Factorial and cumulant moments in e+e− → hadrons at the Z0 resonance
 K. Abe
 I. Abt
 C.J. Ahn
 T. Akagi
 N.J. Allen
 W.W. Ash
 D. Aston
 K.G. Baird
 C. Baltay
 H.R. Band
 M.B. Barakat
 G. Baranko
 O. Bardon
 T. Barklow
 A.O. Bazarko
 R. BenDavid
 A.C. Benvenuti
 G.M. Bilei
 D. Bisello
 G. Blaylock
 J.R. Bogart
 T. Bolton
 G.R. Bower
 J.E. Brau
 M. Breidenbach
 W.M. Bugg
 D. Burke
 T.H. Burnett
 P.N. Burrows
 W. Busza
 A. Calcaterra
 D.O. Caldwell
 D. Calloway
 B. Camanzi
 M. Carpinelli
 R. Cassell
 R. Castaldi
 A. Castro
 M. CavalliSforza
 A. Chou
 E. Church
 H.O. Cohn
 J.A. Coller
 V. Cook
 R. Cotton
 R.F. Cowan
 D.G. Coyne
 G. Crawford
 A. D'Oliveira
 C.J.S. Damerell
 M. Daoudi
 R. De Sangro
 P. De Simone
 R. Dell'Orso
 M. Dima
 P.Y.C. Du
 R. Dubois
 B.I. Eisenstein
 R. Elia
 E. Etzion
 D. Falciai
 C. Fan
 M.J. Fero
 R. Frey
 K. Furuno
 T. Gillman
 G. Gladding
 S. Gonzalez
 G.D. Hallewell
 E.L. Hart
 A. Hasan
 Y. Hasegawa
 K. Hasuko
 S. Hedges
 S.S. Hertzbach
 M.D. Hildreth
 J. Huber
 M.E. Huffer
 E.W. Hughes
 H. Hwang
 Y. Iwasaki
 D.J. Jackson
 P. Jacques
 J. Jaros
 A.S. Johnson
 J.R. Johnson
 R.A. Johnson
 T. Junk
 R. Kajikawa
 M. Kalelkar
 H.J. Kang
 I. Karliner
 H. Kawahara
 H.W. Kendall
 Y. Kim
 M.E. King
 R. King
 R.R. Kofler
 N.M. Krishna
 R.S. Kroeger
 J.F. Labs
 M. Langston
 A. Lath
 J.A. Lauber
 D.W.G.S. Leith
 M.X. Liu
 X. Liu
 M. Loreti
 A. Lu
 H.L. Lynch
 J. Ma
 G. Mancinelli
 S. Manly
 G. Mantovani

 T.W. Markiewicz
 T. Maruyama
 R. Massetti
 H. Masuda
 E. Mazzucato
 A.K. McKemey
 B.T. Meadows
 R. Messner
 P.M. Mockett
 K.C. Moffeit
 B. Mours
 D. Muller
 T. Nagamine
 S. Narita
 U. Nauenberg
 H. Neal
 M. Nussbaum
 Y. Ohnishi
 L.S. Osborne
 R.S. Panvini
 H. Park
 T.J. Pavel
 I. Peruzzi
 M. Piccolo
 L. Piemontese
 E. Pieroni
 K.T. Pitts
 R.J. Plano
 R. Prepost
 C.Y. Prescott
 G.D. Punkar
 J. Quigley
 B.N. Ratcliff
 T.W. Reeves
 J. Reidy
 P.E. Rensing
 L.S. Rochester
 P.C. Rowson
 J.J. Russell
 O.H. Saxton
 S.F. Schaffner
 T. Schalk
 R.H. Schindler
 B.A. Schumm
 A. Seiden
 S. Sen
 V.V. Serbo
 M.H. Shaevitz
 J.T. Shank
 G. Shapiro
 S.L. Shapiro
 D.J. Sherden
 K.D. Shmakov
 C. Simopoulos
 N.B. Sinev
 S.R. Smith
 J.A. Snyder
 P. Stamer
 H. Steiner
 R. Steiner
 M.G. Strauss
 D. Su
 F. Suekane
 A. Sugiyama
 S. Suzuki
 M. Swartz
 A. Szumilo
 T. Takahashi
 F.E. Taylor
 E. Torrence
 A.I. Trandafir
 J.D. Turk
 T. Usher
 J. Va'vra
 C. Vannini
 E. Vella
 J.P. Venuti
 R. Verdier
 P.G. Verdini

 S.R. Wagner
 A.P. Waite
 S.J. Watts
 A.W. Weidemann
 E.R. Weiss
 J.S. Whitaker
 S.L. White
 F.J. Wickens
 D.A. Williams
 D.C. Williams
 S.H. Williams
 S. Willocq
 R.J. Wilson
 W.J. Wisniewski
 M. Woods
 G.B. Word
 J. Wyss
 R.K. Yamamoto
 J.M. Yamartino
 X. Yang
 S.J. Yellin
 C.C. Young
 H. Yuta
 G. Zapalac
 R.W. Zdarko
 C. Zeitlin
 Z. Zhang
 J. Zhou
ABSTRACT We present the first experimental study of the ratio of cumulant to factorial moments of the chargedparticle multiplicity distribution in highenergy particle interactions, using hadronic Z0 decays collected by the SLD experiment at SLAC. We find that this ratio, as a function of the momentrank q, decreases sharply to a negative minimum at q = 5, which is followed by quasioscillations. These features are insensitive to experimental systematic effects and are in qualitative agreement with expectations from nexttonexttoleadingorder perturbative QCD.
 [Show abstract] [Hide abstract]
ABSTRACT: This paper contains a review of the main results of a search of regularities in collective variables properties in multiparticle dynamics, regularities which can be considered as manifestations of the original simplicity suggested by QCD. The method is based on a continuous dialog between experiment and theory. The paper follows the development of this research line, from its beginnings in the 1970's to the current state of the art, discussing how it produced both sound interpretations of the most relevant experimental facts and intriguing perspectives for new physics signals in the TeV energy domain.International Journal of Modern Physics A 01/2012; 20(17). · 1.09 Impact Factor  SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: The theory of strong interactions, quantum chromodynamics (QCD), is quite successful in predicting and describing many features of multiparticle production processes at high energies. In this paper, the general perturbative QCD approach to these processes (primarily e+eannihilation) is briefly formulated, and associated problems are discussed. It is shown that analytical calculations at the parton level using a lowmomentum cutoff are surprisingly adequate in describing experimental data on the final hadronic state in multiparticle production processes at high energies  even though the perturbative expansion parameter is not very small. More importantly, the perturbative QCD not only describes existing data but also predicts many qualitatively new and intriguing phenomena.PhysicsUspekhi 05/2002; 45(5):507. · 1.89 Impact Factor  SourceAvailable from: Andreas Dewanto[Show abstract] [Hide abstract]
ABSTRACT: We study the moments of multiplicty distribution and its relation to the LeeYang zeros of the generating function in electronpositron and hadronhadron 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 electronpositron (e+e) multiplicity data from hadronhadron (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 LeeYang 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 quarkdominated soft scattering events to gluondominated semihard scattering events.International Journal of Modern Physics A 01/2012; 24(18n19). · 1.09 Impact Factor
Page 1
arXiv:hepex/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. BenDavid,(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. CavalliSforza,(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, I40126 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, I44100 Ferrara, Italy
(13)INFN Lab. Nazionali di Frascati, I00044 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, Chikusaku, Nagoya 464 Japan
(20)University of Oregon, Eugene, Oregon 97403
(21)INFN Sezione di Padova and Universit` a di Padova, I35100 Padova, Italy
(22)INFN Sezione di Perugia and Universit` a di Perugia, I06100 Perugia, Italy
(23)INFN Sezione di Pisa and Universit` a di Pisa, I56100 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 chargedparticle multiplicity distribution in highenergy particle interac
tions, using hadronic Z0decays collected by the SLD experiment at SLAC. We find that
this ratio, as a function of the momentrank q, decreases sharply to a negative mini
mum at q = 5, which is followed by quasioscillations. These features are insensitive
to experimental systematic effects and are in qualitative agreement with expectations
from nexttonexttoleadingorder perturbative QCD.
3
Page 4
1Introduction
One of the most fundamental observables in highenergy 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 nonrandom particleproduction
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 finalstate 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 partonhadron duality (LPHD) [3], then such calcu
lations may be expected to reproduce some features of experimental data. An early
calculation [4] in the leading doublelogarithmic 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 higherorder 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 higherorder 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 nexttoleading logarithm approximation (NLA) predicts [11] a
minimum in Hq at q ≈ 5 and a positive constant value for q ≫ 5, while the next
tonexttoleading logarithm approximation (NNLA) predicts [12] that this minimum
is negative and is followed by quasioscillations 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 chargedparticle multiplicity distribution in highenergy
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 wellmeasured 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
thrustaxis [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 chargedparticle 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 initialstate 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 twostage 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 eventselection 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 eventselection cuts and of initialstate 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
initialstate radiation and Psel(ng) is the normalized distribution for those events in
the fullysimulated 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 quasioscillation 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 negativebinomial 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 quasioscillations at higher q.
Experimental systematic effects were also investigated. An important issue is the
simulation of the trackreconstruction 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 quasioscillation.
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 quasioscillation 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 offdiagonal elements of the
matrix. The effect on the Hqof variation of the parameters of the threeGaussian 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 nonhadronic background, which appears predominantly in the lowmultiplicity
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 quasioscillations 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 quasioscillation of
∓0.2 units of rank. There is an uncertainty on the amplitude of the quasioscillation
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 quasioscillation about zero for q≥8 are wellestablished 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
quasioscillations 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 nexttonexttoleading 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
Page 9
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 lognormal distribution
(LND)
?n+1
where µ, σ, and c are free parameters, is predicted [23] by models in which the particles
result from a scaleinvariant stochastic branching process, which might be related to
Pn(µ,σ,c) =
n
N
n′+ cexp
?
−(ln(n′+ c) − µ)2
2σ2
?
dn′, (7)
9
Page 10
the parton branchings in a QCD cascade.
Considering statistical errors only, we performed leastsquares 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
largen tail of the multiplicity distribution due to finite datasample size could lead
to quasioscillations 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 quasioscillation 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 chargedparticle 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 nexttonexttoleadinglogarithm 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 doublelogarithm
10
Page 11
and nexttoleadinglogarithm approximations are not sufficient to describe the data.
The Poisson and negative binomial distributions do not predict these features. The
lognormal 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: DEFG02
91ER40676 (BU), DEFG0392ER40701 (CIT), DEFG0391ER40618 (UCSB), DE
FG0392ER40689 (UCSC), DEFG0393ER40788 (CSU), DEFG0291ER40672 (Col
orado), DEFG0291ER40677(Illinois), DEAC0376SF00098 (LBL), DEFG0292ER40715
(Massachusetts), DEAC0276ER03069 (MIT), DEFG0685ER40224 (Oregon), DE
AC0376SF00515 (SLAC), DEFG0591ER40627 (Tennessee), DEAC0276ER00881
(Wisconsin), DEFG0292ER40704 (Yale); U.S. National Science Foundation grants:
PHY9113428 (UCSC), PHY8921320(Columbia), PHY9204239 (Cincinnati), PHY
8817930 (Rutgers), PHY8819316 (Vanderbilt), PHY9203212 (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
JapanUS Cooperative Research Project on High Energy Physics (Nagoya, Tohoku).
References
[1] For reviews, see, for instance, G. Giacomelli, Int. J. Mod. Phys. A5 (1990) 223;
Hadronic Multiparticle Production, ed. P. Carruthers, World Scientific, Singapore
(1990).
[2] H. Fritzsch, M. GellMann, 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.
11
Page 12
[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.
[14] R. Ugoccioni, A. Giovannini, S. Lupia. Phys. Lett. B342 (1995) 387.
[15] SLD Design Report, SLAC Report 273 (1984).
[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.
[18] C.J.S. Damerell et al., Nucl. Inst. Meth. A288 (1990) 288.
[19] E. Farhi, Phys. Rev. Lett. 39 (1977) 1587.
[20] T. Sj¨ ostrand and M. Bengtsson, Comp. Phys. Comm. 43 (1987) 367.
[21] SLD Collab., K. Abe, et al., Phys. Rev. Lett. 72 (1994) 3145.
[22] See e.g. A. Giovannini and L. Van Hove, Z. Phys. C30 (1986) 391.
[23] See e.g. S. Carius and G. Ingelman, Phys. Lett. B252 (1990) 647.
[24] ALEPH Collab., D. Decamp et al., Phys. Lett. B273 (1991) 181.
12
Page 13
Figure captions
1. Functional form of perturbative QCD predictions of the ratio Hqof cumulant to
factorial moments in the leading doublelogarithm (solid line), nexttoleading
logarithm (dotted line) and nexttonexttoleadinglogarithm (dashed line) ap
proximations. The vertical scale and relative normalizations are arbitrary.
2. a) The corrected chargedparticle multiplicity distribution. The open circles at
n= 2, 4 are the predictions of the JETSET Monte Carlo. The solid and dashed
lines represent fitted negativebinomial and lognormal 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 lognormal (dotdashed line) distributions.
13
Page 14
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
View other sources
Hide other sources
 Available from T. Markiewicz · May 20, 2014
 Available from arxiv.org
 Available from ArXiv