Dependence of multiplicity on energy in high energy pp collisions

Volume 42B, number 1

PHYSICS LETTERS

13 November 1972

D E P E N D E N C E O F M U L T I P L I C I T Y ON E N E R G Y IN H I G H E N E R G Y p - p C O L L I S I O N S S.N. GANGULI and P.K. MALHOTRA Tara Institute o f Fundamental Research, Bornbay-5, India Received 15 August 1972 Revised manuscript received 16 September 1972

The available data on multiplicity of charged particles in the range 4 to 104 GeV/c has been analysed. The following conclusions have been drawn: (i) prediction of statistical and hydrodynamical model is surprisingly well borne out by the data; (ii) the prediction of isobar-pionisation type of models disagrees with the data, and Off) assuming an asymptotic behaviour of multiplicity as log s, the approach to this seems to be characterized by s -a, with ~ 0.2-0.3.

An important aspect which distinguishes different models of multiparticle production is the dependence of average multiplicity of particles on the primary energy. A number of attempts [e.g. I, 2] have been made in the past to ascertain this dependence. However, most of these attempts suffer from a lack of reliable data at low energies, below 30 GeV/c, as well as at high energies, particularly in the region of 100-- 10000 GeV/c. During the last few years considerably more reliable hydrogen bubble chamber data has become available at energies below 200 GeV/c. More recently, the ISR has provided reliable, though so far only moderately precise, data in the region 240-1500 GeV/c. This has prompted us to re-examine the entire available data, from 4 GeV/c to 104 GeV[c, to see whether they favour any of the popular models of the multiple particle production. Let us first briefly outline the predictions of the models. The statistical model of Fermi [3] and hydrodynamical model of Belenjki and Landau [4] predict that average multiplicity should increase with energy as

(n)=as 1/4

(1)

where s = E2m = 2 M(EL+M) for p - p collisions. In the isobar-pionization model of Pal and Peters [5] and the random fragmentation model of Narayan [5], the multiplicity obeys the following relation

(n) = a s 1/2 + b.

(2)

The multiperipheral models [e.g. 6] and Feynman's scaling [7] lead to asymptotic expressions for multiplicity of the form (n) = a(In s) + b.

(3)

Recently, on simple phase space considerations and the constancy of(PT) , Satz [8] and Berger and Krzywicki [8] have shown that in the accelerator energy region the multiplicity should follow the power law (n) = a s 1/3

(4)

and in the high energy region it should asymptotically approach the logarithmic relation (eq. (3)). We now proceed to examine the available experimental data which is presented in table 1 and plotted in fig. 1. (nch)is the average multiplicity of charged particles (including the leading baryons). As far as possible we have tried to restrict to the best possible data. In the accelerator energy region, 4 - 2 0 0 GeV/e, we have included data only from the hydrogen bubble chamber investigations. The errors, and in some cases even the average multiplicities, have been calculated by us from the partial cross sections and errors for different prongs given by the authors. The data of Smith et al. [12] were appropriately corrected for the omitted inelastic two-prong events, by interpolations of the Oel and the total two-prong cross section

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investigations [ 17-19]. For obvious reasons, we have excluded all the cosmic ray investigations in which carbon was used as the target. However, we have used the calorimeter data (300 GeV/c) where LiH was employed as the target. The point at 3000 GeV/c is from an emulsion experiment in which the authors imposed a severe selection criterion, namely restricted to collisions with N h = 0, i.e. events were accepted only if there were no associated black and grey prongs. This minimizes contamination due to collisions with emulsion nuclei. In an attempt to get some reasonably reliable information at the highest possible energy from cosmic ray data we have resorted to the following procedure. Firstly, we restrict ourselves to only those systematic investigations in which the events were located either by first recording the resulting high energy 'cascade' with total energy content greater than about 300-400 GeV and then following them back to the primary interactions, or the interactions were obtained by 'along the track' following of the singly charged fragments resulting from the break-up of the high energy multiply charged nuclei [21]. In both cases reasonably reliable estimates of the primary energy are possible. In the former case the primary energy is estimated from a combination of, (i) Castagnoli method, (it) evaluating the energy carried by the created charged particles on the basis of the observed near constancy of transverse momentum and (iii) measuring the energy in the soft cascade In the latter case primary energy is estimated reliably from the emission angles of the break-up products. In order to minimize the inclusion of collisions with heavy nuclei of the emulsion, we enforce a relatively

Table 1 Momentum (GeV/c)

s (GeV) 2

Target

4.0 5.5 10.0 12.9 18.0 19.0 21.1 24.1 28.4 50.0 69.0 200.0 240 300 484 484 1060

9.5 12,2 20,6 26.0 35.6 37,5 41.4 47.0 55.1 95.6 131.3 377 452 565 910 910 1991

Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen LiH Hydrogen Hydrogen Hydrogen

1060

1991

Hydrogen

1490 1490

2798 2798

3000

5631

10000

18767

Hydrogen Hydrogen Emulsion Nh=O Emulsion N h _< 2


Ref

2.43 ± .03 2.70-+ .02 3.11 ± .06 3.53 ± .05 3.91 ± .04 4.02 -+ ,03 4.17 ± .05 4.30 ± .04 4.56 ± .04 5.47 ± .20 5.81 ± .15 7.65 ± .17 7.0 ± 1.1 9.0 ± 1.0 9.3 -+ 1.4 10.5 ± 1.5 10.9 ± 1.6 11.8 +- 1,6 (11.2 ± 1.5) 12.2 12.4

9 10 11 12 12 13 12 12 12 14 14 15 17 16 17 18 17 18 19

± 1.8 ± 1.8

17 18

15.2 ±2.0

20

16.3 ± 1.1

This survey

available from other investigations. It now seems certain, particularly after the 200 GeV/c p - p experiment [ 15] that the charged particle multiplicities from the Echo Lake cosmic ray experiment have been underestimated. In view of this, we have not used the Echo Lake data in this work. Except for a point [ 16] at pL = 300 GeV/c, all the data in the region 240-1500 GeV/c is from the ISR 2

A

V

I0

o

13 November 1972

H2 BUBBLE CHAMBER

/-A

• ISR

~

U EMULSION (Nn<-- 2]

~

-

'

~

___

-

-

B

5

I

I0

2

.5

102

I I'l]

. . . . . .

i03

I

I

s(GeV) 2

i04

Fig. 1. Average charge multiplicity
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severe selection criterion , n a m e l y N h _< 2, where N h is the n u m b e r o f black and grey (highly ionizing) prongs. In this way, a collection o f worl data was carried out by one of us [1]. To this we have n o w added the available data published later [21]. This procedure gives us a total of 5 4 j e t s in the primary energy range 2 - 4 5 TeV with a m e a n energy o f 10 T e V t . The charged particle multiplicity (including leading baryons) is inch) = 16.3 -+ 1.1"* for p ~ p collisions. The error given is D/X/~ where D = ((nch)--(nch)2) 1/2 = ,7.95 a n d N, the n u m b e r o f jets is 54. It may be p e r t i n e n t to discuss possible biases in the above estimation of multiplicity. Although, we have imposed a severe selection criterion, N h < 2, there may be a small residual admixture of collisions involvin,g two or more n u c l e o n s from the target nucleus ~. This would tend to overestimate the multiplicity. On the other h a n d , since one d e m a n d s a minim u m energy o f 3 0 0 - 4 0 0 GeV in the cascade to be able to record the event, fluctuations will favour events with high (n~ro) and low inch) , resulting in an u n d e r e s t i m a t i o n of (nch). There would therefore be some cancellation and we feel that the n e t bias, if any, will be small. There is, however, an i n d e p e n d e n t indication that the overall bias, if any, must be small

* It is necessary to apply a severe N h selection criterion since there is a fairly strong dependence of (nch) on N h for N h ~> 3. Our world survey yields (n s) = 15.1 -+ 2.9, 15.8 -+ 3.8, 17.3 -+4.2 and 27.9 -+4.2 forN h = 0, 1,2 and 3 - 5 respectively. It is therefore not surprising that one often comes across in the literature a figure somewhat higher than given here for N h < 2. This is because some of the authors in the past have used a relaxed criterion N h <_ 5; our own survey yields, for p-nucleon collisions, (nch)= 19.2 -+ 1.0 for 106 jets with N h < 5 which is somewhat higher than (nch) = 15.8 -+ 1.1 obtained here forN h < 2. 1" Since these jets have been obtained from several experiments with different thresholds for cascade energies, one cannot deduce the average energy using the known exponent of the primary cosmic ray energy spectrum (which would have given a value of 5 TeV for the mean energy). It is indeed difficult to estimate the error on the mean energy. However, from considerations such as the energy radiated in the cascades, we estimate that the error on the mean energy of 10 TeV could be as high as -+ 2 TeV. ** Since nearly 50% of the observed jets are collisions against peripheral neutrons, we have added 0.5 to the observed (nch) to obtain charge multiplicity pertaining to p - p collisions

13 November 1972

indeedtt. We have a t t e m p t e d to fit the following expressions to the data*** on inch) for m o m e n t u m ranges 4--69 GeV/c, 6 9 - 1 0 4 GeV/c and 4 - 1 0 4 GeV/c:

inch) = a I s~ 1

(5)

inch) = a 2 + b 2 Eaa2

(6)

inch) = a 3 + b 3 In s

(7)

inch) = a 4 + b 4 In s + c 4 s -a4

(8)

inch) = a 5 + b 5 l n s + c 5 s a5 I n s

(9)

where Eav = X/s-2M is the 'available energy' in the C.M. system (M is the mass of proton). For s >> M 2, i.e. at high energies, Eav ~ V~. The relations (5), (6) and (7) are in the spirit of models discussed above• Fits to (8) and (9) have been carried out because of the reasons m e n t i o n e d later. F o r minimizing the chi-square, we have used the CERN programme MINUIT. The results are summarized in table 2. First of all, we would like to see h o w well the fit to (5) agrees with the prediction (4). A t first glance t~1 = 0.348 -+ 0.01 for (nch) agrees well with the prediction o f Berger and Krzywicki [8]. In our view, this is just fortuitous, because the relation (4) should really be applicable only if (pT) ~ ( p t ) holds. In the low energy region this is valid for the 'created' particles, b u t it certainly does n o t hold for the leading baryons, which are k n o w n to retain on the average about 50% of the energy in the C.M. system. We therefore feel that in the low energy region the pert i n e n t variable is E a r , the available energy in the C.M. system a n d n o t s. The Satz, Berger and Krzywicki prediction should then be modified to ( n c h ) = a 2 + b 2 E2v/3

(10)

i.e. eq. (6) above with a 2 = 2/3. Moreover one would

t t One finds that the ratio (nch)/D, decreases from 2.3 at 10 GeV/c to 2.1 in the region of 50-200 GeV/c (hydrogen bubble chamber data), i.e. a very weak dependence on $. Our survey at 104 GeV/c yields (nch)/D = 2.0. *** In carrying out these fits, we have ignored the 10 GeV/c points as it invariably yields an abnormally large x 2. 85

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13 November 1972

Table 2 Relation fitted, =

Momentum range (GeV/c)

als

Results of fits Data points

b

4 -69

1.130 -+ .02

a2 + b2Eav

4 -69 4 -104

1.10 -+ .01 .348 -+ .01

a 3 + b 3 Ins

12.9-104 69 -104

-1.60 -+ .25 -4.02 -+ .22

1.55 -+.07 1.99 -+ .03

-8.18 -+ .1

2.36 -+ .07

9.93-+ .3

-5.59 -+.05

2.18 -+.03

8.12-+.2

~2

a 4 + b 4 Ins

4

-10 4

1.184 -+ .005 1.883 ± .01

x2

0.348 -+ .01

10

9.5

0.622 ± .01 0.464 -+ .03

10 21

2.6 15.9

19 12

21.7 4.8

0.28 -+ .02

21

6.5

0.79 -+0.05

21

6.1

-~4

+C 4 $

as +bs Ins +Cs s

-~5

4 -10'*

Ins

expect the value o f a 2 to lie between 1 and 2 (leading charged baryons). Our fit yields a 2 = 0.622 and a 2 = 1.1, in good agreement with the eq. (10). For exactly the same reasons as mentioned in the above paragraph, the prediction of statistical model as well as hydrodynamical model "~.ould be modified to (rich) = a 2 + b 2 E l v [2

(11)

i.e. eq. (6) with ol2 = 0.5. As seen from the tabel 2, the eq. (6) fits the data, in the entire region o f 4 - 1 0 4 GeV/c, rather well and a 2 = 0.46 -+ 0.03. Thus, we would like to conclude that as far as multiplicity is concerned the agreement with the predictions o f the above models is rather good. The deduced value o f a 2 = 0.46 is in disagreement with the isobar-pionization models which require a2 ~ 1 ( a f i t o f the type (nch)= a + bs a to the 4 - 1 0 4 GeV/c data yields a = 0.16(×2=7.0) as compared to the expected value of 0.5 on the above model). The simple log s (and even log Ear ) expression, eq. (7), does n o t give a good fit when the data in the entire range 4 - 1 0 4 GeV/c is used (×2=78); there is considerable improvement if the fit is restricted to 12.9-104 GeV/e, X2 = 21.7 for 17 degrees of freedom (curve B in fig. 1). The eq. (7) fits the high energy data 6 9 - 1 0 4 GeV/c quite well, e.g., X2 = 4.8 for 10 degrees of freedom, and the best fit is

86

(nch)= 1.99 lns -- 4.02.

(12)

Since simple log s dependence is expected to hold in the asymptotic region only, we have therefore considered it worthwhile to carry out fits to the expressions (8) and (9), both o f which imply this asymptotic behaviour and incorporate two different ways in which it is being approached. In support o f relation (9) we may give the following crude justification. Following Mueller approach [22], we can write the invariant inclusive cross sections as

E d3ft/dp 3 ~ A ( P T , X ) + B(PT,X)S (~eff-1)

(13)

where PT and x -~ 2PL/X/~ are in the C.M. system and o%ff is the effective intercept of the secondary Regge trajectories. Integrating over p, we obtain (n) ~ a + b In s + c s -as In s

(9)

where a 5 = 1 - o%ff. As seen from the table 2, both eq. (8) and eq. (9) fit the data very well indeed. The best fit corresponding to eq. (9) is also shown as curve A in fig. 1. The fitted value o f a 5 = 0.79 suggests that the effective intecept of the secondary trajectories is o f the order of 0.21. Thus, we conclude from this analysis of charged particle multiplicities: (i) prediction of statistical and hydrodynamical model is surprisingly well borne out

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by the data; (ii) the prediction of isobar-pionization type of models disagrees with the data (iii) the prediction of Satz [8] and Berger and Krzywicki [8] for the multiplicity in the low energy region is in good agreement with the data (we would like to stress that for such a comparison, one should use Eav rather than x/s), (iv) assuming an asymptotic behaviour of multiplicity as log s, the approach to this seems to be characterized by s -a, with a ~ 0 . 2 - 0 . 3 . It is a pleasure to thank Dr. K.V.L. Sarma and Dr. D.S. Narayan for interesting discussions.

References [1] P.K. Malhotra, Nucl. Phys. 46 (1963) 559. [2] P.K.F. Grieder, Nuovo Cim. 7A (1972) 867. [3] E. Fermi, Prog. Theor. Phys. 5 (1950) 570 and Phys. Rev. 81 (1951) 683. [4] S.Z. Belenjki and L.D. Landau, Suppl. Nuovo Cim. 3 (1956) 15. [5] Y. Pal and B. Peters, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (1964), No. 15; D.S. Narayan, Nucl. Phys. B34 (1971) 386. [6] C.E. DeTar, Phys. Rev. D3 (1971) 128.

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[7] R.P. Feynman, Phys. Rev. Lett. 23 (1969) 1415 and High energy collisions (Gordon and Breach, N.Y. 1969) p. 237. [8] H. Satz, Nuovo Cim. 37 (1965) 1407; E.L. Berger and A. Krzywicki, Phys. Lett. 36B (1971) 380. [91 L. Bodini et al., Nuovo Cim. 58A (1969) 475. [10] G. Alexander et al., Phys. Rev. 154 (1967) 1284. [11] S.P. Almeida et al., Phys. Rev. 174 (1968) 1638. [12] D.B. Smith, R.J. Sprafka and J.A. Anderson, Phys. Rev. Lett. 23 (1969) 1064, and UCRL-20632, 1971. [ 13 ] H. Boggild et al., Nucl. Phys. 27B (1971) 285. [14] Soviet French Collaboration results reported at Int. Conf. on H.E. Collisions, Oxford, 1972. [15] G. Charlton et al., ANL/HEP 7225, 1972. [16] V.V. Guseva et al., J. Phys. Soc. of Japan, Suppl. A-Ill, 17 (1962) 375. [17] M. Breidenback et al., Phys. Lett. 39B (1972) 654. [181 F. Neuhoffer et al., Phys. Lett. 37B (1971) 438; 38B (1972) 51; K.R. Schubert, Vile Rencontre de Moriond sur les Interactions electromagnetiques (1972). [19] S.N. Ganguli and P.K. Malhotra, Phys. Lett. 39B (1972) 632. [ 20] E. Lohrmann and M.W. Teucher, Nuovo Cim. X, 25 (1962) 957. [21] T. Nozki and M. Koshiba, J. Phys. Soc. Japan 30 (1971) 1. [22] A.H. Mueller, Phys. Rev. D2 (1970) 2963.

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