a v i s , Shu-Yuan Chu, B.R. D e s a i , P. Kauss,

Phys. Rev. D20, 743 (1979)

7 ) J. Rosner, P r o g r e s s i n t h e D e s c r i p t i o n o f Heavy Quarkonium, U n i v e r s i t y o f Chicago p r e p r i n t , EFI 83/17 (1983)

and r e f e r e n c e s q u o t e d t h e r e i n

8 ) S.N. Gupta, S.F. Radford, W.W. Repko, Phys. Rev. D26, 3305 (1982) 9 ) M. Bander, D. S i l v e r m a n , B. Klima, U . Maor,

Phys. L e t t . 134B, 258 (1984)

10) M. Bander, D. S i l v e r m a n , B. Klima, U . Maor, U n i v e r s i t y o f

C a l i f o r n i a of I r v i n e r e p o r t UCI No. 83-22 ( 1 9 8 3 ) , t o be p u b l i s h e d i n Phys. Rev.

11) See f o r i n s t a n c e : A. M a r t i n , Phys. L e t t . IOOB, 511 (1983) 12) C r y s t a l B a l l C o l l a b o r a t i o n , R e s u l t s p r e s e n t e d by U. V o l l a n d a t

t h e X X I I . I n t e r n a t i o n a l Conference on High Energy P h y s i c s , L e i p z i g , J u l y 19-25, 1984

13) See f o r instance:V.A. Novikov e t a l . , Phys. Rep. 41C, 1 (1978) 14) ARGUS C o l l a b o r a t i o n , R e s u l t s p r e s e n t e d a t t h e X X I I . I n t e r n a t i o n a l

Conference on High Energy P h y s i c s , L e i p z i g , J u l y 13-25, 1984 15) CLEO C o l l a b o r a t i o n , S. Behrends e t a l . , Phys. Rev. L e t t . 52, 799

(1 984)

16) CUSB C o l l a b o r a t i o n , C . K l o p f e n s t e i n e t a l . , Phys. Rev. L e t t . 51, 160 (1983)

17) CUSB C o l l a b o r a t i o n , K. Han e t a l . , Phys. Rev. L e t t . 49, 1012 (1 982)

18) W. Buchmuller and S.-H. Tye, Phys. Rev. D24, 132 (1981) 19) See a l s o t h e d i s c u s s i o n i n " I n t r o d u c t i o n t o T Quarkonium

P h y s i c s " , A. Fridman, DESY report, DESY 83-731 (1983) 20) C r y s t a l B a l l P r o p o s a l a t t h e DESY PRC, F e b r u a r y 2 , 1984

21) See f o r i n s t a n c e t h e d i s c u s s i o n i n : A . Fridman, P r o c e e d i n g s o f t h e 6 t h European Symposium on Nucleon-Antinucleon I n t e r a c t i o n s , Anales d e F i s i c a , S e r i e A , Vol. 79, 224 (1983)

22) See f o r i n s t a n c e : J . G a i s e r , PhD t h e s i s , S t a n f o r d U n i v e r s i t y (1982) 23) Y.P. Kuang, T.M. Yan, Phys. Rev. D24, 2874 (1981)

24) See f o r i n s t a n c e Ref. 4

25) L.S. Brown, R.N. Cahn, Phys. Rev. D13, 1195 (1976)

26) A. Fridman, A . Schwarz, " F u r t h e r I n v e s t i g a t i o n s on t h e S p i n s o f t h e xb S t a t e s " , C r y s t a l B a l l Note, May 1 4 , 1984