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Peripheral dynamics of inclusive reactions in the fragmentation region
Volume 37B, number 4
P HY SI C S L E T T E R S
PERIPHERAL
DYNAMICS OF INCLUSIVE REACTIONS THE FRAGMENTATION REGION
13 December 1971
IN
F. DUIMIO and G. MARCHESINI Istituto di Fisica dell'UniversitY, Parma and Istituto Nazionale di Fisica Nucleate, Sezione di Milano, Italy Received 26 October 1971 Peripheral dynamics is studied for the limiting fragmentation of protons into protons and pions. The calculated distributions are discussed and compared with experimental data.
A r e m a r k a b l e flow of i n t e r e s t has been r e c e n t l y focused on the e x p e r i m e n t a l and t h e o r e t i c a l a s p e c t s of the i n c l u s i v e r e a c t i o n s a +b ~ c + anything and m o r e s p e c i f i c a l l y on the a n a l y s i s of the h i g h - e n e r g y b e h a v i o u r of the s i n g l e p a r t i c l e d i s t r i b u t i o n s , i n v a r i a n t l y defined by Oc(S,x,q2) -
2E d3ec Crab dq 3 '
(1)
w h e r e E, q a r e the energy and m o m e n t u m of p a r t i c l e c, ~ab is the total a - b c r o s s s e c t i o n and x = 2 q * / s 1-/2 is the F e y n m a n s c a l i n g v a r i a b l e . M~iller [1] p r o p o s e d to extend the ideas of R e g g e pole ex ch an g es to the study of such r e a c t i o n s . In p a r t i c u l a r , M i i l l e r ' s a n a l y s i s shows that in a s m a l l r e g i o n of the p h ase sp ace (x ~ 1, q i ~ 0) w h e r e p a r t i c l e c is r e l a t i v i s t i c with r e s p e c t to all the o t h e r outgoing p a r t i c l e s , the d i s t r i b u t i o n function Pc should exhibit a T r i p l e Regge (TR) behaviour. T he continuation of the T R p a r a m e t r i z a t i o n to a l a r g e r r e g i o n of p h as e s p a c e heavily depends on an e x t e n s i v e exploitation of the duality notion. C l e a r l y a d e t a i l e d knowledge of the m u l t i p a r t i c l e hadron r e a c t i o n s at high e n e r g y , i.e. a s a t i s f a c t o r y t h e o r y of hadron d y n a m i c s would be needed to c l a r i f y M~iller p a r a m e t r i z a t i o n and eventually its continuation, v i a duality, away f r o m the r e g i o n x ~ 1. In a b s e n c e of a c o m p l e t e t h e o r y such a n a l y s i s can be e f f e c t e d in t e r m s of one of the m o d e l s which have been p r o p o s e d in o r d e r to get s o m e insight in p r o b l e m s connected with p r o d u c t i o n phenomena. P u r p o s e of this note is a study of the d y n a m i c s of i n c l u s i v e r e a c t i o n s within the f r a m e w o r k of the m u l t i p e r i p h e r a l (MP) model of Fubini and c o l l a b o r a t o r s [2], in the f r a g m e n t a t i o n r e g i o n (x not n e a r to 0). In p a r t i c u l a r , we c o n s i d e r the r e a c t i o n s p+p -~p + anything and p+p -~ ~ + anything, f o r which ex t e n s i v e data have b e e n c o l l e c t e d [3,4]. In view of the e x p e r i m e n t a l fact that the d i s t r i b u t i o n s s e e m to be a l r e a d y l i m i t i n g at p r e s e n t e n e r g i e s [4, 5], we study the MP d i s t r i b u t i o n s in t h e i r l i m i t i n g f o r m : this g i v e s us the p o s s i b i l i t y of working with a r a t h e r s i m p l i f i e d l i m i t k i n e m a t i c s . We a r e mainly i n t e r e s t e d in the r e g i o n of l a r g e x, i.e., in t e r m s of the MP chain, we want to a n a l y s e the c o n t r i b u t i o n s of the p r o t o n and of the pion c o m i n g out f r o m the r i g h t m o s t blob. Thus we a r e led to c o n s i d e r e s s e n t i a l l y the d i a g r a m s of fig. l a and fig. lb f o r the proton d i s t r i b u t i o n pp and only the diag r a m of fig. l c f o r the pion d i s t r i b u t i o n s py+. In o r d e r to compute the co n t r i b u t i o n s of t h e s e g r a p h s to the p d i s t r i b u t i o n s , one has to sum o v e r a ll the p a r t i c l e s p r o d u c e d in the l e f t - h a n d blobs so that t h e r e is no need of s p e c i f y i n g t h e i r d e t a i l e d s t r u c t u r e . In t h i s way the p r o b l e m is e s s e n t i a l l y r e d u c e d to the e v a l u a t i o n of s i m p l e one pion exchange graphs. Since the b a s i c idea of the pion exchange m o d e l s is the d o m i n a n c e of pion p o l e s in the low m o m e n t u m t r a n s f e r r eg i o n , we e x a m i n e the k i n e m a t i c a l r e l a t i o n s f o r t in fig. l a and t' in figs. lb and l c in o r d e r to have an a p r i o r i e s t i m a t e of the p h a s e - s p a c e r e g i o n w h e r e the model is justified. T o fix the ideas l et us r e q u i r e It l < 0.5 GeV 2 and c o n s i d e r f i r s t the proton case. In fig. l a the m o m e n t u m t r a n s f e r t, in the l i m i t k i n e m a t i c s is given by
427
Volume 37B, number
4
PHYSICS
LETTERS
M M
a
\til
I. .
.
.
I.
1971
C
S"
S"
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3"
\ ii <,\
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JoL
.
S
M
b
S ,r
13 D e c e m b e r
S
S
Fig. 1. Peripheral diagrams for proton fragmentation into protons (a,b) and into pions (e) in proton-proton interactions. Solid lines refer to protons and dotted lines to pions. t = - [ ( 1 - x)2m 2+q~] ~ /x,
(2)
with m = proton m a s s and q± = t r a n s v e r s e m o m e n t u m of the proton in the c e n t e r - o f - m a s s frame. It follows that the p e r i p h e r a l amplitude of fig. l a will be r e l e v a n t in the region x ~> ½ for s m a l l q2. The k i n e m a t i c s for the other two d i a g r a m s is obviously m o r e complicated. F o r t' and s' appearing in the amplitude of fig. lb one obtains, in the l i m i t s ~ ¢o, the following e x p r e s s i o n s : v
t' = t o - y
(3)
with
t o, = r t - 1 -rr
U2 '
(4)
y = ½M2(l-r)(Z-z),
(5)
and s':[1-(1
x)r]
(1-x)(1-r)
+xrn 2
+~_x[xY+7(1-r)+2q.(y(1-r))l/2
cos~]
,
(6)
where # = p i o n m a s s , r = s " / M 2, z, q~ = a n g u l a r v a r i a b l e s of q' in the m i s s i n g m a s s r e s t f r a m e (their range is over the whole sphere), and t is still given by eq. (2). F o r m u l a e (3)-(6) apply also to the case of fig. l c , ff one takes c a r e of i n t e r c h a n g i n g m and ~ and with t not given by eq. (2) but by: t = (1-x)m 2
1 - x U2 _ q-~. X
(2')
X
C l e a r l y an e s t i m a t e of the region of validity in x of the a m p l i t u d e s of fig. l b and p a r t i c u l a r l y of fig. lc is not so straightforward. However let us o b s e r v e that at qz = 0 the k i n e m a t i c a l r e l a t i o n s s i m plify; in p a r t i c u l a r one gets y = ( 1 - r ) q ~2 , i
where q . is the c e n t e r - o f - m a s s t r a n s v e r s e m o m e n t u m of q' both in figs. lh and lc. Since the p h y s i c a l t a m p l i t u d e s a r e strongly bounded to s m a l l values of q'x2, i m p o r t a n t c o n t r i b u t i o n s come only for t' ~ t o. In the case of the proton d i s t r i b u t i o n It~l < ½ GeV 2 for r not close to 1 and for x > ½. At q~_ ¢ 0 but s m a l l such as e s t i m a t e would not change radically. F o r the pion d i s t r i b u t i o n , i m p o r t a n t c o n t r i b u t i o n s to the amplitude come from a much m o r e l i m i t e d range of r , say r l e s s than 0.3-0.5 and f r o m x belonging to the full range, except obviously the region x~-.0. 428
Volume 37B, number 4
P H Y SI C S L E T T E R S
13 December 1971
In view of t h e s e g e n e r a l k i n e m a t i c a l c o n s e q u e n c e s of the pion exchange d o m i n a n c e in the model and of the fact that we d i s r e g a r d c o n t r i b u t i o n s to py f r o m pions coming out from blobs located in the middle of the MP chain, the r e l i a b i l i t y of our r e s u l t s (especially for p~) will be a d e c r e a s i n g function of 1 - x, falling to z e r o at x ~ 0, i.e. in the p i o n i z a t i o n region. P r o t o n distribution. The c o n t r i b u t i o n of the amplitude of fig. l a to the p p d i s t r i b u t i o n is calculated s u m m i n g o v e r all the p a r t i c l e s coming out f r o m the l e f t - h a n d blob into which we have collapsed the MP chain, to give the total yN c r o s s section. Its l i m i t i n g f o r m t u r n s out to be:
%-N
- t(1
-
x)
(7)
p.L.)(a _ 2~ 2 (~NN ( t - ~2)2 ' where g r is the r a t i o n a l i z e d p i o n - n u c l e o n coupling constant, t is given by eq. (2) and CyN, CNN a r e the a s y m p t o t i c pion p r o t o n and p , p c r o s s sections. F o r ayN we a s s u m e the o n - s h e l l value. H e r e and in the following we always ignore off-shell c o r r e c t i o n s as long as the i n t e g r a l s converge, in a c c o r d with the s u g g e s t i o n of Chew and c o l l a b o r a t o r s [6] that the smooth v a r i a t i o n of f o r m f a c t o r s in the region of s m a l l t (dominating the p e r i p h e r a l amplitudes) can be ignored, as long as the MP i n t e g r a l equation r e m a i n s of the F r e d h o l m type. The d i s t r i b u t i o n p ~ ) , as calculated f r o m eq. (7), is plotted in fig. 2 (curve a), together with the exp e r i m e n t a l values, ~t q . = 0.1 GeV/c It is s t r i k i n g that this c o n t r i b u t i o n for x not close to 1 is of the o r d e r of half the e x p e r i m e n t a l value.
o • • v
Po" 14.25 GeV/c] Po=19.2 GeV/c I A L L A B Y et a11971 Po=24.0 GeV/c) Po=19.2 GeV/c A L L A B Y e t a11970
10 2 4.0 Ge V/c 14.25 GeV./c~_;,j
ql 0 1 GeV/c (D
1 •
•
0 VO
I 0.2
0.I
a b /i
0.4
I X
I ;//i//"i 0.8
0.6
Fig. 2. Distribution functions for protons: curve a, contributions from the graph of fig. la; curve b, contributions from the graph of fig. lb with s' < 10 GeV2; curves 1 and 2, contributions from the graph of fig. lb with s' > 10GeV2, and X in the damping factor respectively equal to 1 and 2 GeV2. In the s a m e way, with a m o r e e l a b o r a t e calculationT one obtains the c o n t r i b u t i o n to p p f r o m the a m plitude of fig. lb:
'
/;
/
P ) _ aTrN 1 (I - x) f r d r dr' aNN 2(27r)6 0 -~o 0
dq~
(8) (t'-/~2)2
'
z* b e i n g the c o s i n e of the c e n t e r - o f - m a s s s c a t t e r i n g angle of the ITN s y s t e m in the r i g h t - h a n d blob. Isotopic spin can be taken into account by s u m m i n g over the p o s s i b l e states of the exchanged pion. In eq. (8) A1rN(S' , z*) is the e l a s t i c 7r-N amplitude. In the n u m e r i c a l evaluation of eq. (8) the data [8] for the d i f f e r e n t i a l 7TN c r o s s s e c t i o n have b e e n used. It is i n t e r e s t i n g to c o n s i d e r s e p a r a t e l y the c o n t r i b u t i o n s to p ~ ) f r o m A~RN anA~TTNAHE,with ARN The calculation has been done in the limit kinematics, and the p h a s e - s p a c e integrations have been c a r r i e d out in the missing mass r e s t f r a m e . The rather simple form of our resulting eq. (8) can be t r a c e d b a c k to limit forms of formulae in the paper of F e r r a r i and Selleri [7]. 429
Volume 37B, number 4
P HYSI CS L E T T ER S
13 December 1971
A~N(S',Z*) f o r s ' in the r e s o n a n c e r e g i o n s ' < 10 GeV 2 and AHN E = A~N(S',Z* ) f o r s ' in the h i g h - e n e r g y region, s ' > 10 GeV 2. In fig. 2 the c a l c u l a t e d contribution f r o m A R is shown in c u r v e b. Again the s i z e of this contribution is r e m a r k a b l y high. The sum of c u r v e s a and b, the solid c u r v e c in fig. 2, a p p e a r s to fit the data in a r a t h e r l a r g e r e g i o n of x. In p a r t i c u l a r we r e m a r k that the two b r o a d bumps in c u r v e b r e f l e c t i n g the f o r w a r d and backward decay of ~N r e s o n a n c e s , s e e m to fit with a s t r u c t u r e in the e x p e r i m e n t a l data. In addition, the c a l c u l a t e d slope B(x) of the pp d i s t r i b u t i o n in the q2 v a r i a b l e
B(x) = " ~2_2d lnpp(x, q2) dq ~
,.)
q~ = 0
is a r a t h e r smoothly i n c r e a s i n g function of x and f o r x > 0.5 it is of the o r d e r of 10 GeV - 2 , in f a i r a g r e e m e n t with the a v a i l a b l e e x p e r i m e n t a l i n f o r m a t i o n [3,4]. As f o r the contribution to p p f r o m the ~N amplitude A HE- the i n t e g r a t i o n o v e r t' in eq. (8) is d i v e r gent. It follows f r o m the p r e v i o u s c o n s i d e r a t i o n s that a damping f a c t o r F(t') in the integrand is now n e c e s s a r y : we a s s u m e it to be of the f o r m Fit') = (X/(X- t')) z. "b" In fig. 2 c u r v e s 1 and 2 give the c a l c u l a t e d contribution to p~ ) f o r typical v a l u e s X = 1 GeV 2 and X = 2 GeV 2 r e s p e c t i v e l y . As expected f r o m k i n e m a t i c s this contribution is i m p o r t a n t only f o r x ~ 1. Both d i f f r a c t i o n ( p o m e r o n exchange) and Regge b e h a v i o u r a r e contained in the amplitude A HE so that this c a l c u l a t e d contribution to p p can be r e p h r a s e d in t e r m s of the T r i p l e P o m e r o n (TP) and T r i p l e Regge (TR) language. At x = 1 the TR f o r m u l a e a r e finite, while the T P one, as our c a l c u l a t e d c u r v e , is d i v e r g e n t :~. B e f o r e d i s c u s s i n g the r e l a t i o n with e x p e r i m e n t at x ~ 1 one has to c o n s i d e r that the d y n a m i c s in this region is highly complex, due to e f f e c ts of d if f r a c t i o n d i s s o c i a t i o n . In p a r t i c u l a r the d i f f r a c t i v e p r o d u c tion of r e s o n a n c e s , which is not d e s c r i b e d in the p r e s e n t f o r m of the model, is r e s p o n s i b l e of the bumps at the end of the e x p e r i m e n t a l d i s t r i b u t i o n (due to r e s o n a n t excitation of the p r o j e c t i l e ) : at infinite e n e r g y they should c o n c e n t r a t e at x = 1. The " s y m m e t r i c " r e s o n a n t excitation of the t a r g e t (Deck effect), g i v e s contributions in the region x ~ 0.9 which h o w e v e r do not c o l l a p s e to one at a s y m p t o t i c ene r g i e s [9]. F r o m t h e s e o b s e r v a t i o n s it follows that a d i r e c t c o m p a r i s o n with the data in this region is at p r e s e n t infeasible. Many au t h o r s (e.g. [10]) have t r i e d to fit the data in the r e g i o n of l o w e r x by a s s u m i n g the validity of a duality extension to such r e g i o n of the TR p a r a m e t r i z a t i o n with h i g h - r a n k i n g Regge p o l e s , eventually including a l s o T P contributions. In our model this would c o r r e s p o n d to c o n s i d e r i n g the full contribution of the graph of fig. lb with the ~rN amplitude p a r a m e t r i z e d in t e r m s of p u r e R e g g e - p o l e exchanges, including diffraction. The r e s u l t i n g c u r v e would e s s e n t i a l l y coincide with the c u r v e b of fig. 2. The fact that the graph of fig. la, which can be a s s o c i a t e d with a l o w - l a y i n g pole (the pion) coupled to the p o m e r o n g i v es a s i z a b l e contribution does not s e e m to fit with an approach b a s e d on the d o m i nance of h i g h - r a n k i n g Regge p o l e s , coupled to the p o m e r o n . Pion distribution. T h e single p a r t i c l e d i s t r i b u t i o n f o r the p e r i p h e r a l am p l i t u d e of fig. l c is still give n by eq. (8), w h e r e t is defined by eq. (2') and t ' , s ' by eqs. (3)-(6), with m ~ p, and in the e l a s t i c ~rN amplitude z* has to be r e p l a c e d by - z * . Again one has to s u m o v e r the p o s s i b l e isotopic s t a t e s of the exchanged pion. In figs. 3 and 4 the c a l c u l a t e d l i m i t i n g p~ functions (solid cu r v es) a r e c o m p a r e d with the data f o r 7r+ and ~- r e s p e c t i v e l y , at q~ = 0.1 G e V / c . H e r e the c a l c u l a t i o n shows s o m e a g r e e m e n t with the data only f o r high values of x. T h i s fact is not unexpected and can be u n d e r s t o o d c o n s i d e r i n g that pions p r o d u c e d in the c e n t r a l blobs of the MP chain a r e i m p o r t a n t f o r x not l a r g e ; m o r e o v e r d i f f r a c t i v e p r o c e s s e s (Deck effect) a r e e x p e c t e d [9] f r o m k i n e m a t i c s to be r e l e v a n t f o r x > 0.5. It is n e v e r t h e l e s s i n t e r e s t i n g to o b s e r v e that the c a l c u l a t e d c u r v e s p r e s e n t a s h o u l d e r s t r u c t u r e , as a r e f l e x i o n of the r e s o n a n c e s t r u c t u r e of the ~N amplitude. In o r d e r to c l a r i f y this connection we have plotted in fig. 3 as an e x a m p l e the s e p a r a t e d contributions f r o m four r e g i o n s of m* = ( s ' ) l / 2 ( c u r v e s a, b, c, d). S t r u c t u r e in the p~ d i s t r i b u t i o n is a l s o expected by Deck effect contributions, as d e s c r i b e d in ref. [10]. We b e l i e v e that a d e t a i l e d d i s c u s s i o n of such s t r u c t u r e s has to be postponed to an a n a l y s i s of the contributions coming f r o m pions all along the MP chain. $ It is well known (see for instance ref. [6]) that, in order not to violate unitarity, an intercept of the pomeron slightly smaller than 1 is required. This corresponds in our case to a slight change of the power of 1 - x in eq. (8). This would not practically affect curves 1 and 2 as drawn in fig. 2. 430
Volume 37B, n u m b e r 4
P HY SI C S L E T T ER S
13 D e c e m b e r 1971
10
ALLABY
e t al. 1971
APo-14.25 GeV/c
. . . . . . . .
Po'24.04 GeV/c
,A
P
q~ =0.1 Ge V/c
i
10 -I 10 -I
7o ~
ld; +
.%/..).i"
..... ~" '\ ":.
10
~.• --.4,.,
\
153 os~,~cco~.-o~Po
..... G~v/~ \'.
eAN DE RSON et al67 T5 rrtr Po=30 GeVi/c I 0 r"
I 0-'
0.2
O.
X
0.6
\ l
qI =0"! eev//c
T
0.8
Fig. 3. Distribution function for 1r+: Solid curve, total contribution from the g r a p h of fig. lc; dotted c u r v e s , p a r t i a l contribution for: m* < 1.35 GeV (curve a), 1.35 < rn* < 1.8 GeV (curve b), 1.8 < m * < 2.1 GeV (Curve c), 2.1 < ra* GeV (curve d).
I0 5
,
oi
L
0'.3
,
,
o5
~
o'.7
~
t ; I
o.9
X
Fig. 4. Distribution function for ~-: total contribution from the graph of fig. lc.
Finally, let us add a brief c o m m e n t on the slope B(x) as calculated from eq. (8): B(x) shows a s t r u c ture in c o r r e s p o n d e n c e with the e n h a n c e m e n t s of p~ in x. For y+ this fact has been o b s e r v e d [4] in the region of high x. In addition the calculated B(x) has an average value of 10 GeV - 2 for y+ in the region of l a r g e x ( x ~ 0.5) and i n c r e a s e s at lower v a l u e s . Enlightening d i s c u s s i o n s with J. S. Ball and K. Schllipmann are gratefully acknowledged.
R efeT"eRCFS
[1] A. Mf/ller, Phys. Rev. D2 (1970) 2963. [2] D. Amati, A. Stanghe|lini and S. Fubini, Nuovo Cimento 26 (1962) 896; L. B e r t o e c h i , S. Fubini and M . T o n i n , Nuovo Cimento 25 (1962) 626. [3] H. B~ggild, K.H. Hansen and M. Suk, Nucl. Phys. B27 (1971) 1; E. W. Anderson, E . J . B l e s e r , G.B. Collins, T. Fujii, J. Menes, F. Turkot, R. A. C a r r i g a n J r . , R. M. Edelstein, W. C. Hien, T . J . MeMahon and I. Nadelhaft, Phys. Rev. L e t t e r s 19 (1967) 198; C. W. Akerlof, D.G. C r ~ b , J. L. Day, P. Johnson, P. Kalbach, A. D. K r i s c h , M . T . Lin, M. L. M a r s h a k , J.K. Randolph, P. SehmUser, A. L. Read, K. W. Edwards, J. G. Asbury, G.J. M a r m e r and L. G. R a t h e r , Phys. Rev. D3 (1971) 645; J. V. Allaby, F. Binon, A. N. Diddens, P. Duteil, A. Klovning, R. M e u n i e r , J . P . Peigneux, E . J . S a c h a r i d i s , K. SehB/pmann, M. Spighel, J. P. Stroot, A.M. Thorndike and A.M. Wetherell, CERN 70-12 (1970). [4] J. V. Allaby, A. N. Diddens, R. W. Dobinson, A. Klovning, J. Litt, L. S. R o c h e s t e r , K. Schl~pmann, A.M. Wethe r e U , U. Amaldi, R. B i a n c a s t e l l i , C. Bosio and G. Matthiae, Intern. Conf. on E l e m e n t a r y p a r t i c l e s , A m s t e r d a m (1971). [5] B. C a r a z z a and G. M a r c h e s i n i , Phys. L e t t e r s 35B (1971) 453. [6] G. F. Chew, T. Rogers and D.R. Snider, Phys. Rev. D2 (1970) 765; G. F. Chew and D.R. Snider, Phys. Rev. D3 (1971) 420; H. D. I. A b a r b a n e l , G. F. Chew, M. L. Goldberger and L. M. Saunders, Diffractive dissociation within m u l t i p e r i p h e r a l dynamics (Princeton University p r e p r i n t - M a r c h 1971). [7] E. F e r r a r i and F. S e l l e r i , Suppl. Nuovo Cimento 24 (1962) 453. [8] G. Giacomelli, P. Pini and S. Stagni, HERA Group. A compilation of pion-nucleon s c a t t e r i n g data, CERN/HERA 69.1. [9] G. M a r c h e s i n i , Deck effect in inclusive r e a c t i o n s , L e t t e r e al Nuovo Cimento, in p r e s s . [10] R. D. P e c c e i and A. Pignotti, Phys. Rev. L e t t e r s 26 (1971) 1076; P. D. Ting and H.J. Yesian, Phys. L e t t e r s 35B (1971) 321; R. M. Edelstein, V. R~ttenberg and H. R. Rubinstein, Phys. L e t t e r s 35B (1971) 408.
431
P HY SI C S L E T T E R S
PERIPHERAL
DYNAMICS OF INCLUSIVE REACTIONS THE FRAGMENTATION REGION
13 December 1971
IN
F. DUIMIO and G. MARCHESINI Istituto di Fisica dell'UniversitY, Parma and Istituto Nazionale di Fisica Nucleate, Sezione di Milano, Italy Received 26 October 1971 Peripheral dynamics is studied for the limiting fragmentation of protons into protons and pions. The calculated distributions are discussed and compared with experimental data.
A r e m a r k a b l e flow of i n t e r e s t has been r e c e n t l y focused on the e x p e r i m e n t a l and t h e o r e t i c a l a s p e c t s of the i n c l u s i v e r e a c t i o n s a +b ~ c + anything and m o r e s p e c i f i c a l l y on the a n a l y s i s of the h i g h - e n e r g y b e h a v i o u r of the s i n g l e p a r t i c l e d i s t r i b u t i o n s , i n v a r i a n t l y defined by Oc(S,x,q2) -
2E d3ec Crab dq 3 '
(1)
w h e r e E, q a r e the energy and m o m e n t u m of p a r t i c l e c, ~ab is the total a - b c r o s s s e c t i o n and x = 2 q * / s 1-/2 is the F e y n m a n s c a l i n g v a r i a b l e . M~iller [1] p r o p o s e d to extend the ideas of R e g g e pole ex ch an g es to the study of such r e a c t i o n s . In p a r t i c u l a r , M i i l l e r ' s a n a l y s i s shows that in a s m a l l r e g i o n of the p h ase sp ace (x ~ 1, q i ~ 0) w h e r e p a r t i c l e c is r e l a t i v i s t i c with r e s p e c t to all the o t h e r outgoing p a r t i c l e s , the d i s t r i b u t i o n function Pc should exhibit a T r i p l e Regge (TR) behaviour. T he continuation of the T R p a r a m e t r i z a t i o n to a l a r g e r r e g i o n of p h as e s p a c e heavily depends on an e x t e n s i v e exploitation of the duality notion. C l e a r l y a d e t a i l e d knowledge of the m u l t i p a r t i c l e hadron r e a c t i o n s at high e n e r g y , i.e. a s a t i s f a c t o r y t h e o r y of hadron d y n a m i c s would be needed to c l a r i f y M~iller p a r a m e t r i z a t i o n and eventually its continuation, v i a duality, away f r o m the r e g i o n x ~ 1. In a b s e n c e of a c o m p l e t e t h e o r y such a n a l y s i s can be e f f e c t e d in t e r m s of one of the m o d e l s which have been p r o p o s e d in o r d e r to get s o m e insight in p r o b l e m s connected with p r o d u c t i o n phenomena. P u r p o s e of this note is a study of the d y n a m i c s of i n c l u s i v e r e a c t i o n s within the f r a m e w o r k of the m u l t i p e r i p h e r a l (MP) model of Fubini and c o l l a b o r a t o r s [2], in the f r a g m e n t a t i o n r e g i o n (x not n e a r to 0). In p a r t i c u l a r , we c o n s i d e r the r e a c t i o n s p+p -~p + anything and p+p -~ ~ + anything, f o r which ex t e n s i v e data have b e e n c o l l e c t e d [3,4]. In view of the e x p e r i m e n t a l fact that the d i s t r i b u t i o n s s e e m to be a l r e a d y l i m i t i n g at p r e s e n t e n e r g i e s [4, 5], we study the MP d i s t r i b u t i o n s in t h e i r l i m i t i n g f o r m : this g i v e s us the p o s s i b i l i t y of working with a r a t h e r s i m p l i f i e d l i m i t k i n e m a t i c s . We a r e mainly i n t e r e s t e d in the r e g i o n of l a r g e x, i.e., in t e r m s of the MP chain, we want to a n a l y s e the c o n t r i b u t i o n s of the p r o t o n and of the pion c o m i n g out f r o m the r i g h t m o s t blob. Thus we a r e led to c o n s i d e r e s s e n t i a l l y the d i a g r a m s of fig. l a and fig. lb f o r the proton d i s t r i b u t i o n pp and only the diag r a m of fig. l c f o r the pion d i s t r i b u t i o n s py+. In o r d e r to compute the co n t r i b u t i o n s of t h e s e g r a p h s to the p d i s t r i b u t i o n s , one has to sum o v e r a ll the p a r t i c l e s p r o d u c e d in the l e f t - h a n d blobs so that t h e r e is no need of s p e c i f y i n g t h e i r d e t a i l e d s t r u c t u r e . In t h i s way the p r o b l e m is e s s e n t i a l l y r e d u c e d to the e v a l u a t i o n of s i m p l e one pion exchange graphs. Since the b a s i c idea of the pion exchange m o d e l s is the d o m i n a n c e of pion p o l e s in the low m o m e n t u m t r a n s f e r r eg i o n , we e x a m i n e the k i n e m a t i c a l r e l a t i o n s f o r t in fig. l a and t' in figs. lb and l c in o r d e r to have an a p r i o r i e s t i m a t e of the p h a s e - s p a c e r e g i o n w h e r e the model is justified. T o fix the ideas l et us r e q u i r e It l < 0.5 GeV 2 and c o n s i d e r f i r s t the proton case. In fig. l a the m o m e n t u m t r a n s f e r t, in the l i m i t k i n e m a t i c s is given by
427
Volume 37B, number
4
PHYSICS
LETTERS
M M
a
\til
I. .
.
.
I.
1971
C
S"
S"
\ill
3"
\ ii <,\
/<,"
JoL
.
S
M
b
S ,r
13 D e c e m b e r
S
S
Fig. 1. Peripheral diagrams for proton fragmentation into protons (a,b) and into pions (e) in proton-proton interactions. Solid lines refer to protons and dotted lines to pions. t = - [ ( 1 - x)2m 2+q~] ~ /x,
(2)
with m = proton m a s s and q± = t r a n s v e r s e m o m e n t u m of the proton in the c e n t e r - o f - m a s s frame. It follows that the p e r i p h e r a l amplitude of fig. l a will be r e l e v a n t in the region x ~> ½ for s m a l l q2. The k i n e m a t i c s for the other two d i a g r a m s is obviously m o r e complicated. F o r t' and s' appearing in the amplitude of fig. lb one obtains, in the l i m i t s ~ ¢o, the following e x p r e s s i o n s : v
t' = t o - y
(3)
with
t o, = r t - 1 -rr
U2 '
(4)
y = ½M2(l-r)(Z-z),
(5)
and s':[1-(1
x)r]
(1-x)(1-r)
+xrn 2
+~_x[xY+7(1-r)+2q.(y(1-r))l/2
cos~]
,
(6)
where # = p i o n m a s s , r = s " / M 2, z, q~ = a n g u l a r v a r i a b l e s of q' in the m i s s i n g m a s s r e s t f r a m e (their range is over the whole sphere), and t is still given by eq. (2). F o r m u l a e (3)-(6) apply also to the case of fig. l c , ff one takes c a r e of i n t e r c h a n g i n g m and ~ and with t not given by eq. (2) but by: t = (1-x)m 2
1 - x U2 _ q-~. X
(2')
X
C l e a r l y an e s t i m a t e of the region of validity in x of the a m p l i t u d e s of fig. l b and p a r t i c u l a r l y of fig. lc is not so straightforward. However let us o b s e r v e that at qz = 0 the k i n e m a t i c a l r e l a t i o n s s i m plify; in p a r t i c u l a r one gets y = ( 1 - r ) q ~2 , i
where q . is the c e n t e r - o f - m a s s t r a n s v e r s e m o m e n t u m of q' both in figs. lh and lc. Since the p h y s i c a l t a m p l i t u d e s a r e strongly bounded to s m a l l values of q'x2, i m p o r t a n t c o n t r i b u t i o n s come only for t' ~ t o. In the case of the proton d i s t r i b u t i o n It~l < ½ GeV 2 for r not close to 1 and for x > ½. At q~_ ¢ 0 but s m a l l such as e s t i m a t e would not change radically. F o r the pion d i s t r i b u t i o n , i m p o r t a n t c o n t r i b u t i o n s to the amplitude come from a much m o r e l i m i t e d range of r , say r l e s s than 0.3-0.5 and f r o m x belonging to the full range, except obviously the region x~-.0. 428
Volume 37B, number 4
P H Y SI C S L E T T E R S
13 December 1971
In view of t h e s e g e n e r a l k i n e m a t i c a l c o n s e q u e n c e s of the pion exchange d o m i n a n c e in the model and of the fact that we d i s r e g a r d c o n t r i b u t i o n s to py f r o m pions coming out from blobs located in the middle of the MP chain, the r e l i a b i l i t y of our r e s u l t s (especially for p~) will be a d e c r e a s i n g function of 1 - x, falling to z e r o at x ~ 0, i.e. in the p i o n i z a t i o n region. P r o t o n distribution. The c o n t r i b u t i o n of the amplitude of fig. l a to the p p d i s t r i b u t i o n is calculated s u m m i n g o v e r all the p a r t i c l e s coming out f r o m the l e f t - h a n d blob into which we have collapsed the MP chain, to give the total yN c r o s s section. Its l i m i t i n g f o r m t u r n s out to be:
%-N
- t(1
-
x)
(7)
p.L.)(a _ 2~ 2 (~NN ( t - ~2)2 ' where g r is the r a t i o n a l i z e d p i o n - n u c l e o n coupling constant, t is given by eq. (2) and CyN, CNN a r e the a s y m p t o t i c pion p r o t o n and p , p c r o s s sections. F o r ayN we a s s u m e the o n - s h e l l value. H e r e and in the following we always ignore off-shell c o r r e c t i o n s as long as the i n t e g r a l s converge, in a c c o r d with the s u g g e s t i o n of Chew and c o l l a b o r a t o r s [6] that the smooth v a r i a t i o n of f o r m f a c t o r s in the region of s m a l l t (dominating the p e r i p h e r a l amplitudes) can be ignored, as long as the MP i n t e g r a l equation r e m a i n s of the F r e d h o l m type. The d i s t r i b u t i o n p ~ ) , as calculated f r o m eq. (7), is plotted in fig. 2 (curve a), together with the exp e r i m e n t a l values, ~t q . = 0.1 GeV/c It is s t r i k i n g that this c o n t r i b u t i o n for x not close to 1 is of the o r d e r of half the e x p e r i m e n t a l value.
o • • v
Po" 14.25 GeV/c] Po=19.2 GeV/c I A L L A B Y et a11971 Po=24.0 GeV/c) Po=19.2 GeV/c A L L A B Y e t a11970
10 2 4.0 Ge V/c 14.25 GeV./c~_;,j
ql 0 1 GeV/c (D
1 •
•
0 VO
I 0.2
0.I
a b /i
0.4
I X
I ;//i//"i 0.8
0.6
Fig. 2. Distribution functions for protons: curve a, contributions from the graph of fig. la; curve b, contributions from the graph of fig. lb with s' < 10 GeV2; curves 1 and 2, contributions from the graph of fig. lb with s' > 10GeV2, and X in the damping factor respectively equal to 1 and 2 GeV2. In the s a m e way, with a m o r e e l a b o r a t e calculationT one obtains the c o n t r i b u t i o n to p p f r o m the a m plitude of fig. lb:
'
/;
/
P ) _ aTrN 1 (I - x) f r d r dr' aNN 2(27r)6 0 -~o 0
dq~
(8) (t'-/~2)2
'
z* b e i n g the c o s i n e of the c e n t e r - o f - m a s s s c a t t e r i n g angle of the ITN s y s t e m in the r i g h t - h a n d blob. Isotopic spin can be taken into account by s u m m i n g over the p o s s i b l e states of the exchanged pion. In eq. (8) A1rN(S' , z*) is the e l a s t i c 7r-N amplitude. In the n u m e r i c a l evaluation of eq. (8) the data [8] for the d i f f e r e n t i a l 7TN c r o s s s e c t i o n have b e e n used. It is i n t e r e s t i n g to c o n s i d e r s e p a r a t e l y the c o n t r i b u t i o n s to p ~ ) f r o m A~RN anA~TTNAHE,with ARN The calculation has been done in the limit kinematics, and the p h a s e - s p a c e integrations have been c a r r i e d out in the missing mass r e s t f r a m e . The rather simple form of our resulting eq. (8) can be t r a c e d b a c k to limit forms of formulae in the paper of F e r r a r i and Selleri [7]. 429
Volume 37B, number 4
P HYSI CS L E T T ER S
13 December 1971
A~N(S',Z*) f o r s ' in the r e s o n a n c e r e g i o n s ' < 10 GeV 2 and AHN E = A~N(S',Z* ) f o r s ' in the h i g h - e n e r g y region, s ' > 10 GeV 2. In fig. 2 the c a l c u l a t e d contribution f r o m A R is shown in c u r v e b. Again the s i z e of this contribution is r e m a r k a b l y high. The sum of c u r v e s a and b, the solid c u r v e c in fig. 2, a p p e a r s to fit the data in a r a t h e r l a r g e r e g i o n of x. In p a r t i c u l a r we r e m a r k that the two b r o a d bumps in c u r v e b r e f l e c t i n g the f o r w a r d and backward decay of ~N r e s o n a n c e s , s e e m to fit with a s t r u c t u r e in the e x p e r i m e n t a l data. In addition, the c a l c u l a t e d slope B(x) of the pp d i s t r i b u t i o n in the q2 v a r i a b l e
B(x) = " ~2_2d lnpp(x, q2) dq ~
,.)
q~ = 0
is a r a t h e r smoothly i n c r e a s i n g function of x and f o r x > 0.5 it is of the o r d e r of 10 GeV - 2 , in f a i r a g r e e m e n t with the a v a i l a b l e e x p e r i m e n t a l i n f o r m a t i o n [3,4]. As f o r the contribution to p p f r o m the ~N amplitude A HE- the i n t e g r a t i o n o v e r t' in eq. (8) is d i v e r gent. It follows f r o m the p r e v i o u s c o n s i d e r a t i o n s that a damping f a c t o r F(t') in the integrand is now n e c e s s a r y : we a s s u m e it to be of the f o r m Fit') = (X/(X- t')) z. "b" In fig. 2 c u r v e s 1 and 2 give the c a l c u l a t e d contribution to p~ ) f o r typical v a l u e s X = 1 GeV 2 and X = 2 GeV 2 r e s p e c t i v e l y . As expected f r o m k i n e m a t i c s this contribution is i m p o r t a n t only f o r x ~ 1. Both d i f f r a c t i o n ( p o m e r o n exchange) and Regge b e h a v i o u r a r e contained in the amplitude A HE so that this c a l c u l a t e d contribution to p p can be r e p h r a s e d in t e r m s of the T r i p l e P o m e r o n (TP) and T r i p l e Regge (TR) language. At x = 1 the TR f o r m u l a e a r e finite, while the T P one, as our c a l c u l a t e d c u r v e , is d i v e r g e n t :~. B e f o r e d i s c u s s i n g the r e l a t i o n with e x p e r i m e n t at x ~ 1 one has to c o n s i d e r that the d y n a m i c s in this region is highly complex, due to e f f e c ts of d if f r a c t i o n d i s s o c i a t i o n . In p a r t i c u l a r the d i f f r a c t i v e p r o d u c tion of r e s o n a n c e s , which is not d e s c r i b e d in the p r e s e n t f o r m of the model, is r e s p o n s i b l e of the bumps at the end of the e x p e r i m e n t a l d i s t r i b u t i o n (due to r e s o n a n t excitation of the p r o j e c t i l e ) : at infinite e n e r g y they should c o n c e n t r a t e at x = 1. The " s y m m e t r i c " r e s o n a n t excitation of the t a r g e t (Deck effect), g i v e s contributions in the region x ~ 0.9 which h o w e v e r do not c o l l a p s e to one at a s y m p t o t i c ene r g i e s [9]. F r o m t h e s e o b s e r v a t i o n s it follows that a d i r e c t c o m p a r i s o n with the data in this region is at p r e s e n t infeasible. Many au t h o r s (e.g. [10]) have t r i e d to fit the data in the r e g i o n of l o w e r x by a s s u m i n g the validity of a duality extension to such r e g i o n of the TR p a r a m e t r i z a t i o n with h i g h - r a n k i n g Regge p o l e s , eventually including a l s o T P contributions. In our model this would c o r r e s p o n d to c o n s i d e r i n g the full contribution of the graph of fig. lb with the ~rN amplitude p a r a m e t r i z e d in t e r m s of p u r e R e g g e - p o l e exchanges, including diffraction. The r e s u l t i n g c u r v e would e s s e n t i a l l y coincide with the c u r v e b of fig. 2. The fact that the graph of fig. la, which can be a s s o c i a t e d with a l o w - l a y i n g pole (the pion) coupled to the p o m e r o n g i v es a s i z a b l e contribution does not s e e m to fit with an approach b a s e d on the d o m i nance of h i g h - r a n k i n g Regge p o l e s , coupled to the p o m e r o n . Pion distribution. T h e single p a r t i c l e d i s t r i b u t i o n f o r the p e r i p h e r a l am p l i t u d e of fig. l c is still give n by eq. (8), w h e r e t is defined by eq. (2') and t ' , s ' by eqs. (3)-(6), with m ~ p, and in the e l a s t i c ~rN amplitude z* has to be r e p l a c e d by - z * . Again one has to s u m o v e r the p o s s i b l e isotopic s t a t e s of the exchanged pion. In figs. 3 and 4 the c a l c u l a t e d l i m i t i n g p~ functions (solid cu r v es) a r e c o m p a r e d with the data f o r 7r+ and ~- r e s p e c t i v e l y , at q~ = 0.1 G e V / c . H e r e the c a l c u l a t i o n shows s o m e a g r e e m e n t with the data only f o r high values of x. T h i s fact is not unexpected and can be u n d e r s t o o d c o n s i d e r i n g that pions p r o d u c e d in the c e n t r a l blobs of the MP chain a r e i m p o r t a n t f o r x not l a r g e ; m o r e o v e r d i f f r a c t i v e p r o c e s s e s (Deck effect) a r e e x p e c t e d [9] f r o m k i n e m a t i c s to be r e l e v a n t f o r x > 0.5. It is n e v e r t h e l e s s i n t e r e s t i n g to o b s e r v e that the c a l c u l a t e d c u r v e s p r e s e n t a s h o u l d e r s t r u c t u r e , as a r e f l e x i o n of the r e s o n a n c e s t r u c t u r e of the ~N amplitude. In o r d e r to c l a r i f y this connection we have plotted in fig. 3 as an e x a m p l e the s e p a r a t e d contributions f r o m four r e g i o n s of m* = ( s ' ) l / 2 ( c u r v e s a, b, c, d). S t r u c t u r e in the p~ d i s t r i b u t i o n is a l s o expected by Deck effect contributions, as d e s c r i b e d in ref. [10]. We b e l i e v e that a d e t a i l e d d i s c u s s i o n of such s t r u c t u r e s has to be postponed to an a n a l y s i s of the contributions coming f r o m pions all along the MP chain. $ It is well known (see for instance ref. [6]) that, in order not to violate unitarity, an intercept of the pomeron slightly smaller than 1 is required. This corresponds in our case to a slight change of the power of 1 - x in eq. (8). This would not practically affect curves 1 and 2 as drawn in fig. 2. 430
Volume 37B, n u m b e r 4
P HY SI C S L E T T ER S
13 D e c e m b e r 1971
10
ALLABY
e t al. 1971
APo-14.25 GeV/c
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Fig. 3. Distribution function for 1r+: Solid curve, total contribution from the g r a p h of fig. lc; dotted c u r v e s , p a r t i a l contribution for: m* < 1.35 GeV (curve a), 1.35 < rn* < 1.8 GeV (curve b), 1.8 < m * < 2.1 GeV (Curve c), 2.1 < ra* GeV (curve d).
I0 5
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oi
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Finally, let us add a brief c o m m e n t on the slope B(x) as calculated from eq. (8): B(x) shows a s t r u c ture in c o r r e s p o n d e n c e with the e n h a n c e m e n t s of p~ in x. For y+ this fact has been o b s e r v e d [4] in the region of high x. In addition the calculated B(x) has an average value of 10 GeV - 2 for y+ in the region of l a r g e x ( x ~ 0.5) and i n c r e a s e s at lower v a l u e s . Enlightening d i s c u s s i o n s with J. S. Ball and K. Schllipmann are gratefully acknowledged.
R efeT"eRCFS
[1] A. Mf/ller, Phys. Rev. D2 (1970) 2963. [2] D. Amati, A. Stanghe|lini and S. Fubini, Nuovo Cimento 26 (1962) 896; L. B e r t o e c h i , S. Fubini and M . T o n i n , Nuovo Cimento 25 (1962) 626. [3] H. B~ggild, K.H. Hansen and M. Suk, Nucl. Phys. B27 (1971) 1; E. W. Anderson, E . J . B l e s e r , G.B. Collins, T. Fujii, J. Menes, F. Turkot, R. A. C a r r i g a n J r . , R. M. Edelstein, W. C. Hien, T . J . MeMahon and I. Nadelhaft, Phys. Rev. L e t t e r s 19 (1967) 198; C. W. Akerlof, D.G. C r ~ b , J. L. Day, P. Johnson, P. Kalbach, A. D. K r i s c h , M . T . Lin, M. L. M a r s h a k , J.K. Randolph, P. SehmUser, A. L. Read, K. W. Edwards, J. G. Asbury, G.J. M a r m e r and L. G. R a t h e r , Phys. Rev. D3 (1971) 645; J. V. Allaby, F. Binon, A. N. Diddens, P. Duteil, A. Klovning, R. M e u n i e r , J . P . Peigneux, E . J . S a c h a r i d i s , K. SehB/pmann, M. Spighel, J. P. Stroot, A.M. Thorndike and A.M. Wetherell, CERN 70-12 (1970). [4] J. V. Allaby, A. N. Diddens, R. W. Dobinson, A. Klovning, J. Litt, L. S. R o c h e s t e r , K. Schl~pmann, A.M. Wethe r e U , U. Amaldi, R. B i a n c a s t e l l i , C. Bosio and G. Matthiae, Intern. Conf. on E l e m e n t a r y p a r t i c l e s , A m s t e r d a m (1971). [5] B. C a r a z z a and G. M a r c h e s i n i , Phys. L e t t e r s 35B (1971) 453. [6] G. F. Chew, T. Rogers and D.R. Snider, Phys. Rev. D2 (1970) 765; G. F. Chew and D.R. Snider, Phys. Rev. D3 (1971) 420; H. D. I. A b a r b a n e l , G. F. Chew, M. L. Goldberger and L. M. Saunders, Diffractive dissociation within m u l t i p e r i p h e r a l dynamics (Princeton University p r e p r i n t - M a r c h 1971). [7] E. F e r r a r i and F. S e l l e r i , Suppl. Nuovo Cimento 24 (1962) 453. [8] G. Giacomelli, P. Pini and S. Stagni, HERA Group. A compilation of pion-nucleon s c a t t e r i n g data, CERN/HERA 69.1. [9] G. M a r c h e s i n i , Deck effect in inclusive r e a c t i o n s , L e t t e r e al Nuovo Cimento, in p r e s s . [10] R. D. P e c c e i and A. Pignotti, Phys. Rev. L e t t e r s 26 (1971) 1076; P. D. Ting and H.J. Yesian, Phys. L e t t e r s 35B (1971) 321; R. M. Edelstein, V. R~ttenberg and H. R. Rubinstein, Phys. L e t t e r s 35B (1971) 408.
431