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High–Energy Multiparticle Distributions and a Generalized Lattice Gas Model

arXiv:hep-ph/9307306v1 20 Jul 1993High–Energy Multiparticle Distributions and a Generalized Lattice Gas ModelA. D. Jackson(1),(2), T. Wettig(1),(2) and N. L. Balazs(2)(1)NORDITA, Blegdamsvej 17, DK–2100 Copenhagen Ø, Denmark(2)Department of Physics, State University of New York, Stony Brook, NY 11794-3800(November 21, 2018)A simple lattice gas model in one dimension is constructed in which each site can be occupiedby at most one particle of any one of D species.

Particles interact with a randomly drawn nearestneighbor interaction. This model is capable of reproducing the factorial moments observed in high–energy scattering.

In the limit D →∞, the factorial moments of the negative binomial distributionare obtained naturally.PACS numbers: 13.85.Hd,13.65.+i,24.60.Lz,25.75.+rFactorial moments have provided a useful tool for the analysis of high–energy scattering data as obtained in e+e−scattering [1], p¯p scattering (at energies up to 900 GeV) [2] or the scattering of protons or heavy ions by heavy nucleiat a projectile energy of 200 GeV/A [3]. One considers a range of some variable (usually the rapidity) for whichone knows the average multiplicity, ⟨n⟩, and its dispersion, ⟨∆n2⟩.

The data is broken into M equal bins, and oneconstructs the factorial moments asFq(M) ="1MMXi=1⟨n(n −1)(n −2) . .

. (n −q + 1)⟩i# ,"1MMXi=1⟨n⟩i#q.

(1)Empirical factorial moments found in such high–energy scattering experiments are well reproduced by the ‘negativebinomial distribution’ (NB) for whichF NBq(M) = (1 + cM)(1 + 2cM) . .

. (1 + [q −1]cM)(2)withc = ⟨∆n2⟩−⟨n⟩⟨n⟩2.

(3)The form of Eq. (2) invites the consideration of ‘universal’ plots of Fq versus F2 which permit the comparison of datafrom very different processes [4,5].

Such plots have been made over the available range 1 < F2 < 1.8. They revealboth a remarkably universal behaviour and striking agreement with the negative binomial distribution.

Giovanniniand Van Hove [6] suggested that these results could be understood ‘in terms of partial stimulated emission of bosons,or of a simple form of cascade process, or (more artificially) with both mechanisms’.The most interesting feature of Eq. (2) and the data which it fits is the growth of the factorial moments withincreasing cM.

This has sometimes been regarded as evidence for the presence of fluctuations on many differentscales [7] with the related possibility that this may indicate the presence of critical phenomena. Chau and Huang [8]have offered an alternate view.

They identified the full range of rapidity with the N sites of a one–dimensional Ising(or lattice gas) model. They constructed the factorial moments in this model analytically (for N →∞) and set thetwo parameters of the Hamiltonian by fixing the global values of ⟨n⟩and ⟨∆n2⟩.

The resulting factorial momentsagree through O(M) with Eq. (2) and underestimate higher terms.

Agreement in the constant and linear term ofFq(M) is actually a trivial consequence of (i) the fixing of ⟨n⟩and ⟨∆n2⟩and (ii) the fact that the Ising model predictsmany–body correlations of a finite range [9].Our purpose here is to demonstrate the surprising result that an elementary extension of the lattice gas modelpermits exact replication of F NBq(M) for all q and M. This extension is simply stated: Construct a one dimensionallattice gas model (in the N →∞limit) where each site can either be empty or occupied by one particle which can be ofany of D species. Each species has a chemical potential, µd, and each pair of species has a nearest neighbor interactionof strength ǫdd′.

This model can be solved analytically for the q–body correlation functions and factorial momentsusing textbook techniques. As usual, this involves the construction and diagonalization of a matrix, M, related tothe partition function for one pair of adjacent sites.

The parameters in this model can be chosen to reproduce theresults of Chau and Huang for any D. Another choice of parameters will be shown to reproduce the NB results ofEq. (2).Solution of this model involves a (D + 1)–dimensional real, symmetric matrix, M:1

M00 = 1M0d = exp [−µd/2](4)Mdd′ = exp [−ǫdd′ −µd/2 −µd′/2]where matrix indices run from 0 to D [10].We also require the number operator at site i, ni = 11 −T , withTdd′ = δd0δd′0. We shall arbitrarily set all chemical potentials equal to µo.

In the thermodynamic limit (both Nand µo →∞), it is sufficient to consider the D–dimensional submatrix,¯M, obtained by neglecting the 0–th row andcolumn of M. The coupling of¯M to the remaining elements of M can then be treated exactly using first orderperturbation theory. Correlation functions and factorial moments can be determined in terms of the diagonal form,¯Md, and the related orthogonal matrix θ.

(Where¯M is given as θ ¯MdθT .) For example, the two–body correlationfunction for microscopic sites i and i + j is⟨nini+j⟩= ⟨ni⟩[⟨ni⟩+DXk=1a2k¯λjk](5)where the ¯λk are the eigenvalues of¯M [11] and the a2k are normalized coefficients which follow from the eigenvectorsof¯M asak = N11 −¯λkDXi=1θik(6)where N is chosen such that P a2k = 1.

The second factorial moment is then given asF GLG2(M) = 1 + cM(7)withc =2⟨n⟩DXk=1a2k¯λk1 −¯λk. (8)After considerable algebraic effort, one finds the following expression for the factorial moments of the generalizedlattice gas model:F GLGq(M) =q−1Xk=0q!k!

(q −k)! "dkdzk [1 +∞Xµ=1zµsµ](q−k)#z=0M k .

(9)Here, we have introduced the definitionsµ =1⟨n⟩µDXk=1a2k¯λk1 −¯λkµ(10)which implies s1 = c/2.Eqs. (9) and (10) represent the primary results of our generalized lattice gas model.

There are two special cases ofinterest. First, ifsµ =c2µ,(11)one immediately obtains the results of Chau and Huang for any D. This can be realized either when the sums ofEq.

(10) contain only one term or when all the eigenvalues are equal. The second special case corresponds to thechoicesµ =cµµ + 1 .

(12)With this choice, Eqs. (9) and (10) reduce to the NB results of Eq.

(2) [12].2

It remains to be seen if the parameters in¯M can be selected so that the constraints of Eq. (12) are satisfied.

Thefollowing simple prescription works: Set all the off–diagonal elements of¯M equal to zero. Make a draw of D randomnumbers, xd, from the interval [0, 1].

Set¯Mdd equal to xdL. For each draw, choose L such that s1 = c/2.

(Thissets the dispersion to its empirical value for each draw.) The physical content of this prescription is clear.

Identicalparticles experience a nearest neighbor interaction which ranges from ≈−µo to +∞. Inequivalent particles experiencea nearest neighbor interaction in the range −µo ≪ǫdd′ ≤+∞.

More ‘democratic’ schemes can also be constructed.They share the feature that¯M is sparse with roughly D (randomly selected) elements non–zero [13].Our prescription meets the remaining conditions of Eq. (12) with increasing accuracy as D →∞.

We shall illustratethis with a numerical study of the first seven factorial moments. We take ⟨n⟩= 20 and ⟨∆n2⟩= 110 as would beappropriate for the description of p¯p scattering at 200 GeV.

For these data, c⟨n⟩= 4.5. (Qualitatively similar resultsare obtained for 900 GeV p¯p scattering.) We consider the ratiosrq =(q + 1) PDk=1 a2k¯λk1−¯λkq[c⟨n⟩]q.

(13)These ratios should be rq = 1 to reproduce the negative binomial distribution. Our constraint ensures r1 = 1.

Foreach value of D, we have drawn 105 matrices according to the prescription above. In Table I we report the ‘ensembleaverage over theories’, ⟨⟨rq⟩⟩, and its dispersion for 2 ≤q ≤6 [14] as obtained for D = 1, 2, 4, .

. .

, 512. Since thereare no parameters to adjust, this is an extremely stringent test of our prescription.

It succeeds.Several comments are in order. For fixed q, the value of ⟨⟨rq⟩⟩approaches 1 like 1/D as D →∞.

The dispersionalso vanishes (like 1/√D). Thus, as D becomes large, our simple prescription converges to the results of the NBfor any fixed q.

For fixed D, the error in ⟨⟨rq⟩⟩and its dispersion grow as q →∞. The value of D = 16 results insufficiently small errors and dispersions that randomly drawn dynamics have a high probability of reproducing theempirical factorial moments for p¯p scattering at 200 GeV within existing experimental uncertainties.

This value ofD = 16 is also sufficient to provide a quantitative description of factorial moments for 900 GeV p¯p scattering and,indeed, of all other high–energy scattering experiments for which factorial moments are known. Our point here isthat there exists at least one simple prescription for satisfying Eq.

(12). Other more efficient prescriptions may wellexist.We have shown that a simple one–dimensional lattice gas model with D species and randomly drawn nearest neigh-bor interactions between equivalent species can reproduce the factorial moments of the negative binomial distributionas D becomes large.

Given the limited range of cM covered by current experimental data, a remarkably small numberof species is sufficient to provide a quantitative description of the empirical factorial moments. This offers someunderstanding for both the success of cascade calculations and the anecdotal observation that the results of suchcalculations are often surprisingly insensitive to the details of the model.

In the present picture, any of our randomlydrawn theories would also be likely to succeed (at least at the level of the factorial moments).The empirical observation that factorial moments grow like powers of cM in p¯p, e+e−and relativistic heavy ioncollisions (the phenomenon of intermittency) has often be taken as evidence of the existence of fluctuations on ‘alllength scales’. As such, it is sometimes seen as an indicator of the presence of a non–equilibrium, critical phenomenon.While we do not deny the possibility that intermittency can be a signature of critical phenomena, we have shownthat a simple but highly heterogeneous (equilibrium) system can also lead to intermittency.

In short, intermittencyis not a unique signature of critical phenomena. Given the small number of species required by our model to fit thefactorial moments obtained in high–energy p¯p scattering, a critical phenomenon does not even appear to be the ‘mostplausible’ cause of intermittency.Two of us (ADJ and TW) would like to acknowledge the hospitality of NORDITA.

This work was partially supportedby the U.S. Department of Energy under grant no. DE-FG02-88ER 40388.

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Phys. A 2, 1447 (1987); P. Carruthers, H. C. Eggers and I. Sarcevic, Phys.Lett.

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B 214, 617 (1988); W. Ochs, Z. Phys. C 50, 339 (1991).

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B273, 703 (1986). [8] L. L. Chau and D. W. Huang, Phys.

Lett. B 283, 1 (1992); Phys.

Rev. Lett.

70, 3380 (1993). [9] This second feature is common to all one–dimensional models.

[10] Without loss of generality, we have set the inverse temperature, β, equal to 1. [11] We consider only the physically interesting case where |¯λk| < 1 for all k.[12] Reproduction of all factorial moments of the NB requires that we take the limit as the number of species, D, approachesinfinity.

The proof that Eq. (12) leads to the NB results will be given in Ref.

[13]. [13] A. D. Jackson, T. Wettig and N. L. Balazs, to be published.

[14] Only five factorial moments are currently available experimentally. Therefore, only r2 to r4 are relevant for comparisonwith data.TABLE I.

The ensemble average and dispersion of rq as defined in Eq. (13) for 2 ≤q ≤6 and various values of the numberof particle species, D. The average was taken over 105 randomly drawn theories with c⟨n⟩= 4.5 for each draw.

The case D = 1corresponds to the ordinary lattice gas (Ising) model [8] and has no dispersion.D⟨⟨r2⟩⟩⟨⟨r3⟩⟩⟨⟨r4⟩⟩⟨⟨r5⟩⟩⟨⟨r6⟩⟩10.750.50.31250.18750.10937520.812 ± 0.0230.598 ± 0.0350.417 ± 0.0370.280 ± 0.0330.184 ± 0.02740.881 ± 0.0460.723 ± 0.0780.569 ± 0.0910.435 ± 0.0910.326 ± 0.08380.941 ± 0.0670.851 ± 0.1320.751 ± 0.1790.652 ± 0.2070.558 ± 0.218160.981 ± 0.0760.951 ± 0.1730.916 ± 0.2700.879 ± 0.3600.842 ± 0.441320.998 ± 0.0690.999 ± 0.1701.005 ± 0.2951.017 ± 0.4421.039 ± 0.612641.002 ± 0.0501.007 ± 0.1281.019 ± 0.2291.040 ± 0.3551.071 ± 0.5151281.001 ± 0.0351.005 ± 0.0881.013 ± 0.1541.026 ± 0.2331.044 ± 0.3272561.001 ± 0.0241.003 ± 0.0611.007 ± 0.1051.013 ± 0.1551.023 ± 0.2105121.000 ± 0.0171.002 ± 0.0421.004 ± 0.0721.007 ± 0.1061.012 ± 0.1424

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