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Analysis of the annihilation process pp − π+ π−ω at rest using Veneziano-type amplitudes
~
Nuclear Physics B30 (1971) 525-534. North-Holland Publishing Company
A N A L Y S I S OF THE A N N I H I L A T I O N PROCESS ~p -- u + u - o A T REST U S I N G VENEZIANO-TYPE AMPLITUDES $. U. CHUNG *, L. M O N T A N E T a n d S. R E U C R O F T
CERN, Geneva Received 22 March 1971 A b s t r a c t : An analysis of ~p --.~+~-~0 at r e s t is presented using Veneziano-type a m plitudes involving the p - , f - and B - t r a j e c t o r i e s to r e p r e s e n t the 1S0 and ~S 1 i n i tial states of the annihilation. A reasonable fit is obtained which gives a good d e scription of the Dalitz plot as well as the C0-decay angular distributions. It is shown that most of the discrepancy between fit and experiment is at the high ~Tr effective m a s s .
1. ~ T R O D U C T I O N T h e a n n i h i l a t i o n p r o c e s s a t r e s t pp ~ ~+~-w h a s b e e n e x t e n s i v e l y s t u d i e d [1, 2] u s i n g t h e r e s o n a n c e m o d e l , in w h i c h v a r i o u s p o s s i b l e r e s o n a n c e s a r e i n t r o d u c e d into t h e f i n a l s t a t e to f i t t h e d a t a . T h i s t y p e of a n a l y s i s i n d i c a t e s t h a t t h e d a t a r e q u i r e n o t o n l y t h e m o r e o b v i o u s p ° ( 7 6 5 ) , f(1260) a n d B(1230) r e s o n a n c e s , b u t a l s o a n a d d i t i o n a l ~w r e s o n a n c e a t a p p r o x i m a t e l y t h e B - m a s s b u t w i t h q u a n t u m n u m b e r s c o r r e s p o n d i n g to e i t h e r p (the p' m e s o n ) o r ~ (the o ' m e s o n ) a n d a s t r o n g s - w a v e ~ i n t e r a c t i o n i n c o r p o r a t i n g p o l e s a t 850 a n d 1115 M e V , r e s p e c t i v e l y , a n d a z e r o a t 940 M e V . In t h i s p a p e r , w e p r e s e n t a n a n a l y s i s of t h e s a m e pp ~ ~ w d a t a [2] u s i n g V e n e z i a n o - t y p e f o u r - p o i n t f u n c t i o n s . T h e a m p l i t u d e s we h a v e u s e d to d e s c r i b e t h e s i n g l e t and t r i p l e t s t a t e s of t h e a n n i h i l a t i o n , c o r r e s p o n d i n g to t h e p r o c e s s e s s h o w n in fig. 1 a r e b a s e d on t h o s e s u g g e s t e d b y V e n e z i a n o [3] a n d C a p e l l a e t al. [4]. A p a r t f r o m t h e ~ s - w a v e b e h a v i o u r , a l l t h e r e s o n a n c e s w h i c h w e r e f o u n d t o b e e s s e n t i a l in t h e f i n a l - s t a t e - i n t e r a c t i o n m o d e l a r e an a u t o m a t i c f e a t u r e of o u r a m p l i t u d e s . W e h a v e o b t a i n e d a f i t to t h e d a t a w h i c h g i v e s a g o o d d e s c r i p t i o n of t h e ~yw D a l i t z p l o t a n d t h e w - d e c a y a n g u l a r d i s t r i b u t i o n s . An i n t e r e s t i n g r e s u l t of t h e f i t i s t h a t t h e ~ e n h a n c e m e n t in t h e p - r e g i o n c o n s i s t s p r e d o m i n a n t l y of an i s o s c a l a r JP= 0 + r e s o n a n c e (the ~ - m e s o n ) . A n o t h e r f e a t u r e of t h e f i t i s t h a t t h e ~w e n h a n c e m e n t w i t h e f f e c t i v e m a s s ~ 1220 MeV c o m e s m o s t l y f r o m t h e B - t r a j e c t o r y a s s o c i a t e d with t h e t r i p l e t i n i t i a l s t a t e a n d n o t f r o m t h e p - t r a j e c t o r y w h i c h m a y b e a s s o c i a t e d with b o t h t h e s i n g l e t a n d t r i p l e t i n i t i a l s t a t e s . T h e w o r s t f e a t u r e of t h e f i t i s in t h e h i g h ~ e f f e c t i v e * CERN Visiting Scientist from Brookhaven National L a b o r a t o r y , USA.
526
S. U. Chung et al., The process ~p --~r+Tr-¢o
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Fig. 1. S c h e m a t i c d i a g r a m for the p r o c e s s ~p --*~+y-co for the (a) s i n g l e t initial s t a t e , and (b) t r i p l e t initial s t a t e , with definition of the v a r i a b l e s used. m a s s r e g i o n ; t h e f o r m u l a e w e h a v e u s e d a r e i n c a p a b l e of r e p r o d u c i n g d i p b e t w e e n 900 a n d 1000 M e V .
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2. THE E X P E R I M E N T A L D A T A
F r o m a p p r o x i m a t e l y 20 000 annihilations of stopped p in hydrogen into 2~+2~-~ o [2], we have obtained a s a m p l e of 7487 events of the type ~p -~Ir~co. T h e s e events have been s e l e c t e d if one of the four p o s s i b l e ~+~-yo e f f e c t i v e m a s s - s q u a r e d c o m b i n a t i o n s is in the c0-region, defined by 0.570 < M 2 ( ~ % - ~ o) < 0.655 GeV 2. If m o r e than one of the v+~-~o c o m b i n a tions s a t i s f i e s this c r i t e r i o n , one of the c o m b i n a t i o n s has been s e l e c t e d at r a n d o m . (Approximately 10% of the events have m o r e than one (3~) ° c o m bination in the co-region.) I
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Fig. 3a shows the e x p e r i m e n t a l s c a t t e r p l o t s of M 2 ( ~ ) v e r s u s M2(lrco) f o r the s e l e c t e d events, and figs. 2a and 2b show the M2(~00) and M 2 ( ~ ) d i s t r i S The plot has been l i m i t e d to one rfco combination p e r event, the ~+co and 7r-co c o m bination being chosen f r o m a l t e r n a t e events on the data s u m m a r y tape,
S. U. Chung et al., The process ~p-~?r+rr-a) i
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Fig. 4. S c a t t e r plot of cos a v e r s u s ¢P (the definition of cos a and qg, the ¢0d e c a y a n g l e s , a r e given in the text): {a) e x p e r i m e n t a l distribution; (b) best fit; (c) X2 calculated f r o m s c a t t e r plots (a) and (b) (the n e g a t i v e signs i n dicate c e l l s w h e r e the e x p e r i m e n t a l content is l e s s than the t h e o r e t i c a l content).
528
5. U. Chung et al., The process pp--~ y+~-co
b u t t o n s . T h e r e a r e c l e a r e n h a n c e m e n t s in the D - r e g i o n and in the p - r e g i o n . Also, in the ~y d i s t r i b u t i o n , t h e r e is a s i g n i f i c a n t dip with e f f e c t i v e - m a s s s q u a r e d n e a r 0.90 GeV 2. T h e e x p e r i m e n t a l s c a t t e r plot f o r the w - d e c a y a n g l e s ~ , c o s 0 v e r s u s q~, is given in fig. 4a and the p r o j e c t i o n s in cos and (P a r e given in figs. 5a and 5b. We h a v e i n v e s t i g a t e d the influence of non-co e v e n t s in the s c a t t e r plots of figs. 3a and 4a by s t u d y i n g the co c o n t r o l r e g i o n s , and find that the n o n - w b a c k g r o u n d b e h a v e s p r e d o m i n a n t l y like p h a s e s p a c e . It m a y be concluded, t h e r e f o r e , that the s t r u c t u r e s o b s e r v e d in both s c a t t e r p l o t s a r e a s s o c i a t e d with genuine ~lrco e v e n t s . I
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3. T H E O R E T I C A L F O R M U L A E T h e pp annihilation p r o c e s s , at r e s t , p r o c e e d s t h r o u g h both 1S 0 and 3S 1 initial s t a t e s . F o r the r e a c t i o n ~p -~ ~+~-co, t h e s e s t a t e s h a v e the I G ( J P ) q u a n t u m n u m b e r s 1-(0") and 0 - ( 1 - ) , which c o r r e s p o n d to t h o s e of the ~- and w - m e s o n s , r e s p e c t i v e l y . In o t h e r w o r d s , the a m p l i t u d e f o r p p - ~ + ~ ° c o can be c o n s i d e r e d a s an i n c o h e r e n t addition of the a m p l i t u d e s c o r r e s p o n d i n g to the p r o c e s s e s '~' -~ y~co and 'co' -~ ~yco, r e s p e c t i v e l y , i.e. A(Is0) ~ A('~' ~ ~ w ) and A(3SI) - A('co' ~ 7r~co) , where the apostrophes indicate that the particle mass is displaced at twice the proton mass. The amplitude for ~ -~ ~co has been derived by Veneziano [3] and is given by
A(ZS0 ) = e~tvp~ e 3 P 1"P 2PP 3
A(s,t
' u) '
(1)
In the ¢0 rest frame, the angles ~ and q~ are the spherical angles describing the direction of the normal to the co-decay plane with respect to the co-ordinate s y s tem where the z-axis is chosen along the normal to the y+~r-w plane and the x-axis along the direction of the co in the ~+Tr-co rest system.
S. U. Chung et al., The process ffp---*y+~r-W
529
e 3 r e p r e s e n t i n g the ¢0 p o l a r i z a t i o n v e c t o r and Pi the f o u r - m o m e n t u m of p a r t i c l e i (see fig. lap. Also s, t and u a r e the i n v a r i a n t s given by (P2 +P3) 2, (P1 + P 2 )2 and (P1 + P3 )2, r e s p e c t i v e l y . Using the notation
v(ijk, al,~2)
F(i- al)F(j=
r(k-~l-~2)
a2) '
(2)
t,u), as s u g g e s t e d by Veneziano, m a y be w r i t t e n A 0 = Y(l12, ap(S), ap(U))+ V(l12, ap(S), ~p(t))+ y ( l 1 2 , ap(U), ~p(t)), (3)
the e x p r e s s i o n f o r A 0 ( s ,
where, ap is the p - t r a j e c t o r y . This e x p r e s s i o n for A 0 is not, of c o u r s e , umque, and we have found a significant i m p r o v e m e n t in the d e s c r i p t i o n of the data by including s a t e l l i t e t e r m s . The e x p r e s s i o n for A 0 a s u s e d in the p r e s e n t a n a l y s i s m a y be written
Ao(s , t, u) = f30B0 +/31B 1 + ~2B2 + ~3B3 ,
(4)
w h e r e El, fi2 and/33 a r e u n d e t e r m i n e d p a r a m e t e r s (fl0 d e t e r m i n e s the n o r m a l i s a t i o n ) , and B 0 = V(l12,
ap(S), ap(U)) + V(l12, ap(S), ap(t)) + V(l12, ap(U), a p ( t ) ) ,
B 1 = V(223, ap(S), ap(U)) + V(223,
ap(S), ap(t)) + V(223, ap(U), e p ( t ) ) ,
B 2 = V(224,
ap(S), a p ( t ) ) + V(224, Olp(U), ap(t)),
B 3 = V(224,
ap(S), ap(U)).
(5)
The a m p l i t u d e f o r the p r o c e s s ¢0-~ ~ w is m o r e c o m p l i c a t e d . It m a y be written in t e r m s of the five i n v a r i a n t s "fAi(s , t ,up A(3S1 ) = e4/1 e*3v
[ - g " V A 1 - P ~ P 2 A 2 - P~PVlA3+P~ PfA4+P~ P 2 A s ] ,
the Pi and e i being defined in fig. lb. A g e n e r a l f o r m f o r the A i h a s been d e r i v e d by C a r r u t h e r s and L a s l e y [5] within the f r a m e w o r k of the Veneziano model; unfortunately, it involves too m a n y u n d e t e r m i n e d c o n s t a n t s to be of p r a c t i c a l u s e . C a p e l l a et al. [4] h a v e p r e v i o u s l y given a m o r e p r a c t i c a l , if l e s s g e n e r a l , f o r m f o r t h e A i that r e t a i n s m o s t of the i m p o r t a n t p r o p e r t i e s of the g e n e r a l e x p r e s s i o n s ( m o r e r e c e n t l y , D o r r e n et al. [6] have d e r i v e d e x p r e s s i o n s f o r the Ai s i m i l a r to those of the C a p e l l a et al. v e r s i o n ) . In the a n a l y s i s d e s c r i b e d h e r e , we have u s e d e x p r e s s i o n s f o r the A i b a s e d on the C a p e l l a et al. v e r s i o n , but modified to account f o r the offm a s s - s h e l l c h a r a c t e r of the initial s t a t e '¢o' . The explicit f o r m f o r the inv a r i a n t s is
J" The signs of the first three terms are not the same as those given by Capella et al. the first minus sign is due to the difference in the definition of the Lorentz metric, and the second and third minus signs result from the fact that one of the pions in the process considered by Capella et al., has to be line-reversed to obtain the process treated in this paper.
530
S. U. Chung et al., The process ~p---* Tr+Tr-w A I ( S , t , u ) = 70C 0 - 61D 1 + 62D 2 , A 2 ( s , t , u ) = 72C 2 + 61D 4 + 63D3, A3(s , t ,u) = A2(u , t, s), A 4 ( s , t , u ) = 71C 1 + 73C 3 - 51D 5 - 63D 6, As(s,t
(7)
,u) =A4(u,t ,s),
where a'C 0 = V(l12, aB(S), aB(U) ) + [V(l12, aB(S), af(t)) + (s ~ u ) ] ,
C 1 = V(l12, aB(S), aB(U)) + [V(123, aB(S), af(t)) + (s ~ u ) ] , C 2 = V(133, aB(S) , aB(U)) - V(133, aB(S), af(t)) - V(335, aB(U)), af(t)), C 3 = V(223, aB(S), aB(U)) + [V(234, aB(S), af(t)) + (s ~ u ) ] , D 1 = ½a' g(s) [Y(145, ap(S), ap(U)) + V(134, ap(S), af(t))] + (s ~ u ) , ot'D 2 = V(313, ap(S), af(t)) + (s ~ u ) , D 3 = V(313, ap(S), a f ( t ) ) + V(414, ap(U), a f ( t ) ) , D 4 = V(133, ~ p ( s ) , af(t)) - V(144, ap(S), ap(U)) + 2a'u [V(145, ap(U)), a o ( S ) ) + V(134, ao(u), a f ( t ) ) ] , D 5 = a'(s + #2 _ m2)[r(134, Otp(S), ap(u)) + V(134, a p ( s ) , af(t))] + (s ~-~u and m ~ M).,
D 6 =½[D 3 + (s ~ u ) ] , g(x) = x 2 - x ( s - t + u) + (m2 - u 2 ) ( M 2 - u2) ,
x = s or u.
(8)
Here ~, m and M are the m a s s e s corresponding to the ~, w and 9p s y s t e m s , respectively (see fig. lb) a n d 7 1 , 72, 73, 61, 52 and 53 a r e undetermined p a r a m e t e r s (70 is determined by the normalisation). In the p r e s e n t analysis the p - , f- and B - t r a j e c t o r i e s have been fixed at a p ( t ) = 0.39 + 1.06t + 0.18/ 4t- - t o ,
(9a)
a f (t) = 0.39 + 1.06t + 0.17/ 4t- - t o ,
(9b)
S. U. Chung et al., The process ~p--o 7r+Tr-co
531
Otp(X)= 0 . 3 9 + 1 . 0 6 x + 0 . 3 8 / ~ f x - Xo, x = s o r u ,
(9c)
~B(X) = - 0.56 + 1.06x + 0.28i v~x - x0, x = s or u ,
(9d)
where xo(to) is the x(t) channel threshold. T h e s e t r a j e c t o r y functions c o r r e s p o n d to the following r e s o n a n c e m a s s e s and widths: Mp = 0.76 GeV, F p = 0.17 GeV, M E =0.76GeV, F¢ =0.17GeV, Mf
= 1.23 GeV, F f
= 0.16 GeV,
Mp' = 1.23 GeV, r p ' = 0.23 GeV, M B = 1 . 2 1 G e V , FB =0.18GeV. The u n i v e r s a l slope of the t r a j e c t o r i e s ~ ' has been fixed at 1.06 f r o m a study of the r e l a t i v e position of the p in the ~ e f f e c t i v e - m a s s - s q u a r e d d i s tribution and the B in the ~co distribution. The i n t e r c e p t s and i m a g i n a r y p a r t s of all the t r a j e c t o r i e s involved have been d e t e r m i n e d f r o m an e x a m i nation of the ~Tr and 7rco e f f e c t i v e - m a s s - s q u a r e d distributions (the p- and f t r a j e c t o r i e s a r e a s s u m e d degenerate).
4. THE FITTING PROCEDURE The t h e o r e t i c a l e x p r e s s i o n we have u s e d to fit the data can be written
T = f l IABI 2 + (1 - f l ) ~ f 2 1 A ( 1 S o ) l 2 + (1 - f2)IA(3S1)12},
(10)
where lAB 12, IA(1S0) I 2, and IA(3S1) I 2 r e p r e s e n t the non-c0 background (assumed constant in the p r e s e n t analysis) and the singlet and triplet initial s t a t e s , r e s p e c t i v e l y , each n o r m a l i s e d to the total n u m b e r of events, f l is the f r a c t i o n of the b a c k g r o u n d (fixed at 0.33 in all the fits ¢) and f2 is the f r a c t i o n of the 1S0 initial state. T is a function of the four v a r i a b l e s M2(rrco), M2(lrTr), cos 0 and ~o, and the ten f r e e p a r a m e t e r s , f2, /31, t32, t33, r l , r2, ~3, 51, 52 and 53. It is possible to r e d u c e the n u m b e r of f r e e p a r a m e t e r s to five by imposing the A d l e r s e l f - c o n s i s t e n c y conditionSt, which gives the r e l a t i o n s
¢ The value of f l was determined from a study of the rr+Tr-Tr° effective mass distribution. Preliminary fits with f l as a free parameter have shown that the results obtained are relatively insensitive to the value of f l in the range 0.3 to 0.35. The Adler condition states that the amplitude must go to zero if one of the external pions goes soft. In our formulae, this affects the amplitudes A1, A4 and A5.
532
S. U. Chung et al., The process ~p--~ lr+~-w
62
- 0.170=0, 63- 0.172 ~ 0 . 5 7 3 : 0 ,
(11)
and the B - t r a j e c t o r y f a c t o r i s a t i o n conditions [4] 71 = ± 4 7 0 , 72 = 8 7 0 ,
73 = 2 7 2 ,
(12)
or, a l t e r n a t i v e l y , 71 = 7 2 = 7 3 = 0. T h e r e is a n o t h e r r e l a t i o n a m o n g 70, 71 and 72 which can be d e r i v e d by cons i d e r i n g the r e s i d u e of the B - m e s o n pole. The r e s i d u e should involve two a r b i t r a r y p a r a m e t e r s c o r r e s p o n d i n g to the s - and d - w a v e decay into the ~w s y s t e m ; however, with the a m p l i t u d e s given in eq. (8) the B - p o l e r e s i d u e involves the t h r e e p a r a m e t e r s 70, 71 and Y2. This i m p l i e s that the 7 ' s m u s t satisfy an equation, viz. 2 71 = 27072. (13) Note that this r e l a t i o n is contained in the f a c t o r i s a t i o n conditions (12). As a f i r s t a t t e m p t at d e s c r i b i n g the data, we have p e r f o r m e d X2 fits to the Dalitz plot of fig. 3a with the Adler and f a c t o r i s a t i o n conditions i m posed; however, we find that it is i m p o s s i b l e to achieve good fits to the d a t a with so few p a r a m e t e r s (fit ×2 > 500 evaluated o v e r 240 c e l l s of the Dalitz plot). On the o t h e r hand, if we a t t e m p t to fit the Dalitz plot with a11 ten p a r a m e t e r s f r e e , we find that t h e r e exist m a n y solutions with c o m p a r a b l e goodness of fit; f u r t h e r m o r e , a11 t h e s e fits give widely differing p r e d i c t i o n s for the w - d e c a y a n g u l a r distributions. In o r d e r to o v e r c o m e t h e s e difficult i e s , we have f i r s t fitted the Dalitz plot of fig. 3a alone, with five f r e e p a r a m e t e r s (imposing the Adler and f a c t o r i s a t i o n conditions). We have then u s e d the values of the p a r a m e t e r s thus obtained as s t a r t i n g v a l u e s f o r a ten p a r a m e t e r fit to both the Dalitz plot and co-angle s c a t t e r plot of fig. 4a. In this way we have found a unique m i n i m u m which gives a good d e s c r i p t i o n of both s c a t t e r plots. The t h e o r e t i c a l Dalitz obtained in the b e s t fit is shown in fig. 3b and the distribution in ×2 for this fit is shown in fig. 3c (X2 = 284 o v e r 240 cells). Fig. 4b shows the r e s u l t i n g t h e o r e t i c a l s c a t t e r plot in the w - d e c a y angles and fig. 4c the ×2 evaluated o v e r this plot (X2 = 173 o v e r 120 cells). The p r o j e c t i o n of the two s c a t t e r plots with fitted c u r v e s a r e shown in figs. 2 and 5 (the dotted and dashed c u r v e s show the s e p a r a t e contributions f r o m the singlet and t r i p l e t s t a t e s ) . The fitted v a l u e s of the p a r a m e t e r s a r e GO = 1.0 (fixed), fll = - 3 9
~:4
, fi2 = - 1 2
~i
, /33 = 5.0 ±0.5 ,
7 O= 1.0 (fixed), 71 = - 4 . 2 ~ 0 . 4 ,
72 =
7.5:~0.8,
81 = 0 . 0 6 ± 0 . 0 1 ,
53 =
2.5±0.1,
52 =
1.5±0.I,
f l = 0.33 (fixed)
and
73 = 0 . 4 2 * 0 . 0 5 ,
f2 = 0.09±0.01.
We have i n v e s t i g a t e d the dependence of the cos 0 and ~ distributions as a function of M 2 ( ~ ) and M2(~w) both e x p e r i m e n t a l l y and f o r the b e s t fit and
S. U. Chung et al., The process ~p--~ rr+Tr-w
533
find that the p r e d i c t i o n s of the fit a r e compatible with the data. It is r e m a r k a b l e that, w h e r e a s the fitted p a r a m e t e r s do not satisfy the Adler and f a c t o r i s a t i o n conditions (eqs. (11) and (12)), they a r e in good a g r e e m e n t with condition (13) ¢. Another i n t e r e s t i n g f e a t u r e of this fit is that the ~ enhancement at the m a s s of the p c o n s i s t s entirely of the J P = 0 + e - m e s o n (see fig. 2b). We have s e a r c h e d for an a l t e r n a t i v e solution which admits a l a r g e contribution f r o m the J P = 1- p - m e s o n . The best fit we could obtain with this p r o p e r t y has for the f r a c t i o n of the singlet state f 2 = 0.40 ± 0.05 (with ~ 90% of the p - b u m p being p - m e s o n ) ; however, this fit gives a significantly w o r s e d e s c r i p t i o n of the data than the fit d i s c u s s e d above (×2 = 310 over 240 cells of the Dalitz plot and ×2 = 187 over 120 cells of the w - d e c a y angles s c a t t e r plot). M o r e o v e r , this fit does not r e p r o d u c e the e x p e r i m e n t a l c o r r e l a t i o n s between the co-decay angles and the Dalitz plot, and the fitted p a r a m e t e r s do not satisfy condition (13). In o r d e r to investigate the i m p o r t a n c e of the 1S 0 state satellite t e r m s , we have t r i e d to fit the data with/31, /32 and/3~ fixed at z e r o ; the fit we obtain is c o n s i d e r a b l y w o r s e than the best fit (×2 = 342 over 240 cells of fig. 3a and ×2 = 188 over 120 cells of fig. 4a). This r e s u l t is in a g r e e m e n t with s i m i l a r Veneziano a n a l y s e s of the p r o c e s s e s ~p (at rest) -~ rT~+~- (ref. [7]) and ~n (at rest) ~ y+Tr-Tr- (ref. [8]), which have also shown that satellite t e r m s a r e n e c e s s a r y to adequately d e s c r i b e the data.
5. CONCLUSIONS It has been shoffn that a good fit to both the Dalitz plot and the w-decay angular distributions can be obtained using V e n e z i a n o - t y p e amplitudes to r e p r e s e n t the t r i p l e t and singlet states of the annihilation at r e s t ~p ~ ~+~-¢o. The d e s c r i p t i o n of the data is as good as that obtained with the f i n a l - s t a t e i n t e r a c t i o n model, except for the dip-bump s t r u c t u r e in the high ~ effect i v e - m a s s regien; if those cells with ~+~- e f f e c t i v e - m a s s s q u a r e d g r e a t e r than 0.9 GeV2 a r e not included in the c o m p a r i s o n between the best fit and the data, the ×2 r e d u c e s to 214 over 203 cells f r o m 284 over 240 cells. We have investigated the possibility that the dip may be a consequence of a z e r o in the Veneziano amplitude. Although such a z e r o can indeed be i n t r o duced n e a r the m a s s of the dip by a p r o p e r choice of the t r a j e c t o r y p a r a m e t e r s , it is not " s h a r p " enough to r e p r o d u c e the e x p e r i m e n t a l ~ m a s s s p e c t r u m . We have also found that the f - m e s o n pole is too far outside the limit of phase s p a c e to account f o r the ~ bump n e a r 1.08 GeV 2. The f r a c t i o n of 1S0 state we have found in the best fit is (9 ± 1)% to be c o m p a r e d with 20 to 25% found with the f i n a l - s t a t e - i n t e r a c t i o n model. This apparent d i s c r e p a n c y is not s u r p r i s i n g , as the singlet to t r i p l e t r a t i o is a bit should be noted that the amplitudes given in eq. (8) assume implicity that the s to d-wave ratio at the B-pole is identical for both the 'w'?r and wlr systems. The fact that condition (13) is automatically satisfied by the best fit justifies, a posteriori, this assumption.
534
S. U. Chung et al., The process ~p --* ~+~-w
s t r o n g l y m o d e l - d e p e n d e n t quantity. The p r o c e s s a s s o c i a t e d with the 3S 1 initial state is of sufficient complexity (the g e n e r a l f o r m for the 3S 1 a m p l i tude involves five independent invariant a m p l i t u d e s ) t o simulate any f e a t u r e s of the p r o c e s s a s s o c i a t e d with the 1S 0 initial state; this explains why a fit with only 9% contribution f r o m the singlet initial state can r e p r o d u c e all the f e a t u r e s of the data. The Veneziano-type amplitudes we have used have been built out of the following t r a j e c t o r i e s : those c o r r e s p o n d i n g to the p - and f - m e s o n s in the ~r~ channel and those c o r r e s p o n d i n g to the p - and B - m e s o n s in the It00 channel. The amplitudes have been c o n s t r u c t e d to satisfy signature at the p a r e n t level, but not at the daughter level. In this way, the best fit we have obtained naturally introduces for the 3S 1 state the J P = o + i s o s c a l a r 7r~ r e s o n a n c e at the p - m a s s (the e - m e s o n ) induced by the f - t r a j e c t o r y and the J P = 1+ and 0i s o v e c t o r 7rw r e s o n a n c e s at the B : m a s s (the B- and 7r'-mesons) induced by the B - t r a j e c t o r y . In principle, the B - e n h a n c e m e n t can also contain a c o n t r i bution f r o m the p - t r a j e c t o r y in the t r i p l e t state, but the p a r a m e t e r s of the b e s t fit are such that this is negligible. A c c o r d i n g to our best fit, the only contribution f r o m the singlet initial state is the J P = 1- i s o v e c t o r 7rw r e s o n a n c e n e a r the B - m a s s (the p' meson) induced by the p - t r a j e c t o r y . Note that, b e cause of parity conservation, the model does not admit any s p i n - z e r o daughter at the s a m e m a s s in the 1S0 state. In figs. 2a and 2b a r e shown (as dotted and dashed c u r v e s ) the s e p a r a t e contributions to the M2(~w) and M2(vn) s p e c t r a f r o m the 1S0 and 3S 1 initial states obtained f r o m the best fit. It can be seen that indeed the p - b u m p cons i s t e s entirely of the e - m e s o n f r o m the 3S 1 state and that the B-bump has 90% contribution f r o m the 3S 1 state (predominantly B- and ~ ' - m e s o n s ) . In conclusion, it may be r e m a r k e d that the p r e s e n t a n a l y s i s has p r o v e d to be significantly m o r e complicated than a s i m i l a r a n a l y s i s of events of the type ~p --* 7r+n-77 [7]. It is somewhat s u r p r i s i n g that m e r e l y replacing a spinz e r o p a r t i c l e (7) by a spin one p a r t i c l e (w) can introduce such extensive complications. We would like to thank Dr. A. Capella for many helpful d i s c u s s i o n s and suggestions.
REFERENCES [1] R. Bizzari et al., Nucl. Phys. B14 {1969) 169. [2] J. Diaz et al., presented at the 15th Int. conf. on High Energy Physics, Kiev (1970). [3] G. Veneziano, Nuovo cimento 57A {1968) 190. [4] A. Capella, B. Diu, J. M. Kaplan and D. Schiff, Nuovo Cimento 64A {1969) 361. [5] P. Carruthers and E. Lasley, Phys. Rev. D1 {1970) 1204. [6] I. D. Dorren, V. Rittenberg and H. R. Rubinstein, Nucl. Phys. B20 (1970) 663. [7] S.U. Chung, L. Montanet, S. Reucroft and O. Witt-Hansen, Nucl. Phys., to be published. [8] G. Altare[li and H. R. Rubinstein, Phys. Rev. 183 {1969) 1469; G. P. GopaL R. Migneron and A. Rothery, Imperial College preprint {September, (1970).
Nuclear Physics B30 (1971) 525-534. North-Holland Publishing Company
A N A L Y S I S OF THE A N N I H I L A T I O N PROCESS ~p -- u + u - o A T REST U S I N G VENEZIANO-TYPE AMPLITUDES $. U. CHUNG *, L. M O N T A N E T a n d S. R E U C R O F T
CERN, Geneva Received 22 March 1971 A b s t r a c t : An analysis of ~p --.~+~-~0 at r e s t is presented using Veneziano-type a m plitudes involving the p - , f - and B - t r a j e c t o r i e s to r e p r e s e n t the 1S0 and ~S 1 i n i tial states of the annihilation. A reasonable fit is obtained which gives a good d e scription of the Dalitz plot as well as the C0-decay angular distributions. It is shown that most of the discrepancy between fit and experiment is at the high ~Tr effective m a s s .
1. ~ T R O D U C T I O N T h e a n n i h i l a t i o n p r o c e s s a t r e s t pp ~ ~+~-w h a s b e e n e x t e n s i v e l y s t u d i e d [1, 2] u s i n g t h e r e s o n a n c e m o d e l , in w h i c h v a r i o u s p o s s i b l e r e s o n a n c e s a r e i n t r o d u c e d into t h e f i n a l s t a t e to f i t t h e d a t a . T h i s t y p e of a n a l y s i s i n d i c a t e s t h a t t h e d a t a r e q u i r e n o t o n l y t h e m o r e o b v i o u s p ° ( 7 6 5 ) , f(1260) a n d B(1230) r e s o n a n c e s , b u t a l s o a n a d d i t i o n a l ~w r e s o n a n c e a t a p p r o x i m a t e l y t h e B - m a s s b u t w i t h q u a n t u m n u m b e r s c o r r e s p o n d i n g to e i t h e r p (the p' m e s o n ) o r ~ (the o ' m e s o n ) a n d a s t r o n g s - w a v e ~ i n t e r a c t i o n i n c o r p o r a t i n g p o l e s a t 850 a n d 1115 M e V , r e s p e c t i v e l y , a n d a z e r o a t 940 M e V . In t h i s p a p e r , w e p r e s e n t a n a n a l y s i s of t h e s a m e pp ~ ~ w d a t a [2] u s i n g V e n e z i a n o - t y p e f o u r - p o i n t f u n c t i o n s . T h e a m p l i t u d e s we h a v e u s e d to d e s c r i b e t h e s i n g l e t and t r i p l e t s t a t e s of t h e a n n i h i l a t i o n , c o r r e s p o n d i n g to t h e p r o c e s s e s s h o w n in fig. 1 a r e b a s e d on t h o s e s u g g e s t e d b y V e n e z i a n o [3] a n d C a p e l l a e t al. [4]. A p a r t f r o m t h e ~ s - w a v e b e h a v i o u r , a l l t h e r e s o n a n c e s w h i c h w e r e f o u n d t o b e e s s e n t i a l in t h e f i n a l - s t a t e - i n t e r a c t i o n m o d e l a r e an a u t o m a t i c f e a t u r e of o u r a m p l i t u d e s . W e h a v e o b t a i n e d a f i t to t h e d a t a w h i c h g i v e s a g o o d d e s c r i p t i o n of t h e ~yw D a l i t z p l o t a n d t h e w - d e c a y a n g u l a r d i s t r i b u t i o n s . An i n t e r e s t i n g r e s u l t of t h e f i t i s t h a t t h e ~ e n h a n c e m e n t in t h e p - r e g i o n c o n s i s t s p r e d o m i n a n t l y of an i s o s c a l a r JP= 0 + r e s o n a n c e (the ~ - m e s o n ) . A n o t h e r f e a t u r e of t h e f i t i s t h a t t h e ~w e n h a n c e m e n t w i t h e f f e c t i v e m a s s ~ 1220 MeV c o m e s m o s t l y f r o m t h e B - t r a j e c t o r y a s s o c i a t e d with t h e t r i p l e t i n i t i a l s t a t e a n d n o t f r o m t h e p - t r a j e c t o r y w h i c h m a y b e a s s o c i a t e d with b o t h t h e s i n g l e t a n d t r i p l e t i n i t i a l s t a t e s . T h e w o r s t f e a t u r e of t h e f i t i s in t h e h i g h ~ e f f e c t i v e * CERN Visiting Scientist from Brookhaven National L a b o r a t o r y , USA.
526
S. U. Chung et al., The process ~p --~r+Tr-¢o
(b)
(a) /'Tr'
C ~ P3 ' e3
e4'M "S°J'
o j ~ P3' e3 ,m
Fig. 1. S c h e m a t i c d i a g r a m for the p r o c e s s ~p --*~+y-co for the (a) s i n g l e t initial s t a t e , and (b) t r i p l e t initial s t a t e , with definition of the v a r i a b l e s used. m a s s r e g i o n ; t h e f o r m u l a e w e h a v e u s e d a r e i n c a p a b l e of r e p r o d u c i n g d i p b e t w e e n 900 a n d 1000 M e V .
the
2. THE E X P E R I M E N T A L D A T A
F r o m a p p r o x i m a t e l y 20 000 annihilations of stopped p in hydrogen into 2~+2~-~ o [2], we have obtained a s a m p l e of 7487 events of the type ~p -~Ir~co. T h e s e events have been s e l e c t e d if one of the four p o s s i b l e ~+~-yo e f f e c t i v e m a s s - s q u a r e d c o m b i n a t i o n s is in the c0-region, defined by 0.570 < M 2 ( ~ % - ~ o) < 0.655 GeV 2. If m o r e than one of the v+~-~o c o m b i n a tions s a t i s f i e s this c r i t e r i o n , one of the c o m b i n a t i o n s has been s e l e c t e d at r a n d o m . (Approximately 10% of the events have m o r e than one (3~) ° c o m bination in the co-region.) I
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Fig. 3a shows the e x p e r i m e n t a l s c a t t e r p l o t s of M 2 ( ~ ) v e r s u s M2(lrco) f o r the s e l e c t e d events, and figs. 2a and 2b show the M2(~00) and M 2 ( ~ ) d i s t r i S The plot has been l i m i t e d to one rfco combination p e r event, the ~+co and 7r-co c o m bination being chosen f r o m a l t e r n a t e events on the data s u m m a r y tape,
S. U. Chung et al., The process ~p-~?r+rr-a) i
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3~
I
I
I 0.8
I 1.4 MASS2(~'n
I 2.0 "±)
*1.0 0
I. 2 6 , GeV 2
Fig. 3. Dalitz plot of M2(?r¢o) v e r s u s M 2 0 r ~ : (a) E x p e r i m e n t a l distribution; (b) distribution r e s u l t i n g f r o m the b e s t fit; (c) X2 c a l c u l a t e d for c e l l s for which both (a) and (b) have m o r e than 10 e n t r i e s ( the n e g a t i v e signs indicate c e l l s w h e r e the e x p e r i m e n t a l content is l e s s than the t h e o r e t i c a l content).
Fig. 4. S c a t t e r plot of cos a v e r s u s ¢P (the definition of cos a and qg, the ¢0d e c a y a n g l e s , a r e given in the text): {a) e x p e r i m e n t a l distribution; (b) best fit; (c) X2 calculated f r o m s c a t t e r plots (a) and (b) (the n e g a t i v e signs i n dicate c e l l s w h e r e the e x p e r i m e n t a l content is l e s s than the t h e o r e t i c a l content).
528
5. U. Chung et al., The process pp--~ y+~-co
b u t t o n s . T h e r e a r e c l e a r e n h a n c e m e n t s in the D - r e g i o n and in the p - r e g i o n . Also, in the ~y d i s t r i b u t i o n , t h e r e is a s i g n i f i c a n t dip with e f f e c t i v e - m a s s s q u a r e d n e a r 0.90 GeV 2. T h e e x p e r i m e n t a l s c a t t e r plot f o r the w - d e c a y a n g l e s ~ , c o s 0 v e r s u s q~, is given in fig. 4a and the p r o j e c t i o n s in cos and (P a r e given in figs. 5a and 5b. We h a v e i n v e s t i g a t e d the influence of non-co e v e n t s in the s c a t t e r plots of figs. 3a and 4a by s t u d y i n g the co c o n t r o l r e g i o n s , and find that the n o n - w b a c k g r o u n d b e h a v e s p r e d o m i n a n t l y like p h a s e s p a c e . It m a y be concluded, t h e r e f o r e , that the s t r u c t u r e s o b s e r v e d in both s c a t t e r p l o t s a r e a s s o c i a t e d with genuine ~lrco e v e n t s . I
5001
o) '
[
1
i~ /
5oo 4 o o ~ J ~ 3oo!
200
.. .- . . . . . . '--..P .
"- . . . . .
I O0
"t
'"'" 0 __'""t ........ ]
-I.O
i
,.,,'"'I""'"'""
cosOO
I.O
100~ O f - " Y ± [ ................l........ 0 90" 180°
Fig. 5(a) Cos ~ distribution; (b) ~P distribution. The solid curves show the results of the best fit described in the text. The dotted and dashed curves show the singlet and triplet contribution.
3. T H E O R E T I C A L F O R M U L A E T h e pp annihilation p r o c e s s , at r e s t , p r o c e e d s t h r o u g h both 1S 0 and 3S 1 initial s t a t e s . F o r the r e a c t i o n ~p -~ ~+~-co, t h e s e s t a t e s h a v e the I G ( J P ) q u a n t u m n u m b e r s 1-(0") and 0 - ( 1 - ) , which c o r r e s p o n d to t h o s e of the ~- and w - m e s o n s , r e s p e c t i v e l y . In o t h e r w o r d s , the a m p l i t u d e f o r p p - ~ + ~ ° c o can be c o n s i d e r e d a s an i n c o h e r e n t addition of the a m p l i t u d e s c o r r e s p o n d i n g to the p r o c e s s e s '~' -~ y~co and 'co' -~ ~yco, r e s p e c t i v e l y , i.e. A(Is0) ~ A('~' ~ ~ w ) and A(3SI) - A('co' ~ 7r~co) , where the apostrophes indicate that the particle mass is displaced at twice the proton mass. The amplitude for ~ -~ ~co has been derived by Veneziano [3] and is given by
A(ZS0 ) = e~tvp~ e 3 P 1"P 2PP 3
A(s,t
' u) '
(1)
In the ¢0 rest frame, the angles ~ and q~ are the spherical angles describing the direction of the normal to the co-decay plane with respect to the co-ordinate s y s tem where the z-axis is chosen along the normal to the y+~r-w plane and the x-axis along the direction of the co in the ~+Tr-co rest system.
S. U. Chung et al., The process ffp---*y+~r-W
529
e 3 r e p r e s e n t i n g the ¢0 p o l a r i z a t i o n v e c t o r and Pi the f o u r - m o m e n t u m of p a r t i c l e i (see fig. lap. Also s, t and u a r e the i n v a r i a n t s given by (P2 +P3) 2, (P1 + P 2 )2 and (P1 + P3 )2, r e s p e c t i v e l y . Using the notation
v(ijk, al,~2)
F(i- al)F(j=
r(k-~l-~2)
a2) '
(2)
t,u), as s u g g e s t e d by Veneziano, m a y be w r i t t e n A 0 = Y(l12, ap(S), ap(U))+ V(l12, ap(S), ~p(t))+ y ( l 1 2 , ap(U), ~p(t)), (3)
the e x p r e s s i o n f o r A 0 ( s ,
where, ap is the p - t r a j e c t o r y . This e x p r e s s i o n for A 0 is not, of c o u r s e , umque, and we have found a significant i m p r o v e m e n t in the d e s c r i p t i o n of the data by including s a t e l l i t e t e r m s . The e x p r e s s i o n for A 0 a s u s e d in the p r e s e n t a n a l y s i s m a y be written
Ao(s , t, u) = f30B0 +/31B 1 + ~2B2 + ~3B3 ,
(4)
w h e r e El, fi2 and/33 a r e u n d e t e r m i n e d p a r a m e t e r s (fl0 d e t e r m i n e s the n o r m a l i s a t i o n ) , and B 0 = V(l12,
ap(S), ap(U)) + V(l12, ap(S), ap(t)) + V(l12, ap(U), a p ( t ) ) ,
B 1 = V(223, ap(S), ap(U)) + V(223,
ap(S), ap(t)) + V(223, ap(U), e p ( t ) ) ,
B 2 = V(224,
ap(S), a p ( t ) ) + V(224, Olp(U), ap(t)),
B 3 = V(224,
ap(S), ap(U)).
(5)
The a m p l i t u d e f o r the p r o c e s s ¢0-~ ~ w is m o r e c o m p l i c a t e d . It m a y be written in t e r m s of the five i n v a r i a n t s "fAi(s , t ,up A(3S1 ) = e4/1 e*3v
[ - g " V A 1 - P ~ P 2 A 2 - P~PVlA3+P~ PfA4+P~ P 2 A s ] ,
the Pi and e i being defined in fig. lb. A g e n e r a l f o r m f o r the A i h a s been d e r i v e d by C a r r u t h e r s and L a s l e y [5] within the f r a m e w o r k of the Veneziano model; unfortunately, it involves too m a n y u n d e t e r m i n e d c o n s t a n t s to be of p r a c t i c a l u s e . C a p e l l a et al. [4] h a v e p r e v i o u s l y given a m o r e p r a c t i c a l , if l e s s g e n e r a l , f o r m f o r t h e A i that r e t a i n s m o s t of the i m p o r t a n t p r o p e r t i e s of the g e n e r a l e x p r e s s i o n s ( m o r e r e c e n t l y , D o r r e n et al. [6] have d e r i v e d e x p r e s s i o n s f o r the Ai s i m i l a r to those of the C a p e l l a et al. v e r s i o n ) . In the a n a l y s i s d e s c r i b e d h e r e , we have u s e d e x p r e s s i o n s f o r the A i b a s e d on the C a p e l l a et al. v e r s i o n , but modified to account f o r the offm a s s - s h e l l c h a r a c t e r of the initial s t a t e '¢o' . The explicit f o r m f o r the inv a r i a n t s is
J" The signs of the first three terms are not the same as those given by Capella et al. the first minus sign is due to the difference in the definition of the Lorentz metric, and the second and third minus signs result from the fact that one of the pions in the process considered by Capella et al., has to be line-reversed to obtain the process treated in this paper.
530
S. U. Chung et al., The process ~p---* Tr+Tr-w A I ( S , t , u ) = 70C 0 - 61D 1 + 62D 2 , A 2 ( s , t , u ) = 72C 2 + 61D 4 + 63D3, A3(s , t ,u) = A2(u , t, s), A 4 ( s , t , u ) = 71C 1 + 73C 3 - 51D 5 - 63D 6, As(s,t
(7)
,u) =A4(u,t ,s),
where a'C 0 = V(l12, aB(S), aB(U) ) + [V(l12, aB(S), af(t)) + (s ~ u ) ] ,
C 1 = V(l12, aB(S), aB(U)) + [V(123, aB(S), af(t)) + (s ~ u ) ] , C 2 = V(133, aB(S) , aB(U)) - V(133, aB(S), af(t)) - V(335, aB(U)), af(t)), C 3 = V(223, aB(S), aB(U)) + [V(234, aB(S), af(t)) + (s ~ u ) ] , D 1 = ½a' g(s) [Y(145, ap(S), ap(U)) + V(134, ap(S), af(t))] + (s ~ u ) , ot'D 2 = V(313, ap(S), af(t)) + (s ~ u ) , D 3 = V(313, ap(S), a f ( t ) ) + V(414, ap(U), a f ( t ) ) , D 4 = V(133, ~ p ( s ) , af(t)) - V(144, ap(S), ap(U)) + 2a'u [V(145, ap(U)), a o ( S ) ) + V(134, ao(u), a f ( t ) ) ] , D 5 = a'(s + #2 _ m2)[r(134, Otp(S), ap(u)) + V(134, a p ( s ) , af(t))] + (s ~-~u and m ~ M).,
D 6 =½[D 3 + (s ~ u ) ] , g(x) = x 2 - x ( s - t + u) + (m2 - u 2 ) ( M 2 - u2) ,
x = s or u.
(8)
Here ~, m and M are the m a s s e s corresponding to the ~, w and 9p s y s t e m s , respectively (see fig. lb) a n d 7 1 , 72, 73, 61, 52 and 53 a r e undetermined p a r a m e t e r s (70 is determined by the normalisation). In the p r e s e n t analysis the p - , f- and B - t r a j e c t o r i e s have been fixed at a p ( t ) = 0.39 + 1.06t + 0.18/ 4t- - t o ,
(9a)
a f (t) = 0.39 + 1.06t + 0.17/ 4t- - t o ,
(9b)
S. U. Chung et al., The process ~p--o 7r+Tr-co
531
Otp(X)= 0 . 3 9 + 1 . 0 6 x + 0 . 3 8 / ~ f x - Xo, x = s o r u ,
(9c)
~B(X) = - 0.56 + 1.06x + 0.28i v~x - x0, x = s or u ,
(9d)
where xo(to) is the x(t) channel threshold. T h e s e t r a j e c t o r y functions c o r r e s p o n d to the following r e s o n a n c e m a s s e s and widths: Mp = 0.76 GeV, F p = 0.17 GeV, M E =0.76GeV, F¢ =0.17GeV, Mf
= 1.23 GeV, F f
= 0.16 GeV,
Mp' = 1.23 GeV, r p ' = 0.23 GeV, M B = 1 . 2 1 G e V , FB =0.18GeV. The u n i v e r s a l slope of the t r a j e c t o r i e s ~ ' has been fixed at 1.06 f r o m a study of the r e l a t i v e position of the p in the ~ e f f e c t i v e - m a s s - s q u a r e d d i s tribution and the B in the ~co distribution. The i n t e r c e p t s and i m a g i n a r y p a r t s of all the t r a j e c t o r i e s involved have been d e t e r m i n e d f r o m an e x a m i nation of the ~Tr and 7rco e f f e c t i v e - m a s s - s q u a r e d distributions (the p- and f t r a j e c t o r i e s a r e a s s u m e d degenerate).
4. THE FITTING PROCEDURE The t h e o r e t i c a l e x p r e s s i o n we have u s e d to fit the data can be written
T = f l IABI 2 + (1 - f l ) ~ f 2 1 A ( 1 S o ) l 2 + (1 - f2)IA(3S1)12},
(10)
where lAB 12, IA(1S0) I 2, and IA(3S1) I 2 r e p r e s e n t the non-c0 background (assumed constant in the p r e s e n t analysis) and the singlet and triplet initial s t a t e s , r e s p e c t i v e l y , each n o r m a l i s e d to the total n u m b e r of events, f l is the f r a c t i o n of the b a c k g r o u n d (fixed at 0.33 in all the fits ¢) and f2 is the f r a c t i o n of the 1S0 initial state. T is a function of the four v a r i a b l e s M2(rrco), M2(lrTr), cos 0 and ~o, and the ten f r e e p a r a m e t e r s , f2, /31, t32, t33, r l , r2, ~3, 51, 52 and 53. It is possible to r e d u c e the n u m b e r of f r e e p a r a m e t e r s to five by imposing the A d l e r s e l f - c o n s i s t e n c y conditionSt, which gives the r e l a t i o n s
¢ The value of f l was determined from a study of the rr+Tr-Tr° effective mass distribution. Preliminary fits with f l as a free parameter have shown that the results obtained are relatively insensitive to the value of f l in the range 0.3 to 0.35. The Adler condition states that the amplitude must go to zero if one of the external pions goes soft. In our formulae, this affects the amplitudes A1, A4 and A5.
532
S. U. Chung et al., The process ~p--~ lr+~-w
62
- 0.170=0, 63- 0.172 ~ 0 . 5 7 3 : 0 ,
(11)
and the B - t r a j e c t o r y f a c t o r i s a t i o n conditions [4] 71 = ± 4 7 0 , 72 = 8 7 0 ,
73 = 2 7 2 ,
(12)
or, a l t e r n a t i v e l y , 71 = 7 2 = 7 3 = 0. T h e r e is a n o t h e r r e l a t i o n a m o n g 70, 71 and 72 which can be d e r i v e d by cons i d e r i n g the r e s i d u e of the B - m e s o n pole. The r e s i d u e should involve two a r b i t r a r y p a r a m e t e r s c o r r e s p o n d i n g to the s - and d - w a v e decay into the ~w s y s t e m ; however, with the a m p l i t u d e s given in eq. (8) the B - p o l e r e s i d u e involves the t h r e e p a r a m e t e r s 70, 71 and Y2. This i m p l i e s that the 7 ' s m u s t satisfy an equation, viz. 2 71 = 27072. (13) Note that this r e l a t i o n is contained in the f a c t o r i s a t i o n conditions (12). As a f i r s t a t t e m p t at d e s c r i b i n g the data, we have p e r f o r m e d X2 fits to the Dalitz plot of fig. 3a with the Adler and f a c t o r i s a t i o n conditions i m posed; however, we find that it is i m p o s s i b l e to achieve good fits to the d a t a with so few p a r a m e t e r s (fit ×2 > 500 evaluated o v e r 240 c e l l s of the Dalitz plot). On the o t h e r hand, if we a t t e m p t to fit the Dalitz plot with a11 ten p a r a m e t e r s f r e e , we find that t h e r e exist m a n y solutions with c o m p a r a b l e goodness of fit; f u r t h e r m o r e , a11 t h e s e fits give widely differing p r e d i c t i o n s for the w - d e c a y a n g u l a r distributions. In o r d e r to o v e r c o m e t h e s e difficult i e s , we have f i r s t fitted the Dalitz plot of fig. 3a alone, with five f r e e p a r a m e t e r s (imposing the Adler and f a c t o r i s a t i o n conditions). We have then u s e d the values of the p a r a m e t e r s thus obtained as s t a r t i n g v a l u e s f o r a ten p a r a m e t e r fit to both the Dalitz plot and co-angle s c a t t e r plot of fig. 4a. In this way we have found a unique m i n i m u m which gives a good d e s c r i p t i o n of both s c a t t e r plots. The t h e o r e t i c a l Dalitz obtained in the b e s t fit is shown in fig. 3b and the distribution in ×2 for this fit is shown in fig. 3c (X2 = 284 o v e r 240 cells). Fig. 4b shows the r e s u l t i n g t h e o r e t i c a l s c a t t e r plot in the w - d e c a y angles and fig. 4c the ×2 evaluated o v e r this plot (X2 = 173 o v e r 120 cells). The p r o j e c t i o n of the two s c a t t e r plots with fitted c u r v e s a r e shown in figs. 2 and 5 (the dotted and dashed c u r v e s show the s e p a r a t e contributions f r o m the singlet and t r i p l e t s t a t e s ) . The fitted v a l u e s of the p a r a m e t e r s a r e GO = 1.0 (fixed), fll = - 3 9
~:4
, fi2 = - 1 2
~i
, /33 = 5.0 ±0.5 ,
7 O= 1.0 (fixed), 71 = - 4 . 2 ~ 0 . 4 ,
72 =
7.5:~0.8,
81 = 0 . 0 6 ± 0 . 0 1 ,
53 =
2.5±0.1,
52 =
1.5±0.I,
f l = 0.33 (fixed)
and
73 = 0 . 4 2 * 0 . 0 5 ,
f2 = 0.09±0.01.
We have i n v e s t i g a t e d the dependence of the cos 0 and ~ distributions as a function of M 2 ( ~ ) and M2(~w) both e x p e r i m e n t a l l y and f o r the b e s t fit and
S. U. Chung et al., The process ~p--~ rr+Tr-w
533
find that the p r e d i c t i o n s of the fit a r e compatible with the data. It is r e m a r k a b l e that, w h e r e a s the fitted p a r a m e t e r s do not satisfy the Adler and f a c t o r i s a t i o n conditions (eqs. (11) and (12)), they a r e in good a g r e e m e n t with condition (13) ¢. Another i n t e r e s t i n g f e a t u r e of this fit is that the ~ enhancement at the m a s s of the p c o n s i s t s entirely of the J P = 0 + e - m e s o n (see fig. 2b). We have s e a r c h e d for an a l t e r n a t i v e solution which admits a l a r g e contribution f r o m the J P = 1- p - m e s o n . The best fit we could obtain with this p r o p e r t y has for the f r a c t i o n of the singlet state f 2 = 0.40 ± 0.05 (with ~ 90% of the p - b u m p being p - m e s o n ) ; however, this fit gives a significantly w o r s e d e s c r i p t i o n of the data than the fit d i s c u s s e d above (×2 = 310 over 240 cells of the Dalitz plot and ×2 = 187 over 120 cells of the w - d e c a y angles s c a t t e r plot). M o r e o v e r , this fit does not r e p r o d u c e the e x p e r i m e n t a l c o r r e l a t i o n s between the co-decay angles and the Dalitz plot, and the fitted p a r a m e t e r s do not satisfy condition (13). In o r d e r to investigate the i m p o r t a n c e of the 1S 0 state satellite t e r m s , we have t r i e d to fit the data with/31, /32 and/3~ fixed at z e r o ; the fit we obtain is c o n s i d e r a b l y w o r s e than the best fit (×2 = 342 over 240 cells of fig. 3a and ×2 = 188 over 120 cells of fig. 4a). This r e s u l t is in a g r e e m e n t with s i m i l a r Veneziano a n a l y s e s of the p r o c e s s e s ~p (at rest) -~ rT~+~- (ref. [7]) and ~n (at rest) ~ y+Tr-Tr- (ref. [8]), which have also shown that satellite t e r m s a r e n e c e s s a r y to adequately d e s c r i b e the data.
5. CONCLUSIONS It has been shoffn that a good fit to both the Dalitz plot and the w-decay angular distributions can be obtained using V e n e z i a n o - t y p e amplitudes to r e p r e s e n t the t r i p l e t and singlet states of the annihilation at r e s t ~p ~ ~+~-¢o. The d e s c r i p t i o n of the data is as good as that obtained with the f i n a l - s t a t e i n t e r a c t i o n model, except for the dip-bump s t r u c t u r e in the high ~ effect i v e - m a s s regien; if those cells with ~+~- e f f e c t i v e - m a s s s q u a r e d g r e a t e r than 0.9 GeV2 a r e not included in the c o m p a r i s o n between the best fit and the data, the ×2 r e d u c e s to 214 over 203 cells f r o m 284 over 240 cells. We have investigated the possibility that the dip may be a consequence of a z e r o in the Veneziano amplitude. Although such a z e r o can indeed be i n t r o duced n e a r the m a s s of the dip by a p r o p e r choice of the t r a j e c t o r y p a r a m e t e r s , it is not " s h a r p " enough to r e p r o d u c e the e x p e r i m e n t a l ~ m a s s s p e c t r u m . We have also found that the f - m e s o n pole is too far outside the limit of phase s p a c e to account f o r the ~ bump n e a r 1.08 GeV 2. The f r a c t i o n of 1S0 state we have found in the best fit is (9 ± 1)% to be c o m p a r e d with 20 to 25% found with the f i n a l - s t a t e - i n t e r a c t i o n model. This apparent d i s c r e p a n c y is not s u r p r i s i n g , as the singlet to t r i p l e t r a t i o is a bit should be noted that the amplitudes given in eq. (8) assume implicity that the s to d-wave ratio at the B-pole is identical for both the 'w'?r and wlr systems. The fact that condition (13) is automatically satisfied by the best fit justifies, a posteriori, this assumption.
534
S. U. Chung et al., The process ~p --* ~+~-w
s t r o n g l y m o d e l - d e p e n d e n t quantity. The p r o c e s s a s s o c i a t e d with the 3S 1 initial state is of sufficient complexity (the g e n e r a l f o r m for the 3S 1 a m p l i tude involves five independent invariant a m p l i t u d e s ) t o simulate any f e a t u r e s of the p r o c e s s a s s o c i a t e d with the 1S 0 initial state; this explains why a fit with only 9% contribution f r o m the singlet initial state can r e p r o d u c e all the f e a t u r e s of the data. The Veneziano-type amplitudes we have used have been built out of the following t r a j e c t o r i e s : those c o r r e s p o n d i n g to the p - and f - m e s o n s in the ~r~ channel and those c o r r e s p o n d i n g to the p - and B - m e s o n s in the It00 channel. The amplitudes have been c o n s t r u c t e d to satisfy signature at the p a r e n t level, but not at the daughter level. In this way, the best fit we have obtained naturally introduces for the 3S 1 state the J P = o + i s o s c a l a r 7r~ r e s o n a n c e at the p - m a s s (the e - m e s o n ) induced by the f - t r a j e c t o r y and the J P = 1+ and 0i s o v e c t o r 7rw r e s o n a n c e s at the B : m a s s (the B- and 7r'-mesons) induced by the B - t r a j e c t o r y . In principle, the B - e n h a n c e m e n t can also contain a c o n t r i bution f r o m the p - t r a j e c t o r y in the t r i p l e t state, but the p a r a m e t e r s of the b e s t fit are such that this is negligible. A c c o r d i n g to our best fit, the only contribution f r o m the singlet initial state is the J P = 1- i s o v e c t o r 7rw r e s o n a n c e n e a r the B - m a s s (the p' meson) induced by the p - t r a j e c t o r y . Note that, b e cause of parity conservation, the model does not admit any s p i n - z e r o daughter at the s a m e m a s s in the 1S0 state. In figs. 2a and 2b a r e shown (as dotted and dashed c u r v e s ) the s e p a r a t e contributions to the M2(~w) and M2(vn) s p e c t r a f r o m the 1S0 and 3S 1 initial states obtained f r o m the best fit. It can be seen that indeed the p - b u m p cons i s t e s entirely of the e - m e s o n f r o m the 3S 1 state and that the B-bump has 90% contribution f r o m the 3S 1 state (predominantly B- and ~ ' - m e s o n s ) . In conclusion, it may be r e m a r k e d that the p r e s e n t a n a l y s i s has p r o v e d to be significantly m o r e complicated than a s i m i l a r a n a l y s i s of events of the type ~p --* 7r+n-77 [7]. It is somewhat s u r p r i s i n g that m e r e l y replacing a spinz e r o p a r t i c l e (7) by a spin one p a r t i c l e (w) can introduce such extensive complications. We would like to thank Dr. A. Capella for many helpful d i s c u s s i o n s and suggestions.
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