46 Part Ó. Cumulative processes Z-SCALING AND FRACTAL CHARACTER OF HADRON-HADRON AND HADRON-NUCLEUSCOLLISIONS ÀÒ HIGH P.L 1. Zborovskylt, Ì.Ó. Òîøåó2~, Óè.À. Panebratsev2 G.P. Skoro3 (1) Nèc/ear Physics Institète, Academyî/ Sciences î/ the Czech Repèb/ic,Rei, Czech Repèb/ic (2) Labomtory î/ High Eneryies, Joint Institète /îò Nèc/ear Research, 41980, Dèbna, Moscowregion, Russia (3) Institète î/ Nèc/ear Sciences"Vinca", Facè/ty î/ Physics, University î/ Be/grade, Be/grade, yègos/avia t E-mai/: zborovsky@èjf.cas.cz Q Å-òàè: tokarev@sènhe.jinr.ru Report: http://re/np.jinr.ru/ishepp/xiv/trj032/ Abstract New scaling,z-scaling, in the inclusiveparticle production in ðÀ collisionsis stud- ied. The conceptof z-scalingis based îï the fundamentalprinciples such as self- similarity, locality, scalerelativity and fractality reflecting the generalfeaturesof particle interactions. The scaling function HA(z) is expressed via the invariant cross section ÅdÇî-jdqÇ and the average multiplicity dellsity dNjd1/ of particles produced at pseudorapidity 7) = î in the corresponding nucleon-nucleon interac- tion. It is shownthat the available experimental data îï cross sections in ðÀ col- lisions confirm the scaling properties îÑ the function HA(z). We suggestto èâå z-scaling as an instrument for search new ðhånîùånà in various inclusive pro- cesses at SppS(CERN), Tevatron(Fermilab) , DESY(Hamburg) , SLAC(Stanford) , RHIC(BNL); and LHC(CERN). 1 Introduction Common featères of the particle prodèction at' high energy /S and high transverse momenta (q.L> 1 GeV jc) indicate the local characterof hadron interactions. It leadsto the conclèsion aboèt dimen8ionless of the constitèents taking part in the interactions. The fact that the interaction is local foènds its natèral manifestation in the scale-invariance of the hadron interactions' crosssections. The invariance is an expression of self-similarity principle [1, 2]. This principle ref\ectsthe dropping of certain dimensionalqèantities or parameters oèt of the physicalpictère of the processes. Up to date, the investigations of properties of high energy nèclear interactions have revea\ed widely knownscaling laws. Some of the most popèlar and famoès are the Feynman scaling [3] for inclusiveparticle prodèction, y-scalingobserved in deep inelastic scattering îï nèclei [4], limiting fragmentationfoènd for nèclei fragmentation [5], ò.L scaling [6, 7], scalingin cèmèlative particle prodèction [1, 8, 9], KNO scaling [10] and others. However, detailedexperimental stèdy hasshown certain violations of these. It ñàï Üåconnected with the dynamicsconcerning the transition from the pertèrbative QCD qèarks and glèons to the observed hadrons.
Transcript
46 Part Ó. Cumulative processes
Z-SCALING AND FRACTAL CHARACTER OF HADRON-HADRON ANDHADRON-NUCLEUS COLLISIONS ÀÒ HIGH P.L
AbstractNew scaling, z-scaling, in the inclusive particle production in ðÀ collisions is stud-ied. The concept of z-scaling is based îï the fundamental principles such as self-similarity, locality, scale relativity and fractality reflecting the general features ofparticle interactions. The scaling function HA(z) is expressed via the invariantcross section ÅdÇî-jdqÇ and the average multiplicity dellsity dNjd1/ of particlesproduced at pseudorapidity 7) = î in the corresponding nucleon-nucleon interac-tion. It is shown that the available experimental data îï cross sections in ðÀ col-lisions confirm the scaling properties îÑ the function HA(z). We suggest to èâåz-scaling as an instrument for search new ðhånîùånà in various inclusive pro-cesses at SppS(CERN), Tevatron(Fermilab) , DESY(Hamburg) , SLAC(Stanford) ,RHIC(BNL); and LHC(CERN).
1 Introduction
Common featères of the particle prodèction at' high energy /S and high transversemomenta (q.L > 1 GeV jc) indicate the local character of hadron interactions. It leads tothe conclèsion aboèt dimen8ionless of the constitèents taking part in the interactions. Thefact that the interaction is local foènds its natèral manifestation in the scale-invariance ofthe hadron interactions' cross sections. The invariance is an expression of self-similarityprinciple [1, 2]. This principle ref\ects the dropping of certain dimensional qèantities orparameters oèt of the physical pictère of the processes.
Up to date, the investigations of properties of high energy nèclear interactions haverevea\ed widely known scaling laws. Some of the most popèlar and famoès are the Feynmanscaling [3] for inclusive particle prodèction, y-scaling observed in deep inelastic scatteringîï nèclei [4], limiting fragmentation foènd for nèclei fragmentation [5], ò.L scaling [6, 7],scaling in cèmèlative particle prodèction [1, 8, 9], KNO scaling [10] and others. However,detailed experimental stèdy has shown certain violations of these. It ñàï Üå connected withthe dynamics concerning the transition from the pertèrbative QCD qèarks and glèons tothe observed hadrons.
In the paper we exploit the concept based îï the self-similarity of the elementary in-teractions complemented Üó considerations about fractal structure of the colliding object~.ÒÜå ideas are implcmented into the construction of new scaling, the z-scaling, for the de-scription of inclusive particle production in ðÀ interactions at high energies. ÒÜå scalingwas applied for the analysis of ðð and ÐÐ collisions in the energy range .;s > 23 Ge V inRef. [11]. ÒÜñ scaling function H(z) is expressed via the invariant inclusive cross sectipnEdçè / dqÇ and the multiplicity density of charged particles dN / dl] = ð( à) produced at thepseudorapidity 1] = î. It was found that the H(z) is independent of colliding energy .;sand angle ä of the inclusive particle. In the ñàâå of hadron production the scaling functionH(z) is interpreted as the probability to form hadrons with à formation length z. ÒÜåuniversality of H(z) means that the hadronization mecllanism is of universal nature. Wesuggest that the difference between the H(z) for ðð and/or HA(z) for ðÀ collisions îï înåside and thc HAA(Z) for ÀÀ interactions îï thc other side ñàï give definite evidence aboutthe character of nuclear matter influence îï the proces.., of particlc production. We proposethat the dependence of Í ÀÀ ( Z) îï z for hadronic and QG Ð phases of nuclear matter ñàï Üå
quantitatively distinguished.
2 General principles of z-scalingWe start with the investigation of the inclusive process
Ì1 + Ì2 -+ m\ + Õ, (1)
where M1 and Ì2 are masses of the colliding nuclei (or hadrons) and ò" is the mass of theinclusivc particle. In accordance with Stavinsky's ideas [8] the gross features of the inclusiveparticle distributions for the reaction (1) at high energies ñàï Üå described in ternlS of t.hecorresponding kinematical characteristics of the exclusive subprocess
ÒÜå parameter m2 is introduced in connection with internal conservation laws (for isospin,baryon number, and strangeness). The Õ\ IUld Õ2 are the scale-invariant fractions of theincoming four-momenta Ð, and Ð2 of the colliding objects. ÒÜå energ.y of the part.on sub-process defined as
S~/2 = V(XIPI + Õ2Ð2)2 (3)
represents the center-of-mass energy of the constituents ta.king part in the col1ision, Thecross section for the production of the inclusive pal,ticle is governed Üó the minimal E'nergyof col1iding partons
dO'/dt"" 1/s?".n(:cl,x2), (4)
The corresponding energy s~.~ is fixed l\S minimum of Eq. (3) which is n()CCSSary for crt'utionof the secondary particle with mass ò\ and à four-momentum q.
2.1 Momentum fractions Õl and Õà
Let us consider the elementary parton-parton collision l\S à binnry subproccss \\'hich isà subject to the condition
(XIP\ + Õ2Ð2 - q)2 = (õ\Ì\ + ;r2M2 + ò2)2. (5)
48 Part v. Cumulative processes
ÒÜå relationship between Õ\ and Õ2 ñàï Üå conveniently written in the form
Considering the process (2) as à parton-parton collision, we introduce the coefficientn whichconnects kinematical with dynamical characteristics of the interaction. ÒÜå coefficient ischosen in the form
f!(Xl,X2)=ò(1-Xl)6'(1-X2)62, (.8)
where ò is à màââ constant and 81 and 82 are factors relating the fractal structure of thecolliding objects. We determine the fractions Õ1 and Õ2 in à way to maximize the value of
ÙÕ1,Õ2),
dÏ(i!,Õ2)!dõ! =0, (9Ósimu!taneous!y fu!fi!!ing the condition (6);
The variabIes Õ! and Õ2 satisfy the hypothesis [8] of minimum recoi! mass (5) in thee!ementary constituent interaction. Both are equa! to unity a!ong the phase space !imit.From the conditions Õ; ~ 1 we get the restriction
(Ì! + Ì2 + ò2f + Å2 - ò~ ~ (V/S"A'- Ef. (10)
The symbo! VSA stands for the center-of-mas energy of the ðÀ system. The !ast inequa!itybounds kinematica!!y the maximal possibIe energy Å of the inc!usive partic!e ò! in thec.m.s. of the reaction (1). The meaning of the parameter ò2 as the thresho!d for theproduction of the inc!usive partic!e ò! emerges here in the äàturà! way.2.2 Scaling variable z and scaling function H(z)
In accordance with the self-similarity princip!e we search for the so!utlon depending îïà sing!e sca!ing variabIe z in the form
1 dO'ô(z) == -. (11)< N > Uinel dz
Here O'inel is the ine!astic ctoss section, < N > is the average mu!tip!icity and ô(z)has tbÜå à scaling function. The quantities refer to the ðÀ interactions. The invariant differentia!cross section for the production of the inc!usive partic!e ò! ñàï Üå expressed via the sca!ingfunction H(z) in the way [12]
H(z) ~ -2 ô(z) ~( \ \ -
) E-d ;, (12)1TZ 9 Ël.' Ë2 PAUinel q
where, the factor 9 is given Üó .'
g(Ël, Ë2) ~ (SA)-1 Z (ë;;läz/äËl + Ël1äz/äË2) . (13)
Here, the s is square of the center-of-mass energy of the corresponding N N system, À isatomic number, and PA(S,1J) ~ d < N > /d1J is the multiplicity density.
where Eiin is the transverse kinetic energy of the subprocess (2), ð(â) is the multiplicitydensity in N N collisions at ò/ = Î.2.3 Fractality and scale relativity
ÒÜå asymmetry of the incoming objects is characterized Üó the parameter à = 82/81
which is the ratio of the fracta! dimensions 82 and 81. In the considered case of proton-nucleus interactions, the va\ue of à is chosen to Üå atomic number À. ÒÜå re!ation ref\ectsessentia! feature of the fracta! structure of nuc!ei in the z-sca\ing scheme. ÒÜå fracta\dimensions ratio describes the re!ative reso!ution (ve!ocity) of one fracta\ structure withrespect to the other.
Ve!ocity variable p!ays particular ro!e in respect of the fracta! dimensions. In the con-struction of se!f-consistent and se!f-simi!ar so!ution of the problem we have to re!y îï specia\re!ativity which yie!ds the !imitation of any ve!ocity. Besides the !aws of motion, the re!a-tivity princip!e app!ies to the !aws of sca\e [13], as we!!. It states that the !aws of Natureàãå identica\ in à!! scale-inertial systems. In our case à p!ays the òî!å of à parameter which!àÜål sing!e sca!e-inertia! frames.
The ve!ocities have their origin in the asymmetry of the prob\em and vanish in theco!!isions of objects which have equa! fractal structures (à = 1). Physica\!y, the asymmetry
îññèòý as à consequence of the richer parton content of nuc!eus in comparison with that ofthe sing!e nuc!eon. Relative reso!ution ve!ocity of one fracta\ structure with respect to theother is given Üó [12]
a-lv--- a+l'
For transformations of the velocities it ñàï Üå written
provided
(õ = (Õ'(Õ2. (17)
This leads us to conclude that while the composition of velocities follows Einstein-Lorenzlaw, the composition of the corresponding fractal dimensions follows the multiplicative grouplaw. The relative resolution (õ màó Üå defined as à relative state of scale of reference system[13]. In the considered perspective, the principle of scale relativity states that Einstein-Lorenz composition law of velocities applies to the systems of reference whatever their stateof scale.
3 Z-scaling in pA-collisions
Before analyzing the results îï z-scaling in ðÀ systems, we would like to remind mønfeatures of the scaling concerning the inclusive particle production in nucleon-nucleon in-teractions. In Fig. 1(à) we present the function H(z) for charged hadrons produced in thecentral region of ðð and ÐÐ collisions at .j8 = 19 - 1800 GeV. The result demonstrates
the independence of scaling function H(z) îï colliding energy.j8. Note that the data at.j8 = 630 GeV cover the kinematic range of the transverse momenta of secondary particles
z = i:kjn /(ï. p(s»,
Vl+V2V= ,
1 + VIV2
50 Part v. Cumulative processes.
èð to qJ. = 24 GeV /ñ. For the comparison with the z-presentation, the òJ. -dependence ofthe same data is shown in Fig. I(Ü). We wou!d !ike to note that the òJ. -presentation is tra-ditionally used to describe the particle spectra in à restricted transverse momentum range(say pJ. < 1 - 2 GeV /ñ). Înå ñàï âåå that the invariant cross section does not indicate ànóuniversality as à function of òJ. when p!otted for different energies VS. Simi!ar dependenceof H(z) for 1T--meson production at vs = 53 GeVand () = 2.86° - 90° c.m.s. is shown inFig. 2(à). Here we have used the value of ð,,(2.860) = 0.3 for the ang!e () = 2.86° in Eq. (12)which corresponds to the data [14] îï mu!tip!icity densities in the fragmentation region.The approximate angu!ar independence of the scaling function is in contrast with Fig. 2(Ü)where the âàòå data àãå p!otted as à function of the transverse mass òJ.. Distinctive dif-ferences between the fragmentation and the centra! region represent various norma!izationsand s!opes of the spectra. Let us stress that the energy and the angu!ar universa!ity of thez-sca!ing for ðð and ÐÐ co!!isions was achieved with the same óà!èå of the fracta! dimension01 "" 0.8. We ì!! use, therefore, this number in the anà!óâø of the z scaling in the ñàâå ofthe ðÀ systems.
We have examined the experimenta! data îï four nèñ!åì targets D, Âå, Ti, and Wcovering à wide range of atomic weight [12]. In the considered experiment [15] the measure-ments were made at à !aboratory ang!e of 77 mrad, which corresponds to the ang!es nåàã900 in the center-of-mass system of the corresponding nuc!eon - nucleon co!!isions. First,we exp!oit the data îï the inc!usive 1T-meson production in the ð + d --+ 1Ò+ + Õ processat incoming momentum ð = 400 GeV /ñ. The data àãå expressed in terms of the functionHd(Z) whicll depends îï the scaling variable z as presented in Fig. 3(à). The resu!t showsthat the description in terms of z-representation coincides with good accuracy with the scal-ing function for proton-proton co!!isions. We have checked the sensitivity of such universalbehaviour with respect to the ratio of fracta! dimensions é. The situation is depicted Üótwo dashed !ines in the figure. The value of é = 2 is distinguished with regard to the zscaling universality. The obtained resu!ts confirm z-scaling in pd co!!isions at high energies.
The dependence of the function HBe(z) îï Z for the process ð + Âå --+ 1Ò+ + Õ at theincident proton momenta ð = 200, 300, and 400 Ge V / ñ is presented in Fig. 3(Ü). Simi!arbehaviour for heavier nuclei as titanium (À=48) and tungsten (À=184) àãå obtained [12].As we ñàï âåå from Fig. 4, the function Hw(z) exhibits energy independence in contrast tothe behaviour of Ed30'/dq3 as à function of òJ..
Thus, besides the energy independence, the A-universa!ity of the sca!ing functions isfound. The obtained resu!ts give us strong argument to use z-scaling forma!ism for theanalysis of experimental data îï inc!usive cross sections in ðÀ interactions.
In order to construct the scaling function HA(z) for partic!e production in ðÀ inter-actions, it is necessary to know the values of the average mu!tip!icity density PA(S, TJ) ofsecondaries produced in ðÀ co!!isions. At present there àãå noexperimental data îï ÐÀ forðÀ collisions at high enough energies (vs > 20 GeV). Therefore, the Monte Car!o simu!a-tions were used to determine the energy dependence of the mu!tip!icity densities of chargedpartic!es for different nuclei [12]. The obtained values ñàï Üå parametrized Üó the formu!a
PA(s)~0.67.Ao.18'SO.I05, À;::: 2. (18)
In the ñàâå ofpp or ÐÐ co!!isions, the fit p(s) = 0.74s01O5 is used [16,17].
4 Results and discussion
We focus our study to the regime of !îcal parton interactions of incident hadrons and
nuc!ei. It manifests itse!f in the production of partic!es with high q.L at high energies. In thisregime the parton distribution functions of incoming objects àãå separated and, therefore,the sca!ing function H(z) refiects the features of the fragmentation process of producedpartons into the observabIe hadrons.
Our approach is based îï the princip!es of se!f-simi!arity, !oca!ity and fracta!ity. Fracta!character in the initia! state regards the parton composition of hadrons and nuc!ei and revea!sitse!f with òîãå reso!ution at high energies. Leading Üó these princip!es, we construct thevariabIe z according to Eq. (14). The cross section ñàï Üå expressed in therms of z asfollows
~Î-EdQ3 '"" g(-\.I,-\.2). H(z). (19)
The scaling properties of the function H(z) give èý ýîmå arguments to relate it with thefragmentation function Dh(Zq),
H(z) '"" Dh(Zq). (20)
The relation reflects universality of hadronization mechanism with the variable z consideredas à hadronization parameter.
Figure 1: (à) Scaling function H,,(z) {îã the charged hadrons in central region produced inðð îã ÐÐ interactions at .;s = 19 - 1800 GeV. Detection angle  is 900 c.m.s except the dataat .;s = 63 GeV, where  = 500; (Ü) The corresponding inclusive differential cross sections as
functions ofthe transverse mass ò-L. Experimental data àãå taken {ãîò Refs. [15. 18. 19.20,21].
Really, z ñàï Üå interpreted in terms of parton-parton collision with the subsequentformation of à string stretched Üó the leading quark out of which the inclusive hadronis formed. The energy of the colliding constituents 5;/2 is just the cnergy of the stringwhich connects the two objects in the final state of the subproccss (2). The string cvolvesfurther and splits into pieces. The resultant number of the string pieces is proportional tonumber or density of the final hadrons measured in experiment. It is experimentally knownthat particle multiplicity is proportional to the excitation of transverse degrees of freedom.Therefore, the string transverse energy is à measure of multiplicity. Such ideas allow us tointerpret the ratio
Figure 2: (à) Scaling function Hp(z) for rr--meson production in the central and fragmentationregions at JS = 53 GeV; (Ü) The corresponding inclusive differential cross sections as functions
of m.L. Experimental data are taken from Refs [20. 22].
as à quantity proportionaI to the transverse energy of à string piece VISh, which does notsplit already, but during the hadronization converts into the observed hadron. The processof string splitting is self-similar in the sense that the leading piece of the string forgetsthe string history and its hadronization does not depend îï the number and behaviour ofother pieces. The factor n in the definition of z reflects fractaI structure of the collidingobjects and represents degree of"softness" ofthe initial partons participating the elementaryinteraction. MaximaI softness corresponds to the maximal tension of the generated stringwhat is expressed Üó the condition (9). Then we write following relation
JSh= n. z. (22)
80, in the inclusive hadron production, we consider the variable z as à quantity proportionaIto the length of the elementary string, or to the formation length, îï which the inclusivehadron is formed from its QCD ancestor.
10" 10' 10" 10'Z Z
Figure 3: The scaling functions Hp(z), Hd(Z) (à) and HBe(z) (Ü) for 1T+-meson production. Thecurves (à) illustrate sensitivity of the deuteron z scaling function with respect to the parameterà. Experimental data are taken from Ref. [15].
Figure 4: (à) The normalized inclusive differential cross sections for 1[+-meson production inpW interactions for various incoming proton momenta as à function of the transverse mass ò1-.Experimental data àãå taken from Ref. [15]. (Ü) The corresponding scaling function Hw(z).
The comp!ementary interpretation of the physical meaning of the variable z is basedîï ideas concerning fractality in high energy co!!isions. The fractal objects are usual!ycharacterized Üó power !aw dependence of their fractal measures [13]. The fractal measure,considered in our ñàçå, is given Üó aI! possible configurations of e!ementary interactions that!ead to the production of the inc!usive partic!e. It has the fo!!owing form
ÙÕ1,Õ2) ""' (1- Xl)6'(1- Õ2)6,.
The formu!a expresses the factorization of the fracta! measure with respect to the fractalmeasures of co!!iding objects. Both are described Üó power !aw dependence in the spaceof fractions {Xl,X2}, The sing!e measure reflects number of constituent configurations inthe colliding object taking part in production of the inc!usive partic!e. The measure ischaracterized Üó the fractal dimension Î. Fractal dimensions ñàï Üå different for variousco!!iding objects. Resu!ts of our analysis show that the fractal dimension of nuc!eus ÎÀ isre!ated to the nuc!eon fractal dimension Îí Üó the fo!!owing simp!e form
The relation reflects the additivity of fractal dimensionspicture, the number of initial configurations (8) is maximized and the variabIe
z=Ålin/(ï.ð(s))
ñàï Üå interpreted àç the energy of elementary constituent collision per one initial configu-ration and per one produced particle.
The situation in nucleus-nucleus systems òàó Üå different. In the ñàçå, the excitation ofnuclear medium is realized in an extended volume which ñàï significantly influence particleproduction mechanism. We consider that the dependence of HAA(Z) îï Z for hadron andQCD ðhàçes can Üå quantitatively distinguished. Possible violations of the scaling, especiallyin the region of high transverse momenta., could Üå very interesting. Here one ñàï expect àmanifestation of the transition of nuclear ma.tter to parton ðhàçå. The transition correspondsto the joining of partons from different nucleons of nuclei known àç cumulative process
p+W-ï++X
1-4.4' ;" 100., 400 GeV/c, 300 G.V/c, 200 GeV/c
p+w-,,++xI
ð+ð-ï++Õ1-4.4' in lob. 400 GeV/". 300 GeV/". 200 GeV/"
î
~.....~
,Î...
.. :
[1,8, 9]. The higher stage of cumu!ation corresponds the !arger va!ues of the variable z andñàï manifest itse!f more prominent just in the high momentum tai! of the spectrum. Asà resu!t the enhance of the sca!ing function HAA(Z) for the nuc!eus-nuc!eus interactions incomparison with the scaling found in ðð and ðÀ co!!isions màó Üå expected. We suggest,therefore, that the comparison of the z-sca!ing for ðð and ðÀ co!!isions with data îï ÀÀinteractions ñàï give valuable information regarding exotic physica! phenomena such asquark-g!uon p!asma formation and others.
5 Conclusions
Inclusive particle production in ðÀ collisions at high energies in terms îÑ the z-scaling isconsidered. The scaling Cunction HA(z) is expressed via the invariant inclusive cross sectionÅdÇè / dt and is normalized to the multiplicity density îÑ particles produced in ðÀ collisions.The definition îÑ the scaling variable z includes fractal properties îÑ the colliding objects.The dynamical ingredient îÑ the scaling relates to the energy dependent scale which is theaverage multiplicity density îÑ charged particles produced in the central pseudorapidityregion in the corresponding N N interaction.
ÒÜå observed A-dependence îÑ available experimental data Cor different nuclei (D, Âå,Ti, and W) does not violate the general Ceatures îÑ the z-scaling. It was shown that thefractal dimensions îÑ nuclei are expressed via the fractal dimension îÑ nucleon áÀ = À. áN.Our analysis confirms the energy independence îÑ the scaling functions HA(z). ÒÜèâ, thescaling function H(z) demonstrates møn Ceatures îÑ the hadronization process in terms îÑthe Cormation length z.
ÒÜå z-scaling Cound in ðð and ðÀ collisions reflects general properties îÑ the particleproduction mechanism âèñÜ as selC-similarity, locality, scale-relativity and fractality. It ñàïserve as an effective tool in searching Cor new physical ðÜånîmånà at existing acceleratorsand in Cuture experiments planed at RHIC (BNL) and LHC (CERN).
Acknow ledgments
This work Üàâ Üåån partially supported Üó the Grant îÑ the Czech Àñìñmó îÑ Sciences No.1048703.
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