We demonstrate that the apparently ad hoc parametrization of the particle
production spectra discussed in the literature and used in description of
cosmic ray data can be derived from the information theory approach to
multiparticle production processes. In particular, the violation of the Feynman
scaling law can be interpreted as a manifestation of nonextensivity of the
production processes.Comment: gz-compressed .tar file containing LaTeX file and 3 PS files with
  figures (and 2 PS files with figure captions), 5 pages altogether (Nuovo
  Cimento cimento.cls style file attached) Presented at Chacaltaya Meeting On
  Cosmic Ray Physics, La Paz - Bolivia, 23-27 July 2000. To be published in
  Nuovo Cimento (Proc. Suppl.

Navarra, F. S.

Utyuzh, O. V.

Wilk, G.

Wlodarczyk, Z.

English

arXiv

We demonstrate that the apparently ad hoc parametrization of the particle production spectra discussed in the literature and used in the description of cosmic ray data can be derived from the information theory approach to multiparticle production processes. In particular, the violation of the Feynman scaling law can be interpreted as a manifestation of nonextensivity of the production processes

Navarra, F.S.

Utyuzh, O.V.

Scientific Open-access Literature Archive and Repository

IL NUOVO CIMENTO Vol. 24 C, N. 4-5 Luglio-Ottobre 2001Violation of the Feynman scaling law as a manifestationof nonextensivity(∗)F. S. Navarra(1)(∗∗), O. V. Utyuzh(2)(∗∗∗), G. Wilk(2)( ∗∗∗),and Z. Wlodarczyk(3)(∗∗∗ )(1) Instituto de F´ısica, Universidade de Sa˜o PauloC.P. 66318, 05315-970 Sa˜o Paulo, SP, Brazil(2) The Andrzej Soltan Institute for Nuclear Studiesul. Hoz˙a 69, 00-681 Warsaw, Poland(3) Institute of Physics, Pedagogical Universityul. Konopnickiej 15, 25-405 Kielce, Poland(ricevuto il 14 Settembre 2000; approvato il 12 Febbraio 2001)Summary. — We demonstrate that the apparently ad hoc parametrization of theparticle production spectra discussed in the literature and used in the description ofcosmic ray data can be derived from the information theory approach to multiparticleproduction processes. In particular, the violation of the Feynman scaling law canbe interpreted as a manifestation of nonextensivity of the production processes.PACS 96.40 – Cosmic rays.PACS 01.30.Cc – Conference proceedings.1. – IntroductionThe shape of the x = E/E0 spectra of secondaries is of great importance in allinvestigations concerning developments of cosmic ray cascades [1] (cf. also [2]). Thecrucial problem of practical importance is the existence or nonexistence of the Feynmanscalling, which says that x-spectra of secondaries are energy independent. Because incosmic ray applications one is sensitive essentially to the large x region (of, say, x > 0.1),(∗) Paper presented at the Chacaltaya Meeting on Cosmic Ray Physics, La Paz, Bolivia,July 23-27, 2000.(∗∗) E-mail: navarra@if.usp.br(∗∗∗) E-mail: utyuzh@fuw.edu.pl( ∗∗∗) E-mail: wilk@fuw.edu.pl(∗∗∗ ) E-mail: wlod@pu.kielce.plc© Societa` Italiana di Fisica 725726 F. S. NAVARRA, O. V. UTYUZH, G. WILK, ETC.the commonly used formula [1]dNdx= Da(1 − a′x)4x(1)stresses this fact by the power-like form with exponent obtained from the x → 1 limitof fragmentation data [1]. Here a′ is the parameter responsible for the Feynman scalingviolation, whereas D and a are additional parameters obtained from the fit to data —all are energy dependent (cf. [1] for details).We shall demonstrate that (1) emerges in a natural way from the information theoryapproach to hadronization processes presented in [3] and extended to allow for the possi-ble nonextensivity of the hadronization process by applying Tsallis entropy [4]. (See [5]for details of the notion of nonextensivity and [6] for its application to high energy reac-tions.)2. – Particle production spectra from information theoryIn order to describe the hadronization process in information theory, one uses [3]the least biased and most plausible single particle distribution f(y) = 1NdNdy (where yis the rapidity) resulting from a hadronization process in which a mass M hadronizesinto N secondaries of mean transverse mass µT =√µ2 + 〈pT〉2 each (for simplicity weconsider only one-dimensional hadronization with limited transverse momenta which can,however, depend on M and N). This is done maximalizing the Shannon (or Boltzmann-Gibbs) information entropy, S = − ∫ dy f(y) ln f(y), under constraints of normalization(∫dyf(y) = 1 ) and energy conservation (∫dyµT cosh yf(y) = M/N ), which leads tothe following (extensive) distribution function (cf. [3] for details):f(y) =1Z(M,N)exp [−β(M,N) · µT cosh y] .(2)Here Z(M,N) comes from the normalization of f(y), whereas the Lagrange multiplierβ(M,N) is to be calculated from the energy conservation constraint. Notice that thereis no free parameter here. This should be contrasted with the popular use of eq. (2) asa “thermodynamical parametrization” with inverse “temperature” 1/β = T being a free,positively defined parameter. In [3] it was shown that, in a wide range of energies andmultiplicities, β(M,N) ∼ NM = 1√s NK .However, comparison with data cannot be done in such model-independent way, be-cause of the fluctuations of the inelasticity K (given by the inelasticity distributionχ(K) [7]) causing fluctuations of M and because of fluctuations in the number of parti-cles N = N(M) produced from a massM given by multiparticle distribution P (N ;M) [8](its actual shape accounts here for the possible multisource structure of the productionprocess). One has thereforedNdy=∫ 10dKχ(K)∑NP (N ;K√s)N1Z(K√s,N)exp[−β(K√s;N)µT cosh y] ,(3)where β(K√s,N) is still calculated from the energy conservation constraint, but now fora given event, i.e., it is a fluctuating quantity with fluctuations resembling gamma-likedistribution (and characterised by mean value β¯ and normalised variance ωβ).VIOLATION OF THE FEYNMAN SCALING LAW AS A MANIFESTATION OF NONEXTENSIVITY 727Fig. 1. – Comparison of eq. (5) with UA5 [10] data for q = 1 and q = 072.It has been shown in [9,6] that such fluctuations of the parameter of the exponentialdistribution convert it to a power-like distribution: exp[−xˆ/x0] → expq[−xˆ/x0] = (1 −xˆ/x0/α)α with α = ±1/ωβ in our case, where xˆ = 2µT√s cosh y and x0 is a parameterconnected with the mean value of β ∼ NK .This is precisely the form corresponding to the distribution obtained using the non-extensive Tsallis q-entropy [4, 5], instead of the Shannon one (and reproducing it in thelimit q → 1), Sq = −(1 −∫dy [f(y)]q)/(1 − q) → Sq=1 = −∫dyf(y) ln f(y), togetherwith the modified constraint equation∫dyµT cosh y[f(y)]q = M/N . Using it, one getsinstead of (2) its nonextensive version:fq(y) =1Zq(M,N)[1 − (1− q)βq(M,N) · µT cosh y]11−q .(4)Nonextensivity means that the entropy of the composition (A + B) of two independentsystems A and B is equal to [5] S(A+B)q = S(A)q + S(B)q + (1− q)S(A)q · S(B)q proceedingto its usual additive form only in the q = 1 limit. It arises, whenever in the systemone encounters long-range correlations, memory effects or fractal structure of the cor-responding space-time or phase space. Such situation is expected to occur also in thehadronization processes [6].728 F. S. NAVARRA, O. V. UTYUZH, G. WILK, ETC.Fig. 2. – Comparison of eq. (5) with P238 [11] and UA7 [12] data for rapidity distributions.3. – Comparison with experimental dataWe make therefore the following conjecture: the convolution (3) can, in practice, bereplaced by the following simple one-parameter formula of the type of eq. (1) whichshould be used to describe the existing data of [10-12]:dNdy= 〈N(s)〉 1Zq[1 − (1− q)βq(√s,32〈N(s)〉)µT cosh y] 11−q.(5)The only free parameter (characterising the strenght of fluctuations in the system ac-cording to the previous discussion) is the nonextensivity parameter q. Here Zq =∫dy expq[−βqµT cosh y] and 〈N(s)〉 is the mean (single non-diffractive) charged mul-tiplicity at a given energy√s. Notice that, since βq is calculated from the energy con-servation constraint which involves all produced particles (i.e., total multiplicity), it iscalculated here for 32 〈N(s)〉 = 〈Ntotal(s)〉 particles. For the same reason care must betaken when one addresses data at√s = 630 GeV, as part of them is for charged and partfor neutral particles only. In both cases the βq must be the same (calculated for totalmultiplicity at a given energy) whereas the multiplicity in front of the formula has to bechosen according to the actual situation.In fig. 1 we show our results both for q = 1 and our best fit with q = 0.72. In all calcu-lations the experimentally observed variation of µT with energy has also been accountedfor by using the following simple interpolation formula: µT = 0.3+0.044 ln(√s/20) GeV.The results are reasonable, especially for 53 and 200 GeV. For higher energies our dis-tributions start to be broader than data and this cannot be improved by changing q, asdiminishing its value, in order to make distributions narrower, will spoil the agreementwith data for small rapidities. It turns out that UA7 and P238 data cannot be fittedtogether with UA5 data, as they demand a slightly bigger value of q = 0.85, cf. fig. 2.VIOLATION OF THE FEYNMAN SCALING LAW AS A MANIFESTATION OF NONEXTENSIVITY 729Fig. 3. – The examples of distribution (4) for mass M = 100 GeV hadronizing into N = 20secondaries of (transverse) mass µT = 0.4 GeV each.Notice that P238 data are for charged and UA7 data for neutral particles, therefore theymust be described with, respectively, 23 〈Ntotal〉 and 13 〈Ntotal〉 in eq. (5), this leads todifferences clearly seen in fig. 2.4. – Summary and conclusionsOur approach consists in noticing the striking similarity of eq. (4) to eq. (1) of[1] (dy = dx/x). Figure 3 shows the characteristic features of fq(y) for the case ofhadronization of fireball of mass M = 100 GeV into N = 20 particles of transverse massµT = 0.4 GeV. Notice that for q < 1 one obtains an increase of particle densities inthe central region connected with its decrease at the edges of rapidity range — clearlyresembling the characteristic pattern of Feynman scaling violation. We have replacedtherefore the “exact” formula (3) by a one-parameter fit represented by (5) where qsummarizes the action of the averaging over the fluctuations caused by the reaction initialconditions which are represented by the inelasticity and multiplicity distributions. Wefind it very amazing that such simple approach coincides practically with the empiricalformula (1) and with only one parameter q describes all data fairly well. Notice thatq = 0.72 is not very far from q = 0.75, which gives power 1/(1−q) = 4 in (1). Notice alsothat our general formula allows for the description of the small x region as well. It wouldcertainly be interesting to connect q directly to the parameters describing inelasticity andmultiplicity distributions or to descriptions of leading particles, for example of the typeof that presented in [13]. Our results seem to indicate that most probably the parameterq should be x-dependent reflecting the different character of fluctuations of the quantityN/M [9] in the central (mostly pionization) and fragmentation regions. This problemwill be addressed elsewhere.∗ ∗ ∗Authors ZW and GW are grateful to the organizers of the Chacaltaya Meeting onCosmic Ray Physics for their support and hospitality. The partial support of the Polish730 F. S. NAVARRA, O. V. UTYUZH, G. WILK, ETC.Committee for Scientific Research (grants 2P03B 011 18 and 621/E-78/SPUB/CERN/P-03/DZ4/99) is acknowledged.REFERENCES[1] Ohsawa A., Prog. Theor. Phys., 92 (1994) 1005 and Ohsawa A. et al., this volume p. 705.[2] Wdowczyk J. and Wolfendale A. W., J. Phys. G, 10 (1984) 257; 13 (1987) 411.[3] Wilk G. and Wlodarczyk Z., Phys. Rev. D, 43 (1991) 794.[4] Tsallis C., J. Stat. Phys., 52 (1988) 479; for updated bibliography on this subject seehttp://tsallis.cat.cbpf.br/biblio.htm.[5] Special issue of Braz. J. Phys., 29 (1999) (available also at http://sbf.if.usp.br/WWW pages/ Journals/ BJP/ Vol29/Num1/index.htm).[6] Wilk G. and Wlodarczyk Z., preprint The imprints of nonextensive statisticalmechanics in high energy collisions, hep-ph/0004250.[7] Wlodarczyk Z., J. Phys. G, 21 (1995) 281; cf. also: Dura˜es F. O., Navarra F. S. andWilk G., Phys. Rev. D, 50 (1994) 6804.[8] Cf., for example, Hegyi S., Phys. Lett. B, 466 (1999) 380; 467 (1999) 126.[9] Wilk G. and Wlodarczyk Z., Phys. Rev. Lett., 84 (2000) 2770.[10] Baltrusaitis R. M. et al., Phys. Rev. Lett., 52 (1993) 1380.[11] Harr R. et al., Phys. Lett. B, 401 (1997) 176.[12] Pare E. et al., Phys. Lett. B, 242 (1990) 531.[13] Dura˜es F. O., Navarra F. S. and Wilk G., Phys. Rev. D, 55 (1997) 2708; 58 (1998)0940034.

Violation of the Feynman scaling law as a manifestation of non-extensivity

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Violation of the Feynman scaling law as a manifestation of nonextensivity

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