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L]~TT]~R:E AL I~UOVO CIMENTO VOL. 11, N. 14 7 D i c e m b r e 1974
Geometrical Interpretation of Hadronic Collision at High Energy.
G. CECCHET (*), R . D O L F I N I a n d F . IMPELLIZZERI
I s t i t u t o d i F i s i c a dell ' U n i v e r s i t d - P a v i a
I s t i t u t o iVaz ionale d i F i s i c a iVucleare - Sez ione d i P a v i a
( r i c e v u t o 1'8 Lugl io 1974)
I n v i e w of a c o m p o u n d s t r u c t u r e of t h e h a d r o n two f e a t u r e s r equ i r e p a r t i c u l a r con- s i d e r a t i o n in h i g h - e n e r g y i n t e r a c t i o n processes :
i) t h e r e is no e x p e r i m e n t a l r e su l t w h i c h p r o v e s t h a t t h e poss ib le h a d r o n costi- t u e n t s (quarks , p a t t o n or w h a t e v e r } ex i s t r ea l ly in t h e f inal s t a t e ;
ii) t h e n o r m a l i z e d t opo log ica l cross-sect ions
(1) : _ ~n ~ ffn ~ n
n = final cha rged m u l t i p l i c i t y ,
a r e we l l desc r ibed b y a c o m p o u n d p r o b a b i l i t y d i s t r i b u t i o n of t h e t y p e (x.A)
(2) o'. = X P(iV) Qiv(~), iV
w h e r e P ( N ) a n d Qiv(n) a re p r o b a b i l i t y dens i ty f u n c t i o n s . W e i n t e r p r e t t h e f o r m e r as t h e p r o b a b i l i t y of p r o d u c i n g iV com pl ex c o m p o u n d s (clusters , f i reballs etc.) a n d t h e l a t t e r as t h e p r o b a b i l i t y for t h e w h o l e se t of c o m p o u n d s to decay in to n pa r t i c les .
W e assume, in a g r e e m e n t w i t h i), t h a t in a n y e l e m e n t a r y i n t e r a c t i o n t h e on ly way to p r o d u c e c o m p o u n d s is t r o u g h (~ a n n i h i l a t i o n )~ of c o s t i t u e n t s of t h e or ig ina l h a d r o n s . F o l l o w i n g t h i s hypo t he s i s , l e t b e K ( s ) t h e a v e r a g e n u m b e r of eos t i t uen t s t h a t m a y come i n t o i n t e r a c t i o n in a h a d r o n s c a t t e r i n g a t s q u a r e d c.m. ene rgy S, a n d we t a k e t h i s K as a p a r a m e t e r of t h e model .
(*) On leave of absence from the University of Pavia. (1) A. G I o v A ~ I : Nuovo Cimento, 1@A, 713 (1972). (2) ~r LE BELLAC, J. L. MEUI~IER and G. PT.AUT: Nice University prcprint NTH 73/3. (8) D. LEVY: Nucl. Phys., 59 B, 583 (1973). (4) H. B. NIELSE~ and P. OLESE~: Phys. Left., 43 B, 37 (1973).
622
GEOMETRICAL INTERPRETATION OF HADRONIC COLLISION AT HIGH ENERGY 6 2 ~
If p is the probabil i ty for an annihi la t ion into a compound, the probabil i ty to pro- duce N compounds is expressed by the binomial formula
(3) P ( N ) = ( K N ) p z ~ ( 1 - - p ) x -~ ,
where clearly the number of produced compounds N is never bigger than the number K of the interacting components.
If we assume the usual hypothesis tha t the number of decay products n i of each compound is Poisson distributed around a mean number /~, i.e.
(4) Q ~ ( n ~ ) = nil
and sum over i, equations (3) and (4) give the following probabil i ty distribution ( a = number of ini t ial charged particles):
(5) , & (Nt,) ~-~/K\
which has the general form of equation (2).
I 0.8
0./-.
0 10 ~
t a)
, , , . . . . I , , , , , J J , I ,
+ o,
c)
14
K12
10
8
I I I I I t l 6
4
2
L I I I I I J 0
10 3 10 0 PL~(GeV/c)
101 10 2 101 10 2 10 3
Fig. 1. - a) :~[ea]x n u m b e r /t of produced" par t ic les pe r compound as a func t ion of incident m o m e n t u m for react ions pp, p~, K~p, n~p (ref. (5)). b) P robab i l i t y p of in te rac t ion b e tween e l emen ta ry compo- nents as a func t ion of inc ident m o m e n t u m for reac t ions pp, pp, K ' p , ~• (ref. Q)). c) N u m b e r K of inter- ac t ing components as a funct ion of incident m o m e n t u m for react ions pp, pp, K~p , n~p (ref. (~)). �9 n+, �9 7v-, �9 K-, �9 K +, �9 p, o p.
(~) a) J . W. CHAPMAN, N. GREE2q, B. P. ROE, A. A. SEIDL, D. SINCLAIR, J . C. VANDERVELDE, C. ~V[. BROMBERG, D. COtIE~, W. FERBEL, P. SLATTERY, S. STo-%rE and B. WERkNrER: Phy8. Rev. Left., 29, 1686 (1972); b) G. CI-IARLTO~, Y. CHO, M. DERRICK, R. ENGELMA~, T. FIELDS, L. HYMA~, K. JAEGER, V. ~VIEttTA2qI, B. ~/[IYSGRAVE, Y. OREN, D. RttII~-ES, P. SCHnE1NER, H. YUTA, L. VOYVODICH, ]=~. WALKER, J . ~VHITMORE, t I . B. CRAWLEY, Z. MI~G and R. G. GLASSER: Phys. Rev. Letf., 29, 515 (1972); c) F. T. DAO, D. GORDOn, J . LACtt, E. ~I&LA3IUD, T. MEYER, R. POSTER an d W. SLATER: Phys. Rev., Left.,
6 2 ~ G. CECCH]~T~ R. DOLFINI a n d F. IMPELLIZZERI
Another way to express formula (5) is
(5% a, - - - (N"-~)o, (n--a)'
where (N~-a}a represents the (n - -~ ) - th moment of the distribution
(6) G(N) = F(K + 1)
F(N+ 1)F(K--N + 1) (p exp [--,u])N (1 _p)~-~v.
We take the first three moments of the experimental multiplicity distributions as estimators of the theoretical parameters K, p, #, whose behaviour as a function of the incident momentum P~:b is given in Fig. 1.
The behaviour of the parameter p does not show anything surprising, while the mean number /~ of the charged particles produced in a single compound decay is definitely constant (~ 3). This fact seems to agree with what is expected from POKOl~Sl~I and VA~ HOVE (6) if we interpret # as the proper average number of charged particles in a cluster. The third parameter K shows the remarkable characteristic of a steep raise at the highest analysed energy. Predictions of formula (5) are compared in Fig. 2 with
Dxo, D. GORDON, J . LACrI, E. MALAMITD, T. MEYER, R. POSTER an d W. SLATER: Phys . Rev . Le f t . , 29, 1627 (1972); d) NSRDS, NN and ND interact ions, a compi la t ion L B L 58 (May 1972); e) J . BARTKE: N u c l . P h y s . , 82, 673 (1966); ]) C E R N H E R A 72/1 (May 1972); g) V. V. AMMOSOV, J . P. BATO2r P. BEIL- LIERE, P. BOSETTI, V. A. BUMAZSHNOV, • . CZEJTHEY-BA~TI~, D. P. DALL~IAN, A. I)AUDIN, B. DELER, N. G. DEMIDOV, T. W. DOMBECK, P . DUINKER, P. F. ERMOLOV, A. B. FENYUK, P . A. GORICHEV, F. GRARD, H. GRASSLER, S. A. GUMENYUK, PH. HERQUET, G. KELLNER, J . t~ESTEMAN, E. P. KISTENEV, W. KITTEL, V. M. KUBIK, D. KITHN, K. LANIUS, J . LAITRENT, A. MEYER, B. A. MANYUKOV, A. M. MOISEEV, D. R. O. MORRISON, L. MOSCA, M. NEVEU, V. BE PEREVOZCttIKOV, R. POSE, J . SCHLESINOER, R. ~L SULYAEV, P. R. THORNTON, F. TRIANTIS, F. VERBEURE, A. P. VOROBJEV an d R. WINDMOLDERS: Y u c l . P h y s . , S8 B, 77 (1973); h) 0 . BALEA, V. BOLDEA, S. FELEA, T. PONTA, L. JENIK, D. KISS, S. KRASZNOVSK]i', G. PINTER, B~. POSCH, L. ANIOLA, J . BARTKE, K. ESKREYS, S. KOWALCZYK, A. KWlATKOWSKA, W. ZIELIlqSKI, A. ABURAKHIMOV, N. ANGELOV, V. BELYAKOV, S. ~I. ELISEEY, N. FADEEV, V. GRISHIN, SIt. INOGAMOV, G. JANCSO, W. KANAREK, E. KLADNITSKAYA, J . KOHLY, A. KUZNETSOV, V. LYUVIMOV, N. B[EL~'IKOVA, V. POPOVA, M. SABAU, L. SMIRNOVA, M. SOLOVIEV, KIt . SUPICHAKOV, YU. TEVZADZE, K. VISHNEVSKAYA, B. YULDASHEV, NGUEN ]:)IN TY, V. AMTVIOSOV, I. DANILCHENKO, P. ERMOLOV, A. FENYUK, L. GERDYUKOV, P. GORICHEV, E. KISTENEV, S. KLIMENKO, E. KUZNETSOV, B. ]~[ANYUKOV, V. MAKEEY, S. PARSttIKURA, A. RYBIN, E. SHCHERBAKOV, A. VOROBYOV, P. KERACHEV, V. PENEV, KH. SEMERDJIEV, A. SHKLOVSKAYA, K. ABDULLAEVA, M. AZIMOVA, S. BATRAEV, K. IGAMBERDIEV, KIt . RIZAEV, E. TRUNOVA, T. USMANOV, A. YULDASHEV, S. L. LUTPULLAEV, L. ABESALASHVILI, N. AMA- GLOBELI, M. CHARGEIStIVILI, M. DASAEVA, D. GERSAMIA, C. KHOMASURIDZE, T. KYACHADZE, R. SALU- KVADZE, O. SHAKULASHVILI, I. TULIANI, C. BAATAR, B. CHADRAA, D. TUVDENDORZH, M. BARDADIN- OTWINOWSKA, A. BICZEL, J. GAJEWSKI, R. GOKIELI, M. GORSKI, S. OTWINOWSKI, H. PIOTROWSKA, R. SOSNOWSKI and W. WSJCIK: N u c l . Phys . , 52 B, 414 (1973); i) C E R N H E R A 72/2, (Oct. 1972); k) BONN-HAMBURG-BI['TNCHEN COLLABORATION: Mult ip l ic i t ies , topological cross-sections and s ingle-par t ic le inc lus ive dis tr ibut ions ]rom pp in terac t ions at 12 and 24 GeV/c, paper r u b m i t t e d to the I V In t e rna t i o na l Sy,~posiura on fVIultiparticle Hadrodynamics , Pavia , 1973; l) V. V. AMMOSOV, V. N. BOITSOV, P. F. ER~IOLOV, A. B. FENYUK, P. A. GORICHEV, E. P. KISTENEV, S. V. KLIMENKO, B. A. MANYI/KOV, A. M. ~[OISEEV, t%. )~. SIYLYAEV, S. V. TARASEVICH, A. P. VOROBSEV, I{. BLUME~FELD, $. DERI~E, P . GRA~N'ET, M. A. JABIOL A. LEVEQIYE, ~ . LORET, E. PAIYLI, Y. PONS, J . PREVOST, S. C. SCHEVER, ~VL BORATAV, J . LABERRIGUE, H. K. NGIYYElq a nd S. ORESTEIN: P h y s . Le f t . , ~ B, 519 (1972); m) D. BOGERT, R. HA2~:FT, F. R. I{USON', D. LJUNG, C. PASOAIYD, S. PRUSS, W. M. S:~IART, G. S. ABRAMS, H . H. BINGASI, D. M. CHEW, B. Y. DAUGERAS, W. B. FRETTER, C. E. FRIEDBERG, G. GOLDHABER, W. R. GRAVES, A. I). JOttNSON, J . A. KADYK, L. STU'I"I'E, G. H. TRILLING, F. C. ~rINKELSIANN and G. P. YOST: preprint N A L - P u b - 7 3 / 5 7 - E X P 7200.137 (1973); n) F. T. DAO and ft. LATH: S t u d y of charged m u l t i p l i c i t y and n~utral decay particles ]rom 14.75 GeV/c pp interact ions, paper s u b m i t t e d to the I I A ix -en-Provence In ternat ional Con/erence on E l e m e n t a r y Par t i c l e s (1973). (6) S. POKORSKI and L. VAN HOVE: C E R N repor t T H 1772 (1973).
G E O M E T R I C A L I N T E I ~ P R E T A T I O N O F H A D R O N I C C O L L I S I O N A T H I G I t E N E R G Y ~ 2 ~
the experimentM mult ipl ic i ty dis tr ibut ions; in general the agreement grows with energy and is not sat isfactory up to 100 GeV/c of incident momentum. I t follows that the values calculated for p, K and /~ below this energy may be meaningless.
12.00
8.00
4.00
E ~o
9.60
6.40
3.20
I
a) \\ b)
6.40
3.20
f \ / \
kz
i
4 8 12 16
I I
/ \ z \
i i i i i I i i ~ y ~ -
4 8 12 16 20
4 8
I \ / ~
, \ 12 16
e)
c)
Z . . . . . . ,~-~-..z ~
4 8 12 16
I I
/1 n
I \
I II . . . . . . \\i~Z~.z---~,...__ 8 16 24
I I f )
x l I\~
, / \ ~ z
i i l i i i i i ~ . . . .
8 16 24. 32 /')
Fig, 2. - E x p e r i m e n t a l topological cross-section for pp in terac t ions f r o m 50 to 500 GeV/c of incident m o m e n t u m c o m p a r e d wi th the Poisson 4 is t r ibu t ion (dashed line) and our theore t i ca l predic t ion (5) (continuous line), a) 50 GcV/c, b) 69 GeV/c, e) 102 GeV/c, d) 200 GcV/c, e) 303 GeV/c, ]) 400 GeV/c.
3.0
.;t
2.0
1.0
/,
I J I i I I i I
10 ~
4§ 4
+
+++ §
?
+
0 i i i i i i i i l l i I I I i ~1
10 0 '~0 2 10 3
PLab (GeV/c)
Fig. 3. - Mean n u m b e r ). of compounds as a func t ion of incident m o m e n t u m for react ions pp, pp, K• ~ p (rcf. (5)). �9 ~p, �9 ~:-p, * K - p , �9 pp, o 7~+p, A E+p.
6 2 6 G. CECCttET, R. DOLFINI and F. IMPELLIZZEI~I
I t should be po in ted out t h a t t he mean number of compounds ~ = pK grows wi th energy more s lowly than t h e ave rage charged mul t ip l ic i ty , as one can see in Fig. 3.
I n conclusion we th ink t h a t an in te res t ing fea tu re of th is mode l is the ab i l i ty of ex t r ac t i ng parameters , l ike K, wh ich in pr incip le can descr ibe t he s t ruc ture of t he hadron .
As a m a t t e r of fact , any h in t on the avai lable da t a could be too p rematu re , a l t hough few m o r e po in t s a t energies above 400 GeV should be able to give us a s ta r t ing po in t for a t e n t a t i v e descr ipt ion of t h e hadron s t ructure .
W e should l ike to t hank A. G~OVANNINI and S. RATTI for the i r he lpful cr i t ic ism and suppor t .
