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Tests of vector-dominance in e+e− collisions above 1 GeV
~
Nuclear Physics B15 (1970) 35-44. North-Holland Publ. Comp., Amsterdam
TESTS OF V E C T O R - D O M I N A N C E I N c+e - COLLISIONS ABOVE 1 GeV* PROBIR ROY
Laboratory of Nuclear Studies, Cornell University Ithaca, New York Received 7 July 1969 Abstract: We propose tests of various aspects of the vector-meson-dominance hypothesis that can be made by studying the reactions e+e - --+ 7r~, KK, 7rp, ~W and ~qb above 1 GeV in the c.m.
1. I N T R O D U C T I O N S t u d i e s [1, 2] of e+e - a n n i h i l a t i o n s into h a d r o n s up to ~ 1 GeV in the c . m . h a v e y i e l d e d c l e a n a n d e l e g a n t r e s u l t s on the v e c t o r m e s o n s p, w a n d q$ d u r i n g the p a s t y e a r s . T h e c o n t i n u a t i o n of t h e s e s t u d i e s in the r e g i o n n e a r 1 GeV a n d h i g h e r up i s of s o m e t h e o r e t i c a l i n t e r e s t s i n c e they c a n t e s t c e r t a i n c r u c i a l a s p e c t s of the v e c t o r - m e s o n d o m i n a n c e [3] h y p o t h e s i s in the t i m e l i k e r e g i o n without r e c o u r s e to o t h e r e x p e r i m e n t s . In t h i s p a p e r we s h a l l d i s c u s s how t h e s e t e s t s c a n be p e r f o r m e d by s t u d y i n g f i n a l s t a t e s such a s ~n, KK a n d p~ a n d o t h e r s in e+e - c o l l i s i o n s with ~Ts n e a r a n d a b o v e 1 GeV. T h e u s e of v e c t o r d o m i n a n c e in v a r i o u s e+e - r e a c t i o n s h a s p r e v i o u s l y b e e n m a d e by a n u m b e r of a u t h o r s [4 - 6] i n the n e i g h b o r h o o d of the v e c t o r - m e s o n m a s s e s . H o w e v e r , we shall be c o n c e r n e d also with regions
significantly h i g h e r in energy w h e r e the u s e of the explicitly p o l e - d o m i nated n o n - u n i t a r y v e c t o r - m e s o n propagators (or o f t h e i r partially u n i t a r i z e d modifications) is no longer reliable. In sect. 2 we d i s c u s s the v a r i o u s c o m p o n e n t s of the full s t r u c t u r e of the v e c t o r - m e s o n d o m i n a n c e h y p o t h e s i s a s it i s n o r m a l l y u s e d to c o r r e l a t e d i f f e r e n t s e t s of e x p e r i m e n t a l data. T h e m o s t b a s i c of t h e s e i s the c u r r e n t f i e l d i d e n t i t y . T h e n c o m e s the u n i v e r s a l i t y of v e c t o r - m e s o n c o u p l i n g c o n s t a n t s o r m o r e s t r o n g l y the u n i v e r s a l i t y of the v e c t o r - m e s o n c o n t r i b u t i o n to h a d r o n 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 f o r q2 ¢0. F i n a l l y , t h e r e i s the s m o o t h n e s s a s s u m p t i o n that a l l v e c t o r - m e s o n v e r t e x f u n c t i o n s a r e c o u p l i n g constants. In sect. 3 we d i s c u s s the s i m p l e s t t e s t of the s t r o n g v e r s i o n of u n i v e r s a l i t y (for ~ s i n the r e g i o n of 1.5 GeV) by r e l a t i n g ~tot(e+e - -~ n+~-), ~tot(e+e - ~ K+K -) a n d ~tot(e+e - -~ ,~o,~o~ r~l~2J. * Work
supported
in part by the Office of Naval Research.
3~
P. ROY
In s e c t . 4 we p r o p o s e a t e s t of t h e f u l l VMD h y p o t h e s i s w h i c h r e l a t e s ~ t o t ( e + e - -~ p~), ~tot(e+e - -~ K+K -) a n d ~tot(e+e - -~ K~K~) n e a r s = m ~ . O u r a r g u m e n t s a l s o l e a d to a r e l a t i o n t h a t c a n t e s t VMD a t h i g h e r v a l u e s o f s p r o v i d e d one c a n m e a s u r e ~tot(e+e - -~ ~°w) a n d ~tot(e+e - -~ ~°qS) in t h o s e regions.
2. VMD H Y P O T H E S I S T h e f u l l c o n t e n t of t h e VMD h y p o t h e s i s c a n b e b e s t d e s c r i b e d by t h r e e a s s u m p t i o n s w h i c h a r e l i s t e d b e l o w in t h e o r d e r of d e c r e a s i n g r e l i a b i l i t y . (i) C u r r e n t - f i e l d i d e n t i t y [7 - 9]. A s an o p e r a t o r r e l a t i o n l i n k i n g t h e e l e c t r o m a g n e t i c c u r r e n t j E M = j V + j S to v e c t o r - m e s o n f i e l d s p # , w # a n d t't-~#, this reads
t h e g ' s r e f e r to t h e c o u p l i n g c o n s t a n t s [10] of v e c t o r m e s o n s . E x t e n d e d to t h e f u l l o c t e t , t h e s e i d e n t i t i e s m e a n t h a t t h e h a d r o n i c c u r r e n t s s a t i s f y t h e a l g e b r a of f i e l d s [11]. M o r e o v e r t h e s e e s s e n t i a l l y i m p l y t h a t t h e p h o t o n v e r t e x - f u n c t i o n ( f o r o~ = fl t h i s i s r e l a t e d to t h e o r d i n a r y e . m . f o r m f a c t o r ) F ~ f l (t) with any two p a r t i c l e s c~, fi c a n b e w r i t t e n (with s u i t a b l e C I e b s c h - G o r d a n c o e f f i c i e n t s f a c t o r e d out) a s where
FVfl(t) = gpAp (t) Ypozfl(t)
,
fl( t) = gwaw(t) y wo~fl(t) + g c a~ (t) ~ ~efl(t) .
(2.1)
In eq. (2.1) A v ( t ) (ggu - (q~qu/m2)) i s t h e v e c t o r - m e s o n p r o p a g a t o r (with continuum contributions properly included) carrying four-momentum s q u a r e d t w i t h AV(0~= m ~ 2 a n d t h e v e r t e x f u n c t i o n s ~V~fl(t) a r e a s d e f i n e d in fig. 1. N e a r t = m@, one i n c l u d e s t h e w i d t h F V b y A v ( t ) = 1/[m2r - t -imvFvf(~) ] w h e r e f(t) i s a k n o w n t h r e s h o l d f a c t o r s a t i s f y i n g f(m21= 1. H o w e v e r , we s h a l l not n e e d to u s e a n y s p e c i f i c f o r m f o r AV(t).
Fv
[I) s ~ ~'"
F S It)
I=O
t
B
aB ~ s % 11% %
Fig. 1. Diagrammatic representation of eq. (2.1).
VECTOR DOMINANCE
37
(ii) U n i v e r s a l i t y . T h i s m a k e s the c o u p l i n g b e t w e e n a v e c t o r m e s o n a n d two s i m i l a r p a r t i c l e s , with a l l t h r e e b e i n g on the m a s s - s h e l l , u n i v e r s a l . In o t h e r w o r d s , in fig. 2, Y V a ~ = ~ ¢ so t h a t f o r e x a m p l e 7pN!q = ~ p ~ = YpKK = 7p. A s t r o n g e r v e r s i o n of u n i v e r s a l i t y a s s u m e s that the s a m e i s t r u e e v e n w h e n t h e v e c t o r m e s o n i s off t h e m a s s s h e l l so that y p ~ ( t ) = T p N ~ ( t ) = y p ( t ) a n d s i m i l a r l y for the other cases.
~,
(t) Vaa ~
V
a
....t
Fig. 2. The Vot~ vertex. (iii) S m o o t h n e s s . T h i s s t a t e s that the v e c t o r - m e s o n v e r t e x f u n c t i o n s do not c h a n g e when the v e c t o r m e s o n i s off the m a s s s h e l l , i.e. ~Vap(t)
=
~V~
,
w h e r e YVafl i s a c o n s t a n t . Note that f o r a = ~, the n o r m a l i z a t i o n of 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 F(0) = 1 i m p l i e s g v ~ v a a / m 2 = 1, i.e. Y V a a i s a u n i v e r s a l c o n s t a n t . In o t h e r w o r d s , a s s u m p t i o n s (i) a n d (iii) i m p l y (ii) at l e a s t f o r t = 0. T h e f i r s t of the a b o v e a s s u m p t i o n s c a n b e r i g o r o u s l y c h e c k e d i n e+e c o l l i s i o n s f r o m a k n o w l e d g e of the t o t a l c r o s s s e c t i o n f o r e+e - -~ h a d r o n s f o r a s y m p t o t i c s by t e s t i n g the r e s u l t [12] t h a t f o r s i n g l e p h o t o n e x c h a n g e lim
s 2 Crtot (e+e - -~ h a d r o n s ) = 0 .
O u r i n t e r e s t h e r e i s , h o w e v e r , i n c o n s i d e r i n g t e s t s f o r a s s u m p t i o n s (ii) a n d (iii) w h i c h a r e e x p e c t e d to b e c o m e i n c r e a s i n g l y i n a c c u r a t e a s one m o v e s away f r o m t h e r e s o n a n c e m a s s e s [13]*
3. S I M P L E S T T E S T O F T H E STRONG VERSION O F U N I V E R S A L I T Y W e s t a r t by c o n s i d e r i n g the r e a c t i o n e+e - -~ ~+~-. T h e t o t a l c r o s s s e c t i o n [12] a s a f u n c t i o n of s i s g i v e n b y * * 3
~tot ( e + e - -~ ~+~-) = ~ ~ 2
~_
i F ~ ( s ) 12 ,
S:
* There is already evidence in the studies of nucleon electromagnetic from factors that a straight forward application of assumption (iii} is not valid for large spacelike t [13]. ** We shall always assume single photon exchange.
38
P. ROY
w h e r e the pion 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 FTr(s ) by a s s u m p t i o n (i) is g i v e n by
F~(s) = gpAp(S) :~prr~(s) ,
(3.1)
so that ~tot (e+e- -~ ~+~-) = g1 ~(~2
( s-4rn2+~ ~ 2 2 ~ ~ Z, [ ~ gpl/Xp(S) ~pTrv(s) 82
(3.2)
o o2 have total On the o t h e r hand, the r e a c t i o n s e+e - --* K+K - and e+e - ~ K1K c r o s s s e c t i o n s g i v e n r e s p e c t i v e l y by * 2 (s-4m2+) z 1=1 I:O(s) i2 (~tot (e+e- -~ K+K-) = 13 7roL _~ , IF K (s)+F K $2
(3.3)
and 3
o o
2(s-4m~°
~tot ( e + e - -~ K 1 K 2 / -- ~ ~ro~
_,
f
I
_F/=I(s)+F/K =
0(~)
12
,
(3.4/
S 2
where
~= l(s) = %%(s) ~p~(s) ,
(3.5)
I=O F K (s) : gwAo)(s) ~wK~(s)+gqbAb(S) ydpK~(S) •
(3.6)
and
F o r ~/-s up to 1 . 5 G e V the n e a r b y ~ - p o l e will m a k e A~b(s) >> Ap(s) and Aw(s ~. Since AV(0) = rnV¢~, at s = O, gpVpK~(O)rn~ ~ =gwVwK~(O)rn-w ~"+ g~V~K~(O)rn~b ~. At t h i s s t a g e , we shall m a k e two v e r y p l a u s i b l e a s s u m p t i o n s f o r ~/s in the r e g i o n of 1.5 GeV. 1) T h e p h a s e s of F I= O(s) and F / = l(s) a r e n e g l i b l e at t h e s e v a l u e s of s. 2) FI= O(s) >~K = l(s~ in this r e g i o n , i.e., v a r i a t i o n s in the v e r t e x f u n c tion c a n n o t o v e r c o m p e n s a t e the effect of A s ( s ) >> A^(s). T h e f i r s t a s s u m p t i o n is r e a s o n a b l e s i n c e v~e a r e f a r~ a w a y f r o m the p, ~0 m a s s e s w h e r e a s the ~ - w i d t h is v e r y s m a l l [14]**. T h e s e c o n d a s s u m p t i o n is in the s p i r i t of a s s u m p t i o n (iii) of sect. 2 and should hold so long a s the v i o l a t i o n s of the l a t t e r a r e not c a t a s t r o p h i c . We now obtain f r o m eqs. (3.3), (3.4) and (3.5) * We are ignoring electromagnetic vertex connections which can be estimated to be about 4.2% in the amplitude at the ~b-mass [17]. ** As an order of magnitude estimate, we note that the phase of the p-propagator at ~s= 1.5 GeV is 8 ° in the model of ref. [14].
39
VECTOR DOMINANCE
g2p l Ap (S ) !2 y2pK~(S) 5
3s ~ = 7rot~
+ :
v~-~tot(e+e--~K+K-)
[/s
2 \314
-
-4rnIC )
3:. ~ t2
/(Ztot(e e - ~ K 1 K 2 ) T-~2-x~ ~.s - ~ m K O )
I " )
(3.7)
A c o m p a r i s o n of eqs. (3.2) and (3.7) i m p l i e s that if the s t r o n g v e r s i o n of a s s u m p t i o n (ii) is i n d e e d v a l i d up to 4-s ~ 1.5 GeV, we m u s t h a v e
/
-
(e+e - -~ ~+Tr-) ~t°~/st~ 2 \3/2 =
+
___,
o
-)} o"
2
~/CXtot(e+e- -~ K+K-) ~/~totke e K1K 2 ~,/s 2 ,3/, 7---2--7~ , - 4inK+ )
,
(3,8)
f o r 4-~ a r o u n d 1.5 GeV. T h e c . m . e n e r g y is c h o s e n o p t i m a l l y a r o u n d 1.5 GeV s i n c e if one is b e l o w that the p h a s e of F I= l(s) s t a r t s b e c o m i n g i m p o r tant. On the o t h e r hand, if we go to too high a v a l u e of s, F~=O(s) will no l o n g e r be g r e a t e r t h a n F/K-- l(s). T h e l a t t e r can, of c o u r s e , ~ e e x p e r i m e n t a l l y c h e c k e d by studying ~tot(e+e - ~ K~K~) a c c o r d i n g to eq. (3.4). We e m p h a s i z e that eq. (3.8) c o n s t i t u t e s a t e s t of the s t r o n g u n i v e r s a l i t y a s s u m p tion*.
4. T E S T O F T H E F U L L VMD H Y P O T H E S I S We shall now c o n s i d e r the p r o c e s s e+e - -~ 7rp which is i m p o r t a n t f r o m a n o t h e r point of v i e w since by studying it one c a n m e a s u r e the i s o s p i n of the photon d i r e c t l y if one a s s u m e s single p h o t o n e x c h a n g e . F i r s t let u s look at the 7r77 v e r t e x with o f f - s h e l l p h o t o n s a s d e s c r i b e d in fig. 3. If the o n - s h e l l v e r t e x is e~(q)eV(q')M~v(0 , 0), f o r o f f - s h e l l p h o t o n s we have
F r y y ( q Z q 2 ) .-,./~'/ a2 ~-
I
2
,2
q2,I- I
q ,I'O
2
2
2
q2 i=O
q' , I =I
Fig. 3. /-spin decomposition of the TrT7 vertex.
* At this point one may well ask how crucially does eq. (3.8) test the idea of vectormeson universality. Clearly, the main principle leading to this relation is the universality of the isovector electromagnetic form factors with or without vector mesons. However, the~wo assumptions ~e have made above may not be true for ~/s around 1.5 GeV if F~(= l(s) is not dominated only by the p and FI= l(s) only by the ¢0 and the ~b.
40
P. ROY
Mlau (q2,q,2) = - e- 2 E m 7[o pvafi
q ~pflF
zr 7ryy
(q2, q ,2) .
(4.1)
In eq. (4.1) FTyy (q2, q'2)= Flrr (q2, q2)+ F2y r (q2, q'2) where the vertex functions F 1 and F 2 are given in fig. 3. By symmetry we then have
F1
7ry~
(q2, q ,2) = F 2
7ryy
(q ,2, q2) .
(4.2)
By employing assumption (i) of sect. 2, we can write F 1 (q2,q,2) = F 7ryy
(q2,q,2) A (q2) gp ~rp'y
p
'
and F21ryy (q2, q,2) = FTrvco (q,2, q2) £xw (q2) gco + FTryq5 (q,2, q2) A b (q2) gq5 ' where the definitions of Fvpy(q2,q'2) , FT~ co(q'2,q 2) and F v (q,2, q2) should be transparent in analogy with eq. (4.1). Hence eq. (4.2) implies that
Fvp7 (q2,q ,2) Ap (q2) go = FTTyw (q2,q ,2) Aw (q,2)gw+FTw~ (q2,q,2) A~b (q,2)g~b
(4.3)
By assumption (iii) of sect. 2 0 Fnp7 (q2,q'2) = _ _0 FTr)/w (q2,q,2) = 0
Oq2
Oq,2
Oq,2
Fuv~ (q2,q,2) = 0.
Eq. (4.3) can therefore be rewritten as
{ 2 ,2\ FTrpy ~mp,q )
Ap
~, )/q2\gp
Choosing q2 =0 and writing q ,2 =s, we then have
FTrpy (m2 S) gp P' = F 7r~w (0' m2co) Aco(S)gw +FTr~,~ (O'm2) Aqs(S)gqs" 2 7n
P Near the 4~-mass the 2nd RHS t e r m predominates overwhelmingly and we can w r i t e
2 ~FTrpy (rap m P gp
(4.4)
41
VECTOR DOMI~ANCE On the o t h e r hand, f r o m eqs. (3.5) and (3.6) we have that n e a r the qS-mass ,:0
FK
Hence f r o m eqs. (3.3) and (3.4) we have fftot (e+e- -~ K+K-)s ~ m~ 4mK+)~ o o ~tot(e+e--~K1K2)s
2
-
~
(mq~2 _ 4mKo)~
~m~
=
~
2
2 2
~g~q~Ki~ia~
(s ~m~)]2
(4.5)
3m
E l i m i n a t i n g g~A~p(S ~ m2~) f r o m eq. (4.4) and eq. (4.5) we h a v e * O
O
~tot (e+e- -* K+K-)s ~ m¢2 = (~tot ( e+e- -~ K1K2)s ~ rn~
-3 ~ 4~¢K~
(0,
~
¢4.6)
I~
In the RHS of eq. (4.6), y ~KK is well known f r o m F((p -~ KK) and F u p y can be o b t a i n e d f r o m the r e l a t i o n
e atot (e+e- --" ~OpO) =
IF~ (m ,s)[ 12~ m 2 s ~
-rap
~o] _ m 2 4s
(4.7)
~o
71
by m e a s u r i n g atot (e+e - --" ~OpO) at s ~ m ~ . Fny~)(0, m 2) is in p r i n c i p l e obtainable f r o m the decay ~ -~ ~y which, h o w e v e r , h a s n o t b e e n s e e n yet. But one can c a l c u l a t e it following C r e m m e r and G o u r d i n [6] by using the
* If the smoothness assumption is violated so that YVafl (t) = YVafl (0) (1 + Xv(t/m~¢)), IFTrD./ (rn~, rn~) I in the RHS of eq. {4.6) should be divided by a factor of (1 +Xp). Similarly, eq[ (4.8) should read
and eq. (4.9) should be altered accordingly. For a consideration of off-shell correc tions to the Gell-Mann-Sharp-Wagner model, see ref. [18].
42
P-. ROY
VMD relation
2
_ ~
770~ mTTO
%~
+ ~%~
C¢)
(o,.~
,
D~0
and plugging in go), go [10] with F?7~o~(0, m 2) obtained from F(~o~ 77o~). With current numbers Cremmer and C Gcdin [6] obtain* [F7770(0, m2) l +0.i0 = 0.13_ 0.09 but the e r r o r s will presumably improve with better data. Thus
m e a s u r i n g ~tot ( e+e- -~ K+K-) or etot (e+e - _, K1K2 ) o o and ~tot (e+e - ~ np) at s ~ m~ and using eqs. (4.6) and (4.7) one can test the VMD hypothesis near the 0-mass~ without knowing the details of the 0-propagator. A corresponding test for tirnelike ~-s significantly higher than the v e c t o r - m e s o n m a s s e s is possible provided one is able (and willing) to m e a s u r e ~tot (e+e- -~ vow) and ~tot (e+e - ~ ?700) for such values of ~/s. This can be derived by going back to eq. (4.3) and rewriting it as [
2
,2\
F?Tp7 ~mp, q J
Ap (q2) gP
(q'2) gco+F?T70
=F?7~c° (q2, m 2 ) A
Taking q,2 = 0, q2 = s and using [6] ½ Fv77 eq. (2.1) we obtain
½F?77F (O,O) rn2pAp(S) = F~7co
(0, O) = (gp/m2) FvpT(m 2, 0) F?TV0
m2o)
(q 2 , m 02) A0 ( q ' 2 ) g 0 .
O, %2 ) ~ .
from
(4.8)
m~
+
If by using assumption (iii) of sect. 2, we substitute gp7p?7?7 = m 2 in eq. (3.2) we have
lAp(S)!2----(
Crtot (e+e- ~
3
s2
77o!2 s
Moreover
-4rn
~
?7+?7-)
mp4
½F?7vv(O,O)=IF(n°~ 27)}] 77c~2m °
and in analogy with eq. (4.7)
atot (e+e-
=
-~--5'
12 m s 2 ?7
* See however ref. [15].
4S
- my
VECTOR DOMINANCE
FTr~w(s,m2)
43
~#p
~
Now in eq. (4.8) the p h a s e s of and FTr (s, rn ) a r e the s a m e ( a p a r t f r o m o v e r a l l signs) a c c o r d i n g to the VMD h y p o t h e s i s , e a c h being the p h a s e of ±p(S). H e n c e we c a n r e w r i t e eq. (4.8) a s 4 ~0~2
sF(~ ° -~ 2V) Crtot (e+e - ~ ~+~-) m 3o ( s
:
-
4 r n2o ) z3
1
-
mw 2 2 .2 • -3/4 -Igor2[ l(s-rn~-~smTr°, -rn:° I ~/(~tot (e+eL
e
7,
o,
12 -~ g ° ~ ) [
.
(4.9)
2 FTrV~ and (g~b/m~) F~y~b have In eq. (4.9) we have u s e d the f a c t t h a t (g~/mo~) o p p o s i t e s i g n s [6].
4. CONCLUDING R E M A R K S T h e v a r i o u s p r e d i c t i o n s of the v e c t o r - d o m ~ n c e h y p o t h e s i s a r e at p r e s ent in b r o a d a g r e e m e n t with e x p e r i m e n t f o r ~/q~' n e a r the p - m e s o n m a s s . H o w e v e r , the h y p o t h e s i s is l e s s than well e s t a b l i s h e d a s one m o v e s a w a y f r o m t h i s region. T h e r e a r e a l r e a d y q u e s t i o n s [16] about the v a l i d i t y of the s m o o t h n e s s a s s u m p t i o n in p o l a r i z e d pion p h o t o p r o d u c t i o n . In the s p a c e l i k e r e g i o n s i m p l e v e c t o r d o m i n a n c e of the n u c l e o n f o r m f a c t o r s d o e s not a g r e e with e x p e r i m e n t [13]. H e n c e it is i m p o r t a n t to a s k how well the v a r i o u s c o m p o n e n t s of the h y p o t h e s i s w o r k f o r t i m e l i k e v a l u e s of ~ / ~ a b o v e the p - m a s s . In this p a p e r , we have c o n s i d e r e d s e v e r a l p o s s i b l e t e s t s of the VMD h y p o t h e s i s in t h i s r e g i o n . T h e t e s t p r o p o s e d in sect. 2 a s s u m e s the c u r r e n t - f i e l d identity and v e c t o r - m e s o n u n i v e r s a l i t y but can a c c o m o d a t e r e a s o n a b l e v i o l a t i o n s of the a s s u m p t i o n of s m o o t h n e s s of v e c t o r - m e s o n f o r m f a c t o r s . T h e r e l a t i o n s o b t a i n e d in sect. 3 m a k e u s e of all the t h r e e a s s u m p t i o n s . T h e s e r e l a t i o n s can be t e s t e d only a f t e r an o r d e r of m a g n i t u d e i m p r o v e m e n t in the p r e s e n t s t o r a g e - r i n g s t a t i s t i c s . T h e s e a r e , h o w e v e r , c l e a n p r o b e s on the v a l i d i t y of the v e c t o r - m e s o n - d o m i n a n c e h y p o t h e s i s f o r t i m e l i k e p h o t o n s a b o v e 1 GeV. I would like to thank P r o f e s s o r D. R. Yennie, Dr. J. P e s t i e a u and P r o f e s s o r M. G o u r d i n f o r d i s c u s s i o n s .
REFERENCES [1] V. L.Auslander et al., Phys. Letters 25B (1967) 433 and to be published. J.Augustin et al., Phys. Letters 28B (1969) 508, 513, 518. [2] E. Celeghini and R.Gatto, Nuovo Cimento 57A (1968) 549.
44
[3] [4] [5] [6] [7[ [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
P. ROY
J . J . S a k u r a i , Ann. of P h y s . 11 (1960) 1. G . A l t a r e l l i et al., Nuovo C i m e n t o 47 (1967) 113. A. D o n n a c h i e , P h y s . L e t t e r s 27B (1968) 525. E . C r e m m e r and M . G o u r d i n , Nucl. P h y s . , B10 (1969) 179; M . G o u r d i n , M i s c e l l a n e o u s t o p i c s r e l a t e d to the a n n i h i l a t i o n of e+e - p a i r s ( O r s a y report). R. Haag, K. N i s h i j i m a and R. S c h r o e r , u n p u b l i s h e d (1961). M . G e l l - M a n n and F . Z a c h a r i a s e n , P h y s . Rev. 124 (1961) 953. N . M . K r o l I , T . D . L e e and B. Z u m i n o , P h y s . Rev. 157 (1967) 1376. S. Ting, R a p p o r t e u r ' s talk: E x p e r i m e n t a l e l e c t r o m a g n e t i c i n t e r a c t i o n s , P r o c e e d ings of the 14th I n t e r n a t i o n a l c o n f e r e n c e on high e n e r g y p h y s i c s (ed. J . P r e n t k i and J . S t e i n b e r g e r , CERN 1968) p. 43. T . D . Lee, S . W e i n b e r g a n d B . Z u m i n o , P h y s . Rev. L e t t e r s 18 (1967) 1029. J . J . S a k u r a i , L e c t u r e s on c u r r e n t s and m e s o n s (Univ. of Chicago P r e s s , C h i cago, 1969). G . W e b e r , P r o c e e d i n g s of the 1967 i n t e r n a t i o n a l s y m p o s i u m on e l e c t r o n and photon i n t e r a c t i o n s (ed. S . B e r m a n , SLAC 1967), p. 59; R . E . T a y l o r , ibid, p. 78. G.J.Gounaris andJ.J.Sakurai, P h y s . Rev. L e t t e r s 21 (1968) 244. C . B e m p o r a d et al., P h y s . L e t t e r s 29B {1969) 383. C . G e w e n i g e r et al., P h y s . L e t t e r s 28B (1968) 155; 29B (1969) 41. M.Gourdin, private communication. A. T. F i l l i p o v and Yu. N. J e p i f a n o v , JINR P r e p r i n t , to be p u b l i s h e d .
Nuclear Physics B15 (1970) 35-44. North-Holland Publ. Comp., Amsterdam
TESTS OF V E C T O R - D O M I N A N C E I N c+e - COLLISIONS ABOVE 1 GeV* PROBIR ROY
Laboratory of Nuclear Studies, Cornell University Ithaca, New York Received 7 July 1969 Abstract: We propose tests of various aspects of the vector-meson-dominance hypothesis that can be made by studying the reactions e+e - --+ 7r~, KK, 7rp, ~W and ~qb above 1 GeV in the c.m.
1. I N T R O D U C T I O N S t u d i e s [1, 2] of e+e - a n n i h i l a t i o n s into h a d r o n s up to ~ 1 GeV in the c . m . h a v e y i e l d e d c l e a n a n d e l e g a n t r e s u l t s on the v e c t o r m e s o n s p, w a n d q$ d u r i n g the p a s t y e a r s . T h e c o n t i n u a t i o n of t h e s e s t u d i e s in the r e g i o n n e a r 1 GeV a n d h i g h e r up i s of s o m e t h e o r e t i c a l i n t e r e s t s i n c e they c a n t e s t c e r t a i n c r u c i a l a s p e c t s of the v e c t o r - m e s o n d o m i n a n c e [3] h y p o t h e s i s in the t i m e l i k e r e g i o n without r e c o u r s e to o t h e r e x p e r i m e n t s . In t h i s p a p e r we s h a l l d i s c u s s how t h e s e t e s t s c a n be p e r f o r m e d by s t u d y i n g f i n a l s t a t e s such a s ~n, KK a n d p~ a n d o t h e r s in e+e - c o l l i s i o n s with ~Ts n e a r a n d a b o v e 1 GeV. T h e u s e of v e c t o r d o m i n a n c e in v a r i o u s e+e - r e a c t i o n s h a s p r e v i o u s l y b e e n m a d e by a n u m b e r of a u t h o r s [4 - 6] i n the n e i g h b o r h o o d of the v e c t o r - m e s o n m a s s e s . H o w e v e r , we shall be c o n c e r n e d also with regions
significantly h i g h e r in energy w h e r e the u s e of the explicitly p o l e - d o m i nated n o n - u n i t a r y v e c t o r - m e s o n propagators (or o f t h e i r partially u n i t a r i z e d modifications) is no longer reliable. In sect. 2 we d i s c u s s the v a r i o u s c o m p o n e n t s of the full s t r u c t u r e of the v e c t o r - m e s o n d o m i n a n c e h y p o t h e s i s a s it i s n o r m a l l y u s e d to c o r r e l a t e d i f f e r e n t s e t s of e x p e r i m e n t a l data. T h e m o s t b a s i c of t h e s e i s the c u r r e n t f i e l d i d e n t i t y . T h e n c o m e s the u n i v e r s a l i t y of v e c t o r - m e s o n c o u p l i n g c o n s t a n t s o r m o r e s t r o n g l y the u n i v e r s a l i t y of the v e c t o r - m e s o n c o n t r i b u t i o n to h a d r o n 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 f o r q2 ¢0. F i n a l l y , t h e r e i s the s m o o t h n e s s a s s u m p t i o n that a l l v e c t o r - m e s o n v e r t e x f u n c t i o n s a r e c o u p l i n g constants. In sect. 3 we d i s c u s s the s i m p l e s t t e s t of the s t r o n g v e r s i o n of u n i v e r s a l i t y (for ~ s i n the r e g i o n of 1.5 GeV) by r e l a t i n g ~tot(e+e - -~ n+~-), ~tot(e+e - ~ K+K -) a n d ~tot(e+e - -~ ,~o,~o~ r~l~2J. * Work
supported
in part by the Office of Naval Research.
3~
P. ROY
In s e c t . 4 we p r o p o s e a t e s t of t h e f u l l VMD h y p o t h e s i s w h i c h r e l a t e s ~ t o t ( e + e - -~ p~), ~tot(e+e - -~ K+K -) a n d ~tot(e+e - -~ K~K~) n e a r s = m ~ . O u r a r g u m e n t s a l s o l e a d to a r e l a t i o n t h a t c a n t e s t VMD a t h i g h e r v a l u e s o f s p r o v i d e d one c a n m e a s u r e ~tot(e+e - -~ ~°w) a n d ~tot(e+e - -~ ~°qS) in t h o s e regions.
2. VMD H Y P O T H E S I S T h e f u l l c o n t e n t of t h e VMD h y p o t h e s i s c a n b e b e s t d e s c r i b e d by t h r e e a s s u m p t i o n s w h i c h a r e l i s t e d b e l o w in t h e o r d e r of d e c r e a s i n g r e l i a b i l i t y . (i) C u r r e n t - f i e l d i d e n t i t y [7 - 9]. A s an o p e r a t o r r e l a t i o n l i n k i n g t h e e l e c t r o m a g n e t i c c u r r e n t j E M = j V + j S to v e c t o r - m e s o n f i e l d s p # , w # a n d t't-~#, this reads
t h e g ' s r e f e r to t h e c o u p l i n g c o n s t a n t s [10] of v e c t o r m e s o n s . E x t e n d e d to t h e f u l l o c t e t , t h e s e i d e n t i t i e s m e a n t h a t t h e h a d r o n i c c u r r e n t s s a t i s f y t h e a l g e b r a of f i e l d s [11]. M o r e o v e r t h e s e e s s e n t i a l l y i m p l y t h a t t h e p h o t o n v e r t e x - f u n c t i o n ( f o r o~ = fl t h i s i s r e l a t e d to t h e o r d i n a r y e . m . f o r m f a c t o r ) F ~ f l (t) with any two p a r t i c l e s c~, fi c a n b e w r i t t e n (with s u i t a b l e C I e b s c h - G o r d a n c o e f f i c i e n t s f a c t o r e d out) a s where
FVfl(t) = gpAp (t) Ypozfl(t)
,
fl( t) = gwaw(t) y wo~fl(t) + g c a~ (t) ~ ~efl(t) .
(2.1)
In eq. (2.1) A v ( t ) (ggu - (q~qu/m2)) i s t h e v e c t o r - m e s o n p r o p a g a t o r (with continuum contributions properly included) carrying four-momentum s q u a r e d t w i t h AV(0~= m ~ 2 a n d t h e v e r t e x f u n c t i o n s ~V~fl(t) a r e a s d e f i n e d in fig. 1. N e a r t = m@, one i n c l u d e s t h e w i d t h F V b y A v ( t ) = 1/[m2r - t -imvFvf(~) ] w h e r e f(t) i s a k n o w n t h r e s h o l d f a c t o r s a t i s f y i n g f(m21= 1. H o w e v e r , we s h a l l not n e e d to u s e a n y s p e c i f i c f o r m f o r AV(t).
Fv
[I) s ~ ~'"
F S It)
I=O
t
B
aB ~ s % 11% %
Fig. 1. Diagrammatic representation of eq. (2.1).
VECTOR DOMINANCE
37
(ii) U n i v e r s a l i t y . T h i s m a k e s the c o u p l i n g b e t w e e n a v e c t o r m e s o n a n d two s i m i l a r p a r t i c l e s , with a l l t h r e e b e i n g on the m a s s - s h e l l , u n i v e r s a l . In o t h e r w o r d s , in fig. 2, Y V a ~ = ~ ¢ so t h a t f o r e x a m p l e 7pN!q = ~ p ~ = YpKK = 7p. A s t r o n g e r v e r s i o n of u n i v e r s a l i t y a s s u m e s that the s a m e i s t r u e e v e n w h e n t h e v e c t o r m e s o n i s off t h e m a s s s h e l l so that y p ~ ( t ) = T p N ~ ( t ) = y p ( t ) a n d s i m i l a r l y for the other cases.
~,
(t) Vaa ~
V
a
....t
Fig. 2. The Vot~ vertex. (iii) S m o o t h n e s s . T h i s s t a t e s that the v e c t o r - m e s o n v e r t e x f u n c t i o n s do not c h a n g e when the v e c t o r m e s o n i s off the m a s s s h e l l , i.e. ~Vap(t)
=
~V~
,
w h e r e YVafl i s a c o n s t a n t . Note that f o r a = ~, the n o r m a l i z a t i o n of 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 F(0) = 1 i m p l i e s g v ~ v a a / m 2 = 1, i.e. Y V a a i s a u n i v e r s a l c o n s t a n t . In o t h e r w o r d s , a s s u m p t i o n s (i) a n d (iii) i m p l y (ii) at l e a s t f o r t = 0. T h e f i r s t of the a b o v e a s s u m p t i o n s c a n b e r i g o r o u s l y c h e c k e d i n e+e c o l l i s i o n s f r o m a k n o w l e d g e of the t o t a l c r o s s s e c t i o n f o r e+e - -~ h a d r o n s f o r a s y m p t o t i c s by t e s t i n g the r e s u l t [12] t h a t f o r s i n g l e p h o t o n e x c h a n g e lim
s 2 Crtot (e+e - -~ h a d r o n s ) = 0 .
O u r i n t e r e s t h e r e i s , h o w e v e r , i n c o n s i d e r i n g t e s t s f o r a s s u m p t i o n s (ii) a n d (iii) w h i c h a r e e x p e c t e d to b e c o m e i n c r e a s i n g l y i n a c c u r a t e a s one m o v e s away f r o m t h e r e s o n a n c e m a s s e s [13]*
3. S I M P L E S T T E S T O F T H E STRONG VERSION O F U N I V E R S A L I T Y W e s t a r t by c o n s i d e r i n g the r e a c t i o n e+e - -~ ~+~-. T h e t o t a l c r o s s s e c t i o n [12] a s a f u n c t i o n of s i s g i v e n b y * * 3
~tot ( e + e - -~ ~+~-) = ~ ~ 2
~_
i F ~ ( s ) 12 ,
S:
* There is already evidence in the studies of nucleon electromagnetic from factors that a straight forward application of assumption (iii} is not valid for large spacelike t [13]. ** We shall always assume single photon exchange.
38
P. ROY
w h e r e the pion 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 FTr(s ) by a s s u m p t i o n (i) is g i v e n by
F~(s) = gpAp(S) :~prr~(s) ,
(3.1)
so that ~tot (e+e- -~ ~+~-) = g1 ~(~2
( s-4rn2+~ ~ 2 2 ~ ~ Z, [ ~ gpl/Xp(S) ~pTrv(s) 82
(3.2)
o o2 have total On the o t h e r hand, the r e a c t i o n s e+e - --* K+K - and e+e - ~ K1K c r o s s s e c t i o n s g i v e n r e s p e c t i v e l y by * 2 (s-4m2+) z 1=1 I:O(s) i2 (~tot (e+e- -~ K+K-) = 13 7roL _~ , IF K (s)+F K $2
(3.3)
and 3
o o
2(s-4m~°
~tot ( e + e - -~ K 1 K 2 / -- ~ ~ro~
_,
f
I
_F/=I(s)+F/K =
0(~)
12
,
(3.4/
S 2
where
~= l(s) = %%(s) ~p~(s) ,
(3.5)
I=O F K (s) : gwAo)(s) ~wK~(s)+gqbAb(S) ydpK~(S) •
(3.6)
and
F o r ~/-s up to 1 . 5 G e V the n e a r b y ~ - p o l e will m a k e A~b(s) >> Ap(s) and Aw(s ~. Since AV(0) = rnV¢~, at s = O, gpVpK~(O)rn~ ~ =gwVwK~(O)rn-w ~"+ g~V~K~(O)rn~b ~. At t h i s s t a g e , we shall m a k e two v e r y p l a u s i b l e a s s u m p t i o n s f o r ~/s in the r e g i o n of 1.5 GeV. 1) T h e p h a s e s of F I= O(s) and F / = l(s) a r e n e g l i b l e at t h e s e v a l u e s of s. 2) FI= O(s) >~K = l(s~ in this r e g i o n , i.e., v a r i a t i o n s in the v e r t e x f u n c tion c a n n o t o v e r c o m p e n s a t e the effect of A s ( s ) >> A^(s). T h e f i r s t a s s u m p t i o n is r e a s o n a b l e s i n c e v~e a r e f a r~ a w a y f r o m the p, ~0 m a s s e s w h e r e a s the ~ - w i d t h is v e r y s m a l l [14]**. T h e s e c o n d a s s u m p t i o n is in the s p i r i t of a s s u m p t i o n (iii) of sect. 2 and should hold so long a s the v i o l a t i o n s of the l a t t e r a r e not c a t a s t r o p h i c . We now obtain f r o m eqs. (3.3), (3.4) and (3.5) * We are ignoring electromagnetic vertex connections which can be estimated to be about 4.2% in the amplitude at the ~b-mass [17]. ** As an order of magnitude estimate, we note that the phase of the p-propagator at ~s= 1.5 GeV is 8 ° in the model of ref. [14].
39
VECTOR DOMINANCE
g2p l Ap (S ) !2 y2pK~(S) 5
3s ~ = 7rot~
+ :
v~-~tot(e+e--~K+K-)
[/s
2 \314
-
-4rnIC )
3:. ~ t2
/(Ztot(e e - ~ K 1 K 2 ) T-~2-x~ ~.s - ~ m K O )
I " )
(3.7)
A c o m p a r i s o n of eqs. (3.2) and (3.7) i m p l i e s that if the s t r o n g v e r s i o n of a s s u m p t i o n (ii) is i n d e e d v a l i d up to 4-s ~ 1.5 GeV, we m u s t h a v e
/
-
(e+e - -~ ~+Tr-) ~t°~/st~ 2 \3/2 =
+
___,
o
-)} o"
2
~/CXtot(e+e- -~ K+K-) ~/~totke e K1K 2 ~,/s 2 ,3/, 7---2--7~ , - 4inK+ )
,
(3,8)
f o r 4-~ a r o u n d 1.5 GeV. T h e c . m . e n e r g y is c h o s e n o p t i m a l l y a r o u n d 1.5 GeV s i n c e if one is b e l o w that the p h a s e of F I= l(s) s t a r t s b e c o m i n g i m p o r tant. On the o t h e r hand, if we go to too high a v a l u e of s, F~=O(s) will no l o n g e r be g r e a t e r t h a n F/K-- l(s). T h e l a t t e r can, of c o u r s e , ~ e e x p e r i m e n t a l l y c h e c k e d by studying ~tot(e+e - ~ K~K~) a c c o r d i n g to eq. (3.4). We e m p h a s i z e that eq. (3.8) c o n s t i t u t e s a t e s t of the s t r o n g u n i v e r s a l i t y a s s u m p tion*.
4. T E S T O F T H E F U L L VMD H Y P O T H E S I S We shall now c o n s i d e r the p r o c e s s e+e - -~ 7rp which is i m p o r t a n t f r o m a n o t h e r point of v i e w since by studying it one c a n m e a s u r e the i s o s p i n of the photon d i r e c t l y if one a s s u m e s single p h o t o n e x c h a n g e . F i r s t let u s look at the 7r77 v e r t e x with o f f - s h e l l p h o t o n s a s d e s c r i b e d in fig. 3. If the o n - s h e l l v e r t e x is e~(q)eV(q')M~v(0 , 0), f o r o f f - s h e l l p h o t o n s we have
F r y y ( q Z q 2 ) .-,./~'/ a2 ~-
I
2
,2
q2,I- I
q ,I'O
2
2
2
q2 i=O
q' , I =I
Fig. 3. /-spin decomposition of the TrT7 vertex.
* At this point one may well ask how crucially does eq. (3.8) test the idea of vectormeson universality. Clearly, the main principle leading to this relation is the universality of the isovector electromagnetic form factors with or without vector mesons. However, the~wo assumptions ~e have made above may not be true for ~/s around 1.5 GeV if F~(= l(s) is not dominated only by the p and FI= l(s) only by the ¢0 and the ~b.
40
P. ROY
Mlau (q2,q,2) = - e- 2 E m 7[o pvafi
q ~pflF
zr 7ryy
(q2, q ,2) .
(4.1)
In eq. (4.1) FTyy (q2, q'2)= Flrr (q2, q2)+ F2y r (q2, q'2) where the vertex functions F 1 and F 2 are given in fig. 3. By symmetry we then have
F1
7ry~
(q2, q ,2) = F 2
7ryy
(q ,2, q2) .
(4.2)
By employing assumption (i) of sect. 2, we can write F 1 (q2,q,2) = F 7ryy
(q2,q,2) A (q2) gp ~rp'y
p
'
and F21ryy (q2, q,2) = FTrvco (q,2, q2) £xw (q2) gco + FTryq5 (q,2, q2) A b (q2) gq5 ' where the definitions of Fvpy(q2,q'2) , FT~ co(q'2,q 2) and F v (q,2, q2) should be transparent in analogy with eq. (4.1). Hence eq. (4.2) implies that
Fvp7 (q2,q ,2) Ap (q2) go = FTTyw (q2,q ,2) Aw (q,2)gw+FTw~ (q2,q,2) A~b (q,2)g~b
(4.3)
By assumption (iii) of sect. 2 0 Fnp7 (q2,q'2) = _ _0 FTr)/w (q2,q,2) = 0
Oq2
Oq,2
Oq,2
Fuv~ (q2,q,2) = 0.
Eq. (4.3) can therefore be rewritten as
{ 2 ,2\ FTrpy ~mp,q )
Ap
~, )/q2\gp
Choosing q2 =0 and writing q ,2 =s, we then have
FTrpy (m2 S) gp P' = F 7r~w (0' m2co) Aco(S)gw +FTr~,~ (O'm2) Aqs(S)gqs" 2 7n
P Near the 4~-mass the 2nd RHS t e r m predominates overwhelmingly and we can w r i t e
2 ~FTrpy (rap m P gp
(4.4)
41
VECTOR DOMI~ANCE On the o t h e r hand, f r o m eqs. (3.5) and (3.6) we have that n e a r the qS-mass ,:0
FK
Hence f r o m eqs. (3.3) and (3.4) we have fftot (e+e- -~ K+K-)s ~ m~ 4mK+)~ o o ~tot(e+e--~K1K2)s
2
-
~
(mq~2 _ 4mKo)~
~m~
=
~
2
2 2
~g~q~Ki~ia~
(s ~m~)]2
(4.5)
3m
E l i m i n a t i n g g~A~p(S ~ m2~) f r o m eq. (4.4) and eq. (4.5) we h a v e * O
O
~tot (e+e- -* K+K-)s ~ m¢2 = (~tot ( e+e- -~ K1K2)s ~ rn~
-3 ~ 4~¢K~
(0,
~
¢4.6)
I~
In the RHS of eq. (4.6), y ~KK is well known f r o m F((p -~ KK) and F u p y can be o b t a i n e d f r o m the r e l a t i o n
e atot (e+e- --" ~OpO) =
IF~ (m ,s)[ 12~ m 2 s ~
-rap
~o] _ m 2 4s
(4.7)
~o
71
by m e a s u r i n g atot (e+e - --" ~OpO) at s ~ m ~ . Fny~)(0, m 2) is in p r i n c i p l e obtainable f r o m the decay ~ -~ ~y which, h o w e v e r , h a s n o t b e e n s e e n yet. But one can c a l c u l a t e it following C r e m m e r and G o u r d i n [6] by using the
* If the smoothness assumption is violated so that YVafl (t) = YVafl (0) (1 + Xv(t/m~¢)), IFTrD./ (rn~, rn~) I in the RHS of eq. {4.6) should be divided by a factor of (1 +Xp). Similarly, eq[ (4.8) should read
and eq. (4.9) should be altered accordingly. For a consideration of off-shell correc tions to the Gell-Mann-Sharp-Wagner model, see ref. [18].
42
P-. ROY
VMD relation
2
_ ~
770~ mTTO
%~
+ ~%~
C¢)
(o,.~
,
D~0
and plugging in go), go [10] with F?7~o~(0, m 2) obtained from F(~o~ 77o~). With current numbers Cremmer and C Gcdin [6] obtain* [F7770(0, m2) l +0.i0 = 0.13_ 0.09 but the e r r o r s will presumably improve with better data. Thus
m e a s u r i n g ~tot ( e+e- -~ K+K-) or etot (e+e - _, K1K2 ) o o and ~tot (e+e - ~ np) at s ~ m~ and using eqs. (4.6) and (4.7) one can test the VMD hypothesis near the 0-mass~ without knowing the details of the 0-propagator. A corresponding test for tirnelike ~-s significantly higher than the v e c t o r - m e s o n m a s s e s is possible provided one is able (and willing) to m e a s u r e ~tot (e+e- -~ vow) and ~tot (e+e - ~ ?700) for such values of ~/s. This can be derived by going back to eq. (4.3) and rewriting it as [
2
,2\
F?Tp7 ~mp, q J
Ap (q2) gP
(q'2) gco+F?T70
=F?7~c° (q2, m 2 ) A
Taking q,2 = 0, q2 = s and using [6] ½ Fv77 eq. (2.1) we obtain
½F?77F (O,O) rn2pAp(S) = F~7co
(0, O) = (gp/m2) FvpT(m 2, 0) F?TV0
m2o)
(q 2 , m 02) A0 ( q ' 2 ) g 0 .
O, %2 ) ~ .
from
(4.8)
m~
+
If by using assumption (iii) of sect. 2, we substitute gp7p?7?7 = m 2 in eq. (3.2) we have
lAp(S)!2----(
Crtot (e+e- ~
3
s2
77o!2 s
Moreover
-4rn
~
?7+?7-)
mp4
½F?7vv(O,O)=IF(n°~ 27)}] 77c~2m °
and in analogy with eq. (4.7)
atot (e+e-
=
-~--5'
12 m s 2 ?7
* See however ref. [15].
4S
- my
VECTOR DOMINANCE
FTr~w(s,m2)
43
~#p
~
Now in eq. (4.8) the p h a s e s of and FTr (s, rn ) a r e the s a m e ( a p a r t f r o m o v e r a l l signs) a c c o r d i n g to the VMD h y p o t h e s i s , e a c h being the p h a s e of ±p(S). H e n c e we c a n r e w r i t e eq. (4.8) a s 4 ~0~2
sF(~ ° -~ 2V) Crtot (e+e - ~ ~+~-) m 3o ( s
:
-
4 r n2o ) z3
1
-
mw 2 2 .2 • -3/4 -Igor2[ l(s-rn~-~smTr°, -rn:° I ~/(~tot (e+eL
e
7,
o,
12 -~ g ° ~ ) [
.
(4.9)
2 FTrV~ and (g~b/m~) F~y~b have In eq. (4.9) we have u s e d the f a c t t h a t (g~/mo~) o p p o s i t e s i g n s [6].
4. CONCLUDING R E M A R K S T h e v a r i o u s p r e d i c t i o n s of the v e c t o r - d o m ~ n c e h y p o t h e s i s a r e at p r e s ent in b r o a d a g r e e m e n t with e x p e r i m e n t f o r ~/q~' n e a r the p - m e s o n m a s s . H o w e v e r , the h y p o t h e s i s is l e s s than well e s t a b l i s h e d a s one m o v e s a w a y f r o m t h i s region. T h e r e a r e a l r e a d y q u e s t i o n s [16] about the v a l i d i t y of the s m o o t h n e s s a s s u m p t i o n in p o l a r i z e d pion p h o t o p r o d u c t i o n . In the s p a c e l i k e r e g i o n s i m p l e v e c t o r d o m i n a n c e of the n u c l e o n f o r m f a c t o r s d o e s not a g r e e with e x p e r i m e n t [13]. H e n c e it is i m p o r t a n t to a s k how well the v a r i o u s c o m p o n e n t s of the h y p o t h e s i s w o r k f o r t i m e l i k e v a l u e s of ~ / ~ a b o v e the p - m a s s . In this p a p e r , we have c o n s i d e r e d s e v e r a l p o s s i b l e t e s t s of the VMD h y p o t h e s i s in t h i s r e g i o n . T h e t e s t p r o p o s e d in sect. 2 a s s u m e s the c u r r e n t - f i e l d identity and v e c t o r - m e s o n u n i v e r s a l i t y but can a c c o m o d a t e r e a s o n a b l e v i o l a t i o n s of the a s s u m p t i o n of s m o o t h n e s s of v e c t o r - m e s o n f o r m f a c t o r s . T h e r e l a t i o n s o b t a i n e d in sect. 3 m a k e u s e of all the t h r e e a s s u m p t i o n s . T h e s e r e l a t i o n s can be t e s t e d only a f t e r an o r d e r of m a g n i t u d e i m p r o v e m e n t in the p r e s e n t s t o r a g e - r i n g s t a t i s t i c s . T h e s e a r e , h o w e v e r , c l e a n p r o b e s on the v a l i d i t y of the v e c t o r - m e s o n - d o m i n a n c e h y p o t h e s i s f o r t i m e l i k e p h o t o n s a b o v e 1 GeV. I would like to thank P r o f e s s o r D. R. Yennie, Dr. J. P e s t i e a u and P r o f e s s o r M. G o u r d i n f o r d i s c u s s i o n s .
REFERENCES [1] V. L.Auslander et al., Phys. Letters 25B (1967) 433 and to be published. J.Augustin et al., Phys. Letters 28B (1969) 508, 513, 518. [2] E. Celeghini and R.Gatto, Nuovo Cimento 57A (1968) 549.
44
[3] [4] [5] [6] [7[ [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
P. ROY
J . J . S a k u r a i , Ann. of P h y s . 11 (1960) 1. G . A l t a r e l l i et al., Nuovo C i m e n t o 47 (1967) 113. A. D o n n a c h i e , P h y s . L e t t e r s 27B (1968) 525. E . C r e m m e r and M . G o u r d i n , Nucl. P h y s . , B10 (1969) 179; M . G o u r d i n , M i s c e l l a n e o u s t o p i c s r e l a t e d to the a n n i h i l a t i o n of e+e - p a i r s ( O r s a y report). R. Haag, K. N i s h i j i m a and R. S c h r o e r , u n p u b l i s h e d (1961). M . G e l l - M a n n and F . Z a c h a r i a s e n , P h y s . Rev. 124 (1961) 953. N . M . K r o l I , T . D . L e e and B. Z u m i n o , P h y s . Rev. 157 (1967) 1376. S. Ting, R a p p o r t e u r ' s talk: E x p e r i m e n t a l e l e c t r o m a g n e t i c i n t e r a c t i o n s , P r o c e e d ings of the 14th I n t e r n a t i o n a l c o n f e r e n c e on high e n e r g y p h y s i c s (ed. J . P r e n t k i and J . S t e i n b e r g e r , CERN 1968) p. 43. T . D . Lee, S . W e i n b e r g a n d B . Z u m i n o , P h y s . Rev. L e t t e r s 18 (1967) 1029. J . J . S a k u r a i , L e c t u r e s on c u r r e n t s and m e s o n s (Univ. of Chicago P r e s s , C h i cago, 1969). G . W e b e r , P r o c e e d i n g s of the 1967 i n t e r n a t i o n a l s y m p o s i u m on e l e c t r o n and photon i n t e r a c t i o n s (ed. S . B e r m a n , SLAC 1967), p. 59; R . E . T a y l o r , ibid, p. 78. G.J.Gounaris andJ.J.Sakurai, P h y s . Rev. L e t t e r s 21 (1968) 244. C . B e m p o r a d et al., P h y s . L e t t e r s 29B {1969) 383. C . G e w e n i g e r et al., P h y s . L e t t e r s 28B (1968) 155; 29B (1969) 41. M.Gourdin, private communication. A. T. F i l l i p o v and Yu. N. J e p i f a n o v , JINR P r e p r i n t , to be p u b l i s h e d .