# Factorial and cumulant moments in e+e− → hadrons at the Z0 resonance

**ABSTRACT** We present the first experimental study of the ratio of cumulant to factorial moments of the charged-particle multiplicity distribution in high-energy particle interactions, using hadronic Z0 decays collected by the SLD experiment at SLAC. We find that this ratio, as a function of the moment-rank q, decreases sharply to a negative minimum at q = 5, which is followed by quasi-oscillations. These features are insensitive to experimental systematic effects and are in qualitative agreement with expectations from next-to-next-to-leading-order perturbative QCD.

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**ABSTRACT:**We study the moments of multiplicty distribution and its relation to the Lee-Yang zeros of the generating function in electron-positron and hadron-hadron high energy collision. Our work shows that GMD moments can reproduce the oscillatory behaviour as shown in the experimental data and predicted by quantum chromodynamics at preasymptotic energy, while it can also be used to distinguish electron-positron (e+e-) multiplicity data from hadron-hadron (pp and ) multiplicity. Furthermore, there seems to be a link between the development of shoulder structure in the multiplicity distribution and the development of ear structure in the Lee-Yang zeros. We predict that these structures is going to be very obvious at 14 TeV. We argue that the development of these structures indicates an ongoing transition from quark-dominated soft scattering events to gluon-dominated semihard scattering events.International Journal of Modern Physics A 01/2012; 24(18n19). · 1.09 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In experiment, the multiplicity distributions of inelastic processes are truncated due to finite energy, insufficient statistics, or special choice of events. It is shown that the moments of such truncated multiplicity distributions possess some typical features. In particular, the oscillations of cumulant moments at high ranks and their negative values at the second rank can be considered as ones most indicative of the specifics of these distributions. They allow one to distinguish between distributions of different type.Physics of Atomic Nuclei 12/2003; 67(1):100-107. · 0.60 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Some problems of the jet calculus in perturbative QCD are discussed. The first one is related to specifying the order of perturbation. Because of the cancellation of the leading-order (LO) and next-to-leading-order (NLO) terms in the ratio r of the mean multiplicities in gluon and quark jets, the results presently obtained for this ratio should be associated with the 4NLO approximation. The second problem reveals itself in calculations where corrections to some quantities (in particular, to r′ are greater at present energies than lower order terms. Some features that characterize jets and which do not suffer from this deficiency are proposed. Yet another problem lies in interpreting negative cumulant-moment values, which are considered as an indication of a changeover from attraction to repulsion in sets of specific particle content. Finally, the problem of generalizing QCD equations for generating functions is briefly discussed.Physics of Atomic Nuclei 10/2002; 65(10):1894-1899. · 0.60 Impact Factor

Page 1

arXiv:hep-ex/9601010v1 1 Feb 1996

Factorial and Cumulant Moments in

e+e−→ Hadrons at the Z0Resonance

The SLD Collaboration

K. Abe,(29)I. Abt,(14)C.J. Ahn,(26)T. Akagi,(27)N.J. Allen,(4)W.W. Ash,(27)†

D. Aston,(27)K.G. Baird,(24)C. Baltay,(33)H.R. Band,(32)M.B. Barakat,(33)

G. Baranko,(10)O. Bardon,(16)T. Barklow,(27)A.O. Bazarko,(11)R. Ben-David,(33)

A.C. Benvenuti,(2)G.M. Bilei,(22)D. Bisello,(21)G. Blaylock,(7)J.R. Bogart,(27)

T. Bolton,(11)G.R. Bower,(27)J.E. Brau,(20)M. Breidenbach,(27)W.M. Bugg,(28)

D. Burke,(27)T.H. Burnett,(31)P.N. Burrows,(16)W. Busza,(16)A. Calcaterra,(13)

D.O. Caldwell,(6)D. Calloway,(27)B. Camanzi,(12)M. Carpinelli,(23)R. Cassell,(27)

R. Castaldi,(23)(a)A. Castro,(21)M. Cavalli-Sforza,(7)A. Chou,(27)E. Church,(31)

H.O. Cohn,(28)J.A. Coller,(3)V. Cook,(31)R. Cotton,(4)R.F. Cowan,(16)

D.G. Coyne,(7)G. Crawford,(27)A. D’Oliveira,(8)C.J.S. Damerell,(25)M. Daoudi,(27)

R. De Sangro,(13)P. De Simone,(13)R. Dell’Orso,(23)M. Dima,(9)P.Y.C. Du,(28)

R. Dubois,(27)B.I. Eisenstein,(14)R. Elia,(27)E. Etzion,(4)D. Falciai,(22)C. Fan,(10)

M.J. Fero,(16)R. Frey,(20)K. Furuno,(20)T. Gillman,(25)G. Gladding,(14)

S. Gonzalez,(16)G.D. Hallewell,(27)E.L. Hart,(28)A. Hasan,(4)Y. Hasegawa,(29)

K. Hasuko,(29)S. Hedges,(3)S.S. Hertzbach,(17)M.D. Hildreth,(27)J. Huber,(20)

M.E. Huffer,(27)E.W. Hughes,(27)H. Hwang,(20)Y. Iwasaki,(29)D.J. Jackson,(25)

P. Jacques,(24)J. Jaros,(27)A.S. Johnson,(3)J.R. Johnson,(32)R.A. Johnson,(8)

T. Junk,(27)R. Kajikawa,(19)M. Kalelkar,(24)H. J. Kang,(26)I. Karliner,(14)

H. Kawahara,(27)H.W. Kendall,(16)Y. Kim,(26)M.E. King,(27)R. King,(27)

R.R. Kofler,(17)N.M. Krishna,(10)R.S. Kroeger,(18)J.F. Labs,(27)M. Langston,(20)

A. Lath,(16)J.A. Lauber,(10)D.W.G.S. Leith,(27)M.X. Liu,(33)X. Liu,(7)M. Loreti,(21)

A. Lu,(6)H.L. Lynch,(27)J. Ma,(31)G. Mancinelli,(22)S. Manly,(33)G. Mantovani,(22)

T.W. Markiewicz,(27)T. Maruyama,(27)R. Massetti,(22)H. Masuda,(27)

E. Mazzucato,(12)A.K. McKemey,(4)B.T. Meadows,(8)R. Messner,(27)

P.M. Mockett,(31)K.C. Moffeit,(27)B. Mours,(27)D. Muller,(27)T. Nagamine,(27)

S. Narita,(29)U. Nauenberg,(10)H. Neal,(27)M. Nussbaum,(8)Y. Ohnishi,(19)

L.S. Osborne,(16)R.S. Panvini,(30)H. Park,(20)T.J. Pavel,(27)I. Peruzzi,(13)(b)

M. Piccolo,(13)L. Piemontese,(12)E. Pieroni,(23)K.T. Pitts,(20)R.J. Plano,(24)

R. Prepost,(32)C.Y. Prescott,(27)G.D. Punkar,(27)J. Quigley,(16)B.N. Ratcliff,(27)

T.W. Reeves,(30)J. Reidy,(18)P.E. Rensing,(27)L.S. Rochester,(27)P.C. Rowson,(11)

J.J. Russell,(27)O.H. Saxton,(27)S.F. Schaffner,(27)T. Schalk,(7)R.H. Schindler,(27)

B.A. Schumm,(15)A. Seiden,(7)S. Sen,(33)V.V. Serbo,(32)M.H. Shaevitz,(11)

J.T. Shank,(3)G. Shapiro,(15)S.L. Shapiro,(27)D.J. Sherden,(27)K.D. Shmakov,(28)

1

Page 2

C. Simopoulos,(27)N.B. Sinev,(20)S.R. Smith,(27)J.A. Snyder,(33)P. Stamer,(24)

H. Steiner,(15)R. Steiner,(1)M.G. Strauss,(17)D. Su,(27)F. Suekane,(29)

A. Sugiyama,(19)S. Suzuki,(19)M. Swartz,(27)A. Szumilo,(31)T. Takahashi,(27)

F.E. Taylor,(16)E. Torrence,(16)A.I. Trandafir,(17)J.D. Turk,(33)T. Usher,(27)

J. Va’vra,(27)C. Vannini,(23)E. Vella,(27)J.P. Venuti,(30)R. Verdier,(16)

P.G. Verdini,(23)S.R. Wagner,(27)A.P. Waite,(27)S.J. Watts,(4)A.W. Weidemann,(28)

E.R. Weiss,(31)J.S. Whitaker,(3)S.L. White,(28)F.J. Wickens,(25)D.A. Williams,(7)

D.C. Williams,(16)S.H. Williams,(27)S. Willocq,(33)R.J. Wilson,(9)

W.J. Wisniewski,(27)M. Woods,(27)G.B. Word,(24)J. Wyss,(21)R.K. Yamamoto,(16)

J.M. Yamartino,(16)X. Yang,(20)S.J. Yellin,(6)C.C. Young,(27)H. Yuta,(29)

G. Zapalac,(32)R.W. Zdarko,(27)C. Zeitlin,(20)Z. Zhang,(16)and J. Zhou,(20)

(1)Adelphi University, Garden City, New York 11530

(2)INFN Sezione di Bologna, I-40126 Bologna, Italy

(3)Boston University, Boston, Massachusetts 02215

(4)Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom

(5)California Institute of Technology, Pasadena, California 91125

(6)University of California at Santa Barbara, Santa Barbara, California 93106

(7)University of California at Santa Cruz, Santa Cruz, California 95064

(8)University of Cincinnati, Cincinnati, Ohio 45221

(9)Colorado State University, Fort Collins, Colorado 80523

(10)University of Colorado, Boulder, Colorado 80309

(11)Columbia University, New York, New York 10027

(12)INFN Sezione di Ferrara and Universit` a di Ferrara, I-44100 Ferrara, Italy

(13)INFN Lab. Nazionali di Frascati, I-00044 Frascati, Italy

(14)University of Illinois, Urbana, Illinois 61801

(15)Lawrence Berkeley Laboratory, University of California, Berkeley, California

94720

(16)Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

(17)University of Massachusetts, Amherst, Massachusetts 01003

(18)University of Mississippi, University, Mississippi 38677

(19)Nagoya University, Chikusa-ku, Nagoya 464 Japan

(20)University of Oregon, Eugene, Oregon 97403

(21)INFN Sezione di Padova and Universit` a di Padova, I-35100 Padova, Italy

(22)INFN Sezione di Perugia and Universit` a di Perugia, I-06100 Perugia, Italy

(23)INFN Sezione di Pisa and Universit` a di Pisa, I-56100 Pisa, Italy

(24)Rutgers University, Piscataway, New Jersey 08855

(25)Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX United

2

Page 3

Kingdom

(26)Sogang University, Seoul, Korea

(27)Stanford Linear Accelerator Center, Stanford University, Stanford, California

94309

(28)University of Tennessee, Knoxville, Tennessee 37996

(29)Tohoku University, Sendai 980 Japan

(30)Vanderbilt University, Nashville, Tennessee 37235

(31)University of Washington, Seattle, Washington 98195

(32)University of Wisconsin, Madison, Wisconsin 53706

(33)Yale University, New Haven, Connecticut 06511

†Deceased

(a)Also at the Universit` a di Genova

(b)Also at the Universit` a di Perugia

ABSTRACT

We present the first experimental study of the ratio of cumulant to factorial mo-

ments of the charged-particle multiplicity distribution in high-energy particle interac-

tions, using hadronic Z0decays collected by the SLD experiment at SLAC. We find that

this ratio, as a function of the moment-rank q, decreases sharply to a negative mini-

mum at q = 5, which is followed by quasi-oscillations. These features are insensitive

to experimental systematic effects and are in qualitative agreement with expectations

from next-to-next-to-leading-order perturbative QCD.

3

Page 4

1Introduction

One of the most fundamental observables in high-energy particle interactions is the mul-

tiplicity of particles produced in the final state. A large body of experimentally mea-

sured multiplicity distributions has been accumulated in a variety of hard processes[1].

The Poisson distribution (PD) does not describe the shapes of multiplicity distribu-

tions measured in e+e−, pp, and p¯ p collisions, implying non-random particle-production

mechanisms, but elucidation of the relationship between the measured shapes and the

underlying dynamics has proven to be problematic.

At present the theory of strong interactions, Quantum Chromodynamics (QCD) [2],

cannot be used to calculate distributions of final-state hadrons since the mechanism

of hadron formation has not been understood quantitatively. However, perturbative

QCD can be applied to calculate some properties of the cascade of gluons radiated by

the partons produced in a hard scattering process. If there is a simple relationship

between the distributions of partons and detected final state particles, as follows for

example from the ansatz of local parton-hadron duality (LPHD) [3], then such calcu-

lations may be expected to reproduce some features of experimental data. An early

calculation [4] in the leading double-logarithmic approximation (DLA) was successful

in describing the energy dependence of the average multiplicity, as well as the energy

independence of the “KNO distribution” [5] of n/<n>, the multiplicity scaled by its av-

erage value. However, the width predicted by this calculation is much larger than that

of experimentally observed multiplicity distributions [4]. It has been suggested that

the inclusion of higher-order terms in perturbative QCD calculations should reduce the

predicted width of the multiplicity distribution [6, 7], although no such calculation has

yet been achieved. However, the ratio of cumulant to factorial moments has recently

been proposed [8] as a sensitive measure of the shape of multiplicity distributions and

has been found to be calculable in higher-order perturbative QCD.

Factorial moments have been used to characterize cascade phenomena in various

scientific fields [9]. The factorial moment of rank q is defined [9]

Fq≡<n(n − 1)...(n − q + 1)>

<n>q

, (1)

where n is the particle multiplicity of an event and <n> is the average multiplicity in

the event sample. The cumulant moments Kqare related to the Fqby [10]

Fq=

q−1

?

m=0

(q − 1)!

m!(q − m − 1)!Kq−mFm, (2)

and F0=F1=K1≡ 1. While the DLA QCD calculation predicts [8] that the ratio Hq≡

Kq/Fqdecreases as q−2, the inclusion of higher orders yields more striking behavior.

4

Page 5

A calculation in the next-to-leading logarithm approximation (NLA) predicts [11] a

minimum in Hq at q ≈ 5 and a positive constant value for q ≫ 5, while the next-

to-next-to-leading logarithm approximation (NNLA) predicts [12] that this minimum

is negative and is followed by quasi-oscillations about zero. These predictions are

illustrated in Fig. 1.

In a previous study [13] Hqwere calculated using published multiplicity distribu-

tions from e+e−and p¯ p collisions, and features qualitatively similar to those predicted

by the NNLA calculation were observed. This was a significant result, supporting not

only QCD at the parton level, but also the notion of LPHD. However, no account was

taken of experimental systematic effects or of the correlations, both statistical and sys-

tematic, between values of Hqat different ranks q. In addition, some Hqvalues derived

from data from similar experiments were apparently inconsistent. Furthermore, it was

shown subsequently [14] that the observed features could be induced by the effective

truncation of the multiplicity distribution inherent in a measurement using a finite

data sample.

In this letter we present the first experimental determination of the ratio of cumulant

to factorial moments of the charged-particle multiplicity distribution in high-energy

particle interactions, using hadronic decays of Z0bosons produced in e+e−annihila-

tions. We study systematic effects in detail, in particular the influence of truncation of

the distribution, and investigate the correlations between moments of different rank.

We compare our measurements with the predictions of perturbative QCD, and also

with two widely used distributions predicted by phenomenological models of particle

production.

2 Charged Multiplicity Analysis

Hadronic decays of Z0bosons produced by the SLAC Linear Collider (SLC) were

collected with the SLC Large Detector (SLD) [15]. The trigger and initial selection

of hadronic events are described in [16]. The analysis used charged tracks measured

in the central drift chamber (CDC) [17] and vertex detector (VXD) [18]. A set of

cuts was applied to the data to select well-measured tracks and events well contained

within the detector acceptance. Charged tracks were required to have: a distance of

closest approach transverse to the beam axis within 5 cm, and within 10 cm along

the axis from the measured interaction point; a polar angle, θ, with respect to the

beam axis within |cosθ| < 0.8; and a momentum transverse to the beam axis greater

than 0.15 GeV/c. Events were required to have: a minimum of five such tracks; a

thrust-axis [19] direction within |cosθT| < 0.71; and a total visible energy of at least

20 GeV, which was calculated from the selected tracks assigned the charged pion mass.

5

Page 6

A total of 86,679 events from the 1993 to 1995 SLC/SLD runs survived these cuts and

were included in this analysis. The efficiency for selecting hadronic events satisfying

the |cosθT| cut was estimated to be above 96%. The background in the selected event

sample was estimated to be (0.3 ± 0.1)%, dominated by Z0→ τ+τ−decays, and was

subtracted statistically from the observed multiplicity distribution.

The experimentally observed charged-particle multiplicity distribution was cor-

rected for effects introduced by the detector, such as geometrical acceptance, track-

reconstruction efficiency, and additional tracks from photon conversions and particle

interactions in the detector materials, as well as for initial-state photon radiation and

the effect of the cuts listed above. The charged multiplicity of an event was defined

to include all promptly produced charged particles, as well as those produced in the

decay of particles with lifetime < 3 · 10−10s. A two-stage correction was calculated

using Monte Carlo simulated hadronic Z0decays produced by the JETSET 6.3 [20]

event generator, subjected to a detailed simulation of the SLD and reconstructed in

the same way as the data. Each MC event passing the event-selection cuts yielded a

number of generated tracks ngand a number of observed tracks no, which were used

to form the matrix

M(ng,no) =N(ng,no)

NMC

obs(no), (3)

where N(ng,no) is the number of MC events with nggenerated tracks and noobserved

tracks, and NMC

a sum of three Gaussians was fitted to M(ng,no) and this parametrization was used

in the correction. The effects of the event-selection cuts and of initial-state radiation

were corrected using factors

CF(ng) =Ptrue(ng)

obs(no) is the number of MC events with noobserved tracks. For each no,

Psel(ng), (4)

where Ptrue(ng) is the normalized simulated multiplicity distribution generated without

initial-state radiation and Psel(ng) is the normalized distribution for those events in

the fully-simulated sample that passed the selection cuts.

Both corrections were applied to the experimentally observed multiplicity distribu-

tion Pexp(no) to yield the corrected distribution:

Pcor(n) = CF(n) ·

?

no

M(n,no) · Pexp(no), (5)

which is shown with statistical errors only in Fig. 2a. The factorial moments Fq, cumu-

lant moments Kq, and their ratios Hqwere calculated from this distribution according

to Eqs. 1 and 2. The resulting Hqup to rank q = 17 are shown in Fig. 3 and listed

in Table 1. As q increases, the value of Hq falls rapidly (inset of Fig. 3), reaches a

negative minimum at q = 5, and then oscillates about zero with a positive maximum

6

Page 7

at q = 9 and a second negative minimum at q = 13. The statistical and systematic

errors are strongly correlated between ranks as we now discuss.

3 Statistical and Systematic Errors

Statistical errors and correlations were studied by analyzing simulated multiplicity dis-

tributions. The Hqwere calculated from 10 Monte Carlo samples of the same size as

the data sample and 20 multiplicity distributions generated according to the measured

distribution. For each Hq the standard deviation in these 30 samples was taken as

the statistical error, and is listed in Table 1. In each case the Hqexhibited the same

behavior as those calculated from the data, although the value of H5and the appar-

ent phase of the quasi-oscillation for q ≥ 8 were found to be sensitive to statistical

fluctuations. We investigated the possibility that the observed features might result

from a statistical fluctuation by generating 10,000 multiplicity distributions according

to Poisson and negative-binomial distributions (see below) with the same mean value

as our corrected multiplicity distribution. In no case did any sample exhibit either a

minimum near q= 5 or quasi-oscillations at higher q.

Experimental systematic effects were also investigated. An important issue is the

simulation of the track-reconstruction efficiency of the detector. The Hqwere found

to be sensitive to the global efficiency, which was tuned in the simulation so that our

average corrected multiplicity equalled the value measured in hadronic Z0decays [21].

The Hq resulting from a variation in the global efficiency of ±1.7%, corresponding

to the error on the measured average multiplicity, are shown in Fig. 4. There is an

asymmetric effect on the value of H5and on the apparent phase of the quasi-oscillation.

For each q the difference between the Hqwith increased and decreased efficiency was

assigned as a symmetric systematic uncertainty.

It is important to consider the dependence of the track reconstruction efficiency on

multiplicity. Our simulated efficiency is 91.5% for tracks crossing at least 40 of the 80

layers of the CDC, and is independent of ngwithin ±0.5%. Varying the efficiency for

ng> 20 by ±0.5% caused a change of ±4% in H5, and negligible changes for q > 5.

This change was assigned as a systematic uncertainty.

Variation of the form of the parametrization of the correction matrix M was found

to affect mainly the amplitude of the quasi-oscillation for q ≥ 8. Application of the

unparametrized version of the matrix M(ng,no) produced the largest such effect, which

is shown in Fig. 4. This change was conservatively assigned as a symmetric systematic

uncertainty to account for possible mismodelling of the off-diagonal elements of the

matrix. The effect on the Hqof variation of the parameters of the three-Gaussian fits

to M within their errors increases with increasing q, becoming the dominant uncertainty

7

Page 8

for q ≥16.

The effects on the Hqof wide variations in the criteria for track and event selection

were found to be small compared with those due to the above sources. The effect

of including values of the multiplicity distribution at n=2 and n=4, taken from the

JETSET model, in the calculation of the moments is also small. Varying the estimated

level of non-hadronic background, which appears predominantly in the low-multiplicity

bins, by ±100% produces a negligible change in the Hq.

The uncertainties from the above systematic sources were added in quadrature to

derive a systematic error on each Hq, which is listed in Table 1. All of our studies

showed a clear first minimum in Hqat q = 5 followed by quasi-oscillations for q ≥8.

The value of H5has a total uncertainty of ±13% that is strongly correlated with similar

errors on H6and H7and with an uncertainty in the phase of the quasi-oscillation of

∓0.2 units of rank. There is an uncertainty on the amplitude of the quasi-oscillation

of ±15% that is essentially independent of the other errors. From these studies we

conclude that the steep decrease in Hqfor q<5, the negative minimum at q = 5, and

the quasi-oscillation about zero for q≥8 are well-established features of the data.

4 Comparison of the Hqwith QCD Predictions

We have compared these results with the qualitative predictions of perturbative QCD

discussed in Section 1. Figure 1 shows that the DLA QCD calculation predicts no neg-

ative values of Hqand is inconsistent with the data. The NLA and NNLA calculations

predict [13] a steep decrease in Hqto a minimum at

qmin=

?

96π

121αs(Q2)

?1/2+1

2.

For αs(M2

data. For q > 5, the NLA calculation predicts that Hq increases toward a constant

value, which is not consistent with the data, whereas the NNLA calculation predicts

quasi-oscillations in Hqin agreement with the data.

The moment ratios are thus seen to be a sensitive discriminator between QCD

calculations at different orders of purturbation theory. We conclude that the Hqcalcu-

lated for gluons in the next-to-next-to-leading logarithm approximation of perturbative

QCD describe the shape of the observed multiplicity distribution, whereas the available

calculations at lower order do not.

Z) measured in Z0decays [16] qmin≈ 5. These features are seen in the

8

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Hq

Statistical

error

2.96

1.40

0.74

0.40

0.28

0.20

0.14

0.12

0.10

0.08

0.10

0.09

0.10

0.16

0.19

0.26

Systematic

error

11.13

5.61

0.93

0.51

0.39

0.32

0.10

0.16

0.19

0.09

0.15

0.21

0.13

0.26

0.45

0.37

q

2

3

4

5

6

7

8

9

(10−4)

411.00

54.41

5.15

−4.08

−3.40

−1.40

0.08

0.91

0.84

0.10

−0.66

−0.83

−0.18

0.89

1.50

0.61

10

11

12

13

14

15

16

17

Table 1: Ratio of cumulant to factorial moments, Hq. The errors are strongly correlated

between ranks as discussed in the text.

5Comparison with phenomenological models

Measured multiplicity distributions have been compared extensively with the predic-

tions of phenomenological models. We consider two such predicted distributions. The

negative binomial distribution (NBD)

Pn(?n?,k) = Cn

k+n−1

?

?n?

?n? + k

?n?

k

?n? + k

?k

, (6)

where ?n? and k are free parameters, is predicted [22] by models in which the hard

interaction produces several objects, sometimes identified with the partons in a QCD

cascade, each of which decays into a number of particles. The log-normal distribution

(LND)

?n+1

where µ, σ, and c are free parameters, is predicted [23] by models in which the particles

result from a scale-invariant stochastic branching process, which might be related to

Pn(µ,σ,c) =

n

N

n′+ cexp

?

−(ln(n′+ c) − µ)2

2σ2

?

dn′, (7)

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the parton branchings in a QCD cascade.

Considering statistical errors only, we performed least-squares fits of the NBD and

LND to our corrected multiplicity distribution. These fitted distributions and their

normalized residuals are shown in Figs. 2a and 2b, respectively. Both provide rea-

sonable descriptions of the data, with χ2/ndf of 68.0/24 and 30.5/23, respectively.

Although the NBD has a high χ2and shows structure in the residuals in the core

of the distribution, it is difficult to exclude without a thorough understanding of the

uncorrelated component of the systematic errors. These results are in agreement with

those from a previous analysis [24].

The PD and the phenomenological distributions differ markedly in their moment

structure: for the PD, Hq= 0 for all q; for the fitted NBD, Hqis positive and falls as

q−25; for the fitted LND, Hqfalls with increasing q to a negative minimum at q = 6

and then oscillates about zero. It was recently argued [14] that the truncation of the

large-n tail of the multiplicity distribution due to finite data-sample size could lead

to quasi-oscillations in Hq similar to those observed in the data. We calculated Hq

values from the fitted distributions over the multiplicity range observed in the data,

6 ≤ n ≤ 54, and the results are displayed in Fig. 5. The truncated PD and NBD

are found to produce features similar to those in the data, but with much smaller

amplitudes. The amplitudes are not sensitive to the exact value of the truncation

point and we conclude that the moment ratios predicted by the PD and NDB are

inconsistent with the data. The LND predictions are insensitive to the truncation

point and show the same qualitative features as the data. However, the first minimum

is smaller in amplitude and is at q = 6. The quasi-oscillation for q ≥ 8 has similar

amplitude and period, and is displaced by about one unit from the data. The moment

ratios Hqare thus seen to provide a sensitive test of phenomenological models.

6 Conclusion

In conclusion, we have conducted the first experimental study of the ratio Hqof cu-

mulant to factorial moments of the charged-particle multiplicity distribution in high-

energy particle interactions, using hadronic Z0decays. We find that Hq decreases

sharply with increasing rank q to a negative minimum at q = 5, followed by quasi-

oscillations; we show these features to be insensitive to statistical and experimental

systematic effects.

The predictions of perturbative QCD in the next-to-next-to-leading-logarithm ap-

proximation are in agreement with the features observed in the data, supporting both

the validity of QCD at the parton level and the notion that the observable final state

reflects the underlying parton structure. Calculations in the leading double-logarithm

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and next-to-leading-logarithm approximations are not sufficient to describe the data.

The Poisson and negative binomial distributions do not predict these features. The

log-normal distribution predicts features similar to those of the data, but does not

describe the data in detail. We conclude that the moment ratios Hqof the charged-

particle multiplicity distribution provide a sensitive test both of perturbative QCD and

of phenomenological models.

Acknowledgements

We thank the personnel of the SLAC accelerator department and the technical staffs of

our collaborating institutions for their efforts, which resulted in the successful operation

of the SLC and the SLD. We thank I. Dremin for useful discussions.

This work was supported by U.S. Department of Energy contracts: DE-FG02-

91ER40676 (BU), DE-FG03-92ER40701 (CIT), DE-FG03-91ER40618 (UCSB), DE-

FG03-92ER40689 (UCSC), DE-FG03-93ER40788 (CSU), DE-FG02-91ER40672 (Col-

orado), DE-FG02-91ER40677(Illinois), DE-AC03-76SF00098 (LBL), DE-FG02-92ER40715

(Massachusetts), DE-AC02-76ER03069 (MIT), DE-FG06-85ER40224 (Oregon), DE-

AC03-76SF00515 (SLAC), DE-FG05-91ER40627 (Tennessee), DE-AC02-76ER00881

(Wisconsin), DE-FG02-92ER40704 (Yale); U.S. National Science Foundation grants:

PHY-91-13428 (UCSC), PHY-89-21320(Columbia), PHY-92-04239 (Cincinnati), PHY-

88-17930 (Rutgers), PHY-88-19316 (Vanderbilt), PHY-92-03212 (Washington); the UK

Science and Engineering Research Council (Brunel and RAL); the Istituto Nazionale

di Fisica Nucleare of Italy (Bologna, Ferrara, Frascati, Pisa, Padova, Perugia); and the

Japan-US Cooperative Research Project on High Energy Physics (Nagoya, Tohoku).

References

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Hadronic Multiparticle Production, ed. P. Carruthers, World Scientific, Singapore

(1990).

[2] H. Fritzsch, M. Gell-Mann, and H. Leutwyler, Phys. Lett. B47 (1973) 365;

D.J. Gross and F. Wilczek, Phys. Rev. Lett 30 (1973) 1343;

H.D. Politzer, Phys. Rev. Lett 30 (1973) 1346.

[3] T.I. Azimov, Y.L. Dokshitzer, V.A. Khoze, and S.I. Troyan, Z. Phys. C27 (1985)

65.

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[4] Yu.L. Dokshitzer, V.A. Khoze, S.I. Troyan, in: Perturbative QCD, ed. A.H.

Mueller, World Scientific, Singapore (1989).

[5] Z. Koba, M.B. Nielsen, P. Oleson, Nucl. Phys. B240 (1972) 317.

[6] E.D. Malaza and B.R. Webber, Nucl. Phys. B267 (1986) 702.

[7] Yu.L. Dokshitzer, Phys. Lett. B305 (1993) 295.

[8] I.M. Dremin, Mod. Phys. Lett. A8 (1993) 2747.

[9] A. Bialas and R. Peschanski, Nucl. Phys. B273 (1986) 703.

[10] I.M. Dremin and R.C. Hwa, Phys. Rev. D49 (1994) 5805.

[11] I.M. Dremin, Phys. Lett. B313 (1993) 209.

[12] I.M. Dremin and V.A. Nechitailo, JETP Lett. 58 (1993) 881.

[13] I.M. Dremin et al., Phys. Lett. B336 (1994) 119.

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[16] SLD Collab., K. Abe, et al., Phys. Rev. D51 (1995) 962.

[17] M.D. Hildreth et al., IEEE Trans. Nucl. Sci. 42 (1994) 451.

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[20] T. Sj¨ ostrand and M. Bengtsson, Comp. Phys. Comm. 43 (1987) 367.

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[22] See e.g. A. Giovannini and L. Van Hove, Z. Phys. C30 (1986) 391.

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Figure captions

1. Functional form of perturbative QCD predictions of the ratio Hqof cumulant to

factorial moments in the leading double-logarithm (solid line), next-to-leading-

logarithm (dotted line) and next-to-next-to-leading-logarithm (dashed line) ap-

proximations. The vertical scale and relative normalizations are arbitrary.

2. a) The corrected charged-particle multiplicity distribution. The open circles at

n= 2, 4 are the predictions of the JETSET Monte Carlo. The solid and dashed

lines represent fitted negative-binomial and log-normal distributions, respectively.

The normalized residuals are shown in b). The fits yielded parameter values of

k = 24.9 and ?n? = 20.7 for the NBD and µ = 3.52, σ = 0.175 and c = 13.4 for

the LND. The errors are statistical only.

3. Ratio of cumulant to factorial moments, Hq, as a function of the moment rank

q. The error bars are statistical and are strongly correlated between ranks.

4. Examples of systematic effects on Hq. The data points show the Hqwith sta-

tistical errors derived using the standard correction. The dotted (dashed) line

connects Hqvalues derived with an increase (decrease) of 1.7% in the simulated

track reconstruction efficiency. The solid line connects Hqvalues derived using

the unparametrized correction matrix.

5. Comparison of the Hqmeasured in the data (dots with statistical errors) with the

predictions of truncated Poisson (dotted line joining the values at different q),

negative binomial (dashed line) and log-normal (dot-dashed line) distributions.

13

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0

Moment Rank q

Hq (arbitrary units)

12–958058A1

DLA

NNLA

NLA

5

Page 15

10–5

0

4

–4

10–3

10–1

Charged Multiplicity n

2040 600

Pn

Residual (σ)

12–95 8058A2

SLD

(a)

(b)

Data?

NBD?

LND

NBD?

LND

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