Intermittency and multifractal structures in QCD cascades

Nuclear Physics B328 (1989) 76-84 North-Holland, Amsterdam

1NTERMITTENCY AND MULTIFRACTAL S T R U C T U R E S IN QCD CASCADES Per DAHLQVIST, Bo ANDERSSON and G6sta GUSTAFSON

Department of Theoretical Physics, UniversiO, of Lund, S6lvegatan 14A, S+223 62 Lund, Sweden

Received 6 April 1989 (Revised 3 July 1989)

We show that the concept of intermittency, as defined below, is relevant for high-energy QCD cascades. We also show how this effect will survive fragmentation and that it will be experimentally measurable. This is most easily seen using ordinary multiplicity moments rather than factorial moments. These results are conveniently expressed in terms of multifractal dimensions.

1. Introduction Recently there has been much interest in the notion of intermittency in connection with hadronic multiplicity distributions in high-energy interactions [1]. In particular there are indications from the experimental distributions that there are irregularities in the multiplicities down to very small bins in rapidity [2]. This has been attributed to the occurrence of a cascade mechanism in the production of the final-state hadrons, and it is reasonable to expect that it is due to the well-known Q C D shower cascades. In this note we would like to give a quantitative interpretation of these properties, which in our opinion rather should be classified as unusual scaling properties of the multiplicity distributions with respect to the rapidity binsize 6y under consideration. This is how we will interpret the concept of intermittency. We will in particular try to divide these properties into one part stemming from the QCD cascade and another stemming from the hadronization using some recently found properties [3,4]. For the computer simulations we will use the dipole approximation to the cascades based on results found by the Leningrad group [5] and further developed in Lund [6] and implemented in the computer code A R I A D N E 2 [7]. For the hadronization of the multiparton states we use string fragmentation ~ la Lund as 0550-3213/89/$03.50Cc2Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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implemented in the computer simulation code JETSET 6:3 [8]. We expect, however, that the properties presented here should be very similar, irrespective of the particular framework one uses for the cascades and the hadronization as long as one is infrared-stable and incorporates the interference properties of the QCD cascade. We will start by recalling some notions in connection with the measure function )~, which we introduced in ref. [9] and further developed in refs. [3,4]. The measure )~ is a generalized rapidity length corresponding to the phase space size for hadron emission in a multiparton event. It is well known that the multiplicity is basically given by the available rapidity length (defined by Ay = ln(s/so) ) in longitudinal phase space models and in two-jet events of a qCt-character in the Lund model. In the Lund model it is possible to go a step further and to describe the average final-state hadron distributions by means of a hyperbola spanned by the e n e r g y - m o m e n t u m vectors of the qq-pair [10]. If the energy-momentum vectors of the hadrons are laid out in rank order (the first rank hadron containing the original q-flavor, etc.), then on average they span this hyperbola, which has the length Ay given above. There is a generalization of the hyperbola to a multiparton system, which we called the x-curve in ref. [11] and which has the corresponding length )~. (Actually, the x-curve is defined by means of a differential equation containing the parton energy momenta, and depending upon the precise boundary conditions for the equation one obtains a set of curves. We showed in ref. [4] that these curves are very similar and for our purposes they all provide basically the same results.) In ref. [4] we have shown that the x-curves provide a very good description of the (inclusive) final-state hadronic momentum distributions predicted in the Lurid model. This is obtained if the x-curves stemming from an ensemble of parton states, generated by means of the dipole cascade, are partitioned into "hadrons" each of mass m. The results are better for large energies where there are many gluons emitted and for lower energies in events of a non-two-jet character. The reason for that is that the fluctuations in the hadronization process are relatively larger for events of two-jet character [3,4]. In ref. [4] we used a value of m = 0.8 GeV, i.e. in the centre of the hadronic mass spectrum, which also gives the correct mean multiplicity. The fact that both the mean and the variance of the multiplicity distributions of the final-state hadrons are linear functions of X means that the hadronization process is similar to a poissonian process in the partitioning of the x-curve corresponding to the multiparton state [4]. We feel that this is a realization of the p a r t o n - h a d r o n duality concept as first suggested by the Leningrad group [12]. In sect. 2 we start by showing that our events exhibit intermittency in accordance with the definition in ref. [1]. The authors of that paper used the factorial moments of the multiplicity distributions inside the rapidity range 6y because these quantities have a simple interpretation in terms of the ordinary moments of the weight

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function f in a poissonian representation of the multiplicity distributions:

P y(n) = fdvAy( ) exp(-v)/n!, (n (n - 1 ) . . . (n - j + 1)) =

fdoAy(v)vJ.

(1)

It was thought and also demonstrated in ref. [1] for simple examples that these m o m e n t s m a y exhibit the underlying dynamics behind the intermittency properties. It is evident that in case the particles are emitted in a completely uncorrelated way, i.e. in a true poissonian fashion, then the mean number in a small interval 8y will be proportional to By. In the same way it is also obvious that in a poissonian process the factorial moments all will behave as (By) j with j the order of the moment. The authors of ref. [1] therefore considered the factorial moments for the multiplicity variables scaled with respect to the mean in order to avoid "trivial" poissonian noise. One of our findings will be that the Q C D process itself implies highly nontrivial scaling behaviour with respect to the size of the rapidity interval 6y. We also find that these properties are more easily seen in the ordinary moments of the distributions. In particular, we are able to show that there are strong regularities in these scaling behaviours over a range of sizes of By. This is an important property because it is well known that in models containing the constraints of a physical mass spectrum and e n e r g y - m o m e n t u m conservation, it is difficult to consider very small By-intervals, because of lack of statistics. In order to demonstrate these properties we will make use of the notions of multifractal dimensions, which have been introduced in different other contexts, where there are nontrivial scaling behaviours [13]. We will use this technique both on the hadron distributions as obtained by the dipole cascade followed by hadronization fi la Lund and directly on the x-curve, defined from the partonic state after the cascade. In the latter case we study that part of the x-curve which has a tangent inside the rapidity interval By, i.e. with the rapidity defined as

y=½1n

dE_dpz

.

In this way we are able to separate the effects of the Q C D cascade from those of the hadronization process. We will end this note with a few comments in sect. 3.

2. The scaling behaviour of the multiplicity moments In fig. 1 we show the results of a simulation for two different energies, with and without sphericity cuts, on the factorial moments F 2 and F 3. It is obvious that there

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are irregularities in the sense of ref. [1] down to rapidity sizes 6y at least of the order 0.1, and the more so the larger the sphericity and the energy. We will next show that the levelling-off in fig. 1 is not due to the onset of a poissonian process at small values of 6y. We do that by considering the separate behaviours of the different ordinary moments of the multiplicity distributions for different values of ~y. In order to do that we introduce the multifractal measure F defined in ref. [13]. For a structure with a given density distribution the analysis of different powers of the density can give rise to different non-integer dimensions. These are called multifractal dimensions. Consider a density distribution { p j} ] = 1 , . . . , M for a certain quantity in M boxes of the size 6y. As an example, suppose that we determine the number of hadrons nj in M rapidity bins 3y such

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that the total number of hadrons is )2nj = n tot and that we define nj pj=-

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for all values of 6y in a range. To each value of q one may define such a ~- and then one defines the multifractal dimension Dq by the rescaling

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If there is a ~- implying the scaling behaviour of eq. (4), there must be some self-similarity in the set. In case of a non-integer Dq this implies a nontrivial set. Different values of q will give different weights to the densities. By varying q we will obtain more information on the properties of the set than from the ordinary "geometrical" Hausdorff dimension, which is obtained in the limit q--* 0. The corresponding definitions for pj are obvious in case we study the piece of the x-curve corresponding to the bin 8y as discussed above. In our investigation we use an ensemble of parton states. It is then necessary to modify the prescription given above in the following way. We count the hadronic multiplicities in M bins inside lYc~s{ < 2. Instead of the normalized density in eq. (2) we use a normalization in the mean over the ensemble, i.e. in eq. (2) we make the substitution /'/tot--~ (H tot) "

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If we have scale invariance in a "big" range in M, that is ~'M is independent of M, the corresponding Dq m a y be interpreted as multifractal dimensions. In this way the rapidity binsize 6y is exchanged for 4/M. In figs. 2 a - b we show the behaviour of the locally defined Dq for the multiplicity of the final-state hadrons (for q = 2 and 4) as obtained from the dipole cascade and Lund string fragmentation as a function of the number of bins M. We also show the result of the partitioning of the x-curve into "hadrons" of mass m = 0.8 GeV. Finally we show the result of the Dq in case we use the measure )~ along the x-curve directly. We note that we obtain basically constant values for Dq for all values M from 1 up to 32 in all the cases (which corresponds to 6y down to 0.1). For smaller binsizes the "hadronic" distributions all approach 0, but the )t-results continue to stay essentially flat around 0.6 for q = 2 and around 0.5 for q = 4. We conclude that

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the model predicts well-defined dimensions Dq over a range of binsizes (which should be observable with reasonable statistics) and due to the equality between the X-dimension and the hadronic multiplicity dimensions this must be a property directly of the QCD cascade. The fact that the curves differ below 0.1 units in rapidity is a property well known in different other contexts with unusual scaling behaviour - fractal dimensions in Nature occur only inside the scales relevant to the investigation [14]. We know that the resolution power in hadron physics never goes below the hadronic mass scale and that is the evident reason why we cannot in these models continue further down in 3y-size. It is obvious that a distribution for a finite number of particles must in the mathematical limit 8y ~ 0 have zero multifractal dimension. For a variable like X which takes on a continuous spectrum there is of course not the same constraint. In fig. 3 we show the behaviour of Dq for different values of q (we have plotted the mean of Dq over the rapidity binsizes relevant to hadronic multiplicities, 0.1 to 1). We note that the dimension goes down from the expected Hausdorff dimension 1 to around 0.6 from q = 2 onward. It follows from the continuity of the tangent of the x-curve that the Hausdorff dimension is 1. Thus the QCD cascade described by the x-curve will be of a multifractal but not a fractal nature. It is easy to understand the levelling-off of the factorial moments in fig. 1. For example, for the normalized factorial moment 2 we have

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For a poissonian process both terms are for small 8y of the order 8y 1 while the difference is of order 1. We find instead that as the mean multiplicity in a bin of size

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8y behaves as By, the first term scales according to 8y D2-1~ 8y o4 while the second behaves as 8y-1. As 8y becomes smaller the negative contribution from the second term will start to d o m i n a t e - for even smaller values of 8y there will of course be compensations as the whole expression necessarily must be non-negative. We conclude that there are predictions from the models which may be tested with reasonable statistics and that the scaling properties we have found are due to the properties of the QCD cascade. Although this might be associated with the term "intermittency", we would have preferred the term "unusual scaling behaviour". In sect. 3 we would like to make a few further comments, in particular about the observations in the experiments at present energies.

3. Concluding remarks The properties we have discussed above are basically high-energy phenomena as they stem from the properties of multigluon emission. We have also kept to e+e--annihilation events. There is a nonvanishing amount of gluonic emission also in the P E T R A - P E P regime, in connection with inelastic lepton production and at least in some models for hadronic 1ow-pT physics. We suspect, however, that there m a y be further causes for irregularities in the multiplicities for small rapidity bins, which are more directly related to the fragmentation process of the partonic state. One such possibility is an influence from Bose-Einstein correlations [15], which are not included in the present investigation. We will investigate these effects in a future publication and we will therefore be satisfied with the results from the Q C D cascade at this place. We note that we have found in refs. [3,4] that the fluctuations from the cascade very much dominate the behaviour in the kind of high-energy events we have considered here. We therefore feel that the results should be relevant also in case there are irregularities stemming from different sources in the fragmentation process. One of the reasons that we suspect further contributions to the irregularities in multiplicity for small rapidity bins are the results obtained in an investigation of the H R S - d a t a [16]. These correspond to two-jet like e+e -annihilation events and the results of the analysis indicate intermittent behaviour in the sense of ref. [1] as pointed out in ref. [17]. We have compared the data to the "ordinary" Lurid model in this case just as the authors of ref. [17] did. Although there is some sensitivity to the fragmentation parameters (we are able to obtain a much better agreement with some variations in the parameters), the data seem to indicate significantly larger multiplicity variations than this ordinary Lund model is able to accommodate for small rapidity bins. One of the authors (B.A.) would like to thank I. Dremin from the Eebedev Institute for bringing to his attention the notions of irregularities in multiplicity

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(which he was the first to term intermittency), and for several interesting discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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