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Longitudinal and transverse momentum correlation of the proton spectrum
Volume 39B, n u m b e r 2
PHYSICS
LONGITUDINAL
AND OF
LETTERS
TRANSVERSE THE
PROTON
17 April 1972
MOMENTUM SPECTRUM
CCRRELATION *
R. C. HWA Institute of Theoretical Science and Department of P h y s i c s , University of Oregon, Eugene, Oregon 97403, USA Received 15 F e b r u a r y 1972
The inclusive proton s p e c t r u m in the region 0.5 <~x < 0.9 is d e s c r i b e d by a through-going proton r e coiling against a h i g h - m a s s c l u s t e r . The longitudinal and t r a n s v e r s e m o m e n t u m c o r r e l a t i o n can then be explained in t e r m s of a simple Regge f o r m u l a . Evidence is given for the weakness of the t r i p l e - P o m e r o n coupling.
The proton longitudinal-momentum distribution dcr/dp,, in the c.m. system is known to be flat at 19.2 GeV/c [I], 30 GeV/c [2] and ISR energies [3]. We show in this paper that in the medium-± region not only can the flatness be understood in terms of through-going unexcited protons, but also the p, and p± correlation can be explained by a simple Regge formula. It is therefore natural to speculate that all leading-particle spectra in the medium-x region from about 20 GeV/c to infinity behave similarly. Let us first define variables and regions. In terms of c.m. momenta, we have x = P,,/Pmax and x' = p,, (p2 _p±2)-1/2 We define three regions of x as (a) large x: 0.9 < [xI < i, (b) medium x: 0 . 5 ~~Txa'~ ~< 0.9, and" (c) s m a l l x: t x l < 0 . 5 . In c h e s m a l l - x r e g i o n t h e s p e c t r a of e i t h e r t h e l e a d i n g o r t h e p r o d u c e d p a r t i c l e s c a n b e u n d e r s t o o d in t h e d i f f r a c t i v e e x c i t a t i o n m o d e l [4] a s b e i n g d u e to t h e decay, of a n e x c i t e d s t a t e of o n e of t h e two i n c i d e n t p a r t i c l e s , w h e t h e r o r n o t t h e o t h e r p a r t i c l e i s e x c i t e d T. On t h e o t h e r h a n d , in t h e m e d i u m - x r e g i o n t h e l e a d i n g p a r t i c l e i s u n e x c i t e d a n d t h r o u g h - g o i n g , b u t r e c o i l s a g a i n s t a h i g h l y e x c i t e d s t a t e of t h e o t h e r i n c i d e n t p a r t i c l e [6]. T h i s i n t e r p r e t a t i o n a p p l i e s e v e n w h e n t h e i n c i d e n t e n e r g y i s i n f i n i t e . On t h e b a s i s of t h i s p r o c e s s w e h a v e r e l a t e d in a q u a l i t a t i v e w a y t h e l e a d i n g - p a r t i c l e d i s t r i b u t i o n s of vp a n d pp c o l l i s i o n s [6]. W e now s h o w q u a n t i t a t i v e l y t h a t t h i s d e s c r i p t i o n n o t o n l y w o r k s w e l l f o r t h e i n t e g r a t e d ( o v e r ) x ) c r o s s s e c t i o n , b u t c a n in f a c t a c c o u n t f o r t h e n o n f a c t o r i z a b i l i t y of Pt, a n d Px d i s t r i b u t i o n s . A s i n g l e - e x c i t a t i o n p r o c e s s a + b ~ a + B c a n o c c u r v i a t h e e x c h a n g e of P ( P o m e r o n ) a n d R ( o r d i n a r y t r a j e c t o r i e s : f a n d w) $. If s ' ( w h e r e s ' - M 2 ) is l a r g e e n o u g h to j u s t i f y t r i p l e R e g g e , t h e n t h e two t e r m s c a n b e r e p r e s e n t e d b y t h e two d i a g r a m s in fig. 1, a s s u m i n g V p p p = 0. S i n c e s'/s = 1 - x °,
(1)
x ° = p°/Pmax ~- {x'2 + (4/s)[(I -x'2)p 2 +M2]}I/2
(2)
M being the proton mass, medium x corresponds roughly to 2 < s / s ' < I0. This is not very large s / s ' , but we nevertheless assume Regge behavior on the basis of duality. We thus have d(~
dtds'
- s
-2 2
7 a s p ( t ) V p p R ( t ) 7 b b R ( 0 ) ( - ~ , ) 2 ° t p ( t ) ( s ' ) ~R(0) + s
-2 2
7aaR(t ) VRRp(t ) 7bbp(0)(~_)2aR(t)(s,)ap(0).
(3) Work supported in part by US Atomic Energy Commission Contract No. AT(45-1)-2230. Recently, Jacob and Slansky [5] have suggested a "nova" model based on single excitation p r o c e s s e s . We r e m a r k here that such p r o c e s s e s a r e a subset of the p r o c e s s e s already considered in ref. [4], where the r e c o i l against the excitation of an incident p a r t i c l e has been taken to be e i t h e r through-going or another excitation. The r e c o i l m a s s is integrated over both the d i s c r e t e and the continuum p a r t s . The predictions o n t h e s i n g l e - p a r t i c l e s p e c t r a at s m a l l x a r e independent of the r e c o i l m a s s anyway. Thus the nova model contains no fundamentally new physics a p a r t from the specific assumption that single excitation p r o c e s s e s dominate at l a b o r a t o r y e n e r g i e s < 30 GeV. $ Isospin exchange for pp collisions can be ignored. 251
Volume 39B, number 2
PHYSICS
LETTERS
17 April 1972 ,
,
.
pp._. pX (19.2 , . , ,
,
GeV/c) , ,
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,
IO s 0 ii
D O 13 13 (3 ii1._ / li I II li I 0000
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Fig. 1. Triple-Regge diagrams for a + b ~ a + B.
1,7
°o'-
LIJ
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mmUnO0
b
I00
,
o
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.2
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,
.4
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.;
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.;
'
~:o
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Fig. 2. Proton spectrum measured at CERN (ref. 1) and reanalyzed in ref. [9]. Solid line is calculated d~/dx with scale indicated on the right side.
S e t t i n g u p ( 0 ) = 1 and a R ( 0 ) = 0.5, we s e e that the r a t i o of the f i r s t t e r m to t h e s e c o n d is p r o p o r t i o n a l to ( s / s ' ) ( M ~ / s ' ) 1/2. H e n c e f o r s u f f i c i e n t l y l a r g e s ' , the s e c o n d t e r m , i . e . , fig. l(b), is m o r e i m p o r t a n t than the f i r s t in the m e d i u m - x r e g i o n . How l a r g e s ' m u s t be d e p e n d s on the r e l a t i v e s t r e n g t h s of the c o u p l i n g s . In the f o l l o w i n g we s h a l l c o n s i d e r only the c o n t r i b u t i o n f r o m the s e c o n d t e r m f o r c o l l i s i o n s at e n e r g i e s a b o u t 20 GeV o r a b o v e . W e r e m a r k that a n u m b e r of i n v e s t i g a t o r s [ e . g . , 7] h a v e c o n s i d e r e d f o r m u l a s s i m i l a r to eq. (3), but they a l l c o n f i n e t h e i r c o n s i d e r a t i o n s to the k i n e m a t i c a l b o u n d a r y of the l a r g e - x r e g i o n w h e r e we b e l i e v e the p r o c e s s in fig. l(a) is m o r e i m p o r t a n t . Defining
f ( x ' , p ± ) = x ° d ~ / d x dp 2 = d ~ / d t d ( s ' / s ) we h a v e where and
f(x',p±)
=
(4)
C(t)(1 -xO) - 2 a ' t
(5)
C(t) = 7aaR(t) 2 VRRp(t) Ybbp(0) t = t o - p ±2 / x o ,
to = -M2(1 -x°)2/x °
(6) (7)
and w h e r e e ' is the s l o p e of C~R(t) at t = 0. T o t e s t the v a l i d i t y of eq. (5) we u s e eq. (7) to get L ~ l o g [ f ( x ' , O)/f(x',p±)] = L o + e ' ~ , L o = log {C(to)/C(t)} ,
where
~ = - ( 2 p 2 / x °) log (1 - x ° ) .
(8) (9)
T h e d a t a of r e f . [1] on the p r o t o n s p e c t r u m h a v e b e e n p l o t t e d [8] in the v a r i a b l e s x ' and p 2 T h e y a r e shown in fig. 2. F o r e a c h d a t a p o i n t in the m e d i u m - x r e g i o n 0.5 < x ' < 0.9 we h a v e c o m p u t e d the c o r r e s p o n g i n g v a l u e s of L and (. T h e y a r e shown in fig. 3; p o i n t s w i t h c o m m o n p2 a r e c l u s t e r e d e i t h e r by c o n n e c t i n g l i n e s o r by c o m m o n s y m b o l s . T h e u n i v e r s a l i t y of the c u r v e is s t r i k i n g . L i n e s j o i n i n g p o i n t s of c o m m o n x ' , if d r a w n , would be a l m o s t i n d i s t i n g u i s h a b l e f r o m one a n o t h e r . T o the z e r o t h o r d e r a p p r o x i m a t i o n by n e g l e c t i n g the c u r v a t u r e of the u n i v e r s a l c u r v e in fig. 3, we can c o n c l u d e that L o = 0 and the a v e r a g e c~' ~ I , in p e r f e c t a g r e e m e n t with the s i m p l e R e g g e p a r a m e t r i z a t i o n of the R t r a j e c t o r i e s . T a k i n g the c u r v a t u r e into a c c o u n t i m p l i e s that c~' d e c r e a s e s a s p2 i n c r e a s e s , a l s o in a g r e e m e n t with the u s u a l notion that the e f f e c t i v e R e g g e t r a j e c t o r i e s f l a t t e n out at n e g a t i v e t b e c a u s e of cut c o n t r i b u t i o n s . M o r e o v e r , the t a n g e n t to the u n i v e r s a l c u r v e at a n ~ v a l u e of ~, a s d e s c r i b e d by eq.2(8) , h a s a z e r o i n t e r c e p t L o w h i c h i n c r e a s e s r o u g h l y l i n e a r l y with p±. In f a c t , we m a y w r i t e L o ~ p± in units of ( G e V / c ) 2. We n o t e that the d a t a p o i n t s in fig. 3 c o r r e s p o n d to p2 v a r y i n g f r o m 0.05 to 0.9, a f a c t o r of 18, w h i l e x ' v a r i e s f r o m 0.53 (or g r e a t e r ) to 0.83, c o r r e s p o n d i n g to x ° f r o m a b o u t 0.63 (or g r e a t e r ) to 252
Volume 39B, number 2
PHYSICS 2.0
LETTERS
17 April 1972
p+ p'--~ p + X ¢9.2
1.6
GeV/c
,o: ~o
f (x,P.L) • p o ~
"PIIOP£
0
1.2
2
Pj. ( G e V / c ) t
0,8 $
g'.,
0.4
.15
.6
0
.7
• •
.8 .9
.05 0
Omo 2
i
/ 0.8
i
2xP 2 log ( I - x ' }
t 1.2
i
I 1.6
I
I 2.0
(GeVIc) 2
Fig. 3. Data in fig. 2 replotted, showing universality as suggested by eq. (8). 0.9, l e s s than 50% v a r i a t i o n . T h u s the t a n g e n t s of the c u r v e can a l l be a s s i g n e d s o m e a v e r a g e x ° of a b o u t 0.8. We m a y then w r i t e L o ~ 0.8 p2/x°
(10)
w h i c h in t u r n i m p l i e s on a c c o u n t of e q s . (7) and (9)
C(t) .~ C O exp (0.8t In 10)
(11)
w h e r e C o is a c o n s t a n t **. E q s . (5) and (11) a r e t h e r e f o r e o u r s u m m a r y of the d a t a on p r o t o n s p e c t r u m in the m e d i u m - x r e g i o n . Indeed, the l o n g i t u d i n a l and t r a n s v e r s e m o m e n t u m c o r r e l a t i o n can w e l l be u n d e r s t o o d in t e r m s of a s i m p l e R e g g e p i c t u r e i n v o l v i n g the e x c h a n g e of an o r d i n a r y t r a j e c t o r y . We e m p h a s i z e that t h i s s i m p l e i n t e r p r e t a t i o n would not h a v e b e e n p o s s i b l e if the t r i p l e - P o m e r o n c o u p l i n g had not b e e n a s s u m e d to v a n i s h . If Vpp D ¢ 0, then the f i r s t t e r m of eq. (3) would gain an e x t r a f a c t o r of f ~ , in which c a s e the s e c o n d t e r m wohld n e v e r b e c o m e i m p o r t a n t f o r w h a t e v e r the v a l u e of s ' . T h u s the s u c c e s s of our i n t e r p r e t a t i o n i m p l i e s the w e a k n e s s of the t r i p l e - P o m e r o n c o u p l i n g . F r o m e q s . (1) and (4) we s e e that f(x',p±) is a l s o d c r / d t d x °. We m a y then obtain the i n t e g r a t e d l o n g i tudinal d i s t r i b u t i o n dcr/dx, w h i c h is v e r y n e a r l y dcr/dx ° , by i n t e g r a t i n g eq. (5) o v e r t. We get
to
dxd~ = _oo f
f(x', 0)
](x',p.)dt~ (0.81n 10)[1-2log(1-x°)]
(12)
2 t i m e s the d a t a p o i n t s g i v e n in fig. 2 f o r p± = 0. T h e r e s u l t f o r da/dx is shown by w h e r e f(x', 0), is !;r the s o l i d l i n e in the s a m e f i g u r e . It is c o n s i s t e n t with b e i n g f l a t ; d e v i a t i o n f r o m the c o n s t a n t v a l u e at 27 mb is no m o r e than 5%. T h i s a g r e e s with the q u a l i t a t i v e c o n c l u s i o n m a d e in r e f . [6]. E x t e n s i o n of t h e s e c o n s i d e r a t i o n s to the l e a d i n g pion s p e c t r u m in ~lp c o l l i s i o n s o r the l e a d i n g kaon s p e c t r u m in g p c o l l i s i o n s w o u l d b e n a t u r a l , and should be p u r s u e d a s soon a s r e l i a b l e d a t a a b o v e 20 GeV/c b e c o m e a v a i l a b l e , if they can be s i m i l a r l y e x p l a i n e d , not only can the s i m p l e p i c t u r e of o r d i n a r y R e g g e e x c h a n g e gain f u r t h e r s u p p o r t , but a l s o the v a n i s h i n g of the t r i p l e - P o m e r o n c o u p l i n g w o u l d acquire additional affirmative evidence. ** We remark that the behavior of C(to) can be obtained from eq. (5) and the data for P.L = 0. However, the range in to is too small (<0.2) to test the validity of eq. (11) which is an approximate formula for a much wider range of t values. 253
Volume 39B, number 2
PHYSICS
LETTERS
17 April 1972
References [1] [2] [3] [4] [5] [6]
[7] [8]
254
J . V . A l i a b y e t a[., CERN Report 70-12. E.W. A n d e r s o n e t a l . , Phys. Rev. L e t t e r s 19 (1967) 198. L.G. R a t n e r e t al., Phys. Rev. L e t t e r s 27 {1971) 68, and paper p r e s e n t e d at the R o c h e s t e r Conf., 1971. R . C . Hwa, Phys. Rev. L e t t e r s 26 (1971) 1143; R. C. Hwa and C. S. Lam, Phys. Rev. L e t t e r s 27 (1971) 1098; Phys. Rev. D5 (Feb. 1, 1972). Jacob and Slansky, Phys. L e t t e r s 37B {1971) 408. R. C. Hwa, Phys. L e t t e r s 37B (1971) 405. In eqs. (4), (5) and (6) of that paper, the e x p r e s s i o n d(~/d2p±dx(p.=O) should read da/dtdx(t=O). Also, 1/Trin eq. (4) should be deleted. The t=O point is actually outside the physical region. The minimum value to, defined in eq. (7) of this paper, is, however, small [<0.2(GeV/c) 2] in the m e dium-x region. With the understanding that the values of f y and f p at t = 0 can be approximately obtained by extrapolation from the edge of the physical region, the conclusion of that paper as e x p r e s s e d by eq° (11) therein is unaffected by the change. I am grateful to J. Kasman for pointing out the e r r o r to me. E. L. B e r g e r , Intern. Colloquium on Multiparticle dynamics, May 19'~1. Argonne National Laboratory report HEP 7134. J. Kasman and D. P. Sidhu, Stony Brook p r e p r i n t 1970.
PHYSICS
LONGITUDINAL
AND OF
LETTERS
TRANSVERSE THE
PROTON
17 April 1972
MOMENTUM SPECTRUM
CCRRELATION *
R. C. HWA Institute of Theoretical Science and Department of P h y s i c s , University of Oregon, Eugene, Oregon 97403, USA Received 15 F e b r u a r y 1972
The inclusive proton s p e c t r u m in the region 0.5 <~x < 0.9 is d e s c r i b e d by a through-going proton r e coiling against a h i g h - m a s s c l u s t e r . The longitudinal and t r a n s v e r s e m o m e n t u m c o r r e l a t i o n can then be explained in t e r m s of a simple Regge f o r m u l a . Evidence is given for the weakness of the t r i p l e - P o m e r o n coupling.
The proton longitudinal-momentum distribution dcr/dp,, in the c.m. system is known to be flat at 19.2 GeV/c [I], 30 GeV/c [2] and ISR energies [3]. We show in this paper that in the medium-± region not only can the flatness be understood in terms of through-going unexcited protons, but also the p, and p± correlation can be explained by a simple Regge formula. It is therefore natural to speculate that all leading-particle spectra in the medium-x region from about 20 GeV/c to infinity behave similarly. Let us first define variables and regions. In terms of c.m. momenta, we have x = P,,/Pmax and x' = p,, (p2 _p±2)-1/2 We define three regions of x as (a) large x: 0.9 < [xI < i, (b) medium x: 0 . 5 ~~Txa'~ ~< 0.9, and" (c) s m a l l x: t x l < 0 . 5 . In c h e s m a l l - x r e g i o n t h e s p e c t r a of e i t h e r t h e l e a d i n g o r t h e p r o d u c e d p a r t i c l e s c a n b e u n d e r s t o o d in t h e d i f f r a c t i v e e x c i t a t i o n m o d e l [4] a s b e i n g d u e to t h e decay, of a n e x c i t e d s t a t e of o n e of t h e two i n c i d e n t p a r t i c l e s , w h e t h e r o r n o t t h e o t h e r p a r t i c l e i s e x c i t e d T. On t h e o t h e r h a n d , in t h e m e d i u m - x r e g i o n t h e l e a d i n g p a r t i c l e i s u n e x c i t e d a n d t h r o u g h - g o i n g , b u t r e c o i l s a g a i n s t a h i g h l y e x c i t e d s t a t e of t h e o t h e r i n c i d e n t p a r t i c l e [6]. T h i s i n t e r p r e t a t i o n a p p l i e s e v e n w h e n t h e i n c i d e n t e n e r g y i s i n f i n i t e . On t h e b a s i s of t h i s p r o c e s s w e h a v e r e l a t e d in a q u a l i t a t i v e w a y t h e l e a d i n g - p a r t i c l e d i s t r i b u t i o n s of vp a n d pp c o l l i s i o n s [6]. W e now s h o w q u a n t i t a t i v e l y t h a t t h i s d e s c r i p t i o n n o t o n l y w o r k s w e l l f o r t h e i n t e g r a t e d ( o v e r ) x ) c r o s s s e c t i o n , b u t c a n in f a c t a c c o u n t f o r t h e n o n f a c t o r i z a b i l i t y of Pt, a n d Px d i s t r i b u t i o n s . A s i n g l e - e x c i t a t i o n p r o c e s s a + b ~ a + B c a n o c c u r v i a t h e e x c h a n g e of P ( P o m e r o n ) a n d R ( o r d i n a r y t r a j e c t o r i e s : f a n d w) $. If s ' ( w h e r e s ' - M 2 ) is l a r g e e n o u g h to j u s t i f y t r i p l e R e g g e , t h e n t h e two t e r m s c a n b e r e p r e s e n t e d b y t h e two d i a g r a m s in fig. 1, a s s u m i n g V p p p = 0. S i n c e s'/s = 1 - x °,
(1)
x ° = p°/Pmax ~- {x'2 + (4/s)[(I -x'2)p 2 +M2]}I/2
(2)
M being the proton mass, medium x corresponds roughly to 2 < s / s ' < I0. This is not very large s / s ' , but we nevertheless assume Regge behavior on the basis of duality. We thus have d(~
dtds'
- s
-2 2
7 a s p ( t ) V p p R ( t ) 7 b b R ( 0 ) ( - ~ , ) 2 ° t p ( t ) ( s ' ) ~R(0) + s
-2 2
7aaR(t ) VRRp(t ) 7bbp(0)(~_)2aR(t)(s,)ap(0).
(3) Work supported in part by US Atomic Energy Commission Contract No. AT(45-1)-2230. Recently, Jacob and Slansky [5] have suggested a "nova" model based on single excitation p r o c e s s e s . We r e m a r k here that such p r o c e s s e s a r e a subset of the p r o c e s s e s already considered in ref. [4], where the r e c o i l against the excitation of an incident p a r t i c l e has been taken to be e i t h e r through-going or another excitation. The r e c o i l m a s s is integrated over both the d i s c r e t e and the continuum p a r t s . The predictions o n t h e s i n g l e - p a r t i c l e s p e c t r a at s m a l l x a r e independent of the r e c o i l m a s s anyway. Thus the nova model contains no fundamentally new physics a p a r t from the specific assumption that single excitation p r o c e s s e s dominate at l a b o r a t o r y e n e r g i e s < 30 GeV. $ Isospin exchange for pp collisions can be ignored. 251
Volume 39B, number 2
PHYSICS
LETTERS
17 April 1972 ,
,
.
pp._. pX (19.2 , . , ,
,
GeV/c) , ,
,
,
,
,
IO s 0 ii
D O 13 13 (3 ii1._ / li I II li I 0000
P.L ( G e V / c ) 2 QDO
o ~
~ 0
O
a
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10 4
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b
b
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~
e •o Ù A A ~ • ~>O "~"--
AI ALl OI.2 4' L3
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(b)
Fig. 1. Triple-Regge diagrams for a + b ~ a + B.
1,7
°o'-
LIJ
b
1o)
A a~
A~.
mmUnO0
b
I00
,
o
I
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i
,
.4
,
.;
,
.;
'
~:o
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I
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x'
Fig. 2. Proton spectrum measured at CERN (ref. 1) and reanalyzed in ref. [9]. Solid line is calculated d~/dx with scale indicated on the right side.
S e t t i n g u p ( 0 ) = 1 and a R ( 0 ) = 0.5, we s e e that the r a t i o of the f i r s t t e r m to t h e s e c o n d is p r o p o r t i o n a l to ( s / s ' ) ( M ~ / s ' ) 1/2. H e n c e f o r s u f f i c i e n t l y l a r g e s ' , the s e c o n d t e r m , i . e . , fig. l(b), is m o r e i m p o r t a n t than the f i r s t in the m e d i u m - x r e g i o n . How l a r g e s ' m u s t be d e p e n d s on the r e l a t i v e s t r e n g t h s of the c o u p l i n g s . In the f o l l o w i n g we s h a l l c o n s i d e r only the c o n t r i b u t i o n f r o m the s e c o n d t e r m f o r c o l l i s i o n s at e n e r g i e s a b o u t 20 GeV o r a b o v e . W e r e m a r k that a n u m b e r of i n v e s t i g a t o r s [ e . g . , 7] h a v e c o n s i d e r e d f o r m u l a s s i m i l a r to eq. (3), but they a l l c o n f i n e t h e i r c o n s i d e r a t i o n s to the k i n e m a t i c a l b o u n d a r y of the l a r g e - x r e g i o n w h e r e we b e l i e v e the p r o c e s s in fig. l(a) is m o r e i m p o r t a n t . Defining
f ( x ' , p ± ) = x ° d ~ / d x dp 2 = d ~ / d t d ( s ' / s ) we h a v e where and
f(x',p±)
=
(4)
C(t)(1 -xO) - 2 a ' t
(5)
C(t) = 7aaR(t) 2 VRRp(t) Ybbp(0) t = t o - p ±2 / x o ,
to = -M2(1 -x°)2/x °
(6) (7)
and w h e r e e ' is the s l o p e of C~R(t) at t = 0. T o t e s t the v a l i d i t y of eq. (5) we u s e eq. (7) to get L ~ l o g [ f ( x ' , O)/f(x',p±)] = L o + e ' ~ , L o = log {C(to)/C(t)} ,
where
~ = - ( 2 p 2 / x °) log (1 - x ° ) .
(8) (9)
T h e d a t a of r e f . [1] on the p r o t o n s p e c t r u m h a v e b e e n p l o t t e d [8] in the v a r i a b l e s x ' and p 2 T h e y a r e shown in fig. 2. F o r e a c h d a t a p o i n t in the m e d i u m - x r e g i o n 0.5 < x ' < 0.9 we h a v e c o m p u t e d the c o r r e s p o n g i n g v a l u e s of L and (. T h e y a r e shown in fig. 3; p o i n t s w i t h c o m m o n p2 a r e c l u s t e r e d e i t h e r by c o n n e c t i n g l i n e s o r by c o m m o n s y m b o l s . T h e u n i v e r s a l i t y of the c u r v e is s t r i k i n g . L i n e s j o i n i n g p o i n t s of c o m m o n x ' , if d r a w n , would be a l m o s t i n d i s t i n g u i s h a b l e f r o m one a n o t h e r . T o the z e r o t h o r d e r a p p r o x i m a t i o n by n e g l e c t i n g the c u r v a t u r e of the u n i v e r s a l c u r v e in fig. 3, we can c o n c l u d e that L o = 0 and the a v e r a g e c~' ~ I , in p e r f e c t a g r e e m e n t with the s i m p l e R e g g e p a r a m e t r i z a t i o n of the R t r a j e c t o r i e s . T a k i n g the c u r v a t u r e into a c c o u n t i m p l i e s that c~' d e c r e a s e s a s p2 i n c r e a s e s , a l s o in a g r e e m e n t with the u s u a l notion that the e f f e c t i v e R e g g e t r a j e c t o r i e s f l a t t e n out at n e g a t i v e t b e c a u s e of cut c o n t r i b u t i o n s . M o r e o v e r , the t a n g e n t to the u n i v e r s a l c u r v e at a n ~ v a l u e of ~, a s d e s c r i b e d by eq.2(8) , h a s a z e r o i n t e r c e p t L o w h i c h i n c r e a s e s r o u g h l y l i n e a r l y with p±. In f a c t , we m a y w r i t e L o ~ p± in units of ( G e V / c ) 2. We n o t e that the d a t a p o i n t s in fig. 3 c o r r e s p o n d to p2 v a r y i n g f r o m 0.05 to 0.9, a f a c t o r of 18, w h i l e x ' v a r i e s f r o m 0.53 (or g r e a t e r ) to 0.83, c o r r e s p o n d i n g to x ° f r o m a b o u t 0.63 (or g r e a t e r ) to 252
Volume 39B, number 2
PHYSICS 2.0
LETTERS
17 April 1972
p+ p'--~ p + X ¢9.2
1.6
GeV/c
,o: ~o
f (x,P.L) • p o ~
"PIIOP£
0
1.2
2
Pj. ( G e V / c ) t
0,8 $
g'.,
0.4
.15
.6
0
.7
• •
.8 .9
.05 0
Omo 2
i
/ 0.8
i
2xP 2 log ( I - x ' }
t 1.2
i
I 1.6
I
I 2.0
(GeVIc) 2
Fig. 3. Data in fig. 2 replotted, showing universality as suggested by eq. (8). 0.9, l e s s than 50% v a r i a t i o n . T h u s the t a n g e n t s of the c u r v e can a l l be a s s i g n e d s o m e a v e r a g e x ° of a b o u t 0.8. We m a y then w r i t e L o ~ 0.8 p2/x°
(10)
w h i c h in t u r n i m p l i e s on a c c o u n t of e q s . (7) and (9)
C(t) .~ C O exp (0.8t In 10)
(11)
w h e r e C o is a c o n s t a n t **. E q s . (5) and (11) a r e t h e r e f o r e o u r s u m m a r y of the d a t a on p r o t o n s p e c t r u m in the m e d i u m - x r e g i o n . Indeed, the l o n g i t u d i n a l and t r a n s v e r s e m o m e n t u m c o r r e l a t i o n can w e l l be u n d e r s t o o d in t e r m s of a s i m p l e R e g g e p i c t u r e i n v o l v i n g the e x c h a n g e of an o r d i n a r y t r a j e c t o r y . We e m p h a s i z e that t h i s s i m p l e i n t e r p r e t a t i o n would not h a v e b e e n p o s s i b l e if the t r i p l e - P o m e r o n c o u p l i n g had not b e e n a s s u m e d to v a n i s h . If Vpp D ¢ 0, then the f i r s t t e r m of eq. (3) would gain an e x t r a f a c t o r of f ~ , in which c a s e the s e c o n d t e r m wohld n e v e r b e c o m e i m p o r t a n t f o r w h a t e v e r the v a l u e of s ' . T h u s the s u c c e s s of our i n t e r p r e t a t i o n i m p l i e s the w e a k n e s s of the t r i p l e - P o m e r o n c o u p l i n g . F r o m e q s . (1) and (4) we s e e that f(x',p±) is a l s o d c r / d t d x °. We m a y then obtain the i n t e g r a t e d l o n g i tudinal d i s t r i b u t i o n dcr/dx, w h i c h is v e r y n e a r l y dcr/dx ° , by i n t e g r a t i n g eq. (5) o v e r t. We get
to
dxd~ = _oo f
f(x', 0)
](x',p.)dt~ (0.81n 10)[1-2log(1-x°)]
(12)
2 t i m e s the d a t a p o i n t s g i v e n in fig. 2 f o r p± = 0. T h e r e s u l t f o r da/dx is shown by w h e r e f(x', 0), is !;r the s o l i d l i n e in the s a m e f i g u r e . It is c o n s i s t e n t with b e i n g f l a t ; d e v i a t i o n f r o m the c o n s t a n t v a l u e at 27 mb is no m o r e than 5%. T h i s a g r e e s with the q u a l i t a t i v e c o n c l u s i o n m a d e in r e f . [6]. E x t e n s i o n of t h e s e c o n s i d e r a t i o n s to the l e a d i n g pion s p e c t r u m in ~lp c o l l i s i o n s o r the l e a d i n g kaon s p e c t r u m in g p c o l l i s i o n s w o u l d b e n a t u r a l , and should be p u r s u e d a s soon a s r e l i a b l e d a t a a b o v e 20 GeV/c b e c o m e a v a i l a b l e , if they can be s i m i l a r l y e x p l a i n e d , not only can the s i m p l e p i c t u r e of o r d i n a r y R e g g e e x c h a n g e gain f u r t h e r s u p p o r t , but a l s o the v a n i s h i n g of the t r i p l e - P o m e r o n c o u p l i n g w o u l d acquire additional affirmative evidence. ** We remark that the behavior of C(to) can be obtained from eq. (5) and the data for P.L = 0. However, the range in to is too small (<0.2) to test the validity of eq. (11) which is an approximate formula for a much wider range of t values. 253
Volume 39B, number 2
PHYSICS
LETTERS
17 April 1972
References [1] [2] [3] [4] [5] [6]
[7] [8]
254
J . V . A l i a b y e t a[., CERN Report 70-12. E.W. A n d e r s o n e t a l . , Phys. Rev. L e t t e r s 19 (1967) 198. L.G. R a t n e r e t al., Phys. Rev. L e t t e r s 27 {1971) 68, and paper p r e s e n t e d at the R o c h e s t e r Conf., 1971. R . C . Hwa, Phys. Rev. L e t t e r s 26 (1971) 1143; R. C. Hwa and C. S. Lam, Phys. Rev. L e t t e r s 27 (1971) 1098; Phys. Rev. D5 (Feb. 1, 1972). Jacob and Slansky, Phys. L e t t e r s 37B {1971) 408. R. C. Hwa, Phys. L e t t e r s 37B (1971) 405. In eqs. (4), (5) and (6) of that paper, the e x p r e s s i o n d(~/d2p±dx(p.=O) should read da/dtdx(t=O). Also, 1/Trin eq. (4) should be deleted. The t=O point is actually outside the physical region. The minimum value to, defined in eq. (7) of this paper, is, however, small [<0.2(GeV/c) 2] in the m e dium-x region. With the understanding that the values of f y and f p at t = 0 can be approximately obtained by extrapolation from the edge of the physical region, the conclusion of that paper as e x p r e s s e d by eq° (11) therein is unaffected by the change. I am grateful to J. Kasman for pointing out the e r r o r to me. E. L. B e r g e r , Intern. Colloquium on Multiparticle dynamics, May 19'~1. Argonne National Laboratory report HEP 7134. J. Kasman and D. P. Sidhu, Stony Brook p r e p r i n t 1970.