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Kinematics of momentum transfers in many-body final states
~
Nuclear Physics B7 (1968) 83-90. North-Holland Publ. Comp., A m s t e r d a m
KINEMATICS IN
OF
MOMENTUM
MANY-BODY
FINAL
TRANSFERS STATES
L. LYONS
Nuclear Physics Laboratory, Oxford Received 12 March 1968 Abstract: The kinematics of momentum t r a n s f e r s in many-body final states exhibit some s u r p r i s i n g f e a t u r e s . F o r example, the minimum momentum t r a n s f e r from the beam to a f i n a l - s t a t e p a r t i c l e does not n e c e s s a r i l y occur when the f i n a l - s t a t e p a r t i c l e t r a v e l s forwards with maximum momentum. Some implications of these p r o p e r t i e s relevant to reaction mechanisms a r e discussed.
1. INTRODUCTION
In two-body final states, the phase space for a m o m e n t u m transfer (A2) distribution is isotropic since, at a given b e a m m o m e n t u m , A2 varies linearly with the cosine of the production angle 0. This is a greatly simplifying feature which facilitates the interpretation of experimental A2 distributions. Indeed there are good physics reasons for preferring A2 to cos 0 as a variable. In many-body final states, however, A2 has s o m e surprising properties to which we wish to draw attention. Throughout this article, we use the convention that A2 (a-* b) = - t a b : ( P a - P b ) 2 - ( E a - Eb) 2 Thus for elastic scattering processes,
A2 i s a l w a y s p o s i t i v e o r z e r o .
2. I S O D E L S Consider a specific reaction, say K-p-* K-~°p,
(1)
a t a f i x e d b e a m m o m e n t u m . On a P e y r o u p l o t (a s c a t t e r p l o t of c . m . l o n g i tudirtal against transverse momentum) for a given final-state particle, we can calculate for any final-state momentum vector the momentum transfer f r o m a p a r t i c u l a r i n i t i a l - s t a t e p a r t i c l e . W e c a n t h u s d r a w c o n t o u r s of e q u a l m o m e n t u m t r a n s f e r ( i s o d e l s ) on t h e P e y r o u p l o t . S o m e t y p i c a l e x a m p l e s a r e s h o w n in f i g . 1.
84
L. LYONS
,.6 o,w~
(a) p'r
f
.s
/
A2 K.~IT
S
/ / ! / ! I #
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~T
///
A2p~v
2
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,.
/
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_
, I Pro.
,., o.vlo
(c)
Fig. 1. I s o d e l s on P e y r o u plots. The f i g u r e s next to the contour l i n e s a r e the v a l u e s of the m o m e n t u m t r a n s f e r in (GeV/c) 2. The a s t e r i s k i n d i c a t e s the m o m e n t u m c o r r e sponding to m i n i m u m m o m e n t u m t r a n s f e r , and the dotted s e m i - c i r c l e s h o w s the k i n e m a t i c region for the c o r r e s p o n d i n g t w o - b o d y final s t a t e . The three d i a g r a m s are for 6 G e V / c K-p i n t e r a c t i o n s , and the m o m e n t u m t r a n s f e r s a r e for (a) i n i t i a l - s t a t e proton to f i n a l - s t a t e proton (kinematic region for K-p ~ pK-) (b) i n i t i a l - s t a t e K- to f i n a l - s t a t e ~" (kinematic region for K-p -* ?rA ) and (c) initiaL-state proton to f i n a l state 7r (kinematic region a s in (b}).
MANY-BODY KINEMATICS
85
T h e m i n i m u m v a l u e of A2(i - . f ) o b t a i n s w h e n t h e r e l e v a n t i n i t i a l - a n d f i n a l - s t a t e p a r t i c l e s a r e t r a v e l l i n g w i t h e q u a l v e l o c i t i e s *, a n d i s g i v e n b y : A2min = - ( m i - mf) 2 . Thus the momentum transfer can become negative even when the relevant f i n a l - s t a t e p a r t i c l e i s h e a v i e r t h a n t h e i n i t i a l - s t a t e p a r t i c l e e.g. A2(p _~ A) for the reaction K - p -~A . . . . . . . . . . B a r y o n e x c h a n g e r e a c t i o n s a l s o o f t e n p r o v i d e e x a m p l e s of n e g a t i v e m o m e n t u m t r a n s f e r s , e . g . A2(p-~K) in ~'-p - ' AK ° .
(2)
B e c a u s e of t h e c o n s t r a i n t i m p o s e d b y e n e r g y c o n s e r v a t i o n , t h e k i n e m a t i c a l l y a l l o w e d r e g i o n on a P e y r o u p l o t i s the i n t e r i o r of a c i r c l e , w h o s e r a d i u s i s d e t e r m i n e d b y t h e i n c i d e n t m o m e n t u m a n d the m a s s e s of t h e p a r t i c l e s i n v o l v e d in t h e r e a c t i o n . In s o m e c a s e s , t h e e q u a l - v e l o c i t y c o n f i g u r a t i o n i s k i n e m a t i c a l l y i n a c c e s s i b l e . An e x a m p l e of t h i s i s p r o v i d e d by r e a c t i o n (2), f o r w h i c h A 2 (p -~ A) i s n e v e r n e g a t i v e A s a r e s u l t of t h e equal-velocity r e l a t i o n s h i p , t h e s m a l l e s t v a l u e s of A2(p-~Ir) o r A2(K--.Ir~ a r e o b t a i n e d f o r q u i t e m o d e s t c . m . momenta of t h e p i o n . W e now c o n s i d e r s o m e e f f e c t s of t h i s . %
J
3. I N F L U E N C E O F P E R I P H E R A L I S M ON MASS S P E C T R A C o n s i d e r now t h e r e a c t i o n K - p - ~ ~°Ir+Tr - ,
(4)
w h i c h m a y p r o c e e d b y a m e c h a n i s m s u c h a s s h o w n in fig. 2 (a). If t h e 7r- i s p e r i p h e r a l l y p r o d u c e d , t h e ~oTr+ m a s s s p e c t r u m m a y d e v i a t e f r o m t h e s i m ple phase-space prediction. The naive argument is that peripheralism m e a n s t h a t t h e 7r- t r a v e l s f o r w a r d f a s t , a n d a s t h e LL'°Tr+m a s s s q u a r e d i s l i n e a r l y r e l a t e d t o m i n u s t h e 7r- c . m . e n e r g y t h i s s h o u l d e n h a n c e t h e E°Tr+ m a s s a t l o w v a l u e s . (See fig. 2.) T h e a b o v e a r g u m e n t i s not n e c e s s a r i l y c o r r e c t if b y p e r i p h e r a l i s m we m e a n t h a t t h e lr- i s p r o d u c e d at l o w m o m e n t u m t r a n s f e r s ** w i t h r e s p e c t t o t h e K - . T h i s c a n b e u n d e r s t o o d b y r e f e r e n c e to fig. 1 (b). If t h e p e r i p h e r a l i s m i s s o s t r o n g t h a t o n l y t h e m i n i m u m v a l u e of A 2 ( K ~ 7r-) i s i m p o r t a n t , t h e 7r- m o m e n t u m s p e c t r u m i s a 5 - f u n c t i o n a t a f a i r l y s m a l l m o m e n t u m , a n d h e n c e t h e ~olr+ m a s s s p e c t r u m i s a 6 - f u n c t i o n a t a f a i r l y h i g h m a s s . W h e n s l i g h t l y l a r g e r v a l u e s of A2 a r e a l l o w e d ( s a y , up to + 0 . 5 ( G e V / c ) 2 * This is most e a s i l y shown in the r e s t system of the incident p a r t i c l e . ** If instead we mean that the ?r- is produced at small angles with r e s p e c t to the Kthen, provided t h e r e are no angular effects at the E°?r+ vertex, the E°Tr+ m a s s spectrum is identical with phase space.
86
L. LYONS
(a}
M CT°~ *)
M~E°lf'~
M~.E°'tf ")
M(r°lf~
Fig. 2. ~o?r + effective mass distributions for K-p ~ L'°Tr+?r- and (a) Possible production mechanism for the ~°lr+?r- final state; (b) Phase space; (c) Naive guess at effect of diagram (a); (d) Actual effect of 7r- being produced only at its minimum momentum transfer; (e) Actual effect of a small momentum t r a n s f e r cut off on A2(K-. ?r-)(see, for example, shaded area of fig. 1 (b)) for beam momentum of a few GeV/c.
s h a d e d r e g i o n of fig. 1 (b)) it is the high m o m e n t a w h i c h a r e s e l e c t i v e l y e x c l u d e d , w h i l e the s m a l l m o m e n t a a r e l e s s a f f e c t e d . T h u s t h e ~o~+ i s p e a k e d at high m a s s v a l u e s a s c o m p a r e d with p h a s e s p a c e . In fig. 3, we show the ~,o~+ a n d 3oIr- m a s s d i s t r i b u t i o n s o b t a i n e d b y g e n e r a t i n g M o n t e C a r l o e v e n t s a c c o r d i n g to t h r e e d i f f e r e n t p r o c e s s e s v i z . , (a) lr- p r o d u c e d p e r i p h e r a l l y at the top v e r t e x , (b) ~,o p r o d u c e d p e r i p h e r a l ly at the b o t t o m v e r t e x , a n d (c) b o t h 7r- a n d ~ o p r o d u c e d p e r i p h e r a l l y by a double p e r i p h e r a l p r o c e s s . The weighting factors used were (a) exp(-XA2(K--~ It-)) , (b) exp(-XA2(p --. Zo)) , (c) e x p ( - h A 2 ( K - _~ ~ - ) - k A 2 ( p - ~ O ) ) ,
}, = 6 ( G e V / c ) - 2 , = 6 (GeV/c) -2 , X = 3 (GeV/c) -2 .
T h e e x p e r i m e n t a l ~Tr m a s s d i s t r i b u t i o n s * o b t a i n e d at 6 G e V / c a r e s h o w n f o r c o m p a r i s o n . F o r m o s t r e a c t i o n s it i s d i f f i c u l t to d i s c r i m i n a t e b e t w e e n the d o u b l e p e r i p h e r a l p r o c e s s on the one h a n d , a n d a c o m b i n a t i o n of the two s i n g l e p e r i p h e r a l p r o c e s s e s on the o t h e r . In the ~°Tr+Tr- f i n a l s t a t e , t h e p e c u l i a r m o m e n t u m t r a n s f e r k i n e m a t i c s r e s u l t s i n the ~Tr m a s s s p e c t r a p r o v i d i n g a r a t h e r s e n s i t i v e t e s t of t h e p r o d u c t i o n k i n e m a t i c s . T h e c u r r e n t d a t a r e s u l t i n a p r o b a b i l i t y r a t i o of 2 x 104 : 1 i n f a v o u r of t h e d o u b l e p e r i p h * The data on the reaction K-p ~ ~°Tr+Tr- are discussed more fully by Allison and Lyons [1].
87
MANY-BODY KINEMATICS k°
11'"
k"
~r-
Process
Experimental data
r~°mass
&.off- m a s s
1.5
GEV
3.O
(a)
cb)
¢c)
f.d)
Fig. 3. ~,o~.+ and ~o~.- m a s s s p e c t r a for various models, and experimental data. The squared m a t r i x elements used in the Monte Carlo calculation a r e (a) exp(-6A2(K-~Ir-)) , (b) exp(-6A2 (p -. L,o)) , (c) exp(-3L&2 (K--" ~r-)-3Z~2(p--.~'o)) . The experimental data (shown in (d)) a r e for 6 GeV/c incident momentum, as a r e the Monte Carlo calculations. era/process t; unfortunately the absolute probability for the double per i p h e r a l p r o c e s s i s o n l y 4 x 1 0 - 3 . It w o u l d a p p e a r t h a t a h i g h e r s t a t i s t i c s e x p e r i m e n t on t h e ~.olr+lr- f i n a l s t a t e c o u l d p r o v i d e c o n v i n c i n g e v i d e n c e on the production mechanism.
4. R E A C T I O N M E C H A N I S M S In m o s t t h r e e - b o d y f i n a l s t a t e s , i t i s p o s s i b l e t o d r a w m o r e t h a n one double peripheral diagram which may contribute to the reaction e.g., for t h e r e a c t i o n K * - p l r o, two p o s s i b i l i t i e s a r e s h o w n in fig. 4 (a) a n d (b). O n e i d e a f o r s e p a r a t i n g t h e s e t w o p r o c e s s e s i s to c o m p a r e Z~2(K- - * K * - ) w i t h A2(K- -.Tro). T h e n if t h e s e c o n d i s s m a l l e r t h a n t h e f i r s t , d i a g r a m (b) i s s u p p o s e d t o c o n t r i b u t e [2]. Because of the low s t a t i s t i c a l accuracy of the p r e s e n t data, the values of k in the various fits were not varied, but the goodness of fit is only a slowly varying function of the k.
88
L. LYONS k-
k ~
k-
11'
km
ta)
tb)
(.c)
Fig. 4. Possible reaction mechanisms (a) and (b). Double peripheral processes for the reaction K-p--* K*lrp. (c) Peripheral production of pions in the reaction pp--* pn lr+. T h i s is l i k e l y to b e m i s l e a d i n g s i n c e the s m a l l e s t v a l u e s of A2(K -. ~-) o c c u r f o r quite m o d e s t c . m . m o m e n t a of the pion. T h u s any p r o c e s s w h i c h p r o d u c e s slow p i o n s in the c . m . can p r o d u c e a p e a k of e v e n t s with low A2(K-~ 7r). One s u c h p r o c e s s is c l e a r l y d i a g r a m 4 (a). In fig. 5 (a) we s h o w the d i s t r i b u t i o n o b t a i n e d by weighting Monte C a r l o e v e n t s by a f a c t o r exp(-3 A2(K .-. K*)-5A2(p -. p)) , c o r r e s p o n d i n g to the double p e r i p h e r a l p r o c e s s of fig. 4 (a).
(bi~ b21 k-*l~)
(a)
300.
A2( k.~11, )
ISO SO
I
3
S
I
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S
tO)
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Aa~ ~.~- ,
200
I00
30'
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i
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3
5
A2 (,GcV/¢~
Fig. 5. Various momentum transfer distributions (a) A2(K~ ?r) for 6 GeV/c K-p ~ K*~rp Monte Carlo events generated with a weighting factor exp(-3A2(K-~ K*) ' - 5 A 2 ( p - - . p ) ) . (b) F o r t h e p r o c e s s K - p ---* p K * * (1300); K * * ---* K * 7r, (1) A 2 ¢K--'K" * ) a n d " (ii) A2(K--,Tr ) (c) &2(p__.ff) f o r r e a l 7 G e V / c pp --* pnff + e v e n t ~ .
MANY-BODY KINEMATICS A n o t h e r e x a m p l e of t h e c o n f u s i n g p r o p e r t i e s action sequence: K - p --. K * * ( 1 3 2 0 ) p ,
89
of ,,x2 i s p r o v i d e d b y t h e r e -
K**(1320) --. K*Tr.
For Monte Carlo-generated phase-space events (i.e., the proton is prod u c e d i s o t r o p i c a l l y in t h e o v e r a l l c . m . s y s t e m ) w e h a v e p l o t t e d t~2(K ~Tr) a n d A2(K --. K*) in f i g . 5 (b). F r o m t h e s e d i s t r i b u t i o n s , it i s t e m p t i n g incorrect) to deduce that the pion is produced alone at the meson vertex. E v e n m o r e s t r i k i n g e f f e c t s c a n o c c u r w h e n A2(p _~ 7r) i s c o n s i d e r e d . W e s h o w t h i s in fig. 5 (c) f o r t h e r e a c t i o n [3] pp ~ p n lr+ ,
(5)
a t 7 G e V / c . W e do n o t r e g a r d f i g . 5 (c) a s p r o v i d i n g e v i d e n c e f o r a m e c h a n i s m s u c h a s s h o w n in fig. 4 (c). Thus we see that momentum transfers to final state pions from heavier initial state particles need very, careful interpretation. Indeed for conside r a t i o n s of r e a c t i o n m e c h a n i s m s , i t i s g e n e r a l l y p r e f e r a b l e to a n a l y s e p r o c e s s e s in t e r m s of s o m e o t h e r k i n e m a t i c v a r i a b l e $.
5. N E G A T I V E M O M E N T U M T R A N S F E R S F i n a l l y w e i n v e s t i g a t e a p r o p e r t y of A2 w h i c h a l s o a p p l i e s t o t w o - b o d y final states. F r o m a k i n e m a t i c v i e w p o i n t , z e r o m o m e n t u m t r a n s f e r h a s no s p e c i a l s i g n i f i c a n c e ( e x c e p t t h a t in t h e f o r w a r d d i r e c t i o n , 4 2 = 0 f o r e l a s t i c s c a t t e r i n g a t a l l m o m e n t a , a n d a t h i g h e n e r g y f o r a n y p r o c e s s A2 -~ 0). In s o m e inelastic reactions, the 0o momentum transfer can even change sign as a f u n c t i o n of b e a m m o m e n t u m ( e . g . , A 2 ( p _~ w) f o r t h e r e a c t i o n K - p -~ Aw). S i n c e n e g a t i v e m o m e n t u m t r a n s f e r s a r e p o s s i b l e , i s t h e r e a n y s e t of c i r c u m s t a n c e s u n d e r w h i c h t h e p r o p a g a t o r (t~2 + m 2 e x ) - I b e c o m e s i n f i n i t e ? F o r t h e f r a m e in w h i c h p a r t i c l e a i s a t r e s t , t h e m o m e n t u m t r a n s f e r i s given by A 2 ( a - . b) = - m 2 a - rn2b + 2 m a E b • Thus for A 2 = -rn2ex, m2 a + rn2 b - m 2 e x Eb - __ 2m a .... , w h i c h i s j u s t t h e c o n d i t i o n f o r t h e e n e r g y of a p a r t i c l e of m a s s m b ( w h i c h i s p r o d u c e d t o g e t h e r w i t h a p a r t i c l e of m a s s m e x ) w h e n t h e t o t a l e n e r g y of t h e s y s t e m i s rn a. T h i s i s s i m p l y s a y i n g t h a t if a d e c a y s into b + ex, t h e n 1: Thus to d i s c r i m i n a t e between the two double p e r i p h e r a l p r o c e s s e s of figs. 4 (a) and 4 (b), the 'local c.m. scattering angle' defined as I~-. I~* (the vectors being evaluated in the (K*lr) c.m. system) can be used [4].
90
L. LYONS ~T
-I
-o-s
k~
.o's
/
,i
bu
( GeV/c } LA21K'.~It, ) - - M2 K for 3-S GIVI¢
K"~
Fig. 6. Peyrou plot of the Y+ in the reaction K-p ~ K*°pY-; K*° ~ K - y +, at 6 GeV/c. The curve is theA2(K*-, y+) = -nr2K isodel, corresponding to a beam of 3.5 GeV/c K* t h e e x c h a n g e d p a r t i c l e i s on i t s m a s s s h e l l a n d t h e m o m e n t u m t r a n s f e r f r o m a to b i s e x a c t l y a t t h e p o l e , r e g a r d l e s s of w h a t s u b s e q u e n t l y h a p p e n s to t h e e x c h a n g e d p a r t i c l e . An e x a m p l e of t h i s w o u l d b e K*p -- ~Kp . T h e n , p r o v i d e d t h e f i n a l - s t a t e y c o m e s d i r e c t l y f r o m t h e d e c a y of t h e K* w i t h o u t s c a t t e r i n g , i t i s c o n f i n e d to t h e A2(K* -. y) = - m 2 K i s o d e l on i t s Peyrou plot. In fig. 6, we s h o w t h e P e y r o u p l o t f o r t h e 7;+ in t h e r e a c t i o n K-p-~ K*Oy-p ,
K *° -~ K - y +
(6)
a t 6 G e V / c b e a m m o m e n t u m [5]. T h e i s o d e l d r a w n i s t h a t of A 2 ( K , -. y+) = - m 2 K , f o r 3.5 G e V / c K*. T h i s s c a t t e r p l o t i s t h u s s i m p l y a ( r a t h e r i n f e r i o r ) w a y of s h o w i n g t h a t t h e K* in r e a c t i o n (6) a r e p r o d u c e d w i t h h i g h l a b m o m e n t a in a s m a l l c o n e a r o u n d t h e f o r w a r d d i r e c t i o n .
REFERENCES
[I] [2] [3] [4]
W . W . M . A l l i s o n and L. Lyons, Nuovo Cimento 51B (1967) 404. P . J . D o r n a n et al., Phys, Rev. L e t t e r s 19 (1967) 271. G.Alexander et al., Heidelberg Conf. 1967. K. P a l e r et al., A double p e r i p h e r a l interpretation of some 6 GeV[c K-p i n t e r a c tions, Rutherford Laboratory, preprint R P P / H / 2 2 . [5] K.I.Wilkinson, thesis, Oxford, 1967, unpublished.
Nuclear Physics B7 (1968) 83-90. North-Holland Publ. Comp., A m s t e r d a m
KINEMATICS IN
OF
MOMENTUM
MANY-BODY
FINAL
TRANSFERS STATES
L. LYONS
Nuclear Physics Laboratory, Oxford Received 12 March 1968 Abstract: The kinematics of momentum t r a n s f e r s in many-body final states exhibit some s u r p r i s i n g f e a t u r e s . F o r example, the minimum momentum t r a n s f e r from the beam to a f i n a l - s t a t e p a r t i c l e does not n e c e s s a r i l y occur when the f i n a l - s t a t e p a r t i c l e t r a v e l s forwards with maximum momentum. Some implications of these p r o p e r t i e s relevant to reaction mechanisms a r e discussed.
1. INTRODUCTION
In two-body final states, the phase space for a m o m e n t u m transfer (A2) distribution is isotropic since, at a given b e a m m o m e n t u m , A2 varies linearly with the cosine of the production angle 0. This is a greatly simplifying feature which facilitates the interpretation of experimental A2 distributions. Indeed there are good physics reasons for preferring A2 to cos 0 as a variable. In many-body final states, however, A2 has s o m e surprising properties to which we wish to draw attention. Throughout this article, we use the convention that A2 (a-* b) = - t a b : ( P a - P b ) 2 - ( E a - Eb) 2 Thus for elastic scattering processes,
A2 i s a l w a y s p o s i t i v e o r z e r o .
2. I S O D E L S Consider a specific reaction, say K-p-* K-~°p,
(1)
a t a f i x e d b e a m m o m e n t u m . On a P e y r o u p l o t (a s c a t t e r p l o t of c . m . l o n g i tudirtal against transverse momentum) for a given final-state particle, we can calculate for any final-state momentum vector the momentum transfer f r o m a p a r t i c u l a r i n i t i a l - s t a t e p a r t i c l e . W e c a n t h u s d r a w c o n t o u r s of e q u a l m o m e n t u m t r a n s f e r ( i s o d e l s ) on t h e P e y r o u p l o t . S o m e t y p i c a l e x a m p l e s a r e s h o w n in f i g . 1.
84
L. LYONS
,.6 o,w~
(a) p'r
f
.s
/
A2 K.~IT
S
/ / ! / ! I #
I I
b,. (b)
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~T
///
A2p~v
2
/
,.
/
\
/
\
_
, I Pro.
,., o.vlo
(c)
Fig. 1. I s o d e l s on P e y r o u plots. The f i g u r e s next to the contour l i n e s a r e the v a l u e s of the m o m e n t u m t r a n s f e r in (GeV/c) 2. The a s t e r i s k i n d i c a t e s the m o m e n t u m c o r r e sponding to m i n i m u m m o m e n t u m t r a n s f e r , and the dotted s e m i - c i r c l e s h o w s the k i n e m a t i c region for the c o r r e s p o n d i n g t w o - b o d y final s t a t e . The three d i a g r a m s are for 6 G e V / c K-p i n t e r a c t i o n s , and the m o m e n t u m t r a n s f e r s a r e for (a) i n i t i a l - s t a t e proton to f i n a l - s t a t e proton (kinematic region for K-p ~ pK-) (b) i n i t i a l - s t a t e K- to f i n a l - s t a t e ~" (kinematic region for K-p -* ?rA ) and (c) initiaL-state proton to f i n a l state 7r (kinematic region a s in (b}).
MANY-BODY KINEMATICS
85
T h e m i n i m u m v a l u e of A2(i - . f ) o b t a i n s w h e n t h e r e l e v a n t i n i t i a l - a n d f i n a l - s t a t e p a r t i c l e s a r e t r a v e l l i n g w i t h e q u a l v e l o c i t i e s *, a n d i s g i v e n b y : A2min = - ( m i - mf) 2 . Thus the momentum transfer can become negative even when the relevant f i n a l - s t a t e p a r t i c l e i s h e a v i e r t h a n t h e i n i t i a l - s t a t e p a r t i c l e e.g. A2(p _~ A) for the reaction K - p -~A . . . . . . . . . . B a r y o n e x c h a n g e r e a c t i o n s a l s o o f t e n p r o v i d e e x a m p l e s of n e g a t i v e m o m e n t u m t r a n s f e r s , e . g . A2(p-~K) in ~'-p - ' AK ° .
(2)
B e c a u s e of t h e c o n s t r a i n t i m p o s e d b y e n e r g y c o n s e r v a t i o n , t h e k i n e m a t i c a l l y a l l o w e d r e g i o n on a P e y r o u p l o t i s the i n t e r i o r of a c i r c l e , w h o s e r a d i u s i s d e t e r m i n e d b y t h e i n c i d e n t m o m e n t u m a n d the m a s s e s of t h e p a r t i c l e s i n v o l v e d in t h e r e a c t i o n . In s o m e c a s e s , t h e e q u a l - v e l o c i t y c o n f i g u r a t i o n i s k i n e m a t i c a l l y i n a c c e s s i b l e . An e x a m p l e of t h i s i s p r o v i d e d by r e a c t i o n (2), f o r w h i c h A 2 (p -~ A) i s n e v e r n e g a t i v e A s a r e s u l t of t h e equal-velocity r e l a t i o n s h i p , t h e s m a l l e s t v a l u e s of A2(p-~Ir) o r A2(K--.Ir~ a r e o b t a i n e d f o r q u i t e m o d e s t c . m . momenta of t h e p i o n . W e now c o n s i d e r s o m e e f f e c t s of t h i s . %
J
3. I N F L U E N C E O F P E R I P H E R A L I S M ON MASS S P E C T R A C o n s i d e r now t h e r e a c t i o n K - p - ~ ~°Ir+Tr - ,
(4)
w h i c h m a y p r o c e e d b y a m e c h a n i s m s u c h a s s h o w n in fig. 2 (a). If t h e 7r- i s p e r i p h e r a l l y p r o d u c e d , t h e ~oTr+ m a s s s p e c t r u m m a y d e v i a t e f r o m t h e s i m ple phase-space prediction. The naive argument is that peripheralism m e a n s t h a t t h e 7r- t r a v e l s f o r w a r d f a s t , a n d a s t h e LL'°Tr+m a s s s q u a r e d i s l i n e a r l y r e l a t e d t o m i n u s t h e 7r- c . m . e n e r g y t h i s s h o u l d e n h a n c e t h e E°Tr+ m a s s a t l o w v a l u e s . (See fig. 2.) T h e a b o v e a r g u m e n t i s not n e c e s s a r i l y c o r r e c t if b y p e r i p h e r a l i s m we m e a n t h a t t h e lr- i s p r o d u c e d at l o w m o m e n t u m t r a n s f e r s ** w i t h r e s p e c t t o t h e K - . T h i s c a n b e u n d e r s t o o d b y r e f e r e n c e to fig. 1 (b). If t h e p e r i p h e r a l i s m i s s o s t r o n g t h a t o n l y t h e m i n i m u m v a l u e of A 2 ( K ~ 7r-) i s i m p o r t a n t , t h e 7r- m o m e n t u m s p e c t r u m i s a 5 - f u n c t i o n a t a f a i r l y s m a l l m o m e n t u m , a n d h e n c e t h e ~olr+ m a s s s p e c t r u m i s a 6 - f u n c t i o n a t a f a i r l y h i g h m a s s . W h e n s l i g h t l y l a r g e r v a l u e s of A2 a r e a l l o w e d ( s a y , up to + 0 . 5 ( G e V / c ) 2 * This is most e a s i l y shown in the r e s t system of the incident p a r t i c l e . ** If instead we mean that the ?r- is produced at small angles with r e s p e c t to the Kthen, provided t h e r e are no angular effects at the E°?r+ vertex, the E°Tr+ m a s s spectrum is identical with phase space.
86
L. LYONS
(a}
M CT°~ *)
M~E°lf'~
M~.E°'tf ")
M(r°lf~
Fig. 2. ~o?r + effective mass distributions for K-p ~ L'°Tr+?r- and (a) Possible production mechanism for the ~°lr+?r- final state; (b) Phase space; (c) Naive guess at effect of diagram (a); (d) Actual effect of 7r- being produced only at its minimum momentum transfer; (e) Actual effect of a small momentum t r a n s f e r cut off on A2(K-. ?r-)(see, for example, shaded area of fig. 1 (b)) for beam momentum of a few GeV/c.
s h a d e d r e g i o n of fig. 1 (b)) it is the high m o m e n t a w h i c h a r e s e l e c t i v e l y e x c l u d e d , w h i l e the s m a l l m o m e n t a a r e l e s s a f f e c t e d . T h u s t h e ~o~+ i s p e a k e d at high m a s s v a l u e s a s c o m p a r e d with p h a s e s p a c e . In fig. 3, we show the ~,o~+ a n d 3oIr- m a s s d i s t r i b u t i o n s o b t a i n e d b y g e n e r a t i n g M o n t e C a r l o e v e n t s a c c o r d i n g to t h r e e d i f f e r e n t p r o c e s s e s v i z . , (a) lr- p r o d u c e d p e r i p h e r a l l y at the top v e r t e x , (b) ~,o p r o d u c e d p e r i p h e r a l ly at the b o t t o m v e r t e x , a n d (c) b o t h 7r- a n d ~ o p r o d u c e d p e r i p h e r a l l y by a double p e r i p h e r a l p r o c e s s . The weighting factors used were (a) exp(-XA2(K--~ It-)) , (b) exp(-XA2(p --. Zo)) , (c) e x p ( - h A 2 ( K - _~ ~ - ) - k A 2 ( p - ~ O ) ) ,
}, = 6 ( G e V / c ) - 2 , = 6 (GeV/c) -2 , X = 3 (GeV/c) -2 .
T h e e x p e r i m e n t a l ~Tr m a s s d i s t r i b u t i o n s * o b t a i n e d at 6 G e V / c a r e s h o w n f o r c o m p a r i s o n . F o r m o s t r e a c t i o n s it i s d i f f i c u l t to d i s c r i m i n a t e b e t w e e n the d o u b l e p e r i p h e r a l p r o c e s s on the one h a n d , a n d a c o m b i n a t i o n of the two s i n g l e p e r i p h e r a l p r o c e s s e s on the o t h e r . In the ~°Tr+Tr- f i n a l s t a t e , t h e p e c u l i a r m o m e n t u m t r a n s f e r k i n e m a t i c s r e s u l t s i n the ~Tr m a s s s p e c t r a p r o v i d i n g a r a t h e r s e n s i t i v e t e s t of t h e p r o d u c t i o n k i n e m a t i c s . T h e c u r r e n t d a t a r e s u l t i n a p r o b a b i l i t y r a t i o of 2 x 104 : 1 i n f a v o u r of t h e d o u b l e p e r i p h * The data on the reaction K-p ~ ~°Tr+Tr- are discussed more fully by Allison and Lyons [1].
87
MANY-BODY KINEMATICS k°
11'"
k"
~r-
Process
Experimental data
r~°mass
&.off- m a s s
1.5
GEV
3.O
(a)
cb)
¢c)
f.d)
Fig. 3. ~,o~.+ and ~o~.- m a s s s p e c t r a for various models, and experimental data. The squared m a t r i x elements used in the Monte Carlo calculation a r e (a) exp(-6A2(K-~Ir-)) , (b) exp(-6A2 (p -. L,o)) , (c) exp(-3L&2 (K--" ~r-)-3Z~2(p--.~'o)) . The experimental data (shown in (d)) a r e for 6 GeV/c incident momentum, as a r e the Monte Carlo calculations. era/process t; unfortunately the absolute probability for the double per i p h e r a l p r o c e s s i s o n l y 4 x 1 0 - 3 . It w o u l d a p p e a r t h a t a h i g h e r s t a t i s t i c s e x p e r i m e n t on t h e ~.olr+lr- f i n a l s t a t e c o u l d p r o v i d e c o n v i n c i n g e v i d e n c e on the production mechanism.
4. R E A C T I O N M E C H A N I S M S In m o s t t h r e e - b o d y f i n a l s t a t e s , i t i s p o s s i b l e t o d r a w m o r e t h a n one double peripheral diagram which may contribute to the reaction e.g., for t h e r e a c t i o n K * - p l r o, two p o s s i b i l i t i e s a r e s h o w n in fig. 4 (a) a n d (b). O n e i d e a f o r s e p a r a t i n g t h e s e t w o p r o c e s s e s i s to c o m p a r e Z~2(K- - * K * - ) w i t h A2(K- -.Tro). T h e n if t h e s e c o n d i s s m a l l e r t h a n t h e f i r s t , d i a g r a m (b) i s s u p p o s e d t o c o n t r i b u t e [2]. Because of the low s t a t i s t i c a l accuracy of the p r e s e n t data, the values of k in the various fits were not varied, but the goodness of fit is only a slowly varying function of the k.
88
L. LYONS k-
k ~
k-
11'
km
ta)
tb)
(.c)
Fig. 4. Possible reaction mechanisms (a) and (b). Double peripheral processes for the reaction K-p--* K*lrp. (c) Peripheral production of pions in the reaction pp--* pn lr+. T h i s is l i k e l y to b e m i s l e a d i n g s i n c e the s m a l l e s t v a l u e s of A2(K -. ~-) o c c u r f o r quite m o d e s t c . m . m o m e n t a of the pion. T h u s any p r o c e s s w h i c h p r o d u c e s slow p i o n s in the c . m . can p r o d u c e a p e a k of e v e n t s with low A2(K-~ 7r). One s u c h p r o c e s s is c l e a r l y d i a g r a m 4 (a). In fig. 5 (a) we s h o w the d i s t r i b u t i o n o b t a i n e d by weighting Monte C a r l o e v e n t s by a f a c t o r exp(-3 A2(K .-. K*)-5A2(p -. p)) , c o r r e s p o n d i n g to the double p e r i p h e r a l p r o c e s s of fig. 4 (a).
(bi~ b21 k-*l~)
(a)
300.
A2( k.~11, )
ISO SO
I
3
S
I
3
S
tO)
tbi;}
Aa~ ~.~- ,
200
I00
30'
;
'
"
i
'
~
-,
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3
5
A2 (,GcV/¢~
Fig. 5. Various momentum transfer distributions (a) A2(K~ ?r) for 6 GeV/c K-p ~ K*~rp Monte Carlo events generated with a weighting factor exp(-3A2(K-~ K*) ' - 5 A 2 ( p - - . p ) ) . (b) F o r t h e p r o c e s s K - p ---* p K * * (1300); K * * ---* K * 7r, (1) A 2 ¢K--'K" * ) a n d " (ii) A2(K--,Tr ) (c) &2(p__.ff) f o r r e a l 7 G e V / c pp --* pnff + e v e n t ~ .
MANY-BODY KINEMATICS A n o t h e r e x a m p l e of t h e c o n f u s i n g p r o p e r t i e s action sequence: K - p --. K * * ( 1 3 2 0 ) p ,
89
of ,,x2 i s p r o v i d e d b y t h e r e -
K**(1320) --. K*Tr.
For Monte Carlo-generated phase-space events (i.e., the proton is prod u c e d i s o t r o p i c a l l y in t h e o v e r a l l c . m . s y s t e m ) w e h a v e p l o t t e d t~2(K ~Tr) a n d A2(K --. K*) in f i g . 5 (b). F r o m t h e s e d i s t r i b u t i o n s , it i s t e m p t i n g incorrect) to deduce that the pion is produced alone at the meson vertex. E v e n m o r e s t r i k i n g e f f e c t s c a n o c c u r w h e n A2(p _~ 7r) i s c o n s i d e r e d . W e s h o w t h i s in fig. 5 (c) f o r t h e r e a c t i o n [3] pp ~ p n lr+ ,
(5)
a t 7 G e V / c . W e do n o t r e g a r d f i g . 5 (c) a s p r o v i d i n g e v i d e n c e f o r a m e c h a n i s m s u c h a s s h o w n in fig. 4 (c). Thus we see that momentum transfers to final state pions from heavier initial state particles need very, careful interpretation. Indeed for conside r a t i o n s of r e a c t i o n m e c h a n i s m s , i t i s g e n e r a l l y p r e f e r a b l e to a n a l y s e p r o c e s s e s in t e r m s of s o m e o t h e r k i n e m a t i c v a r i a b l e $.
5. N E G A T I V E M O M E N T U M T R A N S F E R S F i n a l l y w e i n v e s t i g a t e a p r o p e r t y of A2 w h i c h a l s o a p p l i e s t o t w o - b o d y final states. F r o m a k i n e m a t i c v i e w p o i n t , z e r o m o m e n t u m t r a n s f e r h a s no s p e c i a l s i g n i f i c a n c e ( e x c e p t t h a t in t h e f o r w a r d d i r e c t i o n , 4 2 = 0 f o r e l a s t i c s c a t t e r i n g a t a l l m o m e n t a , a n d a t h i g h e n e r g y f o r a n y p r o c e s s A2 -~ 0). In s o m e inelastic reactions, the 0o momentum transfer can even change sign as a f u n c t i o n of b e a m m o m e n t u m ( e . g . , A 2 ( p _~ w) f o r t h e r e a c t i o n K - p -~ Aw). S i n c e n e g a t i v e m o m e n t u m t r a n s f e r s a r e p o s s i b l e , i s t h e r e a n y s e t of c i r c u m s t a n c e s u n d e r w h i c h t h e p r o p a g a t o r (t~2 + m 2 e x ) - I b e c o m e s i n f i n i t e ? F o r t h e f r a m e in w h i c h p a r t i c l e a i s a t r e s t , t h e m o m e n t u m t r a n s f e r i s given by A 2 ( a - . b) = - m 2 a - rn2b + 2 m a E b • Thus for A 2 = -rn2ex, m2 a + rn2 b - m 2 e x Eb - __ 2m a .... , w h i c h i s j u s t t h e c o n d i t i o n f o r t h e e n e r g y of a p a r t i c l e of m a s s m b ( w h i c h i s p r o d u c e d t o g e t h e r w i t h a p a r t i c l e of m a s s m e x ) w h e n t h e t o t a l e n e r g y of t h e s y s t e m i s rn a. T h i s i s s i m p l y s a y i n g t h a t if a d e c a y s into b + ex, t h e n 1: Thus to d i s c r i m i n a t e between the two double p e r i p h e r a l p r o c e s s e s of figs. 4 (a) and 4 (b), the 'local c.m. scattering angle' defined as I~-. I~* (the vectors being evaluated in the (K*lr) c.m. system) can be used [4].
90
L. LYONS ~T
-I
-o-s
k~
.o's
/
,i
bu
( GeV/c } LA21K'.~It, ) - - M2 K for 3-S GIVI¢
K"~
Fig. 6. Peyrou plot of the Y+ in the reaction K-p ~ K*°pY-; K*° ~ K - y +, at 6 GeV/c. The curve is theA2(K*-, y+) = -nr2K isodel, corresponding to a beam of 3.5 GeV/c K* t h e e x c h a n g e d p a r t i c l e i s on i t s m a s s s h e l l a n d t h e m o m e n t u m t r a n s f e r f r o m a to b i s e x a c t l y a t t h e p o l e , r e g a r d l e s s of w h a t s u b s e q u e n t l y h a p p e n s to t h e e x c h a n g e d p a r t i c l e . An e x a m p l e of t h i s w o u l d b e K*p -- ~Kp . T h e n , p r o v i d e d t h e f i n a l - s t a t e y c o m e s d i r e c t l y f r o m t h e d e c a y of t h e K* w i t h o u t s c a t t e r i n g , i t i s c o n f i n e d to t h e A2(K* -. y) = - m 2 K i s o d e l on i t s Peyrou plot. In fig. 6, we s h o w t h e P e y r o u p l o t f o r t h e 7;+ in t h e r e a c t i o n K-p-~ K*Oy-p ,
K *° -~ K - y +
(6)
a t 6 G e V / c b e a m m o m e n t u m [5]. T h e i s o d e l d r a w n i s t h a t of A 2 ( K , -. y+) = - m 2 K , f o r 3.5 G e V / c K*. T h i s s c a t t e r p l o t i s t h u s s i m p l y a ( r a t h e r i n f e r i o r ) w a y of s h o w i n g t h a t t h e K* in r e a c t i o n (6) a r e p r o d u c e d w i t h h i g h l a b m o m e n t a in a s m a l l c o n e a r o u n d t h e f o r w a r d d i r e c t i o n .
REFERENCES
[I] [2] [3] [4]
W . W . M . A l l i s o n and L. Lyons, Nuovo Cimento 51B (1967) 404. P . J . D o r n a n et al., Phys, Rev. L e t t e r s 19 (1967) 271. G.Alexander et al., Heidelberg Conf. 1967. K. P a l e r et al., A double p e r i p h e r a l interpretation of some 6 GeV[c K-p i n t e r a c tions, Rutherford Laboratory, preprint R P P / H / 2 2 . [5] K.I.Wilkinson, thesis, Oxford, 1967, unpublished.