Bandyopadhyay, Prabhat

Deb, Argha

Ghosh, Dipak

Lahiri, Madhumita

Mondal, Mitalt

Patra, Kanchan Kumar

IACS Institutional Repository

/J«»/W j5.78A (3), 353-357 (2004)< U PP io n  fluc tua tion study in  ir - A g B r  interactions at 350 GeV/cDipak Ghosh*, Argha Deb, Prabhat Bandyopadhyay, Mitali Mondal, Madhumita Lahiri and JCanchan Kumar PatraNuclear and Particle Fhyj|cs Research Centre,Department of Physics, Jadavpur Urivcrsiiy, Kolkata-700 032, IndiaE-mail ; dipakgho.shjn@yahoo com R eceived  29 January 2003, act epted 4 July 2003Abstract Fluctuations in pseudorapidily distribution of secondary particles in n -AgBr interactions at 350 GeV/c arc analysed using moments, (i moments and moments. In order to investigate the presence of non-thermal phase transition, the required parameters have been determined from these three moments and are compared among themselves with ordersKc‘ywords . Factorial moment, moment, Takagi moment, non-thermal phase transition PACS Nos. : 13 85.Hd, 25.80.-e.1. IntrcKluctionIn rapidity densities o f  charged secondaries, the study o f non- statistical fluctuations have recently received much attention due to its potential to provide a better understanding of multi­particle production mechanism. The concept o f  intermittency was first introduced in the field o f  multiparticle production by Bialas and Peschanski f 1 -3] from the theory o f  turbulence where It IS used to measure the effects o f  bursts in a turbulent system. Specifically, they suggested that if  intermittency exists, the normalized factorial moments o f multiplicity distribution should exhibit power-law dependence on the width Sri o f  the rapidity window, as S r j - - > 0 -  This point o f  view o f Bialas and Peschanski received a tremendous boost o f  support when intermittent behaviour were observed in <?'*‘e“ annihilation [4], hadron-hadron |5], hadron-nucleus [6 ] and nucleus-nucleus [7J interactions. In i^ich case, a power-law dependence o f  the scaled factorial moments of the multiplicity distribution on the rapidity bin width was observed and this type o f multiplicity fluctuation was called intermittency.The power-law behaviour o f factorial moments reveals self- similarity and in general, indicates the existence o f  fractal Pi^operties [8 ] in m ultip artic le  production  p rocess. For investigating the fractal structure in multiparticle data, various Methods have been suggested. Hwa [9J proposed the moment t^orresponding Authorapproach which has enriched they study o f multifractal analysis in multiparticle production. However, a crucial point to be stressed here is that the power-law behaviour o f is to be analyzed event by event, and not for the event average which suppresses the importance o f llucluation in low multiplicity events. Hwa and Pan [lO] modified the old forms o f the G^  ^moments by introducing a step function which can act as a filter for the low multiplicity events. Although for investigation o f  fractal structure, anothe moment has also been suggested by Takagi 111). In this particular approach, the experimental data present the expected linear behaviour in a log-log plot within the required mathematical limit, the number o f points —> oo . T ^ moment approach is different from both moment and G^  ^moment approach. A crucial difference lies in the fact that the total number o f points (particles) can be made arbitrarily large.The motivation o f this paper is to investigate the signal of non-thermal phase-transition in Ti^-AgBr interactions at 350 GeV/c. In particular, approaches based on the concept o f (Multi) fractals seem to be most interesting as they may be related to phase transitions 112], self-similarity cascade etc. Using these three methods, we can try to gain som e knowledge about possible phase transitions. One should not necessarily imply the transition from a hadron phase to quark-gluon plasma but can also speculate about other possible transitions due to symmetry breaking in the system.© 20041ACS354 Dipak Ghosh et al2, Experimental dataWe study the hadron nucleus interaction data o f rr-AgBr at 350 GcV/c incident energy. The stacks o f G5  nuclear emulsion plates were exposed horizontally to a ;r ‘ beam at CERN with 350 GeV/ c incident energy. The nuclear emulsion covers 471 geometry and provides very good accuracy (even less than 0 . 1  m rad) in angle measurements o f produced particles with respect to the projectile beam axis due to high spatial resolution and thus, is suitable as a detector for the study o f  fluctuations in fine resolution intervals o f the pseudo-rapidity space. The emulsion plates were area scanned with Lcitz Metalloplan Microscope fitted with a semi automatic scanning device, having a resolution along with the X and Y axes o f I jiim while that along with the Z axis is 0.5 )im. A sample o f 569 events o f 7T-AgBr at 350 GcV/c was chosen by the usual emulsion methodology. The details o f  the experimental data set has been described in our earlier paper LI3]. The average number o f shower tracks for the data sample used, is 11.74 ±0.12.The em ission angle 0 o f  each produced secondary was measured by recording the space coordinates (x, y, z) o f a point on each shower track, the point o f interaction and a point on the incident track. Pseudo-rapidity 17 o f all shower particles were determined from measured space angle Q with reference to the beam by the relationr7 = ~ ln t a n ( e / 2 ) .  ( 1 )3, Method o f analysisThe methods based on moment, moment and T  moment are explained in this section.(A )  m o m e n ts  m e t h o d :For the analysis o f fluctuations in pseudorapidity distribution, the pseudorapidity interval Ar\ is divided into M bins o f size St] -  A t ] /  M  * The ^-th order scaled factorial moment is defined b y [l] .' V  N m=l { l y } 'where is the number o f  events in the sample, is the numbero f  showers in bin tti and <N> is the mean multiplicity o f showers in the A r j  interval. The dependence o f  the scaled factorial m oments on bin width Sr} is a measure o f the size o f  the pseudorapidity fluctuations. An intermittent pattern o f  the fluctuations leads to a power-law behaviour o f the moments ( 1  ] given byAtjL<5»j J ' (3)w here the s lo p e  va lu e a ,  characterizes the strength o f  intermittency.Fialkowski e t  a l  [ 14] suggested that scaled factorial momen o f eq. (2 ) should be corrected to take into account of the " uniform shape o f  the single-particle pseudorapidity di.stribui " The corrected scaled factorial moments are given by( ' i L = ( ' ; > / * , •where is the g-th order correction factor given by'  M  { N rWe have evaluated the corrected sealed factorial nionienh( ' i L . -(B ) G^  ^m o m e n ts  m e t h o d :For investigating multiparticle fluctuations, the method of fracta! moment determined to beM{ G n ) = l i ' ‘n ,IN )"  d O i Jw- 1 ( 6 >has been recently suggested by Hwa [9]. Here, an order q is an arbitrary real number, is the produced partncic multipliciiy mthe m-lh bin, Sr)(= A t} / M )  is the pseudorapidity interval .1//divided into M bins and N  is the total m ultiplicity ol iho considered event given byM^  and 0{T ]^ )  is the (Heavi.side) step functionThe modified form o f multifractal multiplicity moments ol order q  proposed by Hwa and Pan [lO] is defined for a situ’lc event by(7)instead o f  eq. (6 ). At very large N ,  the new definition coincides with eq. (6 ). Since in reality, N  is limited, the presence of difleicni 0 -functions gives different results.To get the statistical contribution to N-particIcs ol each event are distributed randomly throughout the considered q  -interval and the value o f  ( G ‘^*  ^ moment is obtained with the redistributed particles.For se lf similar processes, ^G  ^^  are the power-law functions o f the witdh St]oc oc . (8)After the average overall events o f  quantities o f I^n one getsPion fluctuation study in i r - A g B r  interactions at 350 GeV/c 355' A\nM(9)in the range o f values o f  A/ for which the self-sim ilar behaviour(8 )o f ih e  (C ,)  moment exists.The slopes characterize multifractal properties o f  the pseudorapidity distributions and it was revealed that the slopes of dynamical T^ *" and statistical t *' contributions are connected with each other by the relationr f  = T . < + r / - l , no)wheie the slopes include both statistical and dynamical ones. / C ’] T m o m e n ts  m e t h o d :Takagi [II ]  proposed a m ethod to study the process o f  muUiparticIe production at som e incident energy and the disinbiition in the rapidity space. A single event contains n fiadrous distributed in the interval Themultiplicity n changes from event to event according to the distribution P„(T]) where E)ivide the fullfiipidity interval o f  length At] into M  bins o f  equal size  St]^ A r] l  M • The multiplicity distribution for a single bin isdenoted as (5?;) for = 0> 1 ,2 ......... . where Takagi [11] assumedthat the inclusive rapidity distribution d n id r ]  is constant and IS independent o f  the location o f  the bin. Let A"be total number of hadron produced and the multiplicity o f  hadron in (he /-ih bin of the i- ih  event. The theory o f multifractals motivated r.ikagi to consider the normalized density P j^ defined by( 11)Hadron produced in Q  independent events are distributed in  Qh/l bins of size 5r]  and the method o f fractal moment ( S tj)may be determined asM/=!( 12)Lvaluating the double sum o f  P j  for sufficiently large Qand using Q M  ^  K / < n  > , Takagi f i l l  indicated a linear relation likeln{/T') = A , + ( 5 ,  + l ) l n < « > .  (13)where and are constants independent o f  Sr]- l^esults and d iscussionget an insight into the production mechanism at work in adron-nucleus in te ra c tio n s , w e h ave stu d ied  various parameters which serve as signatures for describing the origin ^^ f the observed non-thermal behaviour. These parameters arecalculated from the slope values o f moments, G  moments and r  moments. The errors shown with the data for F  , G  ^and 1 ^  moments are statistical in nature. One should note that it is not easy to calculate these errors because in most cases, the points are correlated fl5J.  However, the inclusion o f  their contribution does not change the main result appreciably as shown inRef. 116).A linear rise of In /^;  ^  ^ with InA^  is observed which exhibits A e presence of intermitted behaviour shown in the Figure 1 .Figure 1. as a funclion of InM for (f ~ 2 lo 7 lanes represcnlthe Jineai fils lo the data.The slope value ctff , a characteristic parameter o f intermittency IS computed by a linear fit o f our data in pseudorapidity space to the relationTable 1 shows the slope values o f  for corrected scaled factorial moments with order q.Table 1. Using F^  momcnls, the filled slope paramelers wilh order q found in n -AgBr interactions at 350 GcV/cOrdci q 2 3 4 5 6 70 637 1 382 2 141 2 «92 3 655 4 408± 0 004 ± 0.009 ± 0 017 ± 0 0.39 ± 0.056 ± 0 090Wc plot as a function o f ln</i> shown in Figure 2.Our experimental data exhibit the linear behaviour in the log-logTable 2, Fitted slope parameter B of moments analysis with order <y found in rr-AgBr interactions at 350 CJeV/c.Order q 2 3 4 5B•i 0.613 1 .141 1.658 2.186± 0.02 ± 0  02 ± 0.03 ± 0,02plot and now there is no problem to calculate the slope values o f the linear fit. Table 2 presents the slope value o f moments.q s 5356 Dipak Ghosh et alq ns 4q = 3q ac 2OSOLllllu0 80 1 30 1 80LLlJ-LiU 2 30ln<n>Figure 2. ln</f/> as a function of ln<«> lor = 2 to 5. Lines represent the linear fit to the dataFigure 3 depicts the variation o f  In^G^^ with InM. A linear dependence o f  In^G^^ on InM is observed indicating se lf­similarity in particle production process. The best fit curves o f  our data are drawn as smooth straight lines in Figure 3. Thefitted slope values T‘ll and are listed in the Table 3.InMFigure 3. ln<G^> as a function of InM for = 2 to 4. Lines repre.sent the linear fit to the dataTabic 3. Fitted slope parameters o f moment analysiswith order q in 7r-AgBr interactions at 350 GeV/c.Order q 2 3 40.66 ± 0.02 1.15 ± 0.04 1.38 ± 0.060.84 ± 0.02 1.74 ± 0.03 2.76 ± 0.08d^yn<^i 0.82 ± 0.03 1.41 ± 0.05 1.62 ± O.IOThe dynamical part o f  has been extracted accordinogto the formula ( 1 0 ) for each order o f  moments and is related lo the generalized fractal dim ension bySimilarly, there is no difficulty in determining the geneialiad fractal dimension d J  with the help o f  the slopes o f the linear eq .(1 3 )b y(15)Multifractality exists if  the generalized fractal dimension increases with the decrease o f  the order. Figure 4 shows the q dependence o f  for each moment satisfies the condition n! multifractality. It is also observed that for moments the pim in with q  is steepy than that o f  other two momentso Using moment •  Using moment it Using G moment0.8 0 7  0 6  0 5  0 4* i-I----1— 1  ^ "__JL_2 3 4 5 6 7qFigure 4. Generalised fractal dimension as a function of ortki <j m different moments.The relation between generalized dimension and anomalous fractal dim ension is given by definitionwhere d,, =  a ,  /(</ - 1 ) .From eqs. (14)-(17), one getsand a l = q - l - B „ .( I 6 l(1 7 i(18)(19)The relationship am ong a,^, and are differen) because the approaches o f  different m oments arc different Figure 5 exhibits that the values o f  a ^ ,  and are noi exact but all o f  them increase with order q.Peschanski has suggested that the self-sim ilar cascade can occur in different phases [17], the normal phase populated by many relatively sm all fluctuations and the spin-glass pha c^ consisting o f  a few  very large fluctuations. The condition tor the coex isten ce o f  these tw o phases o f  the cascade is the existence o f  a minimum o f  the parameter at a certain value ofthe order q  = q^.  The value o f  A^ is defined byPion fluctuation study i n n -  AgBr interactions at 350 GeV/c 357theThe regions q  < and q > q ^  correspond to the normal and spin-glass phase respectively.4 - o4 6qI'iguri’ 5. Comparison of the exponents of moments, moments ,pi)Ilusion Iree) and 7;^  moments.We can easily calculate the parameter with the help o f  (he eq (20) for these three types o f  m oments. The variation o f  with order q  is shown in Figure 6 . Our data with moments :uid T moments show flattening around <7 = 4 , but no clear minimum is seen. On the other hand, a minimum o f  A^  ^ for moments for a certain value <7 = 3 is observed. The observation o| minimum may be an indication o f  the presence o f  non-thermal phase transition.0 80 0 7 5 -  0 70A,, 0 6 5 .0 60 0 55 0 50' ►  ^ ♦ o Using moment•  Using moment♦ Using G moment2 4 6 8q5. ConclusionWe end with the concluding remark that in hadron-nucleus interactions at 350 G eV/c, the observed pattern o f against q  gives no definite indication for the existence o f  two phases.AdcnowlcdgmentThe authors would like to thank Prof. W Lock o f  CERN for providing us with the exposed and developed em ulsion plates U90d tor this work. Authors al.so gratefully acknowledge the fioancial help from the University Grants Comm ission (India) ut^er the CX).S1.ST programme.K l i f e r c n c c si n  A Bialas and K IVschanski Nik I rii\s  B273 703 (1086) ; B.^08 857 (1988)| 2 |  A Bialas in rn>r Quark M ttcr VO, Nurl r h y \  A525 345c (1991)(3) R Pcschanski hit I Mod Phws A6 3681 (1991)[4J B Buschbcck, R Lipa and R Pcschanski Pfiy.s. Lett. B215 788(1988)15 J IV  Ajincnko vt al Phys Hvv Lett 222 306 (1989)16] R Holynski Pt al Pliys Hev Lett 62 733 (1989)[7] A Biala.s and R Pcschanski Phys U tl B207 59 (1988)181 G Paladin and A Vulpiani Phys Hep 156 147 (1987)(9) R (' Hwa Phys Rev. 1)41 1456 (1990) flOJ R C Hwa and P C Pan Phys Hev T)45 1476 (1992)[11] F Takagi Phws Hev U ‘ti 72 32 (1994)[121 B C Hwa and M T Na/irov Phys Hev Ijett 69 741 (1992)[13] I) Ghosh et al Phys Hev 1)46 3712 (1092)[141 K F'lalkowski, B Wosick and J Wosick Acta Phys Polon, B20 639(1989)[15] M l  Adamovich et al Nurl Idiys B38K 3 (1992)[16J N M Agababyan et al Phvs U t^t B382 305 (1996)[17] R Pcschanski Nurl Phys B327 144 (1989)Hgurc 6. Parameter as a function of order q in different moments

Pion Fluctuation Study in π--AgBr Interactions at 350 GeV/C

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Pion Fluctuation Study in π--AgBr Interactions at 350 GeV/C

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