fulltext

Multifractality study in “hot” and “cold” events in

12

C-AgBr interactions at 4.5A GeVOc

D. GHOSH, K. PURKAIT, R. SENGUPTA, S. SARKARand A. DEV

High Energy Labratory, Department of Physics, Jadavpur University - Calcutta 32, India (ricevuto il 19 Giugno 1997; revisionato il 15 Agosto 1997; approvato il 24 Ottobre 1997)

Summary. — The paper presents a multifractal analysis of pions produced in “hot”

and “cold” events as characterised by two temperatures in12_{C-AgBr interactions at}

4.5A GeVOc. The study indicates multifractality in both cases but the pionisation in case of “cold” events seems to be more complex.

PACS 25.70 – Low and intermediate energy heavy-ion reactions.

1. – Introduction

In recent years an important probe has been the analysis of multiplicity distribution in full or in limited phase space intervals in the study of multiparticle production mechanism in relativistic energy hadronic or nuclear collisions. The study of large density fluctuations in rapidity space has already been reported [1]. In order to describe the multiplicity distribution in full phase space [2] or in limited phase space intervals [3] a number of theoretical models have been developed. However, these models have limited success for increasing accelerator energy. Batskovic et al. used the scaling in multiplicity to search for collective effects including a quark-gluon plasma and observed that the scaling in multiplicity holds with good accuracy at rather low energy [4]. At present the study of non-statistical fluctuation with the help of factorial and fractal moments is being undertaken by most workers in this field. Miyamura and co-workers, in their pioneer work, analysed the multiplicity distribution in limited intervals of rapidity, using the normalised factorial moments and emphasized the importance of bin size dependence [5]. Bialas and Peschanski [6] found a power-law behaviour of the factorial moments with decreasing phase space interval size by analysing the JACEE events. This is an indication of self-similar fractal structure of multiparticle production in heavy-ion interactions. The spectrum of fractal dimensions provides us with the information on the scale invariance of the hadronisation process. In particular, approaches based on the concept of (multi)fractals seem to be most interesting as they may be related to phase transition [7], self-similar cascades, chaos, entropy, etc. Thus the above analyses contain two approaches: one is the strong evidence of non-statistical fluctuation; the other is a statistical fluctuation around a

non-smooth fractal distribution. The fact is that the assumed mathematical limit (the number of points tends to Q) is not valid for real experimental data since the number of particles in each event is always finite. As a result, for most methods, experimental data sets do not show linearity in a log-log plot of moment against bin size, as would be expected from the mathematical formulations. To overcome this problem, Takagi developed a new approach [8]; he tried to extract directly a (multi)fractal structure from the observed distributions with no particular assumption on the separation between statistical and non-statistical fluctuations and got successful results [9-11].

This paper presents a successful result of an application of the above method to multiparticle data in “hot” and “cold” classes of events in 12_{C-AgBr interaction at}

4.5A GeVOc. Baumgardt et al. [12] compared the transverse momentum distribution of projectile alpha-particles emitted in 56Fe-emulsion interaction at 1.9A GeV with double Maxwell-Boltzmann distribution. We have performed the experiment for 12_C-emulsion

interaction at 4.5A GeVOc with improved statistics [13], thereby obtaining the two temperatures as 10 MeV and 40 MeV (cold and hot). In our earlier works we have observed that the pion coherence zones are different for “hot” and “cold” events [14]. Our observation on studying the fractorial moments of the medium-energy protons reveals an intermittent pattern in the case of “cold” events, whereas “hot” events do not show such self-similarity property [15]. The study of forward-backward multiplicity correlation of pions and protons for these two temperature events showed some interesting results [16]. There is a positive correlation between forward and backward pions and protons in both classes of events but the nature of the correlations is significantly different for the two classes of events. We have also studied the two-particle correlation among shower particles [17]. Recently this study of “hot” and “cold” events has become extremely interesting due to its possible relevance for liquid-gas phase transition [18].

2. – Methodology

2.1. Method of analysis. – The methodology proposed by Takagi [8] is described as follows. Let us consider a multiparticle production process at some incident energy and the distribution in pseudorapidity (h) space. A single event contains n hadrons distributed in the interval hminE h E hmax. The multiplicity n changes from event to

event according to the distribution Pn(Dh), where Dh 4hmax2 hmin. The selected

pseudorapidity interval of length Dh is divided into m bins of equal size dh 4DhOm. The hadrons produced in V independent events are distributed in Vm bins of size dh . Let k be the total number of hadrons produced in these V events and nai the

multiplicity of hadrons in the i-th bin of a-th event. The theory of multifractals [19] motivates one to consider the normalised density Paidefined by

Pai4

nai

k

(1)

and to consider if the quantity

Tq(dh) 4 ln

!

!

behaves like a linear function of the logarithm of the “resolution” R(dh) which is obviously a function of dh ,

Tq(dh) 4Aq1 Bqln R(dh) ,

(3)

where Aqand Bqare constants independent of dh .

If such a behaviour is observed for a considerable range of R(dh), a generalised dimension may be determined as

Dq4

Bq

q 21 .

(4)

Evaluating the double sum of Pq

aiTakagi showed that for a sufficiently large V

!

!

!

k

l

!

!

As pseudorapidity distribution is assumed to be flat in the considered region, one can write anb 4 k V dhDh . (6) So Tqmay be expressed as Tq(dh) 4 anq_b kq 21_{(kOV Dh) dh} 4 ln an q b 2ln dh1const . (7)

For the simplest choice of the “resolution” R(dh) 4dh, relation (3) becomes

Tq(dh) 4Aq1 Bqln dh .

(8)

Then comparing relations (8) and (7) Takagi obtained the relation ln anq

b 4Aq1 (Bq1 1 ) ln dh 4 Aq1 [ (q 2 1 ) Dq1 1 ] ln dh .

(9)

While analysing real data [9-11] it was observed [8] that the plot of ln anq_{b against dh}

saturates for large-h region. This deviation may be due to the non-flat behaviour of dnOdh in the large-h region. Following Takagi anb will be a better choice of the “resolution” as dnOdanb is flat by definition [8, 20, 21].

Choosing R(dh) 4 anb one has ln anq

b 4Aq1 [ (q 2 1 ) Dq1 1 ] ln anb ,

(10)

which is a simple linear relation between ln anq_{b and lnanb. The generalised dimension}

Dq can be obtained from the slope values. The information entropy, which is entirely

different from the entropy studied by Simak et al. [22], can be determined from a simple relations proposed by Takagi [8]

S(dh) 4 an ln nbOanb1ln K .

This in turn gives the values of information dimension D1from the relation

an ln nb Oanb 4C11 D1ln anb .

(12)

Thus following this method the fractal nature of the particle production process could be understood by analysing the multiplicity data over different selected pseudorapidity bins.

So far this method has been applied to non-overlapping bins. Here we have applied this method of analysis to overlapping pseudorapidity bins, i.e. the multiplicity distribution Pn(dh) was taken for symmetric bins (2hcE h E hc) of the centre-of-mass

pseudorapidity neglecting the information from the outer region.

2.2. Separation of “hot” and “cold” events. – Previously it has been observed that there are two distinct classes of events at two temperatures, namely “hot” (40–60 MeV) and “cold” (8–10 MeV). These observations were made in a study of the projectile fragmentation region by analysing alpha-particles produced from 56Fe-emulsion at 0.9A GeV [23], 1.7A GeV [24], 1.9A GeV [12], and 40_{Ar-emulsion at 2.0A GeV [23]. Our}

earlier work also includes the analysis of transverse momentum of the alpha-particles emitted as projectile fragments in the carbon-emulsion interaction at 4.5A GeVOc [13]. The work was done by following Baumgardt et al. [12]. We calculate the transverse momentum of projectile a-particles using the relation

PT4 [Am0VO

k

where A is the atomic number, m0 the nucleon mass, V the a velocity,

k

Lorentz factor of the projectile fragment and u is the emission angle of a with incident beam. Let the momentum distribution in the projectile rest frame at some temperature

T be a Maxwell-Boltzmann distribution. Then the integral frequency distribution of the

transverse momentum per nucleon squared, Q 4 (PTOA)2, is

ln F(DQ) 42 A

2 m0T

Q .

A cumulative plot of ln F(D Q) as a function of Q showed a non-linear distribution. It was interpreted as a mixture of two Maxwell-Boltzmann distribution components with two distinct temperatures: one is 40 GeV (the “hot” group) and the one is 10 MeV (the “cold” group). Details are given in our earlier work [13].

3. – Experimental details

The required data is obtained from the NIKFI-BR2 emulsion plates exposed to a

12

C beam at 4.5A GeVOc from the Dubna Synchrophasotron. Each interaction was scanned using the “along-the-track” method by the following optics: 100 3objective and 20 3ocular with the help of leitz Ortholux microscopes. We have excluded the events occurring within 20 mm thickness from the top of bottom surface of the pellicle. Thus we have chosen 1200 primary events for analysis. Great care has been taken on the identification of different tracks. We have identified the tracks using the conventional method in the emulsion technique—by their grain density—as follows:

i) projectile fragments, with a constant grain density g Bg0 _{along the track}

length of 2 cm ionization ( g0is the plateau grain density for singly charged particles), and emission angle E 37 (in the laboratory frame);

ii) black track or short-range particles (b) with range l E3 mm; iii) grey black particles (g) with l D3 mm and grain density gD1.47; iv) relativistic particles (s) with g E1.4g0_.

Great care was taken that no contamination took place between the pions and protons emitted as projectile fragments. Relativistic protons emitted within the projectile fragmentation region were identified and were isolated from the sample. This was done by the method of angular cut [25] at an angle of u given by

u 4 0 .2 Elab( GeV )

.

The pseudorapidity values of the relativistic charged secondary particles (shower tracks) were calculated from space angles (u) of the tracks. The space angle is measured by taking the space co-ordinates of a point on the track, the space co-ordinates of the production point and the space co-ordinates of a point on the incident beam track. The pseudorapidity value is related to space angle by the following formula:

h 42ln tan (uO2) .

4. – Results and discussion

We calculated an ln nb Oanb and lnanq_{b for different pseudorapidity intervals. We}

considered the central symmetric overlapping h bins around the peak value of the h distributions. We took the first bin of size equal to one and then increased the bin size symmetrically in steps of one. Figures 1a) and b) show the plots of an ln nb Oanb and ln anq

b vs. lnanb, for “hot” events, respectively. Both an ln nb Oanb and lnanq

b (for q 42, 3, 4 and 5) in “cold” and “hot” events exhibit exact linear behaviour as a function of

Fig. 1. – Behaviour of ln anq

b and an ln nb Oanb with ln anb for q42, 3, 4 and 5 in “cold” (a) and “dot” (b) events, respectively.

Fig. 2. – Plot of generalised dimensions Dq (q 42, 3, 4, 5), vs. q for both “cold” and “hot”

events.

ln anb. From the slope values of the plots lnanq

b vs. lnanb (for q 42, 3, 4 and 5) the generalised dimension Dqare obtained using eqs. (4) and (10). Figure 2 shows the plot

of generalized dimension Dq vs. q for both “cold” and “hot” events. Our observations

are:

i) the values of Dqin both cases are less than unity;

ii) the Dqvalues of “cold” events are less than the Dqvalues of “hot” events for all

q’s (for q 42, 3, 4 and 5), i.e. they follow the rule 1 DDq(hot) D Dq(cold).

5. – Conclusion

i) The multifractal nature (i.e. DqE Dq, for q Dq8) of the produced charged

particles has been shown in both “cold” and “hot” groups of events.

ii) As ( 1 2Dq) is a measure of non-statistical fluctuation, the reaction process of

“cold” events is more complex than that of “hot” events. * * *

We would like to thank Prof. K. D. TOLSTOV, JINR, Russia for providing us with the

emulsion plates. We also gratefully acknowledge the financial assistance sanctioned by the University Grant Commission (Govt. of India) under their COSIST Programme.

R E F E R E N C E S

[1] BURNETTT. H. et al., Phys. Rev. Lett., 50 (1983) 2062; ALNERG. J. et al., Phys. Rep., 154 (1987) 247; ADAMOVICHM. I. et al., Phys. Lett. B, 201 (1988) 397; ADAMUSM. et al., Phys. Lett. B, 185 (1987) 200; SINGHG., SENGUPTAK. and JAINP. L., Phys. Rev. Lett., 61 (1988) 1073. [2] KOBAZ., NIELSENH. B. and OLESENP., Nucl. Phys. B, 40 (1972) 317; NAKAMURAE. R. and

KUDOK., Z. Phys. C, 40 (1988) 81; CHEWC. K., KIANGD. and ZHOUH., Phys. Lett. B, 186 (1987) 411; JAINP. L. et al., Phys. Lett. B, 231 (1988) 548.

[3] GIOVANNINIA. and VANHOVEL., Z. Phys. C, 30 (1986) 391; MORSEW. M. et al., Phys. Rev. D,

15 (1977) 66; BREAKSTONEA. A. et al., Phys. Rev. D, 30 (1984) 528; GHOSHD. et al., Phys. Lett. B, 218 (1989) 431.

[4] BATSKOVICS., KRPICD. and SHABOLSKILYU. M., Yad. Fiz., 54 (1991) 1126.

[5] COOPER A. M., MIYAMURAO., SUZUKI A. and TAKAHASHI K. Phys. Lett. B, 87 (1979) 393; MIYAMURAO., Prog. Theor. Phys., 63 (1980) 203.

[6] BIALASA. and PESCHANSKIR., Nucl. Phys. B, 273 (1986) 703.

[7] BRAXPH. and PESCHANSKIR., Nucl. Phys. B, 346 (1990) 65; ANTONIOUN. G. et al., Phys. Rev. D, 45 (1992) 4034; HWAR. C. and NAZRIROVM. T., Phys. Rev. Lett., 69 (1992) 741.

[8] TAKAGIF., Phys. Rev. Lett., 53 (1984) 427; Phys. Rev. C, 32 (1985) 1799; Phys. Rev. Lett., 72 (1994) 32; XXI International Symposium on Multiparticle Dynamics (1991), edited by W. YUANFANGand L. LIANSHOU.

[9] UA5 COLLABORATION(ANSORGER. E. et al.), Z. Phys. C, 43 (1989) 357. [10] TASSO COLLABORATION(BRAUNSCHWEIGW. et al.), Z. Phys. C, 45 (1989 159. [11] DELPHI COLLABORATION(ABREUP. et al.), Z. Phys. C, 52 (1991) 271. [12] BAUMGARDTH. G. et al., J. Phys. G, 7 (1981) L175.

[13] GHOSHD., ROYJ. and SENGUPTAR., J. Phys. G, 14 (1988) 711.

[14] GHOSHD., ROYJ., SENGUPTAR. and SARKARS., J. Phys. G, 18 (1992) 935.

[15] GHOSHD., DEBA., SENGUPTAR., DASS. and PURKAITK., J. Phys. G, 19 (1993) L19. [16] GHOSHD., ROYJ., SENGUPTAR. and SARKARS., Z. Phys. A, 342 (1992) 191.

[17] GHOSHD., SENGUPTAR. and SARKARS., Hadronic J., 12 (1989) 279. [18] KARNAUKHOVV. A., Dubna preprint 1996.

[19] HENTSCHELH. G. E. and PROCACCIAI., Physica D (Amsterdam), 8 (1983) 435.

[20] TAKAGI F., in Physics of Elementary Interactions, Proceedings of the XIII Warsaw Symposium on Elementary Particle Physics, 1990, Kazimierz, Poland, edited by Z. AJDUK et al. (World Scientific, Singapore) 1991, p. 371.

[21] BIALASA. and GAZDZICKIM., Phys. Lett. B, 252 (1990) 483.

[22] SIMAKV., SUMBERAM. and ZBOROVSKYI., Phys. Lett. B, 206 (1988) 159.

[23] AGGARWALM. M., BHALLAK. B., DASG. and JAINP. L., Phys. Rev. C, 27 (1983) 640. [24] BHALLAK. B. et al., Nucl. Phys. A, 367 (1981) 446.