We review the predictions on angular correlations and their recent
experimental tests at LEP and HERA. Power behaviour of correlation functions
appear for large angles and reflect the underlying fractal structure in jet
evolution. Asymptotic scaling laws work rather well for the low momentum
particles. For angular correlations the predictions work at least
qualitatively, sometimes quantitatively, at LEP and the higher HERA energies.
Some limitations of the approach and possible improvements are discussed.Comment: 5 pages, 1 figures in eps, talk given at XXXI International Symposium
  on Multiparticle Dynamics, Sept 1-7, 2001, Datong China. URL
  http://202.114.35.18

Ochs, Wolfgang

English

arXiv

We review the predictions on angular correlations and their recent experimental tests at LEP and HERA. Power behaviour of correlation functions appear for large angles and reflect the underlying fractal structure in jet evolution. Asymptotic scaling laws work rather well for the low momentum particles. For angular correlations the predictions work at least qualitatively, sometimes quantitatively, at LEP and the higher HERA energies. Some limitations of the approach and possible improvements are discussed

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QCD DESCRIPTION OF ANGULAR CORRELATIONSWOLFGANG OCHSMax-Planck-Institut fu¨r Physik (Werner Heisenberg Institut)Fo¨hringer Ring 6, D80805 Mu¨nchen, GermanyE-mail: wwo@mppmu.mpg.deWe review the predictions on angular correlations and their recent experimen-tal tests at LEP and HERA. Power behaviour of correlation functions appear forlarge angles and reflect the underlying fractal structure in jet evolution. Asymp-totic scaling laws work rather well for the low momentum particles. For angularcorrelations the predictions work at least qualitatively, sometimes quantitatively,at LEP and the higher HERA energies. Some limitations of the approach andpossible improvements are discussed.1 IntroductionThere has been much interest in the last decade in the study of correlationfunctions and their possible power dependence on resolution scale. Such be-haviour is expected, in particular, from the fractal structure of the selfsimilarparton cascade.1 Specic predictions, mainly within the Double Logarithmicapproximation (DLA) of pQCD, have been derived for angular correlationsand multiplicity moments.2,3,4 In recent years these predictions have beentested in various experimental works by the L35, DELPHI6 and ZEUS7 Col-laborations, see also talk at this conference8, so there is now a good time fora critical assessment on what has been learned from these studies.A major aim is the test of \Local Parton Hadron Duality" (LPHD 9) whichsuggests comparing directly the multi-hadron with the multi-parton nal statewithout the interface of a \hadronization model." This approach has beenproposed originally for inclusive spectra but has been applied subsequentlyto many other problems (recent reviews10). Here we adress its application tomulti-particle correlations beyond single inclusive phenomena.2 Correlations in Full and Restricted Angular RangeParticle multiplicity distributions can be characterized by their factorial mo-ments Fq = hn(n−1) : : : (n− q +1)i=hniq which relate to integrals of the den-sities (dn=dp1 : : : dpq)=hniq. Perturbative QCD predictions for the momentsin full phase space (event or jet) improve strongly with increasing accuracyof the logarithmic approximations (see review11) and are entirely satisfactoryfor the exact numerical solution of the pertaining evolution equations.12 The0111459: submitted to World Scientific on December 4, 2001 1predictions depend on the QCD scale  and one non-perturbative parame-ter (k? cut-o Q0); they can be determined from a t to the global eventmultiplicities.For limited phase space dierent observables have been considered:Distributions in relative angle #12 between two particles, both inside the for-ward cone of a jet with half opening angle  and momentum P . Thedistribution is normalized either by the full multiplicity inside this coner^(#12) = (dn=#12)=n() or by the corresponding distribution of uncorrelatedparticles r(#12) = (dn=#12)=(dn=#12)uncorr.Multiplicity moments Fq, from particles inside an angular ring of size 2 atpolar angle  to the jet axis or from inside a cone of half angle  again atpolar angle ; they correspond to dimensions D = 1 and D = 2 respectively.In the theory with xed coupling s all these observables (genericallydenoted by hq) show a universal power behaviour (#12 ! )hq(; ; P )  (=)ϕq (1)for Fq : ’q = (q − 1)D − (q − q−1)γ0; for r^ : ’2 = −3γ0=2 (2)corresponding to the selfsimilar structure of the parton cascade. The poweris given in terms of the \QCD anomalous dimension" γ0 =√6s=.For running coupling this result is retained only at the large angles  O() where the coupling varies as s(P=). However, with decreasingangles  there is an increasing deviation from the power law because of thelogarithmic growth of the coupling for small angular scales k?  P. Theauthors2,3,4 obtain slightly dierent approximations, for example,2hq(; ; P )  exp(2qγ0(P=)!(; q)); " = ln(=)ln(P=) (3)where !(; q) is the solution of an algebraic equation and reproduces the limit(1) for large angles.3 Discussion of Experimental ResultsAngular dependence (on ; #12). In all measurements at LEP and HERA thedata behave qualitatively as predicted: for large angles (small ") there is anapproximate power behaviour and a deviation at small angles according to theprediction. The deviation between predictions and experiment varies between10-20% for lnFq; q  3 and a bit more for F2, the dierential correlations r; r^at the higher HERA energies (Q2 > 2000 GeV2) agree rather well with theprediction but deviate considerably at low Q2, according to what is expectedfor an asymptotic prediction.0111459: submitted to World Scientific on December 4, 2001 2Figure 1. Initial slope ϕq of factorial moments from DELPHI6 in comparison with asymp-totic predictions (1,2) for angular ring (D = 1) and cone (D = 2) at order q, curves connectresults at integer q (parameters nf = 3,  = 0.04 GeV).Initial slope ’q. The powers ’q in (1) have been extracted by DELPHI6 fromthe Fq moments at small " for q  5 and D = 1; 2, see Fig. 1. The overallvariation of the slopes by a factor 20 is roughly met by the prediction (2)and the deviations grow up to 30% at high q. Furthermore it is found,6 thatthe initial slope decreases with increasing  as expected from the runningcoupling s(P=), best results are obtained for  = 0:04 GeV which is abit small but still acceptable for a DLA calculation, also nf = 3 is taken.Dependence on approximation scheme. The approximations (DLA) are jus-tied only at asymptotic energies and could be responsible for the partialdisagreement with experiment. The ZEUS collaboration also compared theirresults on Fq with the predictions of a parton level MC (ARIADNE13) usinga small cut-o parameter Q0 as in the analytic calculations. Then agreementwith data improves considerably, with results comparable, for example, to theLEPTO MC with full hadronization.4 Asymptotic scaling lawsThe DLA results correspond to the high energy asymptotics. These resultscan often be written in scaling form. A well known example is the KNOscaling, which says, that the rescaled probability hniPn approaches an energyindependent (scaling) limit as function of the rescaled multiplicity n=hni. In asimilar way one nds an asymptotic limit for the energy spectra in appropri-ate rescaled variables (\-scaling"14). The data follow this scaling predictiononly for the low momentum particles, a phenomenon understood as a conse-quence of soft gluon coherence.10 Typically, at all available primary energies0111459: submitted to World Scientific on December 4, 2001 3the deviations from the scaling limit increase with momentum (they can betaken into account by the higher order (MLLA) corrections).15For angular correlations Eq. (3) can be rewritten in scaling formln hq(; ; P )=γ0(; P ) = f(; q) (4)with known f , so the rescaled correlations in the variable " become indepen-dent of both the jet energy P and the large angle  (\"-scaling"2).Remarkably, the correlation functions r; r^ approach the predicted scalinglimit within the HERA Q2 range, furthermore, they show the independenceof angle  in the range  = 30 : : : 90, also at LEP. We relate this goodagreement already at present energies to the good scaling properties of lowmomentum particles which dominate the correlation measurements.5 Limitations and ImprovementsA problem has recently been encountered with moments for particles in thecylinder p? < pcut? . A recent measurement by ZEUS7,8 has revealed a rise ofthese moments for small cut-o pcut? contrary to the prediction16 of a decreaseFq ! 1 for pcut? ! 0 (or pcut? ! Q0 in the calculation) corresponding to aPoissonian multiplicity distribution; this prediction was also conrmed by aparton MC calculation.16We take this observation as showing a limitation of the duality approach.For sure, there cannot be a perfect matching between parton and hadron -nal states, as already obvious from the existence of hadronic resonances. Thecorrelations of particles at low p? seem to involve additional non-perturbativeeects. Perhaps one can nd a better treatment of particles near the kine-matic cut-o p?  Q0 as it was possible for single inclusive spectra in thislimit where similar problems occur. Beyond the original discussion,16 wewould expect the perturbative prediction of the Poissonian limit to apply forjets of not too low virtuality ycut  (Q0=Q)2 where the dependence on Q0disappears.On the other hand, the perturbative DLA predictions for correlations inangle are in overall agreement with measurements. A considerable improve-ment of the calculations is not obtained by including MLLA corrections.3 Wewould expect a major improvement in accuracy, especially for the large anglebehaviour, if full matrix element calculations were carried out as it has beendone for azimuthal angle correlations with good success.17 An easier path to-wards higher accuracy is available by running a parton MC (ARIADNE) withnon-standard parameters Q0   as discussed previously.160111459: submitted to World Scientific on December 4, 2001 46 SummaryThe dual connection between parton and hadron nal states is not restrictedto single particle distributions but extends to multi-particle correlations; carehas to be taken with particles near the kinematic border at p?  Q0. Thesimple asymptotic formulae from DLA for angular correlations describe alldata at least at the qualitative level. Near quantitative results follow for globalobservables or from MC calculations. The xed coupling results correspondto a selfsimilar cascade and fractal structure. The realistic QCD cascade withrunning coupling corresponds to this scaling picture approximately but showscharacteristic deviations in angular distributions and with primary energy.References1. B. Andersson, P. Dahlquist, and G. Gustafsson, Phys. Lett. B 214, 604(1988); Nucl. Phys. B 328, 76 (1989).2. W. Ochs and J. Wosiek, Phys. Lett. B 289, 159 (1992); Phys. Lett. B304, 144 (1993); Z. Phys. C 68, 269 (1995);3. Yu.L. Dokshitzer and L. Dremin, Nucl. Phys. B 402, 139 (1993).4. Ph. Brax, J.L. Meunier and R. Peschanski, Z. Phys. C 62, 649 (1994).5. L3 Collaboration, M. Acciari et al., Phys. Lett. B 428, 186 (1998).6. DELPHI Collaboration, P. Abreu et al., Phys. Lett. B 440, 203 (1998);Phys. Lett. B 457, 368 (1999).7. ZEUS Collaboration, J. Breitweg et al., Eur. Phys. J. C 12, 53 (2000);S. Chekanov et al., Phys. Lett. B 510, 36 (2001);Contribution to ICHEP98, Vancouver, Abstract 802.8. L. Zawiejski, this conference.9. Ya.I. Azimov, Yu.L. Dokshitzer, V.A. Khoze and S.I. Troyan, Z. Phys.C 27, 65 (1985); ibid, 31, 213 (1986).10. V.A. Khoze and W. Ochs, Int. J. Mod. Phys. A 12, 2949 (1997);V.A. Khoze, W. Ochs and J. Wosiek, Handbook of QCD, ed. M. Shifman(World Scientic, Singapore, 2001) Vol. 2, p. 1101.11. I.M. Dremin and J.W. Gary, Phys. Rep. 349, 301 (2001).12. S. Lupia, Phys. Lett. B 439, 150 (1998)13. L. Lo¨nnblad, Comp. Phys. Comm. 71, 15 (1992).14. Yu.L. Dokshitzer, V.S. Fadin and V.A. Khoze, Z. Phys. C 18, 37 (1983).15. S. Lupia, W. Ochs, Eur. Phys. J. C 2, 307 (1998).16. S. Lupia, W. Ochs and J. Wosiek, Nucl. Phys. B 540, 405 (1999).17. Yu.L. Dokshitzer, G. Marchesini and G. Oriani, Nucl. Phys. B 387, 675(1992).0111459: submitted to World Scientific on December 4, 2001 5

QCD Description of Angular Correlations

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QCD DESCRIPTION OF ANGULAR CORRELATIONS

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