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SPIN DEPENDENT FORCES IN QUARK MODELS
J. Rosner
To cite this version:
J. Rosner. SPIN DEPENDENT FORCES IN QUARK MODELS. Journal de Physique Colloques,
1985, 46 (C2), pp.C2-77-C2-93. �10.1051/jphyscol:1985208�. �jpa-00224519�
JOURNAL DE PHYSIQUE
Colloque C2, supplément au n°2, Tome k(>, février 1985 page C2-77
SPIN DEPENDENT FORCES IN QUARK MODELS J . L . Rosner
CERN, CH-1211 Geneva, Switzerland and
Enrico Fermi Institute and Department of Physios, University of Chicago, Chicago, II 60637, U.S.A.*
Résumé - Les expériences des récentes années nous ont donné une compréhension qualitative des forces entre les quarks qui dépendent du spin: ?v$2» t ' ^ ' s12- Une perspective : théorique est présentée et d'autres expériences sont suggérées.
A b s t r a c t - Q u a l i t a t i v e understanding o f h y p e r f i n e , s p i n - o r b i t , and tensor forces i n systems of both l i g h t and heavy quarks i s now a v a i l a b l e . The experimental s i t u a t i o n leading t o t h i s understanding i s reviewed, t h e o r e t i c a l perspectives are g i v e n , and some suggestions f o r f u r t h e r study are made.
I - INTRODUCTION
From heavy quarkonium systems (cc and b b ) , much has been learned about forces between quarks. I n i t i a l e f f o r t s led t o the n o t i o n of a f l a v o r - i n d e p e n d e n t p o t e n t i a l whose form i s n e a r l y uniquely s p e c i f i e d between 0.1 and 1 fm / I - 5 / . Concurrent w i t h these s t u d i e s , and j u s t now approaching s i g n i f i c a n t experimental t e s t s , have been attempts t o understand the spin-dependence o f these f o r c e s . (Recent reviews appear i n Refs. / 6 - 1 0 / . )
At s h o r t distances quark-quark and q u a r k - a n t i q u a r k i n t e r a c t i o n s appear t o be gov- erned by the exchange o f a s i n g l e v e c t o r gluon. N o n r e l a t i v i s t i c p o t e n t i a l models, s p i n - s p i n , s p i n - o r b i t , and tensor forces a l l are i n q u a l i t a t i v e accord w i t h t h i s n o t i o n . The question j u s t now being s e t t l e d regards the nature o f the f o r c e a t longer d i s t a n c e s . To the degree t h a t d e s c r i p t i o n o f t h a t force by a p o t e n t i a l makes sense, a consensus i s forming t h a t t h i s p o t e n t i a l behaves as a Lorentz s c a l a r . This a r t i c l e reviews some experiments r e l e v a n t t o spin-dependent forces i n quark models, and some t h e o r e t i c a l approaches t o these f o r c e s . Suggestions f o r f u r t h e r work are g i v e n . A companion review / l l / examines i n more d e t a i l the spin-deoendent forces i n the u p s i l o n (bb) system.
Quark spins are useful i n q u a n t i t a t i v e c a l c u l a t i o n s , p a r t i c u l a r l y o f magnetic moments. The s t a t u s o f baryon moments and some other prospects f o r measuring quark magnetic moments i n hadrons are r e c a l l e d i n Sec. I I . Ue then disqyss the formalism o f spin-dependent e f f e c t s , which are n e c e s s a r i l y r e l a t i v i s t i c (Sec. I l l ) , and t u r n t o d e t a i l s o f s p i n - s p i n i n t e r a c t i o n s (Sec. IV) and s p i n - o r b i t and tensor forces (Sec. V). The s p i n - o r b i t force i s a key t o understanding how quarks i n t e r a c t a t large d i s t a n c e s . Both l i g h t and heavy quarks are needed t o o b t a i n a l l the necessary i n f o r m a t i o n .
Some remarks on spin-independent r e l a t i v i s t i c c o r r e c t i o n s are made i n Sec. V I . These concern the question o f s c a l a r v s . v e c t o r exchange a t long d i s t a n c e s , and the
p o s s i b l e d i s t o r t i o n s of the n o n r e l a t i v i s t i c p o t e n t i a l t h a t can a r i s e i f spurious spin-independent r e l a t i v i s t i c c o r r e c t i o n s are adopted. I t i s shown how toponium ( t t ) and s c a l a r quarkonium (qq) spectra can provide i n f o r m a t i o n i n order t o resolve
Permanent a d d r e s s .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985208
C2-78 JOURNAL DE PHYSIQUE
t h i s ambiguity. A b r i e f summary (Sec. V I I ) , i n c l u d i n g a discussion o f some open questi ons,concludes.
I I - QUARK SPINS IN HADROMS: EXAMPLES
A. Baryon magnetic moments. One cannot ask f o r a n i c e r system i n which t o discuss the e f f e c t s o f quark spins t h a n t h e magnetic moments o f the n e u t r o n , p r o t o n , and r e - l a t e d ground-state baryons. This s u b j e c t has been reviewed e a r l i e r / 1 2 / , so we s h a l l be b r i e f about a l l but the l a t e s t developments.
In the s i m p l e s t form o f the quark model, magnetic moments o f SU(3) o c t e t baryons B composed o f quarks aa@ are given by
vB=3ya - 3ye ( 2 - 1 )
w h i l e yA = us and v ^ A = ( yd - %)/S3.
This reproduces a l l baryon magnetic moments t o w i t h i n ± 0.3 UM. In order t o do b e t t e r , i t seems necessary t o i n t r o d u c e c o n f i g u r a t i o n mixing / T 3 - 1 7 / , a l l o w i n g f o r o r b i t a l angular momentum among the quarks. A p a r a m e t r i z a t i o n which allows f o r t h i s mixing i s / 1 4 , 1 5 /
Vn
=°
+ a ) yu,d "
a»d,u
( 2-
2>
vz ± = ( l + a ' ) u{u} - a ' vs ( 2 . 3 )
u=o = ( l + a ' ) ys - a 'P {u} ( 2 . 4 )
The coefficients a and a' differ because E ' S and H'S can mix with SU(3) decimet mem- bers, while nucleons cannot. The coefficients of ya and yg in baryons <xag still add up to 1, as is true when the net result of admixtures of configurations with nonzero total orbital angular momentum cancels. The deviation from 1 in the sum of these coefficients has been estimated not to exceed 10% /14/.
Eqs. (2.2)-(2.4) contain five unknowns for six quantities. There is one relation among them /16/; defining y E (yp + yn)/2, we find
(uE+ - y )2 - („j._ - y )2 = (yH.o - u )2 - (uH_ - v )2 , (2.5)
TABLE 1 . Baryon magnetic moments 1n units of v^ in naive quark model and with configuration mixing.
Baryon
P n A i*
t°
z~
-.•
= -
u(E° - A)
"u
"d
H
a a1
vJvi
Experiment (From Ref. / 1 4 / )
2.793 -1.913 -0.613iO.005
2.38:0.02
-
- I . l l i 0 . 0 4 -1.250»0.014
-0.69t0.04 ' •8 2- 0 . 2 5
Naive Model
Input input input 2.67 U+ + l")/2
-1.09 -1.44 -0.50 -1.60
1.852 -0.972 -0.613 1/3 (input) 1/3 (input)
-1.91
Mixinq / 1 4 /
input input not predicted
input (£* • r " ) / 2 -1.04s0.06(exp)±0.06(th)
input input -1.74j0.20(exp)±0.03(th)
1.875 -0.995 -0.739 0.32 0.20 -1.88
The values o f a and a ' i n Table 1 f o r t h e mixed s o l u t i o n support t h e idea t h a t mix- i n g affects mainly t h e hyperons: a % 1/3, b u t a ' < 1/3. This l a s t e f f e c t leads t o a "quenching"/l6/ of t h e odd q u a r k ' s moment i n hyperons.
The d e v i a t i o n from uu/ud=-2 may be due t o small quark anomalous moments /13,17/.
One model f o r such moments /13/ envisions a c o n t r i b u t i o n from Y + ( l i g h t quark loops) -+ 3 gluons -+ quark. T h i s ought t o l e a d t o a u n i v e r s a l c o n t r i b u t i o n t o t h e t o t a l moment, e.g.,
!lU= uUO
+
A ; p d = udO+
A ; u c = p c 0 + A . (2.6) If one sets ! J ~ O / ! J ~ ~ = - ~ and takes t h e values o f !JU and u d from Table 1, one e s t i - mates A =- (0.03 t o 0 . 0 4 ) ~ ~ . How e l s e c o u l d such e f f e c t s show up?1 2
B.
~ a ~ n e t i c moment o f charmed quark. k!ith u c O-
gvU0=-5 1111 ( s i n c e mc = 5 mu) t h e c o r r e c t i o n A t o t h e charmed quark moment i n (2.6) can be o f o r d e r 10%. There a r e several i n t e r e s t i n g ways t o measure !lc, though probably n o t t o t h a t accuracy.1. Charmed baryon ~~(2280). J u s t as i n t h e A , t h e u and d quarks i n A, = udc a r e i n a s t a t e o f t o t a l s p i n zero. Thus vhc = u c = 0.2yu I 0 . 4 ~ Measurement o f t h e magnetic moment o f such a s h o r t - l i v e d p a r t i c l e ( r A C = 2 ~ 1 0 - ' ~ ' s ) would be a use- f u l t o u r de f o r c e , n o t t o t a l l y o u t o f t h e q u e s t i o n (K. H e l l e r , p r i v a t e communica- t i o n ) .
2 . Magnetic d i p o l e t r a n s i t i o n s i n charmonium. A systematic study o f M1 t r a n s i - t i o n s such as JI 4 ,,C., JII + n C B Y
,. . .
, has been made by several authors r e c e n t l y 1 8 1 9 I n p r i n c i p l e these t r a n s i t i o n s should proceed j u s t v i a a s p i n f l i p , so t h e i r r a t e s are c a l c u l a b l e : r(3S1 -+ 'So + y ) = 4 ~ 3 ~ ~ 7 - 1 I 2/27n. where I i s anl
o v e r l a p i n t e g r a l which would be 1 i n t h e n o n r e l a t i v i s t i c i m i t , and U i s t h e photon energy. I n f a c t i t appears t h a t
I
112 < 1 both i n charmonium and i n l i g h t - q u a r k /12/ systems. Given t h e experimental r a t e /20/r
(J, -+ y n C ) = 0.5-
1 keV,
and the d e f i n i t i o n o f t h e anomalous moment K , u c = Q c l c l ( l + ~)/2m,, we f i n d ( l + ~ ) (1.5 GeV/mc)l 11 = 0.4 t o 0.6. E i t h e r mc i s s u b s t a n t i a l l y above 1.5 GeV (values above 1.9 EeV are u n l i k e l y i n p o t e n t i a l models), K < 0, o r11 l 2
= 1/4. I t i s t h e conclusion o f Ref. 18 t h a t the r a t e s f o r Q -+ usc can be e x p l a i n e d w i t h o u t recourse t o anomalous moments, b u t t h e c a l c u l a t i o n s o f 11l 2
a r e n o t y e t a t t h e accuracy r e - q u i r e d t o determine K a t an i n t e r e s t i n g l e v e l .3. Magnetic quadrupole t r a n s i t i o n s i n $ ' + yx and -P v$. The dominant t r a n s i - t i o n s i n r a d i a t i v e decays of heavy quark systems a r e e l e c t r i c d i p o l e ( E l ) . Under some circumstances, however, i t may be p o s s i b l e through c a r e f u l study o f photon angular d i s t r i b u t i o n s t o see t h e i n t e r f e r e n c e of M2 t r a n s i t i o n s w i t h the dominant El amplitude. The M2 t r a n s i t i o n amplitude i s p r o p o r t i o n a l t o t h e t o t a l quark magnetic moment.
Present data a r e on t h e verge o f being a b l e t o see t h e M2 t r a n s i t i o n ' s effects.
When p r e d i c t i o n s /21,22/ a r e compared w i t h experiment /23/, one f i n d s (see Eq. (6-7) and Fig. 6-1 of t h e f i r s t o f Refs. 23) a charmed quark t o t a l moment
These numbers are based on 10 6- rl' decays. From Eq. (2.7) one would expect a sub- s t a n t i a l impact from t h e study o f 1 0 7 $' decays. Since M2 e f f e c t s a r e p r o p o r t i o n a l t o py/mS, they are probably b e s t s t u d i e d i n charmonium and n o t b6 systems.
C . Magnetic moment o f b quark. By t h e same reasoning o f the previous subsection,
we estimate ubO = -0.06 UN. The b quark l i v e s l o n g enough (about 10-12s) t h a t i t may n o t be any harder t o study yhb than uhc, given a s u i t a b l e source o f hb z udb.
(Since t h i s p a r t i c l e has n o t y e t been produced, we are being o p t i m i s t i c ! ) An anomaly o f -0.03 !JH would s u b s t a n t i a l l y a f f e c t t h e magnitude o f b.
I f t h e r a t e f o r T + nby exceeds t h a t c a l c u l a t e d w i t h one's b e s t estimates o f t h e o v e r l a p i n t e g r a l I, w h i l e t h a t f o r $ + ncy i s lower than c a l c u l a t e d , i t may be worth t a k i n g t h e i d e a o f anomalous quark moments m o r e i o u s l y . T h i s adds t o t h e impor- tance o f c o n t r o l l i n g r e l a t i v i s t i c c o r r e c t i o n s enough t o e s t i m a t e I r e l i a b l y .
C2-80 JOURNAL DE PHYSIQUE
A. S c a l a r and v e c t o r exchanges. The c o v a r i a n t m a t r i x element f o r quark-antiquark s c a t t e r i n g may be w r i t t e n
where q i s t h e 4-momentum t r a n s f e r . These are t h e two forms t h a t have a s t a t i c l i m i t and hence make sense f o r a n o n r e l a t i v i s t i c p o t e n t i a l . Tensor, a x i a l , and pseudo- s c a l a r i n t e r a c t i o n s o f course are a l s o allowed.
B. Short-and long-range forces.
1. A t s h o r t r a n e, t h e exchange o f a s i n g l e gluon leads t o a v e c t o r form o f t h e
inters-ore
F o u r i e r t r a n s f o r m i s V ( r ) = - ( 4 / 3 ) [ a s ( r ) / r ] . The s t r o n g f i n e - s t r u c t u r e constant a s ( r ) has a slow l o g a r i t h m i c v a r i a t i o n .2. A t l o n g range, l i n e s o f force between a quark and a n t i q u a r k appear t o be c o l l i - mated i n t o a " s t r i n g " o f l i m i t e d cross s e c t i o n a l area. By Gauss' law, t h i s leads t o a constant force, o r a s t a t i c energy ~ ( r ) = k r , where k = 0.18 GeV2. This value o f k i s deduced from t h e spectra o f h i g h o r b i t a l e x c i t a t i o n s i n l i g h t - q u a r k systems, o r l i n e a r Regge t r a j e c t o r i e s . A simple p i c t u r e o f a r e l a t i v i s t i c r o t a t i n g s t r i n g /24/ y i e l d s the r e l a t i o n L = a'M2 between o r b i t a l angular momentum L and resonance mass 11, where t h e slope a ' i s r e l a t e d t o t h e energy d e n s i t y k by a ' = 1/(2*k). The e m p i r i c a l value a ' = 0.9 G ~ V - ~ then leads t o t h e estimate quoted.
The Lorentz t r a n s f o r m a t i o n p r o p e r t i e s o f t h e l o n g range f o r c e a r e n o t known a p r i o r i . I t i s these we seek t o l e a r n from experiment, and deduce from theory.
C. The e f f e c t i v e p o t e n t i a l . L i m i t i n g t h e discussion t o s c a l a r and v e c t o r exchanges, we w r i t e t h e n o n r e l a t i v i s t i c p o t e n t i a l as V = V(r) + S ( r ) . The e f f e c t i v e p o t e n t i a l between a quark and a n t i q u a r k , each o f massN/$, can be w r i t t e n /25,6/
Here
V-
i s aipin-independent pieces
'4 v2vNR
- 'fii
+ P.PTRv.?
Vl
E - b + 4m2
mL,
where R = 1
-
and V acts o n l y on V w h i l e$
a c t s b o t h on V and wave f u n c t i o n l . We s h a l l discuss t h e s i g n i f i c a n c e o f V1 f u r t h e r i n Sec. V I I . The s p i n - o r b i t term i st . 5
v,
= ( 3 V ' - S ' ) (3.4)w h i l e t h e s p i n - s p i n and t e n s o r terms are 252 - 3 S12
v '
"3 = "V
+ m -
V")Here S = 2 [ 3 ( $ . r ) ( f . r )
-
3 2 3 i s t h e t e n s o r operator, which o n l y c o n t r i b u t e s when L > ~ . ' ~ ~ e c h n i q u e s f o r e v a l u a t i n g t h e m a t r i x elements o f t h e t e n s o r o p e r a t o r S12 are given, f o r example, i n Ref. /26/. Values o f V2 and V3 f o r unequal-mass cases are given, f o r example,in Refs. /27/ and /28/.We n o t e several p o i n t s about V2 and V3.
1. The s i n d i f f e r e n c e i n V2 between V and S c o n t r i b u t i o n s a r i s e s from t h e con- t r i b u t i o n ingV o f t r a n s v e r s e and Coulomb gluon exchange which i s absent i n S. Only t h e Thomas precession term i s present i n S. I t i s a l s o present i n V, b u t over- whelmed by t h e o t h e r c o n t r i b u t i o n s .
2. Only V c o n t r i b u t e s t o s p i n - s p i n and t e n s o r forces. The s i g n a l s f o r such forces i n experimental spectra then p o i n t t o exchange o f
some
v e c t o r p a r t i c l e . Thercagnitudes o f such forces appear t o be i n accord w i t h expectations from gluon ex- change.
3. A Coulomb p o t e n t i a l V %
-
l / r c o n t r i b u t e s t o S waves only, s i n c ev - l . I n p a r t i c u l a r f o r such a p o t e n t i a l t h e l P 1 - T s p l i t t i n g i s zero.
D. I s t h e p o t e n t i a l an e f f e c t i v e s c a l a r a t l a r g e r ? An appealing d i s t i n c t i o n be- tween t h e behavior o f t h e p o t e n t i a l i n QCD a t s h o r t and long distances has been drawn by Buchm'u'ller /9,29/ and supported i n more r i g o r o u s language by Gromes /30/.
A t s h o r t distances t h e quark-antiquark system i s e f f e c t i v e l y Coulombic. As i l l u s - t r a t e d i n Fig. l a , an a n t i q u a r k o r b i t i n g a quark crosses f i e l d l i n e s and i s s u b j e c t t o an e f f e c t i v e chromomagnetic f i e l d
B
=3
Xe,
where i s t h e ( r a d i a l ) chromoelec- t r i c f i e l d . T h i s leads t o an e x p l i c i t s p i n - o r b i t f o r c e . The Thomas term w i l l be present i n a d d i t i o n .Fig. 1. Chromoelectric l i n e s o f f o r c e a c t i n g on a quark
4
r o t a t i n g about a sourceQ. (a) a t s h o r t distances; (b) a t l o n g distances (From Ref. /g/.)
A t l o n g distances, an o r b i t i n g anqiquark drags chromoelectric f i e l d l i n e s around w i t h i t ( F i g . I b ) . There i s then no e x p l i c i t
P*$
f o r c e . Because i t i s i n a r o t a - t i n g reference frame, t h e o r b i t i n g a n t i q u a r k i s o f course s t i l l s u b j e c t t o a Thomas f o r c e . Since t h i s i s o f opposite s i g n t o ther-3
f o r c e associated w i t h $ Xp,
the e f f e c t i v e p o t e n t i a l here would be i n t e r p r e t e d as a s c a l a r exchange. The n e tP-$
term i n t h i s p i c t u r e i s thus p o s i t i v e a t small distances and n e g a t i v e a t l a r g e distances. This has some i n t e r e s t i n g consequences.
1. As r grows, one expects a diminished
t.$
e f f e c t as compared w i t h t h e t e n s o r f o r c e . This i s borne o u t by a comparison between c € and b6 systems.-
2. A t v e r l a r e r , one ex e c t s m u l t i l e t i n v e r s i o n , i . e . , M(3P ) > M(3P1) >
M(3P2) /31/.Y We %ave n o t fou:d a suitabee system t o d i s p l a y t h i s i f h e r s i o n , how- ever, as w i l l be mentioned i n Sec. V.
A s c a l a r c o n f i n i n g n o t e n t i a l i n the D i r a c equation, behavina i n t h e same way as t h e mass
1
i $+
m+
V(r)]$ = 0, avoids a K l e i n paradox as ~ ( r ) gets l a r g e 1321. Some e a r l y phenomenological analyses 133,341 viewed s c a l a r confinement favorably, and more r e c e n t treatments, /9,19,25,28,29,35-411 a r e s t i l l c o n s i s t e n t w i t h t h e n o t i o n . The reasons f o r c o n t i n u i n g t o r a i s e t h e q u e s t i o n a r e several, however. Some approaches t o t h e behavior o f QCD a t l a r g e distances /42/ seem t o preserve t h e form o f v e c t o r exchange, b u t w i t h the gluon propagator simply m o d i f i e d a t small q2.Another p o s s i b i l i t y /43-451 i s t h a t t h e i n t e r a c t i o n i s o f v e c t o r form a t s h o r t d i s - tances, b u t t h a t the transverse degrees o f freedom o f t h e exchanged p a r t i c l e do n o t c o n t r i b u t e t o t h e long-range i n t e r a c t i o n . This d e s c r i p t i o n appears phenomenologi- c a l l y adequate, b u t d i f f e r s i n p r i n c i p l e from t h e long-range s c a l a r i n t e r a c t i o n . I t has a l s o been suggested on t h e b a s i s o f f l u c t u a t i o n s i n s t r i n g s t h a t t h e r e may be a long-range t e n s o r i n t e r a c t i o n 1461.
With these t k e o r e t i c a l p o s s i b i l i t i e s t o choose from, what do we l e a r n from t h e data themselves?
JOURNAL DE PHYSIQUE
I V
-
SPIN-SPIN FORCES A. Light-quark systems1. L=.O mesons and baryons. A simple mass formula 127,471 F? = c mi + h C -c&: .&.>/m.m. f i t s 7 meson masses:
i i r j 1 J 1 J
r , K,
1
mu = md = 310 MeVp , U ,
X*,
( I m s = 485 MeVw i t h one value o f h = h M and 8 baryon masses:
N, A , c ,
e
mu = md = 365 MeV* *
A , y 1 ,
e
9 "w i t h a value o f A ,= A i n p r i n c i p l e r e l a t e d by QCD t o h#. Note t h a t t h e values o f quark masses f o r t h e Earyons i n (4.2) are s h i f t e d by 53 MeV w i t h regard t o those f o r t h e mesons i n (4.1) 1481. Thus one may i n t e r p r e t the f i t s i n terms o f 4-6 f r e e parameters t o determine 15 masses. The h y p e r f i n e i n t e r a c t i o n i s t h e n e x t most im- p o r t a n t q u a n t i t y a f t e r t h e strange-nonstrange quark mass d i f f e r e n c e i n determining t h e values o f t h e l i g h t hadron masses.
L - l b a r ons. The s i m p l e s t o r b i t a l l y e x c i t e d baryons c o n s i s t o f a mixed s t a t e o
t i . t l
; a -0 and (a12=0, a 3 = l ) , where a l p i s t h e o r b i t a l angular momentum be- tween quarks and 2, and a3 i s t h e o r b i t a l angular momentum o f quark 3 r e l a t i v e t o t h e 1-2 system.
The s p i n - s p i n i n t e r a c t i o n
z1 .z2
f a v o r s (31+32)2=0, a12=0, which i s allowed i f I12=0 (we consider here f o r s i m p l i c i t y systems o f nonstrange quarks o n l y ) . Then one can understand t h e observed masses o f t h e L = l nonstrange baryon resonances /27,49/:1
*
3 * * 1( S q = $ < ( s 2 = T ) . ) I ,I, (S q = $ A
The h y p e r f i n e i n t e r a c t i o n gives r i s e t o an average s p l i t t i n g o f about 150 f!eV f o r the L=l baryons. T h i s magnitude even t u r n s o u t t o be q u a n t i t a t i v e l y p r e d i c t e d /27/
t o be h a l f t h a t o f t h e N-A s p l i t t i n g , which i s 300 MeV.
B. Mixed l i g h t - h e a v y systems: D, F, B:
1. The D*-D d i f f e r e n c e i s p r e d i c t e d t o be ms/mc)
*
1/3 t i n e s t h e K -K d i f f e r e n c e ,*
which i s % 400 MeV. The observed value I 4 9 1 o f 144 MeV i s i n accord w i t h t h i s pre- d i c t i o n .
2. The F -F d i f f e r e n c e should be about mu/m
*
c 213 times t h e D*-D d i f f e r e n c e . Recent measurements o f 1442957 FleV (ARGUS /50/? o r 139.558.33.7 MeV (TPC 1511) a r e s l i g h t l y l a r g e r than t h i s n r e d i c t i o n . Here, however, one must t a k e account o f t h e v a r i a t i o n o f the wave f u n c t i o n a t t h e o r i g i n , s i n c e AM due t o , t h e term V3 i n Eq.(3.5) ( g e n e r a l i z e d t o unequal masses) i s p r o p o r t i o n a l t o ly(0)62. For h e a v i e r qyarks, A y ( 0 ) / 2 i s b i ger. Taking account o f t h i s , A. M a r t i n 1-21 p r e d i c t s t h a t F -F I 1 0 MeV, and .?F*) = 2110 MeV by i n t e r p o l a t i n g between ss and cc systems.
This l a s t p r e d i c t i o n i s p a r t i c u l a r l y close t o t h e r e c e n t value m ( ~ * ) = 2 1 0 9 ~ 9 ~ 7 MeV quoted by ARGUS /50/.
3. The B*-B d i f f e r e n c e should be mp/mc
*
1/3 times t h e D -D d i f f e r e n c e , o r about*
48 MeV. The CUSB group 1531 sees a s i g n a l f o r B* + By a t h i g h CESR energies Etm 2 11 GeV: Ey = 50
5
6 GeV.C. Heavy quarkonia.
l. The 3S -IS s p l i t t i n g s due t o V3 a r e expected t o r e c e i v e s t r o n g c o r r e c t i o n s : l- 0
where t h e c o e f f i c i e n t o f a /a quoted i s t h a t a p p r o p r i a t e f o r t h e cc system /54/, and m i s t h e quark mass. bne may f i n d 1 ~ ( 0 ) 1 ~ from t h e l e p t o n i c decay r a t e o f the 3S1 s t a t e :
the two-g1 uon decay r a t e o f t h e ISo s t a t e :
o r from t h e geometric mean o f t h e two, f o r which t h e e r r o r would l a r g e l y cancel out. U n f o r t u n a t e l y t h e two-gluon widths o f t h e S s t a t e s a r e n o t known p r e c i s e l y enough y e t . For t h e observed J/$-nC spacing o f ll8+6 MeV /20,49/, one r e q u i r e s a value o f as=0.3-0.4 i n Eq. (4.6) i f I u ( 0 ) 12 i s e x t r a c t e d from ( 4 . 7 ) . The p r e d i c t e d
'
spacing i s then 8 0 3 0 MeV /54/, t o be compared w i t h t h e observed value /20/o f 9&5 MeV.
The T -nb spacing i s n o t so r e p r o d u c i b l y p r e d i c t e d . A best e s t i m a t e from a p e r t u r - b a t i v e QCD c a l c u l a t i o n , w i t h c o r r e c t i o n s , gives /54/ T
-
nb = 30 MeV, b u t values up t o 100 MeV have appeared i n t h e l i t e r a t u r e . T h i s l a s t p r e d i c i i p n can be understood q u a l i t a t i v e l y . E m p i r i c a l l y t h e p a r t i a l decay r a t e s r(3S1 -+ e e ) appear mainly t o depend on quark charges so I y ( 0 ) I2/m 2 i s almost constant over a wide range o f quark masses.The Crystal+Bi11 group /11,55/ has observed a 4-50 peak i n t h e photon spectrum ob- t a i n e d i n e e a n n i h i l a t i o n s a t Ec-m.=M(~). The photon energy i s about 1.07 GeV, correspondi~ng t o a s t a t e a t 8.32 GeV. T h i s s t a t e , c a l l e d t h e " g " , has a t o t a l w i d t h
r
< 80 tleV,narrow compared w i t h t h e d e t e c t o r r e s o l u t i o n , and i s produced w i t h an apparent branching r a t i o B[T -+ a 0.5%. Since T-
= 1.14 GeV i s more than 10 times the h i g h e s t estimated T -nb spacing, we can conclude t h a t 5 i s n o t t h e ~ b - . 2. The I P-
3P S a c i n S are o f i n t e r e s t because they probe non-Coulombic p a r t s o f t h e ~ o h *e$ti:l. more complete discussion o f t h i i p o i n t has been given i n Refs. / l , 6, 1 l, 56/. The mass s p l i t t i n g i s p r o p o r t i o n a l t o the m a t r i x element o f t h e spin-spin+term i n Vg: AM X <Plv2V P> which vanishes i f V s- 1
s i n c e v 2 ( - l / r ) = 4 ~ 6 ~ ( r ) . I t i s s t i l l very sma1
1 i f a ( r ) has o n l y a slow l o g a r i t h m i c v a r i a t i o n . Most models p r e d i c t t h a t t h e lP1 s t a t e l i e s w i t h i n a few MeV o f t h e 3 P ~ c e n t e r - o f - g r a v i t y . As p o i n t e d o u t by A. M a r t i n a t t h i s conference, t h e very l a s t experiment t o , be r u n a t the ISR /57/ detected a few candidate events f o r t h e 'P1 s t a t e o f charmonium a t t h e a p p r o p r i a t e mass, M('P1) = 3.52 GeV.D. What i s t h e G ?
So f a r , t h e 5 has been detected o n l y under r e s t r i c t e d circumstances i n one e x p e r i - ment. I t i s n o t seen i n T' decay: BY ( T ' ) / B ~ ( T ) < 0.22 (90% c.7 . ) , though many i n t e r p r e t a t i o n s r e q u i r e t h i s r a t i o t o be o f o r i e r 1. A t some l e v e l , i t should show up i n T' -t T T T , T -+ gy. F u r t h e r data w i l l be taken both a t DORIS and a t CESR, and by t h e time t h i s a r t i c l e i s i n p r i n t one should know much more about the e x p e r i - mental s i t u a t i o n . Meanwhile, here are some suggested i n t e r p r e t a t i o n s .
1. I f t h e i s a H i S boson /58/, i t s Yukawa c o u p l i n g gy t o b quarks a u s t be about t e n timzs t h e st::dard value: g s t d = mb/v, where v-247 GeV 2 - 1 / 4 ~ ~ - 1 / 2 i s t h e electroweak symmetry-breaking parameter.
For any p o i n t s p i n l e s s p a r t i c l e H coupled w i t h Yukawa c o u p l i n g gy t o a heavy quark
4, one f i n d s
C2-84 JOURNAL DE PHYSIQUE
where V i s the 3~ Q0 bound s t a t e /59/, as l o n g as one can t r u s t a f r e e quark calcu- l a t i o n . I t i s t h j s r e s u l t t h a t l e d t o t h e e x p e c t a t i o n o f
2
1 f o r t h e r a t i oBy5(T')/BY (*). Eq. (4.7) i s independent o f whether H i s s c a l a r (o+ ) o r pseudo- s c a l a r ( 0 - f .
Bound-state m o d i f i c a t i o n s o f (4.7) have been estimated /60/. The r a t i o
B ( T ' ) / B ~ ( T ) i s s t i l l expected t o be % l f o r a pseudoscalar, b u t i s l e s s than 1 bJS20-40% For a s c a l a r w i t h the mass o f
c .
J. D. Jackson and I f i n d t h a t f o r t h e weak-binding and soft-photon l i m i t ( l e v e l spacings) << E << M,
(4.7) i s reproduced both f o r J ~ ( H ) = o - (where t h e r e s u l t i s e s s e n t i a l l y t r i v i a l ) a i d f o r J~(H)=o+, if one performs a sum o v e r 0- o r O+ bound s t a t e s c o u p l i n g t o H, and 1- bound s t a t e s c o u p l i n g t o.
The 0' r e s u l t occurs as a consequence o f t h e r e l a t i o n...
among S- and P-wave r a d i a l wave f u n c t i o n s and d i p o l e m a t r i x elements. This work i s s t i l l i n progress.
2. A g l u i n o - g l u i n o bound s t a t e /61/ can be produced i n r a d i a t i v e T decays. Pre- d i c t e d r a t e s tend t o be t o o low f o r t h e observed 5 s i g n a l , b u t can be r a i s e d w i t h an unusual J~~ f o r t h e 6 (l++) o r i f a l i g h t s c a l a r quark b e x i s t s (J. H. Kuhn, p r i v a t e communication). The 2S/1S r a t i o remains a problem.
3. H y b r i d (qqg) o r g l u e b a l l i n t e r p r e t a t i o n s have been suggested (B.F.L. Ward, unpublished; /62/).
4. S c a l a r quarkonium p r o p e r t i e s were s t u d i e d a few years ago (/63,64/), and t h e 5 has been i n t e r p r e t e d /65/ as t h e ground s t a t e of such a system. We s h a l l r e t u r n
t o t h i s i d e a i n Sec. VI. B.
V
-
SPIN-ORBIT AND TENSOR FORCES A. L i g h t quarks1. Baryons. A example. I n Fig. 2a we show t h e behavior of t h e f i n e s t r u c t u r e f o r A++ = uuu s t a t e s as a f u n c t i o n o f i n c r e a s i n g o r b i t a l angular momentum L. The A system i s p a r t i c u l a r l y clean i n t h a t c o n f i g u r a t i o n mixings /66,67/ are expected t o be a t a minimum.
Fig. 2. Fine s t r u c t u r e in l i g h t - q u a r k systems. a) Baryons ( A example); b) Kesons ( I = l example). Parentheses denote u n c e r t a i n mass o r assignment.
The f i n e s t r u c t u r e f o r A ' S i s small (no more than several tens of HeV f o r L = l ) and appears t o be decreasing w i t h i n c r e a s i n g L. A s a t i s f a c t o r y d e s c r i p t i o n o f baryon spectra has i n f a c t been obtained w i t h o u t any f i n e s t r u c t u r e a t a l l 1671.
2. Mesons. 1=1 example, 3LJ s t a t e s . The l i g h t - q u a r k mesons p r o v i d e s c a t t e r e d h i n t s , b u t n o t a complete p i c t u r e , o f f i n e s t r u c t u r e . The present experimental s i t u a t i o n i s i l l u s t r a t e d i n F i g . 2b. Masses a r e quoted from Ref. /56/. Several p o i n t s remain t o be resolved.
(a) We do n o t know i f t h e ~ ~ ( 1 " ) i s r e a l l y as close as shown t o t h e Az(2
++
). A r e cent Mark I 1 experiment I 6 8 1 sees a ps peak around 1100 GeV i n T -9- p ~ , . Estimates o f s h i f t s i n t h e p n mass from a " t r u e " A1 a t h i g h e r mass are not.enough t o e x p l a i n the e f f e c t 1691.(b) The 6(0+) s t a t e may be a
KR
l?nolecule" r a t h e r than a t r u e qq s t a t e 1701. One i s s t i l l searching, i f so, f o r the 3Po qq 1=1 s t a t e .( c ) No guidance i s p r o v i d e d by 1=0 s t a t e s , among which g l u e b a l l candidates (0"
I 7 1 1 and 2" 1721) are p r o l i f e r a t i n g .
(d) The p ' may be an L=O s t a t e
or
a m i x t u r e o f L=2 and L=O.(e) A 2-- s t a t e i s needed t o complete t h e p i c t u r e f o r L=2 mesons.
For L=2 l i g h t - q u a r k mesons, H. S c h n i t z e r 1401 f i n d s t h a t s p i n - o r b i t e f f e c t s are, again, decreasing w i t h i n c r e a s i n g L.
B. Heavy quarks.
Here t h e p i c t u r e i s much more s a t i s f a c t o r y , as i l l u s t r a t e d i n Fig. 3. The existence o f f l a v o r thresholds, below which 94 s t a t e s are very narrow, i s a boon t o spectros- COPY
1. The cc system has a w e l l - d e f i n e d s e t o f ~ P J l e v e l s . The corresponding ~ D J l e v e l s l i e above DD t h r e s h o l d , and i n p a r t i c u l a r , t h e 3D1 s t a t e , $(3772) i s a copious and clean source o f DD-in e+e- a n n i h i l a t i o n s 1731. I t may be p o s s l b l e t o produce t h e ' ~ 2 and 3 ~ 2 s t a t e s i n pp a n n i h i l a t i o n s . These s t a t e s a r e expected t o be narrow (we thank R. Cester-Regge f o r t h i s suggestion), singe,they cannot decay t o DD; t h e low- e s t a v a i l a b l e channel i s expected t o be DD o r DD
,
w i t h t h r e s h o l d 3.87 GeV. Nearly a l l p o t e n t i a l models e s t i m a t e t h e ID2 and 3D2 masses t o l i e below t h i s value.2. The b6 system has provided x b 1741 and x b ' 1751 s t a t e s , as reviewed i n Ref.
/Ill. The s p l i t t i n g s i n t h e x b ' s t a t e s a r e s m a l l e r than those f o r x b , as expected.
Some o f t h e p r e d i c t e d L=2 s t a t e s may be observable i n accurate searches f o r ( L = l ) +
(L=2) + y t r a n s i t i o n s , o r i n e+e- p r o d u c t i o n o r decays o f 3 ~ 1 s t a t e s . These a r e
Fig. 3.
note f l
0 1 2 0 2
L
Fine s t r u c t u r e i n heavy-quark systems. a) cc; b) b6. H o r i z o n t a l l i n e s de- avor threshold. Dotted l i n e s denote l e v e l s p r e d i c t e d b u t n o t y e t observed.
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expected t o have p a r t i a l ete- decay w i d t h s o f a few eV a t most /44/.
3. Models. The masses o f 3 ~ J mesons a r e described by t h e expression M ( 3 ~ J ) = fi
+
a <P.3
>+
b < S12 >,
where, f o r 3PJ s t a t e s ,
S p e c i f i c values o f a and b a t t a i n e d from experiment and i n v a r i o u s models a r e shown i n Table 2. S a t i s f a c t o r y agreement occurs i n most models. Exceptions a r e the Eichten-Feinberg model, which underestimates t h e t e n s o r f o r c e f o r charmonium, t h e McClary-Byers model, which overestimates t h e t e n s o r f o r c e f o r t h e b6 systems ( p o s s i b l y because o f a l a r g e assumed U, value), and t h e Moxhay-Rosner model, which underestimates t h e
1.3
s p l i t t i n g s i n bg. Generally scalar-confinement models f a r e q u i t e w e l l . However, a) a modest improvement i n accuracy f o r t h e b6 measurements would h e l p exclude some models, and b) t h e t e c h n i c a l d i f f e r e n c e s among various models are a t l e a s t as g r e a t as those due t o u n d e r l y i n g physics. This stresses t h e need f o r a r e l a t i v i s t i c a p r i o r i d e s c r i p t i o n independent o f t h e p o t e n t i a l idea.One p a r t i c u l a r l y simple v e r s i o n o f t h e model w i t h s c a l a r confinement has been d i s - cussed by Buchm'u'ller /9,29/. I f one takes a short-range Coulomb p o t e n t i a l
V ( r ) =- (4/3)?,/r ( n e g l e c t i n g t h e v a r i a t i o n o f w i t h r ) and a long-range s c a l a r p o t e n t i a l S ( r ) = k r , one f i n d s
TABLE 2. Experimental v a l w s and predictions i n MeV f o r s p i n - o r b i t (a) and tensor (b) parameters i n PJ nuss fonnula (5.1)
Experiment
~Refs./11.49,74,75/) 34.8 9.9 12.7 2.2 9.4 1.4
1 . 3 t0.4 1 . 3 i0.9 i2.5 i1.5
Eichten-Feinberg 38 4.4 14 1.7 11 1.3
(Ref. /43/)
Ruchmiiller 35 3.6 15 2.8
(RefS./9,29/) 2.1 8 1.6 C)
Gupta e t a l . 36 10.6 11 2.3 9 1.8 c)
(Ref. /36/)
HcClary-Bye- 33 12.2 17 5.3 15 4.2 c)
(Ref. /25/)
Moxhay-Rosner 27 8.5 8.8 3.0 6.5 2.1
(Ref. /44/)
Bander e t a l . 37 6.8 17 3.0 11 1.8 c). d)
(Ref. /37/)
Carlson e t al. 34 8.0 13 3.1 11 2.4 c)
(Ref. 1391)
Crater-Van A l s t i ne 36 7.7 17 3.1 12 2.2 c)
(Ref. /41/)
Grotch e t al. 24 10.3 14 2.5 10 1.8 c)
(Ref. /19/)
a) @,(l)
-
0.49 ; b) a S f r ) = 0.28c) Scalar confinement a s s m d . Differences a k technical. Not a l l approaches attenpt f i t s t o R.
d) Representative choice of pure scalar confinement w i t h a p a r t i c u l a r cutoff scheme.
Euchmiiller's predictions f o r Table 2 are based on a standard nonrel a t i v i s t i c poten- t i a l /5/ and the choices k = 0.153 GeV2, as($) = 0.53, a ? ( ~ ) = 0.43 (case ( a ) ) or 0.28 (case ( b ) ) . These l a s t two estimates are obtained via d i f f e r e n t extrapolation methods. CJhile t h e value of as($) i s large f o r perturbation theory, t h e agreement i s s a t i s f a c t o r y .
In most models the r a t i o a/b i s predicted t o be l e s s than 6 , as expected from Ens.
(5.4) and (5.5). The value of a would change sign f o r I < l / r z / < l / $ > ) ' 2 (1 fm) in the Buchmijller model. This r a t i o i s only (0.4, 0.2) fm
G
f o r the( C C , b6) s y s t e m considered.
C. Mixed hadrons
Some hope f o r seeing the change of sign in spin-orbit forces due t o s c a l a r exchange a t long distances e x i s t s f o r systems of one l i g h t and one heavy quark /31/. In
Fig. 4, we compare such a system ( b ) with one of two l i g h t quarks ( a ) . A s t r i n g calculation parallel t o t h a t of Ref. 24 shows t h a t the l i g h t quark i s a f a c t o r of 4 7 f a r t h e r from the center-of-mass i n the light-heavy system than i n the l i g h t - l i a h t system. Since one sees l i t t l e fine s t r u c t u r e in the lioht-quark baryons, i n d i c a t i n ~ a
2
0,
One might expect multiplet inversion ( a < 3 ) in the mixed s y s t e m .This expectation i s not borne out in one s p e c i f i c calculation with a s c a l a r confin- ing potential /37/. B. Klima has calculated the soectra f o r mixed ( E an$
B)
mesons and finds the r e s u l t s i l l u s t r a t e d i n Fig. 5. No multiplet inversion occurs. un the other hand, i t i s i n t e r e s t i n g t h a t the 1P1 mass i s perceptibly s h i f t e d from theFig. 4. String pictures of quark-antiquark systems. ( a ) Two l i g h t quarks; ( b ) one l i g h t quark q and heavy quark 0. Here r denotes the distance from the center-of-mass.
Fig. 5. Spectra of rnt'xed mesons (one l i g h t and one heavy quark) calculated in a model /37/ with a s c a l a r confining p o t e n t i a l . Note t h a t multiplet inversion does not occur. a) D mesons (one c3armed quark); b) B mesons (one b auark).
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3Pj center of gravity. The same pattern i s found for the f i r s t radially excited P states. De Rujula ejt aX-, also did not f i n d inverted multiplets /27,76/, but they did not assume scalar confinement.
The d i f f i c u l t i e s of finding P states in D mesons are exacerbated by numerous kine- matic reflections /76/77/ from D*-»• DTT. The search f o r P-wave B = bq (q= u or d) mesons may be easier, since B* cannot decay to Bit but only to By.
VI - SPIN-INDEPENDENT EFFECTS
Here we discuss some possible effects of the spin-independent r e l a t i v i s t i c correc- tion V-| in Eq. (3.9). With a scalar (S) or vector (V) potential the different terms in V-] containing S and V lead to s l i g h t differences in the underlying nonrelativis- t i c potential used to f i t observed cc and bb spectra. Tests of this underlying potential may be provided by t t spectra, f o r which the r e l a t i v i s t i c effects w i l l be less important, and by scalar quarks ( i f they e x i s t ) , for which the r e l a t i v i s t i c corrections ought to provide further d i s t i n c t i o n between scalar and vector confine- ment.
A. Toponium in a nutshell
A popular value for the t quark mass these days i s 30-50 GeV /7S/. Choosing mt_ = 40 GeV, we i l l u s t r a t e in Table 3 the differences among predictions of some potentials for t t spectra. The Martin potential / 5 2 / , V =- A + Br0-1, predicts a pattern of level spacings behaving as m~l/2-1, and hence a 2S-1S spacing smaller than that in charmonium (589 MeV) or bB (563 MeV). In almost a l l potentials with a short-distance Coulonb-like singularity due to gluon exchange, the effects of this singularity should become visible in an increased 2S-1S spacing by the time the quark mass attains 40 GeV. (The short-Compton-wavelength t quarks are able to probe this s i n g u l a r i t y ) . Similar differences occur i n the ratios of 2S and 3S leptonic widths relative to that of the IS state. The stronger the Coulomb-like s i n g u l a r i t y , the larger the 2S-1S spacing and the smaller the 2S/1S and 3S/1S leptonic width r a t i o s . We regard the potential of Ref. /44/ as the most singular credible potential i n the l i t e r a t u r e , but even more singular cases are quoted from time to time. The singularity in the potential of Ref. /44/ was due in some degree to the need to reproduce the observed cc and bb spin-independent spectra in the presence of r e l a t i v i s t i c corrections, and to reproduce t h e i r fine structure s p l i t t i n g s while assuming that transverse degrees of freedom in the gluon f i e l d be- come damped at large distances. The large 2S-1S spacing predcited in Table 3 f o r that model may be a key feature of the assumed physics.
TABLE 3. Examples of differences among predictions for t t spectra (nv=40 GeV) Quantity
2S-IS 3S-2S 4S-3S 2S-2P 3S-3P re e( 2 S ) / re e( l S ) re e( 3 S ) / re e( l S )
Martin ( R e f . / 5 2 / )
520 303 217 145 107 0.55 0.39
Buchmuller-Tye ( R e f . / 5 / )
A(MeV) 200 629 310 209 129 79 0.44 0.27
500 847 344 220 93 63 0.28 0.16
K'lihn- Ono
a) 644 308 217 91 65 0.33 0.21
b) 913 368 221 96 70 0.27 0.15
Moxhay- Rosner ( R e f . / 4 4 / )
958 372 231 105 66 0.27 0.15
a) "Potential T" ; b) Richardson potential. [See f i r s t of Refs. / 7 9 / ]
Many o t h e r i n t e r e s t i n g features o f t h e toponium system are discussed i n more d e t a i l elsewhere /79,80/. b!hen f r e e t quarks a r e produced, weak quark decay w i l l occur so r a p i d l y t h a t i t competes w i t h T* +.Ty. Thus i f a t quark i s "dressed" i n t o a T*, whose p o l a r i z a t i o n i s c o r r e l a t e d w ~ t h t h a t o f t h e t, t h e decay products i n t + b+
...
w i l l c a r r y some i n f o r m a t i o n about t h e i n i t i a l t quark p o l a r i z a t i o n /80/.
B. S c a l a r quarkonium
The bound s t a t e s o f a s c a l a r quark and s c a l a r a n t i q u a r k are i n general d i f f i c u l t t o f i n d /63-65 ; see a l s o appendix F o f Ref. 47/. L e t us suppose t h a t when p r o d u c i n ~ T i n e+e- a n n i h i l a t i o n s , through sheer l u c k we have a l s o e x c i t e d t h e t h i r d P s t a t e o f s c a l a r quarkonium. Then t h e 5 may be i n t e r p r e t e d as the ground s t a t e o f t h i s system, reached by an e l e c t r i c d i p o l e t r a n s i t i o n /65/.
To i l l u s t r a t e t h i s p o s s i b i l i t y we chose a s c a l a r quark mass, m- = 4.34 GeV, so as t o o b t a i n a 3P s t a t e degenerate w i t h T
,
and " f i n e - t u n e d " the p o t 8 n t i a l o f Ref. /44/ by s e t t i n g A= 600 MeV t o o b t a i n a r; a t 8.32 GeV. The r e s u l t i n g l e v e l s ? r e shown i n Fig.6. The p r e d i c t e d l e v e l s would simply be a carbon copy o f those i n bb, s h i f t e d by a constant, i f t h e p o t e n t i a l were o f t h e form /2,81/ V c I n r. Ke have i g n o r e d r e l a - t i v i s t i c and coupled-channel s h i f t s . S i m i l a r r e s u l t s are found i n Ref. 1651. Inde-
pendently o f t h e r;
,
t h e e x e r c i s e i s worth considering as a guide t o t h e search f o r s c a l a r quarkoni um.The p r o p e r t i e s o f s c a l a r quarkonium are a l l c a l c u l a b l e , g i v e n t h e quark charge, quark mass, and p o t e n t i a l /63/.
1. The l e p t o n i c widths o f P wave ( I - - ) s t a t e s are ree = 24 e - 2 a 2 ( R p ' ( 0 ) ( 2 / M 4,
9 (6.1)
where e i s t h e quark charge, R ( r ) i s t h e r a d i a l P wave wave f u n c t i o n , and M i s t h e resonanse mass. For resonance! i n t h e 8-10 GeV range, we o b t a i n re, = (30,120) eV f o r e q = (-1/3, 2/31. The values o f Rpl(0) do n o t vary appreciably w i t h p r i n c i p a l quantum number. The values (6.3) a r e t o be compared w i t h r e e ( ~ ) = 1.2 keV and t h e experimental upper l i m i t s from the LENA group /X?/, re ( r /rt t) S 300 eV f o r any resonance i n ete- a n n i h i l a t i o n s i n t h e ranges E,-.. =
f7.d -
9.481 and (8.67-
9.43) GeV. No s c a l a r quarkonium resonance would have been seen by t h e LENA scan.
S P D F
Fig. 6. R a d i a t i v e t r a n s i t i o n s i n a h y p o t e n t i c a l s c a l a r quarkonium system w i t h t h e t h i r d P s t a t e taken degenerate w i t h t h e T . The h o r i z o n t a l l i n e denotes an estimate o f f l a v o r threshold.
C2-90 JOURNAL DE PHYSIQUE
2. The p r e d i c t e d r a d i a t i v e decay widths o f the 3P s t a t e (the one assumed degener- ate w i t h T ) are shown i n Table 4 . Whatever the branching r a t i o B o f 3P t o
Y +?(1S), one expects equal o r l a r g e r branching r a t i o s t o y + 2S', y + 3S, and Y + 2D. These t r a n s i t i o n s are i l l u s t r a t e d i n F i g . 6. A f u r t h e r photon l i n e o f about 180-190 MeV occurs when the 2D s t a t e decays t o 2P, which i t ' s h o u l d do w i t h about an 85% branching r a t i o .
3. The p r e d i c t e d t h r e e - g l u o n widths o f the P states have been r e - e s t i m a t e d since the Conference. Recent c a l c u l a t i o n s by P. Moxhay, Y. J . Ng, and S.-H. H. Tye f i n d
160 a 3 M
r ( P - 3g) = - gi j rf - | R p ' ( 0 ) |2 I n J , ( 6 . 2 )
where A = 2< r > " . This i s s m a l l e r by a f a c t o r o f 4ir than the value i n published l i t e r a t u r e / 6 3 / . I t e n t a i l s r(3P ->• 3g) = (9+3, 22±7) keV f o r a? = ( 0 . 1 5 , 0 . 2 ) . The sum o f a l l r a d i a t i v e decay modes o f the 3P s t a t e i s about ( 4 4 , 171) keV f o r
eq = ( - 1 / 3 , 2 / 3 ) . Hence when the t h r e e - g l u o n width i s added t o the r a d i a t i v e w i d t h , the n e t . branching r a t i o f o r 3P •*• y+ IS i s about 15%. T h i s , combined w i t h the r a t i o . re e( 3 P ) / re e( T ) = 3%, leads t o a s i g n a l which would be o f acceptable magnitude
( ft 1/2%) f o r t h a t observed f o r the 5.
4. The value o f AR i n e+e " a n n i h i l a t i o n s due t o a s c a l a r quark o f eq = ( 1 / 3 , 2 / 3 ) i s ( 1 / 1 2 , 1 / 3 ) . Experiments a t high energies may j u s t b a r e l y be able t o exclude the l a t t e r p o s s i b i l i t y w i t h improved systematic e r r o r s / 8 3 / . Since a ( e+e " .* eft} * g3, no sharp step i n R w i l l be seen.
To summarize, p r o p e r t i e s o f the s c a l a r quarkonium system are so w e l l - d e f i n e d t h a t i f i t h a s n ' t been discovered a l r e a d y , systematic searches f o r i t are p o s s i b l e w i t h ex- periments o f present-day accuracy. I t would add immeasurably t o our understanding o f the spin-dependent forces between quarks, by p r o v i d i n g the textbook " s p i n l e s s c a s e . "
V I I - CONCLUSIONS
Q u a r k s have s p i n s . The d e s c r i p t i o n o f hadron s p e c t r a , t r a n s i t i o n s , and magnetic moments a l l depend q u a n t i t a t i v e l y on t h i s f a c t . Magnetic moments o f baryons are described t o w i t h i n 0 . 2 - 0 . 3 nuclear magneton i n the simplest model, and more accur- a t e l y i f d e s i r e d .
Hyperfine e f f e c t s i n quark models are w e l l understood, from the l i g h t e s t mesons ( T T , P , . . . ) and baryons ( N , A , . . . ) t o the h e a v i e s t . They are i n accord w i t h the p r e - d i c t i o n s o f a short-range s p i n - s p i n i n t e r a c t i o n mediated by v e c t o r gluon exchange.
With t h i s understanding, one can p r e d i c t w i t h confidence t h a t the e i s not the n^,!
Studies o f f i n e s t r u c t u r e {t-t and tensor e f f e c t s ) p o i n t toward the n o t i o n Of a s c a l a r q u a r k - c o n f i n i n g f o r c e . Many models based on t h i s n o t i o n agree i m p r e s s i v e l y w i t h s p e c t r a , e s p e c i a l l y those observed r e c e n t l y i n the bb system. A note o f
TABLE 4. P r e d i c t e d r a d i a t i v e decay rates o f s c a l a r quarkonium 3P s t a t e s , f o r m- = 4.34 GeV and p o t e n t i a l o f Ref. / 4 4 / , w i t h A * 600 MeV.
Decay
3P * 1S+Y 2S+r 3S+Y 1D+T ZD+r
u (MeV)
- T f J L
1070 520 180 380 92
r(keV)
«o = " 1 / 3
8.5 8.4 14.2 0.7 11.9
eq = 2 / 3 34 33 57 2.8 48
c a u t i o n : Many models based on a long-range s c a l a r f o r c e i n v o l v e e x t r a phenomenolopical parameters, u s u a l l y i n t h e form o f c u t o f f s . These models then cannot be e x t r a n o l a t e d r e l i a b l y t o s h o r t e r distances and h e a v i e r quarks. One o b t a i n s adequate d e s c r i p t i o n s o f most o f t h e phenomena so f a r observed w i t h models o t h e r than those i n v o l v i n g s c a l a r long-range exchange. Modest improvements i n xb, xb' l e v e l measurements, and discovery o f lP1 and D-wave s t a t e s i n quarkonium, would be most welcome.
The " o n e - p a r t i c l e " r e l a t i v i s t i c problem o f o r b i t a t l y e x c i t e d D and B mesons i s worth pursuing experimentally, as i t ought t o p r o v i d e f u r t h e r c o n s t r a i n t s on our under- standing o f t h e s p i n - o r b i t force. One i s s t i l l searching ( b o t h t h e o r e t i a l l y and ex- perimental l y ) f o r a system which would d i s p l a y t h e hypothesized m u l t i p l e t i n v e r s i o n expected i f a long-range s c a l a r f o r c e e x i s t s .
To n a i l down t h e n o n r e l a t i v i s t i c p o t e n t i a l , t h e toponium ( t f ) spectrum wi l 1 be very h e l p f u l . I f s c a l a r quarks e x i s t , they may p r o v i d e valuable clues t o spin-independent r e l a t i v i s t i c e f f e c t s .
What can a r e a l t h e o r y o f hadrons provide? I t would be very s a t i s f y i n g t o v e r i f y t h e i n t u i t i v e f l u x - t u b e p i c t u r e o f s c a l a r confinement. E v e n t u a l l y one hopes t o be a b l e t o c a l c u l a t e s p e c t r a and t r a n s i t i o n r a t e s independent o f t h e use o f a long-range
" p o t e n t i a l " . A t present, l a t t i c e gauge t h e o r i e s have made an encouraging s t a r t i n t h i s d i r e c t i o n . (See, e.g., Refs. /84/.) For t h e moment, p e r t u r b a t i v e QCD a t s h o r t distances, j o i n e d w i t h simple phenomenological p i c t u r e s a t l a r g e r ones, has l e d t o a very s a t i s f a c t o r y d e s c r i p t i o n o f t h e f o r c e s between s p i n n i n g quarks.
ACKNOWLEDGEEIENTS
I wish t o thank M. Bander, L. Brekke, W. Buchmuller, N. Byers, J . D. Jackson, B. Klima, J. H. Klhn, U. Maor, P. Moxhay, M. O r e g l i a and R. G. Sachs f o r f r u i t f u l d i s c u ~ s i o n s , and t h e CERN theory group f o r i t s h o s p i t a l i t y d u r i n g t h e . p r e a r a t i o n o f t h i s r e o o r t . This work was supported i n p a r t by t h e U. D. Department o f &ergy under c o n t r a c t no.
DE-AC02-82ER-40073.
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