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Page 1: A universal scaling function for hadron-hadron interactions

IL NUOV0 CIMENTO VoL. 55 A, N. 4 21 Febbraio 1980

A Universal Scaling Function for Hadron-Hadron Interactions (*)(**).

A. D ' I ~ o c E ~ z o , G. I~G~OSSO and P. ]~0TELLI

Istituto di Fisiea dell' Universit~ - ~ecce

(ricevuto il 21 Novembre 1979)

Summary. - - In a recent two-component fireball model applied to pp multiplicity data the central fireball was expressed as a convolution integral over a KN0 extrapolation of p~ annihilation with a weight func- tion F(z). This analysis is extended here to nip and K~p for machine energies above 10 GeV/c. Scaling functions F(z) for each class of these processes are obtained and their similarity to that for pp inteIactions is noted. A single scaling function is capable of describing satisfactorily all the experiments treated independently of their energy and of the nature of the incoming hadrons.

1 . - I n t r o d u c t i o n .

Hadron-hadron interactions appear intractable unless they are first divided

into subclasses. Thus, apar t f rom the separation based upon the nature of

the incoming particles, it is convenient to distinguish between elastic and

inelastic interactions. These are related by uni tar i ty , but they exhibit quite

distinct characteristics. This distinction leaves diffractive inelastic interac-

tions in an ambiguous situation. Indeed, the so-called classical two-component

multiplici ty models (1) fur ther separate inelastic interactions into diffractive

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (**) Work supported in part by INFN. (1) K. FIALKOWSKI: Phys. ~elt. B, 41, 379 0972); L. VAN Hove: Phys. ~ett. B, 43, 65 (1973); K. FIALKOWSKI and It. I. MI~TTI~]~: .Phys. ~ett. B, 43, 61 (1973); H. HAR~I and E. RABIUOVICI: Phys. Le~t. B, 43, 49 (1973); J. LACtI and E. I~ALA~UD: Phys. ~ett. B, 44, 474 (1973); A. W~OBLEWSKI: Aeta Phys. Pol. B, 4, 857 (1973).

417

Page 2: A universal scaling function for hadron-hadron interactions

418 A. D~II~NOCF, NZO, G. II~GROSSO and P. ROT:ELLI

and nondiffractive parts . Another subdivision occurs in hadron-an t ihadron

interactions, the separation between annihilat ion and nonannihilation. This

process of subdivision need not stop here, one can continue unt i l one arrives a t a class of processes which is interpretable in terms of a set of simple Feymnan-

like diagrams. We wish to l imit ourselves, for the moment , to the four major subdivisions

made above. We shall interrelate them in a way which introduces a simplicity and universal i ty in all s t rong-interact ion physics. To describe our model, we employ the terminology fireballs (*). All strong hadron-hadron interactions are envisioned as occurring in one or other of the three categories indicated in fig. 1-3.

These diagrams are to be in terpre ted as probabi l i ty diagrams. Figure 1 represents annihilation. Figure 2 represents, in first approximation, diffractive scattering (including the elastic term) (~.3). Figure 3 represents the dominant

Fig. 1. - One-fireball diagram (annihilation).

LF

LF

Fig. 2. - Two-fireball diagram (including elastic scattering).

pa r t of high-energy inelastic interactions. I t is characterized by the presence

of a central fireball (CF) in which most of the outgoing mult ipl ici ty is created.

I t is na tura l to assume tha t the so-called leading fireballs (LF) in this diagram

have the same mult ipl ici ty s t ructure as those in fig. 2.

The leading fireballs are called so because they give rise to the leading-

part icle effect in inelastic interactions. This effect occurs, in our model, because

(*) This terminology may become superfluous, once detailed models of the various types of fireballs become available. (~) A. D'INNoC~.NZO, G. INGROSSO and P. ROT~.LLI: N•ovo Cimento A, 44, 375 (1978). (3) A. D'INNOC~NZO, G. INGROSSO and P. ROTELLI: NCWVO Cimento A, 50, 475 (1979). (4) G. I~G~osso and P. ROT~.LLI: Nuovo Cimento A, 41, 233 (1977).

Page 3: A universal scaling function for hadron-hadron interactions

A UNIVERSAL SCALING FUNCTION FOR HADRON-HADRON INTERACTIONS 419

LF

q

~ 1 c i

(

i.F

Fig. 3. - Three-fireball diagram (dominating high-energy inelastic interactions).

in the majority of interactions the LF is synonymous with single outgoing elementary particle. This fact, coupled to the kinematic feature that the LFs carry away, on the average, the major part of the incoming momentum in the 3FB diagram, reproduces the leading-particle effect (4). This necessi- tates the inclusion of the elastic term as the dominant part of fig. 2. The exchange legs in fig. 2 have yet to be identified, but it is natural to use Regge language, since we are dealing here with two-body or quasi-two-body processes. Thus we expect the Regge amplitude corresponding to this diagram to be dominated by pomeron exchange at high energy. However, it is to be noted that this diagram also includes a nondiffractive inelastic part, nor is the dif- fractive inelastic part necessarily limited to the two-firebaU term. In multi- plicity studies of processes such as pp, in which annihilation is absent, our model reduces to a new type of two-component model

The most important of our promised simplifications occurs with the identi- fication of the CF as an average over the single-fireball structure of fig. 1 (an- nihilation). For our present purposes, fig. 1 will be synonymous with p~ anni- hilation (5). More precisely, we shall use a Koba, Nielsen and Olesen (6) fit to p~ annihilation (extrapolated both below the p~ threshold and above the existing annihilation data) as the input for the afore-mentioned averaging in the CF. The above hypothesis assumes that the CF is factorizable. Con- sequently, it links annihilation and nonannihflation inelastic interactions. Our model also relates fragmentation with two-body and quasi-two-body interac- tions, and links elastic and inelastic interactions in a way distinct from uni- rarity. An unexpected unification is the possibility of fitting satisfactorily

(5) A. D'INNOCENZO, G. INGROSSO and P. ROTELLI: ~ett. _7~',~ovo Gimento, 25, 393 (1979) (e) Z. KOBA, H. B. NIELSEN and P. OLESEN: Nu¢I. Phys. B, 40, 317 (1972); J. G. RUSHBROOKE and B. R. W ~ B ~ R : Phys. Rep., 44, No. 1 (1978).

Page 4: A universal scaling function for hadron-hadron interactions

~20 A. D'I I~NOCENZO, G. INGROSSO and P. ROTELLI

all the experiments available on 7:±p~ K+p and pp multiplicities f rom 10 GeV/c

up to 405 GeV/c (including a pp exper iment a t 60 GeV/c previously neglected

by us) with the use of a universal weight funct ion F(z). The small bu t systematic mult iplici ty differences between the CF of pp and 7:p interactions at the same

incoming s-value (found in our previous exp-Gauss model) are accounted

for by the dependence of the scaling variable z upon the masses of the incoming

particles.

2. - The t h r e e - c o m p o n e n t m o d e l .

In this section we wish to describe in some detail the s t ructure of our model and in par t icular the simplifying assumptions we made. We shall, for example,

discuss the significance of ba ryon exchange in the 2FB diagram for 7:-p, not

because it influences significantly our fits, bu t because it is per t inent to the

question of whether this process contains an (( annihilation ~ mode. We can divide our discussion into three parts , each par t under the subti t le of one of

the three components of the model.

The single-fireball annihilation. In processes such us p~ the pract ical defini- t ion of annihilat ion is the absence of baryons in the final state (*). Nevertheless, the number of direct measurements of annihilation above a few GeV/c is very limited. W ha t is more, experimenters often measure only the outgoing pionic channels.

I n our model, annihilat ion can formally be defined as those contributions in which a L F is absent. However , it is to be noted tha t for pp interactions

baryon exchange in fig. 2 also leads to annihilation (outgoing mesonic) final states. Fur thermore , in these cases the (( LFs ~) would no longer be dominated by single-particle production, since each ver tex represents off-mass-shell baryon- an t ibaryon annihilat ion (fig. 4). We shall accordingly assign this class of pro-

_ ~ - - ~ mesons

~ = ~ - = ~ mesons

Fig. 4. - A contribution to pp annihilation with two off-mass-shell annihilation vertices. This term is associated to fig. 1 and not to fig. 2 where only meson exchange legs are considered.

(*) In theory, annihilation can also produce outgoing baryons-antibaryons as long as these are disconnected from the incoming baryon-antibaxyon.

Page 5: A universal scaling function for hadron-hadron interactions

A U N I V E R S A L S C A L I N G F U N C T I O N F O R H A D R O N - H A D R O N I N T E R A C T I O N S 421

cesses (baryon exchange) to the one-fireball term, fig. 1. In practice, this assign- ment is equivalent to limiting the exchanges in fig. 2 and 3 to meson l~egge poles (including the pomeron) and the redefinition of the corresponding LF mul- tiplicity structure. In this logic, we must, for consistency, conclude that baryon exchange in r~(K)p interactions is also to be collocated in a single-fireball diagram; even though these diagrams describe backward elastic scattering and its generalization. In the following fits these contributions of fig. 1 to ~(K)p interactions, and thus their backward scattering, will be neglected.

The two-]ireball term. Annihilation as we have defined it above is not to be confused with (~ charge annihilation ~. A charged mesonic ]~egge exchange in fig. 2 can result in a charge annihilation such as 7:-1) ~> ~:°7:°n, without destroying the LF structure involved. In a past analysis (3) it was pointed out that the main contributions to charge ~nnihil~tion come from fig. 3. Thus, without excessive loss of generality, we can consider only neutral exchanges in fig. 2. From previous studies we know that the three-fireball diagram dominates inelastic interactions at high energy. The inelastic part of fig. 2 must, there- fore, be small. Consequently, the lowest (single particle) multiplicity channel dominates the multiplicity structure of the LF's. We parametrize this mul- tiplicity structure with a single-parameter exponential form in the number of extra charged particles created:

(])

The index on the parameter a~ distinguishes between proton LF, pion LF etc. The proportionality factor is determined by the normalization condition

LF LF P~+I : 1. The 1 in the subscript of P~+l represents the charge of the

l = 0

incoming particle, which exists for all the processes considered in this paper. In practice only the first few terms in the power expansion of eq. (1) are im- portant, so that the ad hoc choice of an exponential form may approximate well enough a large range of more complex probability structures.

The three-]ireball term. In these interactions (fig. 3) the creation of hadrons occurs in three distinct regions, each represented by one of the fireballs (as- sumed faetorizable). I t is natural to call these regions the target fragmentation, projectile fragmentation and central regions, even if these names are normally associated with over-simplified kinematic regions. The overall 3FB multi- plicity structure is a double sum over the product of the multiplicity structures of the three fireballs:

(2) P~F~ ~ ~ P ~ / ' ~ . n = 2 ~+1 2 m + l

l=O m=O

cv c pc~ ~1 c+d_-- c_d+)] • [ P ~ _ ~ _ ~ , ( +d_ + e_d+) ÷ , , _ ~ , , , _ ~ , _ ~ , -

Page 6: A universal scaling function for hadron-hadron interactions

~2 9. A. D'INNOCENZ0, G. INGROSSO aI ld P. ROTELLI

where we assume (for simplicity) the same LF structure as that in the 2FB term. The probabilities e+, d etc. refer to the probability that the corresponding + -- charge is exchanged as indicated in fig. 3. I t is convenient to define

=_ c+d_ ~- c_d+. The CF term in eq. (2) is written as the sum of two terms. One corresponds to those events in which the incoming charge to the CI~ is plus-minus and may, therefore, annihilate. These CF terms are directly sub- stituted by the corresponding multiplicity structure for p~ annihilation. All the other 3FB events give rise to incoming CF charges which must necessarily be conserved in the CF interaction. These charges are simply added to the multi- plicity structure which would occur in neutral-neutral strong annihilations. Since we have no direct experimental information on this latter class of anni- hilations, we shall assume that they are identical to p~ annihilations. The first term in square brackets in eq. (2) is, in our model, the only source for a neutral outgoing mode in processes such as 7:-13 interactions, since we have neglected charge exchanges in the 2FB term. The value of ~ is thus uniquely determined by the neutral topological ao mode.

The expression for a general (three component) process is given by

(3) P . ( 1 - f l ) t ' [ ~ + ° " * " ~ = ap.t',, -[- fl(1 -- o:).P~ ~B,

where ~ and fl are energy-dependent weights and where 1 - fl----a~nn/ato t. For two-component fits, where annihilation is absent or neglected, eq. (3)

reduces to

(4) P~ = ~ ' ~ + (1 - ~ ) P ~ .

This is the structure we use in this paper with ~ ~ a.:/ato ~.

3. - A n n i h i l a t i o n and the central f ireball .

The model, as presented up to this point, needs only an explicit input for the multiplicity structure of the single fireball (fig. 1) and of the central fireball (fig. 3) to be complete. In an earlier work the authors used a Gaussian multi- plicity structure for the central fireball (~.3). From a consideration of fig. 1-3 it appears legitimate to ask whether the 1FB and the CF are not, in some sense, equivalent. Both involve bell-like multiplicity structures, and in both the incoming particles (~ annihilate ~. This equivalence was proposed in a recent paper and applied successfully to pp multiplicity data (5). The P ~ , as it appears in eq. (2), is an average over PC.r(so) with different incoming s° energies (centre-of-mass energy squared of the CF). Thus, at a given s value, the cor- responding average so value (~o) will be much smaller. The averaging over s. also results in a broadening of the underlying multiplicity curve, i.e. an increase in the dispersion of the CF (and hence of the inelastic data) compared to p~ an-

Page 7: A universal scaling function for hadron-hadron interactions

A UNIVERSAL SCALING FUNCTION FOR HADRON-HADRON INTERACTIONS ~ S

nihilation. As in our previous works we neglect the possible k inemat ic limi-

ta t ions t ha t the CF energy imposes on the LFs. Thus the mul t ip l ic i ty s t ruc ture

of the L F will be assumed to be the same, indipendent ly of the energy of the LF. This simplification is obviously incorrect in the ex t reme l imit of low L F

energy (mass).

I t is essential for the above hypothesis to be val id t h a t the annihi lat ion

mul t ip l ic i ty da ta be narrower t h a n the inelastic da ta a t a corresponding

(average n) value. Expe r imen ta l ly this is so, as demons t ra t ed b y the K N O

fits to pp and p~ annihi lat ion shown by SALAVA and ~n~hK (7).

Nevertheless , the identification :[FB = CF at the same ]ixed energy value is still nontrivial . I ndeed the two-par t ic le m o m e n t u m correlat ion functions

of annihi lat ion and nonannihi la t ion are ve ry different (s), but , ut least quali ta-

t ively, these differences are ascribable to the effect of the LFs in nonannihflat ion

interactions. Thus e n e r g y - m o m e n t u m (long-range) correlations are more im-

p o r t a n t in annihi lat ion processes for outgoing pion pairs t han in, say, pp in-

teract ions.

The result of this in te rpre ta t ion of p~F is t ha t in eq. (2) we mus t employ

(5) 1

P[~ = f~ . ( z , s)~7(so) dz , o

where F,~(z, s) is a weight function, which in general depends on two variables

and wi th i, j to indicate the incoming particles. The var iable z is defined so

as to run f rom 0 to 1 when so runs over the k inemat iea l ly avai lable range

f rom #5 to (V~ -- ml - - m~) ~, z ---- (s c - - #2)/(s m -- #2), #~<So~Smp, where s m is the m a x i m u m CF s-value s m = (E .... - - 2 m , ) ~.

I n this range # represents the (~ effective ,) mass of the part icles produced

in the CF (an average over s table mesons and resonances) and m~ and m~ the

effective Lie masses. I n our fits we have al lowed/~ to be a free var iable and

set m~ and m2 equal to the incoming-par t ic le m~sses. This procedure is the

s implest which allows us to t ake into account the i m p o r t a n t fea ture of resonance

product ion in the CF (9).

(v) J. SALAVA and V. SIMS.K: Natl. Phys. B, 69, 15 (1974). (s) L. MONTANET: Symposium on Antinueleon-Nucleon Interactions, .Liblice-Prague, June 25-28, 1974, CERN 74-18. (9) See references in (2); R. RAJA, C. T. MURPHY, L. VOYVODIC, R. E. ANSORGE, C. P. BUST, J. R. CARTER, W. W. NEALE, J. G. RUSHBROOKE, D. R. WARD, B. Y. OH, M. PRATAP, G. A. SMITH and J. WHITMORE: Phys. l~ev. D, 16, 2733 (1977); H. KIRK, G. RUDOLPH, lJ. KRIEGEL, H. VOGT, K. B()CKMANN, H. G. ZOBERNIG, H. BOHR, V. T. CoccoNI, M. J. COUNIHAN, P. K. ~[ALHOTRA, D. R. 0. MORR.ISON, H. SAARIKKO, P. SCHMID, D. KISIELEWSKA, K. W. J. BARNHAM, n . M. EASON, F. MANDL, M. MAR- KYTAN, K. DOHOBA and A. PARA: Nuel. Phys. B, 128, 397 (1977); M. MARKYTAN, P. JOHNSON, P. MASON, H. MUIRREAD. G. PATEL, G. WARREN, G. EKSPONG, S. O. HOLM-

Page 8: A universal scaling function for hadron-hadron interactions

~2~ h. D'INNOCENZO, G. INGROSSO &lid P. ROTELLI

At this point we have made the hypothesis t ha t F~j scales, i.e. F~(z, s) =

= F~j(z). The va l id i ty of this hypothes is will be confirmed in this work. Fur-

thermore , our conclusion will pe rm i t us to drop even the i, j indices of F~j.

Thus u l t imate ly we shall proc la im

(6) 1

P~ = f F(z) P:~'(So) az. o

ann S For _P~ (o) we have used a K I l O fit to p~ mul t ip l ic i ty da ta made in the

avai lable range s = (4 -- 19) (GeV/c)2:

(7) ~ P ~ ( s o ) = (a ~- bx) exp [-- (x - - xo)2/22 ~]

0

for so> (n#) 2 ,

for so< (n/~) ~ ,

wi th a - ~ 1 . 3 0 , b = 0 . 9 8 , Xo----0.91, 2~ 2 - - 0 . 2 7 , where x = n / ~ ( s o ) and the

value of ~(so) has been pa ramet r i zed b y

(s) n(so) = Bs~

with B = 2.2 (GeV/c) -~ and fi----0.29. These values give the best fit to the

l imited mul t ip l ic i ty da ta (fig. 5), bu t wi th one point (the arrowed one) excluded.

6.0

fi

5.0

4..0

3.0

2.0 i i 1'0 I 2 5 20 s

50

Fig. 5. - The average number of particles in annihilation vs. s. The solid curve is the best fit (excluding the arrowed point) quoted in the text.

oR~x, S. NILSSON, R. STE~]~ACKA and CII. WALCK: Nucl. Phys. B, 143, 263 (1978); H. KICHIMI, 1~. FUKAWA, S. KABE, F. OCIIIAI, R. SUGAHARA, A. SUZUKI, Y. ~OSHI- MURA, K. TAKAHASHI, T. OKUSAWA, K. TANAHASHI, M. TERANAKA, O. KUSUMOTO, T~ KONISttI, H. OKABt~ and J. YOKOTA: Lett. Nuovo Cimento, 24, 129 (1979).

Page 9: A universal scaling function for hadron-hadron interactions

A U N I V E R S A L S C A L I N G F U N C T I O N F O R H A D R O N - H A D R O N I N T E R A C T I O N S 4 2 5

Our K N O fit to p~ annih i la t ion mul t ip l i c i ty is d i sp layed in fig. 6. The la rge

uncer ta in t ies in the n ~ 0 mode (x = 0) exhibi t expl ic i t ly the dominance of

this mode by nonann ih i l a t i on channels . The fit exh ib i t ed in fig. 6 has a <~ tes t ,~

2.41

2.C

"i.6

1.2

0.8

O.Z,

Y [ I I I 0 O,~- 0.8 1.2 1.6 2.0 2.4 2.8

n/fi

Fig. 6. - The KNO plot for Dp annihilation data up to 9.1 GeV/c. The solid curve is the best fit (arrowed point excluded) quoted in the text.

va lue T = ~ Z2/NDF = 1.16. This n u m b e r sets a s t a n d a r d for our mul t i -

p l ic i ty fits. I t is to be no t ed t ha t , while the pD d a t a used run over a na r row

range , we emp loy the K ~ O fit for s t va lues up to ~ 800 (GeV/c) ~. This is an

enormous ex t r apo la t ion , g iven the u n c e r t a i n t y in the va l id i ty of K • O scaling

over such a vas t ene rgy range. Our jus t i f ica t ion is t h a t we have no a l t e rna t ive

avai lable.

4 . - F i t s to t h e data .

Our fenomenologica l fits begin wi th a conf i rmat ion of the scaling func t ion

p rev ious ly ob ta ined for pp in te rac t ions (10). I n our p rev ious pape r (5) we con-

(lo) For experimental pp multiplicity data see references in (2).

2 8 - I I Nuovo Cimento A.

Page 10: A universal scaling function for hadron-hadron interactions

426 A. D'INNOCENZ0, G. INGROSSO and P. ROTELLI

eluded tha t

(9) Fpp(z) -~ ( 1 - z) exp [ - -3 .85 v ~ ] and /~ ~- 0.43 GeV/c ~ .

I n fig. 7 this scaling funct ion has been tes ted with the pp mul t ip l ic i ty struc-

ture a t 60 GeV/c (11). The fit is per fec t ly acceptable wi th T --~ 0.40. I t is to

i 0 - ~ _

10 -2 _

10 --3 _ _

10-4i I I I I I I I z, 6 8 10 12 14 16 18

D

Fig. 7 . - Total multiplicity probabilities plotted against n for pp interactions a~ 60 GeV/c. The stars are the best-fit values using the Fpp(z) obtained from other pp interactions in ref. (5).

be r emembered t h a t a confronta t ion of the K N O curve for pp~ ~:p and K p mul-

t ipl ic i ty da ta suggests the need of a scale factor--~ 2. Consequently, we con- f irm our self-imposed l imit (") for an acceptable fit of T < 4 .

(zx) C. BROMBERG, T. FERBEL, T. J~NSEN and P. SLATTERY: Phys. Rev. D, 15, 64 (1977).

Page 11: A universal scaling function for hadron-hadron interactions

A UNIVERSAL SCALING FUNCTION FOR H A D R O N - H A D R O N INTERACTIONS 427

Our sea rch for t h e ~:~p a n d K ± p sca l ing f u n c t i o n s was l i m i t e d to t he func-

t i o n a l f o r m p r e d i l e c t e d b y t h e p p d a t a , i.e.

(10) F~(z) oc (1 - - z) ~'J exp [ - - 7~jV/z] .

The b e s t f i ts to 7:~p a n d K i p d a t a (1~) a re shown in t a b l e s I a n d I I . The p p fi ts

a r e r e t a b u l a t e d in t a b l e I I I .

TABLE I. -- Best-lit values of the pacamete~s for r:~p interactions above 10 GeV/c. Excludcd mult ipl ic i ty point indicated in square brackets.

Reaction Pi~b (GeV/e) ~ ~ % a n ~. z2/NDF

~-p l0 0.284 0.127 1.53 1.44 0.20/2

~-p 16 0.264 0.141 1.34 1.33 3.96/4

r¢-p 18.5 0.229 0.120 1.47 1.34 1.18/5

~+p 18.5 0.213 1.46 1.37 0.36/4

~-p 25 0.268 0.097 1.22 1.21 5.56/5

~-p 40 0.297 0.263 1.04 0.65 6.24/8

~-p 50 0.267 0.238 1.06 0.81 4.74/5

¢:+p 50 [8] 0.251 1.05 1.04 6.34/4

r:+p 60 0.280 0.91 0.90 4.21/5

=-p 100 0.262 0.094 0.90 0.89 13.58/8

~+p 100 0.331 0.93 0.92 8.23/7

~:-p 147 0.281 0.062 0.86 0.85 8.84/10

r:-p 205 0.284 0.020 0.87 0.78 4.89/9

~-p 360 0.239 0.029 1.15 1.14 12.96/11

TABLE II . - Best-lit values o] the parameters ]or K±p interactions above 10 GeV/c. Excluded mult ipl ic i ty point indicated in square brackets.

Reaction Pl~b(GeV/c) ~ (~ ap a m ~ ~2/NDF

K+p 12.7 0.392 1.55 1.54 0.11/2

K-p 14.3 [12] 0.236 0.148 1.43 1.42 0.67/2

K-p 32 0.258 0.160 1.18 1.13 3.80/4

K+p 32 [14] 0.312 1.19 1.18 3.91/4

K-p 33.8 0.211 0.216 1.04 1.03 4.48/5

K+p 100 0.384 0.95 0.70 4.45/5

(12) For experimental =p and Kp mult ipl ic i ty da ta see references in (3).

Page 12: A universal scaling function for hadron-hadron interactions

428 A. D'INNOCENZO, G. INGROSSO a n d P. ROTELLI

( ]1 )

a n d

(12)

The overall bes t fits were given b y

F~,(z) oc (1 - - z) exp [ - - 3.95 yr~]

FKgZ) oC (1 -- Z) exp [-- 3.45 V ; ]

with # = 0.38 GeV/c ~

wi th # = 0.41 GeV/c ~ .

10 -I _ _

10 -~ _

I0 -3 _

I 1 I I I I lO-aO 2 4 6 8 10 12 14

/3

Fig. 8. - Total multiplicity probabilities plotted against n for K-p at 14.3 GeV/c. The stars are the best-fit values (arrowed point excluded) of the mliversal/~(z).

The s imilar i ty be tween eqs. (9), (11) and (12) encouraged us to search for a

single universal scaling function. Thus a s imultaneous fit to ~]l the ~:p, K p and pp da ta wus made, while m~inta ining the functionul fo rm of eq. (10). The

Page 13: A universal scaling function for hadron-hadron interactions

A UNIVERSAL SCALING FUNCTION FOR HADI~ON-HADI~ON INTEI:~ACTIONS 429

I

10 - 1 m

10 -2 _

10-'3 _

10 -4

10 -5

t

I I I I I , I I I

4 6 8 10 12 14 16 18 20 n

Fig. 9 . - Total multiplicity probabilities plotted against n for pp interactions at 102 GeV/e. The stars are the best-fit values of the universal F(z).

Page 14: A universal scaling function for hadron-hadron interactions

~ 0 A. D'INNOCENZO, G. INGROSSO and P. ROTELLI

resul t was t h a t an accep tab le fit exists and is g iven b y

(]3) F(z) oc (1 -- z) exp [ - - 3 .9V~] wi th # = 0.40 GeV/c ~ .

&

10 -I

10 -2 __

r-

10 .-3 _ _

10--4 0

I I I I 8 12 16

I

I 20 24

n 28

Fig. 10. - Total multiplicity probabilities plotted against n for 7:-p interactions at 205 GeV/e. The stars are the best-fit values of the universal •(z).

I n tab le I V we display the p a r a m e t e r s and tes t values of this overM1 fit. I n

fig. 8, 9, 10 we exhib i t an example of this fit to each class of in te rac t ions (*).

The var ia t ions of the exponen t iM pa rame te r s are d i sp layed in fig. 11 and 12

(*) These are shown on a In P~ v s . n plot in order to display the validity of the fit at the high-n range, where a linear-linear plot is insufficient.

Page 15: A universal scaling function for hadron-hadron interactions

A UNIVERSAL SCALING FUNCTION FOR I-IADI%ON-HADI~ON INTERACTIONS 431

TABLE I I I . - Best- / i t val~es o/ the parame ter s /o r pp interactions above l0 GeV/c. Excluded mult ipl ic i ty point indicated in square brackets.

Reaction Pl~b (GeV/c) ~ 6 ap a m ~ z2/NDF

pp 10 0.429 1.50 0.07/2

pp 12 [10] 0.421 1.46 0.78/2

pp 19 0.372 1.40 6.20/5

pp 24 [12] 0.407 1.27 10.93/5

pp 28.5 0.356 1.38 11.68/4

pp 5O 0.390 1.03 5 .01 /6

pp 60 0.316 1.16 2.42/6

pp 69 0.402 0.88 8.33/7

pp 102 0.388 0.93 3.10/8

pp 205 0.349 0.87 8.94/11

pp 300 0.324 0.98 11.39/12

pp 405 [16] 0.327 0.95 13.20/13

1.2

1.0

0.8

• A •

B

1.61 - •

• o •

1.41 o A

1.2

1.0

0.8 500 10

Fig. 12.

°%o 5; ' ' ; lO0 50 100 500

Fig. 11.

Fig. 11. - The exponential p~rameter am of the meson LF plotted against P~b for the universal •(z) fit. • z:ip, • K±p.

Fig . 12. - The exponential parameter ap of the proton LF plotted against P,~b for the universal F(z) fit. , pp, , ~ p , • K~p.

for the meson LI~ and proton LI~, respectively. I t is obvious from the non-

physical oscillations tha t these parameters are insensitive to the data and,

Page 16: A universal scaling function for hadron-hadron interactions

432 A. D ' I N N O C E N Z O , G. INGROSSO and P. ROT~ELLI

TABL]~ IV. - The best-fit values o[ the parameters for all r:*p, K i p and pp interactions above 10 GeV/c with a universal T'(z). Excluded mult ipl ic i ty point indicated in square brackets.

Reaction P~b (GeV/c) a ~ a~ a m ~, x2/NDF

~-p 10 0.283 0.123 1.45 1.40 0.58/2

pp l0 0.453 1.58 0.64/2

pp 12 [10] 0.435 1.55 1.83/2

K+p 12.7 0,392 1.55 1.43 0.71/2

K-p 14.3 [12] 0,241 0.144 1.35 1.33 9.13/3

~-p 16 0.256 0.127 1.35 1.30 4.50/4

~ p 18.5 0.219 0.110 1.60 1.25 1.09/5

~+p 18.5 0.191 1.55 1.30 0.43/4

pp 19 0.372 1.49 10.38/5

pp 24 [6] 0.395 1.35 7.35/5

T:-p 25 0.255 0.082 1.25 1.20 5.26/5

pp 28.5 0.368 1.41 13.25/4

K-p 32 0.267 0.147 1.10 1.05 4.20/4

K+p 32 [14] 0,314 1.20 1.05 2,40/4

K-p 33.8 0.214 0.207 1.00 0.95 8.34/5

~-p 40 0.300 0.270 1.00 0.65 6.94/8

~-p 50 0.258 0.239 1.20 0.75 5.20/5

r:+p 50 [8] 0.243 1.20 0.95 5.73/4

pp 50 0.390 1.03 4.08/6

v:+p 60 0.277 1.00 0.85 4.36/6

pp 60 0.346 1.10 2.86/7

pp 69 0.418 0.87 6.59/7

~-p 100 0.261 0.094 1.00 0.85 13.72/8

7:+p 100 0.331 0.95 0.90 8.25/7

K+p 100 0.389 0.85 0.70 5.19/5

pp 102 0.383 0.94 3.14/8

~-p 147 0.283 0.061 0.90 0.85 9.22/10

7:~p 205 0.274 0.019 1.00 0.75 4.21/9

pp 205 0.352 0.86 9.73/11

pp 300 0.328 0.96 12.69/12

~-p 360 [4] 0.236 0.028 1.20 1.15 14.60/11

pp 405 [16] 0,328 0.93 13.84/13

Page 17: A universal scaling function for hadron-hadron interactions

A UNIVERSAL SCALING FUNCTION FOR HADRON-HADBON INTERACTIONS 4 3 3

0.5

0.4

0.3

0.~

0.1

0.4

0.3

0.2

o.)

0 me,,m

0

0

, I

A

A

A

b)

A

A

A A

A

• A • ~ & • •

I 50

c)

I i 50 100

( 0"110 100 500 10 500

Ptab (GeV/c)

Fig. 13. - The weight parameter a p lot ted against ~Plab for the universal F(z) fit. Also shown are the corresponding experimental values of ael/atot used. a) pp interactions, • %]/Otot, o ~; b) ~ p interactions, • (~/atot, A ~; e) K±p interactions, o aoJatot, * ~.

Page 18: A universal scaling function for hadron-hadron interactions

~3~ A. D'INNOCENZO, G. INGROSSO and P. ROTELLI

therefore, poorly determined. ~evertheless , the values of bo th ap and a m show a decline with energy, at low energies, followed by an approximate ly constant

behaviour above a P~b of 50 GeVfc. The energy dependence of a and (~ for this universal fit are displayed in

fig. 13 and 14, respectively. We again observe substantial f luctuations which, however, are a direct consequence of similar fluctuations in the measured ratio

ao~/(~o ~ and (~o/a~ (also shown in fig. 13 and 14). This is principally caused by

the stat ist ical errors in the measurement of the elastic cross-sections and of the neutral-charge mode.

0.30

0.20

0.'IC

0

0

A

• • l i

• A

Al

5OO 0 I I

10 50 100

Fig. 14. - The charge annihilation parameter a (4 7:-p, o K-p) oi the three-fireball term plotted against P~b for the universal F(z) fit. Also shown are the corresponding experimental values of aolai~ n used (i re-p, * K-p).

5 . - C o n c l u s i o n s .

In a previous two-component model applied to pp, ~:p and Kp mult ipl ici ty

data the authors used an exponential (LF)-Gaussian (CF) s t ructure which involved four parameters for each process and each energy. Excel lent phe- nomenological fits were obtained (~,3). However, a systematic difference was found in the best-fit values for the parameters of the CF for meson-proton and pro ton-pro ton interactions at the same incoming Plab value, and even at the same ~ value. Thus these two classes of processes differed not only in the relat ive weight ~ ___ ao~/atot and in the mult ipl ici ty s tructures of one of the

Page 19: A universal scaling function for hadron-hadron interactions

A UNIVERSAL SCALING FUNCTION FOR H A D R O N - H A D R O N INTEI~ACTIONS 4 3 5

LF vertices, bu t also in the mult ipl ici ty s t ructure of the respective central

fireballs. I t is, therefore, a pleasant surprise tha t in the universal fit of our

model this feature is sufficiently compensated for b y the definition of the

scaling parameter z. We recall t ha t

with

while

= (8o - t ~ ) / ( s m - t~ ~)

s m --~ (E .... -- 2mD)~

s m ---- ( E . . . . - - m s - - m . , K ) ~

for pp in teract ions ,

for ~(K)p interact ions.

Thus the same z value corresponds to different so values because of the different incoming masses. Obviously, this leads to the prediction tha t asymptot ical ly (s -> co, ~ -~ co) the systematic differences between pp and

~:(K)p interactions attributable to the CF should become negligible.

I t m ay be objected tha t in our model we do not take into account diagrams

with more than 3FBs. This is not correct. Our CF is almost certainly a com-

posite object. As an average over annihilation processes it contains all (mnl-

tifireball) s tructures tha t annihilation processes contain. Whether a deeper insight into annihilation mult ipart icle product ion follows from a (bare) fire- ball decomposition of fig. 1 or whether , as ma n y believe, annihilation is a suitable subject for statist ical models remains an open question.

The t r ea tmen t of resonance product ion in our fits solely by means of the effective mass /z is a gross simplification. In part icular, it does not take into account the pecul iar i ty of the charged zero mode which cannot receive contri- butions from rho-meson decays. However , since this mode is poorly deter- mined experimental ly, we do not consider it worthwhile developing a more

complicated model at this moment . I t is edifying for us tha t the best-fit value for # ~ 0.4 GeV/c ~ agrees with the hypothesis of basic vector resonance dom- inance (BVRD) which two of the authors have long advocated (la,14). An hypothesis which is now well verified exper imental ly (9).

The confirmation of the val idi ty of a universal F(z) will come f rom an

analysis of p~ interactions in which the single-fireball t e rm (annihilation)

must be considered. Our model may also be applied to the ISR data on pp multiplicities and this is in progress, bu t i t requires a profound fai th in our

KlqO extrapolat ion procedure.

(1s) p. ROTELL:: Nuovo Cimento A, 24, 483 (1974); F. HAYOT: .Lett. Nuovo Cimento, 12, 676 (1975). (14) G. INGROSSO and P. ROTELLI: Lett. Nuovo Cimento, 14, 275 (1975).

Page 20: A universal scaling function for hadron-hadron interactions

436 A. D'INNOCENZO~ G. INGR0880 a r id P. ROTELLI

• R I A S S U N T O

In un reeente modello a due eomponenti applicato ai dati di molteplieit£ pp il fireball certtrale ~ stato espresso come un integrale di convoluzione su u~'estrapolazione K N 0 dei dati di anniehilazione pp con funzione peso F(z). Questa artalisi ~ qui estesa ai pro- cessi r:±p e K±p per energie sopra 10 GeV/c, Sono state ottenute delle funzioni di scala 2~(z) per ciascun tipo di processo ed ~ stata notata la somiglianza con quella otte- nu ta pe~ le interazioni pp. Una singola funzione di scala ~ capace di descrivere soddisfa- centernente tu t t i gli esperimenti t ra t ta t i indipendentemente dalla loro energia e dalla natura degli adroni incidenti.

Pe3ioMe He IIO.rlyqeHO.