The shape of mesons

-]Nuclear Physics B9 (1969) 10-16.

North-Itolland Publ. Comp., Amsterdam

THE SHAPE OF M E S O N S S. Y. LO

Rutherford High Energy Laboratory, Chillon, Didcot, Berkshire, England Received 24 October 1968

Abstract: From Chou-Yang's model for high-energy scatterings numerical values of the form factors 7r, K, p, co, 0 are deduced in analogy to the proton form factor. It is concluded that (i) the 7r, K form factors are incompatible with the naive vector meson dominance model for t ¢ 0, and (ii) the process y+p---~ d)+p is crucial to the validity of the model. F u r t h e r it is pointed out that the qualitative features of the results are valid for other s i m i l a r models as well.

1. I N T R O D U C T I O N Chou and Y a n g [1] have a r g u e d that the a s y m p t o t i c e l a s t i c s c a t t e r i n g a m p l i t u d e s of two h a d r o n s A, B s h o u l d be r e l a t e d to t h e i r f o r m f a c t o r s FA(I) , FB(I) by #2 V~AB(/) = /O.FA(I) FB(/) - ~ FA(/) FB(/) ~) FA(/) FB(/)

~3 + 37- FA(I) FB(/) C VA(t) FB(t) ® FA(t) FB(t) . . . .

(1.1)

w h e r e p is s o m e i n t e r a c t i o n c o n s t a n t , l = - q i s the f o u r - m o m e n t u m t r a n s f e r s q u a r e d , and ® d e n o t e s c o n v o l u t i o n i n t e g r a l s . T h e i r f o r m f a c t o r s can be r e l a t e d to t h e i r e l a s t i c s c a t t e r i n g a m p l i t u d e s by FA(/) FB(/) = c o n s t . [ ~ A B ( / ) + ½~AB(I) ® S A B ( t ) + ½~AB(/)®~AB(/)®

~AB(/) + ...].

(1.25

T h e r e is s t r o n g e v i d e n c e f r o m N - N s c a t t e r i n g [1,2] that the a b o v e f o r m u l a e hold q u i t e well. It i s the p u r p o s e of t h i s p a p e r to d e d u c e the f o r m f a c t o r s of 7r, K, p , co, 4) by eq. (1.2) f r o m e l a s t i c ~p, K p and p h o t o p r o d u c t i o n of vector mesons ~+p ~ V+p. T h e f o r m f a c t o r i s the F o u r i e r t r a n s f o r m of the d i s t r i b u t i o n of the h a d r o n m a t t e r i n s i d e the m e s o n F ( q 2) = f d 3 x p ( x ) e i q ' x = 1 - ~(r)2q 2 + ....

(1.35

MESON SHAPE

11

Any t h e o r y that d e a l s with the s t r u c t u r e of h a d r o n s h o u l d e x p l a i n the b e h a v i o u r of s u c h f o r m f a c t o r s . The p i o n , kaon f o r m f a c t o r s a r e d i s c u s s e d and c o m p a r e d with t h o s e o b t a i n e d f r o m the v e c t o r - m e s o n d o m i n a n c e m o d e l in s e c t . 2. T h e o , c o m e s o n s a r e d i s c u s s e d in s e c t . 3. In s e c t . 4, the ~ m e son is d i s c u s s e d , and it is found that p r e s e n t l o w - e n e r g y y + p ~ d) + p i s i n c o m p a t i b l e with the C h o u - Y a n g m o d e l . F u t u r e h i g h - e n e r g y data a r e c r u c i a l to the v a l i d i t y of the C h o u - Y a n g m o d e l .

2. THE ~ , K FORM F A C T O R S The a s y m p t o t i c c r o s s s e c t i o n s for e l a s t i c 7rp, Kp s c a t t e r i n g s a r e D a r a m e t r i z e d in the f o r m :

de =A e Bl m b / G e V 2 dt

(2.1)

The d a t a u s e d [3] a r e l i s t e d in t a b l e 1. Eq. (1.2) g i v e s us the p r o d u c t of F:r(t) Fp(t). In o r d e r to avoid the a m b i g u i t y of which p r o t o n f o r m f a c t o r s F l P ( / ) o r GMP(t) to u s e , one t a k e s the !oroton f o r m f a c t o r o b t a i n e d d i r e c t l y f r o m p - p s c a t t e r i n g . The a s y m p t o t i c f o r m of pp e l a s t i c s c a t t e r i n g is a l s o l i s t e d in t a b l e 1. T e r m s up to the fifth o r d e r in eq. (1.2) a r e u s e d in c a l c u l a t i n g the f o r m f a c t o r s . H o w e v e r for s m a l l t - v a l u e s I tl --< 1 GeV the f i r s t t e r m a c c o u n t s for m o r e than half of the c o n t r i b u t i o n . In g e n e r a l , the s m a l l e r Itl i s , the m o r e d o m i n a n t is the f i r s t t e r m . T h e r e is a l s o t h e u n c e r t a i n t i e s in the a s y m p t o t i c f o r m of c r o s s s e c t i o n s . U s u a l l y the h i g h e s t e n e r g y d a t a a r e u s e d . The f o r m f a c t o r s a r e s e n s i t i v e to the v a l u e s of B, but r a t h e r i n s e n s i t i v e to the v a l u e of A. The f o r m f a c t o r s Fzr(t), FK(t) , Fp(t) a r e p l o t t e d in fig. 1. The r a d i i @}, d e f i n e d by eq. (1.3), f o r the m e s o n s a r e l i s t e d in t a b l e 1. Both the pion and Droton f o r m f a c t o r s a r e o b t a i n e d f i r s t in ref. [1]. H e r e w e o n l y find that the kaon has a s m a l l e r r a d i u s ~ 0.54 fm. T h e e r r o r of 0.1 fm is a s c r i b e d to t h e s e v a l u e s due to the u n c e r t a i n t y of B in the a s y m p t o t i c f o r m of eq. (2.1). The v e c t o r - m e s o n d o m i n a n c e m o d e l in the f i e l d - c u r r e n t i d e n t i t y [4J v e r s i o n has d e f i n i t e p r e d i c t i o n s a b o u t the e l e c t r o m a g n e t i c f o r m f a c t o r s of 7r, K. They a r e -1 Fr;(t) = ( 1 - ~ t 2 ) fVM(t), P

FK(t ) = ½ _

P

+ ½

_

m~

$

2 i s i n 0 y sin ON rnc° /TfVM(t)M (2.2) + ~ ~os~-y-0N ) _t~2 V co w h e r e f M M ( t ) is the v e c t o r - m e s o n , p s e u d o s c a l a r - m e s o n , p s e u d o s c a l a r m e s o n f o r m f a c t o r s . T h e n a i v e v e c t o r - m e s o n m o d e l [5] is d e f i n e d by

12

S . Y . LO

Table 1 E l a s t i c s c a t t e r i n g c r o s s s e c t i o n s and r a d i i of h a d r o n s . E1 as tic scattering pp

A ( m b / G e V 2)

B (GeV 2)

(fm)

_

79.0

10.3

0.68±0.1

a)

F

rr-p b)

25.34

32.4

9.46±0.71

0.64 ± 0.1

K - p b)

15.91

24.6

7.85 • 1.26

0.54

OP c)

4.5-5.8

44.5

7.9

±0.9

0.51

cop

c)

4.5-5.8

73.9

7.6

±1.2

0.44

Cp

e)

4 . 5 - 5.8

11.4

fy/4rrm~=

P1 ab (GeV/c)

2.88,

f~/4rf

0

3.5

= 2.5, 0 y = 35 ° , 0 N = 22.5 °

0.585 GeV 2, m 2 = 0.614 GeV 2, ~7~) = 1.039 GeV 2.

a) F i t t a k e n f r o m r e f . [11 . b) D a t a t a k e n f r o m r e f s . [3, 7]. c) C a l c u l a t e d f r o m d a t a in r e f . [61 by eq. (3.3).

F(t)

-01

I

2

3 -t

GcV =

F i g . t. T h e f o r m f a c t o r of cO, p, K, p, g a s a f u n c t i o n of f o u r - m o m e n t u m ~,quared - t . T h e c i r c l e d d o t s a r e o b t a i n e d f r o m GpM(t) = {1 It i s p l o t t e d to g i v e a f e e l i n g of a c c u r a c y .

transfer

(t/0.71)~.-2.

MESON SHAPE

I.O

13

,

,

I

2

- -

! "'"i>,.

OI ~t

GcV 2

Fig. 2. C o m p a r i s o n s of the 7r, K f o r m f a c t o r s between the naive v e c t o r meson dominance model (dotted line), and the Chou-Yang model (solid line) as a function of -t.

lY

f l ~ i M ( / ) : 1.

(2.3)

T h e m i x i n g a n g l e s 0 N = 2 2 . 5 ° , 0 y = 35 ° can be o b t a i n e d f r o m v e c t o r - m e s o n d e c a y s [5J. H e n c e t h e v a l u e s f o r Fyr, F K of (2.2), (2.3), a r e c o m p a r e d w i t h t h o s e f r o m t h e C h o u - Y a n g m o d e l in f i g . 2. T h e d i s a g r e e m e n t is a f a c t o r of 3 at -! = 2 G e V 2. It i s f a i r to c o n c l u d e t h a t n a i v e v e c t o r m e s o n m o d e l i s i n c o m p a t i b l e w i t h t h e C h o u - Y a n g m o d e l at t ¢ 0. fn o r d e r to a v o i d t h e u n c e r t a i n t y of t h e f o r m f a c t o r fVMM(q), o n e c a l c u l a t e s t h e r a t i o

R(t) = F~ (t)/FK(I).

(2.4)

The comparison is made on fig. 3. The disagreement is about 30%, and it is difficult to draw a definite conclusion, since there are g r e a t e r uncertainties in the values of asymptotic c r o s s sections. In any case, one can say that the v e c t o r meson dominance model is good at least at q2 = 0. Hence it is possible to obtain elastic pp, wp, (bp scatterings from photoproduction processes.

14

S.Y. LO Io

9

8

7

5

o

i"

•2

4

b

8

I0

1'2

14

16 -t

GcV ~

Fig. 3. The r a t i o R(t) = F~(t)/FK(t ) o b t a i n s f r o m v e c t o r - m e s o n d q m i n a n e e model (dotted line), and f r o m C h o u - Y a n g m o d e l as a f u n c t i o n of - t (solid line). 3. T H F p , co F O R M

FACTORS

The electromagnetic current is given by the field-current identity [4, 5] to be

~,,,2oj. +2~yCOS 1 OYn'2 d);z - -2fy j~ =eLTo -Is Oi n Y

m2 COl"j .

(3.1)

The scattering amplitude for any process ),+p ~ a, where a is any eigenstate of strong interaction is

<~outl~ in) = e

(aoutlpp

in)

e

+

cos 0y
JO e

- --

2& s i n

O y { a o u t Ipco i n ) .

(3.2)

Hence we obtain al~proximately dCrD

dt

47r dcr (y, o) = a fp ;2 ~ (O, O)

dCrD -

'

-

dl

4~

0", ~) = a ~

dcr

cos2 0y ~/ (@, ~),

4fv

d~D 4~ dcr (co, co) dt (y, w) = a 4~y sin2 0y ~

(3.3)

MESON SHAPE

15

w h e r e dCrD(1,, V)/dt d e n o t e s the d i f f r a c t i v e p a r t of the c r o s s s e c t i o n s of the s c a t t e r i n g p r o c e s s e s y + p - - V + p , and d e ( V , V ) / d ! the e l a s t i c V-V s c a t t e r i n g c r o s s s e c t i o n s . The data u s e d [6J a r e l i s t e d in t a b l e 1, and the f o r m f a c t o r s ofp,co a r e p l o t t e d in fig. 1. The r a d i i of P,co a r e a l s o l i s t e d in t a b l e 1. Both co,O a r e s m a l l e r than the p r o t o n , the k a o n , and the pion, and co has the hardest core.

4. THE 0 -MESON, AND AN U P P E R BOUND ON

d(~/dl

rN THE SMALL-Ill REGION F r o m the p o s i t i v e d e f i n i t e n e s s of the r a d i u s @)2 * in eq. (1.3), one can s t a t e an i n e q u a l i t y of the e l a s t i c c r o s s s e c t i o n s of two h a d r o n s A, B at s m a l l

Ill

o.5 aeV2 dcr/{ d(~) <~F2(t) /=0-

(4.1)

dl/\d!

For the 0p ~ySp processes, one has the limit

dld~ --

O.II[dg

1/=0

<

Fp(2 t ) ,

(4.2)

where Fp(/) is the proton form factor. Roughly this is equivalent to a limit on B ~ 5.2 GeV -2. Experimentally for 7 + p - - d)+p B ~ 3.5±0.9. It violates the limit by about two standard deviation. If the B-value should persist in future higher-energy experiments, it would throw serious doubts on the Chou-Yang model. There is evidence at present [8] that B shrinks to ~ 4.5 at 15 GeV/c. Or alternatively, Chou-Yang model predicts that 7 + p ~ ~5+p should shrink at least to limit of (4.2). If it does, one has the result that d) is a very point-like object. This may explain why the ~ couplings to other particles, e.g. 4~NNis smaller than predicted by usual SU(3) symmetry a r guments, since the overlapping wave functions of 0NN would be much smaller than that of say, pNN.

5. CONCLUSIONS A l t h o u g h the C h o u - Y a n g m o d e l is e x c l u s i v e l y u s e d in a r r i v i n g at the n u m e r i c a l v a l u e s , any m o d e l that has the a s y m p t o t i c f o r m for e l a s t i c AB s c a t t e r i n g c r o s s s e c t i o n [9J: act/{ d(~)

=

F2A(t)F2B(I)

(5.1)

s h o u l d a l s o have the s a m e q u a l i t a t i v e f e a t u r e s p r e s e n t e d above. They a r e : (a) v, p, K, P, w all have r a d i i of the s a m e o r d e r of m a g n i t u d e 0.4 ~ 0.7 fro. F o r the m e s o n , the h e a v i e r one a l s o has a s m a l l e r r a d i u s . T h e i r f o r m f a c t o r s d i f f e r m o r e at l a r g e r ]t], o r , in o t h e r w o r d s , they d i f f e r m o r e in the c o r e s t r u c t u r e s . * This is derived from assuming the density PA(X) i> 0.

S.Y.

16

LO

(b) Naive vector meson model is incompatible with (5.1) at f f 0. (c) The upper bound on da/d1 of.eq. (4.2) holds. The nrocess y+p - Q+p is crucial as a test of models that have the form (5.1). The @-meson is most likely a very point-like object. The author uscript.

wishes

to thank

C. Michael

for a critical

reading

of the man-

REFERENCES [l] [2] [3]

[4] [5] [6] [7] [8] [9]

T. T. Chou and C. N. Yang, Phgs. Rev. Letters 20 (1968) 1213; Phys. Rev. (1968) 1591. S. Y. Lo, Phys. Letters 27B (1968) 308. K. J. Foley et al., Phys. Rev. Letters 15 (1965) 45; Aachen-Berlin-CERN-London (I.C.)-Vienna Collaboration, M. Aderholz Phys. Letters 24B (1967) 434. N. M. Kroll, T. D. Lee and B. Zumino, Phys. Rev. 157 (1967) 1376. S.Y.Lo, Nucl.Phys.B7 (1968) 68. R. J. Oakes and J. J. Sakurai, Phys. Rev. Letters 19 (1967) 1266. Aachen-Rerlin-Bonn-Hamburg-Heidelberg-Mtinchen, Phys. Letters 27B 54. preprint. W. G. Jones et al., SLAC-PUB-434 L. Van Hove, Particle interaction at high energies, ed. by T. W. Preist 1. L. J. Vick (Oliver and Boyd 1967) pp. 63.

170

et al.,

(1968)

and