- Email: [email protected]
Factorial moments — A new test of hadronization models
Volume 228, number 2
FACTORIAL
PHYSICSLETTERSB
MOMENTS
-
14 September 1989
A NEW TEST OF H A D R O N I Z A T I O N M O D E L S ~
Maciej JI~DRZEJCZAK
Institute of Physics, WarsawUniversity,Biatystok Branch, Lipowa 41, Biatystok, Poland Received 8 February 1989;revised manuscript received 4 July 1989
Westudythe dependenceof the factorialmomentsof the rapiditydistribution on the rapidityresolutionin the Webber-Marchcsini model and the model based on local parton-hadron duality. Saturation in the first case and intermittent behaviour in the second case is obtained,
During the last few years many efforts have been directed to find new tests of perturbative QCD. From this point of view, the most interesting observables are the ones that are not sensitive to hadronization processes. New results in perturbative QCD - the summation of leading infrared logarithms as well as some next-to-leading corrections [ 1,2 ] - allow for a more detailed comparison between the partonic and hadronic level. Such a comparison can shed new light on the problem of confinement. To study this particular problem, observables that are most sensitive to the nonperturbative part of the process should be considered. There are several coexisting hadronization models. The most important difference between them is the mass scale Qo at which the QCD evolution is broken and nonperturbative effects turned on. In earlier versions of the string model [3 ] the parameter Qo is, for e+e - annihilation as an example, of the order of Q the virtual photon mass. On the other hand, in the class of models [4,5-8] based on the "preconfinement" idea [9 ] this critical scale is reduced to the value at which perturbation calculus ceases to be valid. Finally, it is possible to go even further, reducing Qo down to AQCI~,i.e., beyond the values allowed by perturbation theory [ 10 ]. Experimental e+e - annihilation data indicate that the one-particle inclusive distribution ofhadrons rather closely follows the parton distribution in such a formal limit. This observation, called local parton-hadron duality ~r This research has been supported in part by the RPBRRRI.14. 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(LPHD), reduces nonperturbative effects to a constant factor that relates partonic and hadronic levels. On the basis of inclusive hadron distributions alone it is quite difficult to discriminate between various hadronization models. More fine characteristics are needed. Recently Bialas and Peschanski [ 11,12 ] have proposed to study multiplicity distributions in small rapidity bins and have predicted a power-like growth of factorial moments with resolution in rapidity. Subsequently, such a dependence has been found in experimental data [ 13 ]. These new observables are sensitive to fluctuations of different scales and we show in this letter that it may be used to distinguish possible hadronization mechanisms. The plan of the letter is as follows. In the first part we present the analysis of the factorial moments of the rapidity distribution calculated in the WebberMarchesini ( W - M ) model [6,7], i.e., the model with hadronization based on "preconfinement". We show, that the rapidity resolution dependence of the moments indeed gives some insight into the branching process. Next, we check that the partonic level of the W - M model in the limit Qo=2Aoco is finite and consistent with the LPHD hypothesis. In the last part of the letter we compare the factorial moments calculated for the hadronic final state generated in the W - M model and the partonic state in the LPHD limit. Extended to the case of inclusive distributions, the ith factorial moment for the division of rapidity range to M b i n s is defined [ 11 ] by
~ 1 km(km__l)...(km__i+l)) ' = M i - l ( 1him= 259
Volume 228, n u m b e r 2
PHYSICS LETTERS B
where km is the number of particles in the rnth bin and ( ) denotes the average over a chosen class of events with an average multiplicity n. Let us identify the structures characteristic for the W-M model in the M dependence of ( F i ) . The evolution variable of perturbative QCD branching in this model is (=pl'p2/oglO)2, where Pl.2((.01,2) are fourmomenta (energies) ofpartons 1 and 2 from the same branching. The effects of soft gluon interference is taken into account (to leading order) by a kinematical cut-off imposed on (. In each branching its value is lower than in the previous one. The ( ordering with infrared cut-off Qo implies a shrinkage of the energy fraction phase space in successive branchings [4 ]. The maximum rapidity separation of the daughter partons for given ( reads
14 September 1989
1.6 Qo= 1.0 GeV 1.5
A V o2
1.4
o
Q0=0,65 GeV O o
o
o
• • O o o
"ll 1.3
Q= 1 TeV
$ 1.2
$ . . . . . . .
110
6y
Ay ~ In (wx/~/Qo - 1 ), where o9 is the energy of the parent parton. We define the rapidity as y = ½1n[ (E+p)/(E-p) ] instead of the usual definition of the momentum parallel to the jet axis. It is an entirely inclusive quantity, independent of the overall jet characteristics [ 10]. As long as the rapidity resolution ~y exceeds the separation of the partons Ay, the branching products usually fall into the same bin and the corresponding fluctuations do not contribute to the observed distributions. The finer the resolution is, the smaller rapidity separations contribute and the larger is the part of the branching process that is visible. In particular, changing 8y one can follow the QCD evolution and look for the sign of fluctuations of nonperturbative origin. In fig. 1 the dependence of the fifth factorial moment on the rapidity resolution in the W - M model with Qo = 1.0 GeV and 0.65 GeV is shown. The resuits presented are for charged and neutral pions produced in e+e - annihilation at a virtual photon mass of 1 TeV. We have used here two samples of 10 000 events with multiplicities exceeding 80. An interesting feature of the results is the sharp step for the cut-off Qo = 1 GeV. Its origin is the following. Perturbative branching in the W - M model breaks when the parton virtualities fall to a value of the order of Qo. Then for all final gluons the virtual mass is set exactly to the cut-off value and nonperturbative splitting to a light qcl pair is enforced. It simplifies 260
Fig. 1. The dependence of the factorial m o m e n t ( F s ) on the rapidity resolution in the W - M model (program BIGWIG v.4.3 without weak decays and with correlations due to soft gluon interference taken into account [Azimuthal correlations due to gluon interference have been considered in ref. [ 7 ]. Here we have taken into account only the first two terms of the eikonalized amplitude Fourier expansion in the azimuthal angle 0. The probability distributions of ~ for the soft gluon emitted by parton i under angel O~ in the vicinity of parton j reads (when Oi<
considerably the identification of colour singlet groups of partons (clusters). The splitting is assumed to have an isotropic distribution in the gluon rest-frame. The average rapidity separation of the quarks than reads D(v)=-I [(l+v) ln(l+v)+(1-v)ln(1-v)], v
(1)
where v = x / l - (2mJQo) 2 is the quark velocity in the gluon rest-frame and mq is the quark mass. The D(v) values for both Qo values are marked in fig. 1 by arrows. It is seen that nonperturbative splitting gives its contribution to ( F s ) only for the resolution ~y<~D(v). The step is due to the fact that all final gluons have identical masses and split in the same
Volume 228, number 2
PHYSICS LETTERS B
way. Therefore the presence of this "structure" has particularly strong influence on the fluctuations of the corresponding scale. For the best fit (Qo=0.65 GeV, AQCD=0.2 GeV) proposed in ref. [ 7 ] the cut-off value is very close to the limit <20= 2mq. In this limit the energy separation of quarks tends to zero, and the splitting cannot be resolved, even for finest binning. After gluon splitting, the hadronic phase of the evolution begins. This phase does not contribute to the fluctuations on a scale less than 6y~- 1. Now let us consider the L P H D problem originally formulated [ I0 ] in the framework of the modified leading logarithmic approximation [2 ] which includes all leading logarithms and next-to-leading corrections to infrared sensitive quantities. It has been conjectured that the W - M Monte Carlo algorithm for the QCD evolution also takes all these corrections properly into account. If we take q 2 = z 2 ( 1 - z ) 2k2, as an argument of as, than in the formal limit Qo-~ 2AQCDthe Sudakov form factor is finite, although at the integration limits z--, ½, k---,2Qo the coupling constant diverges. The comparison of the parton rapidity distribution (for Q o = 2 A Q c D = 0 . 3 2 GeV) with the charged pion distribution from e+e - annihilation at 29 GeV calculated in the W - M model is shown in fig. 2. We have adopted the simplest one-parameter fit, entirely ignoring the conversion factor K~ between partonic and hadronic distributions (in ref. [10] K~=I.1 was assumed). From our analysis it follows that instead of breaking the perturbative QCD evolution at some scale and supplementing it by a hadronic phase, as in the W M model, one can prolong the QCD evolution to the "nonperturbative" limit Qo = 2AQcD. The one-particle inclusive distributions obtained in both cases are very similar. Now, let us proceed to the main point of this letter and calculate the factorial moments for the two cases considered above. We have generated in each case a sample of 70 000 events with multiplicities exceeding 12 at Q = 2 9 GeV. In fig. 3a the moments calcu,ted for pure QCD evolution in the L P H D limit with = 0.32 GeV are shown. The slopes of the linear fits mall rapidity bins are presented in fig. 3b. An imately linear growth of the slope with facto,ent rank is observed. The fifth moments for
5.0
14 September 1989
.
.
.
.
t
.
.
.
.
+
'
~
,
,
i
.
.
.
.
i
.
.
.
.
Q=29 GeV 4.0
+ •
3.0
o. Cll
o r~
2.0
1.0 o 1.0
2+0
3.0
4.0
5.0
Fig. 2. Rapidity distribution for pure QCD cascading in the Qo = 2AQcD limit (full circles) compared with W-M model predictions for charged pions (open circles).
both cases at Q = 1 TeV are compared in fig. 4. In the rapidity intervals considered (from ~y--~0.1 to ~y,-~ 1.0 ) the model based on the LPHD manifests an intermittent [ 12 ] type ofbehaviour: for finer binning additional fluctuations of finer scale contribute. The W - M model behaves in a quite different manner. After a short rise for larger values of fly, the ( F 5 ) moment saturates with better and better resolution. (Similar results concerning the W - M model have been also presented in a recent paper [ 14 ]. ) The saturation can be related to the transition from the partonic level, with small scale fluctuations, to the hadronic level, which produces fluctuations of larger scales only. The general behaviour of the moments is approximately energy independent. Thus we arrive at the conclusion that the factorial moment dependence on the resolution can serve as a sensitive test of the multiparticle production models. In particular, it can help in judging whether the LPHD hypothesis refers only to the one-particle inclusive characteristics of the final state or whether it is a deeper feature of confinement. The existing data suggest [ 15 ] that the second possibility is more favorable, but extensive studies are needed to find the final answer.
261
Volume 228, number 2
PHYSICS LETTERS B
.40
I
|
I
I
I
I
14 September 1989
.'q
I
,
I
I 'i
,
!
I
1=5 •
1=4 .30
.2
&
t=3
V
-1--2 ~
.~0
0 • oo
.I
q=29 GeV
~->~=o.a~ e~v a)
All partons .|0
I
I
I
t
I
t
,1
b) t
.0
8.
d~y
*
i
,
l
,
3.
,
,
i
l
l
i
*
4,
,
6.
R a n k of t h e m o m e n t .
Fig. 3. (a) Dependence of factorial moments on the rapidity resolution in the LPHD case. The moments are rescaled to the value of (F2) for ~y~-0.7. (b) Slopes of the log (FD-log ~y dependence versus factorial moment rank.
1.50
•
We are grateful to G. Biatkowski, M. Krawczyk and J. Kalinowski for useful and stimulating discussions. Q= 1 TeV
1.45
I
References Ato v
1.40
o
0 o
(3
• o O
•© 1.35 o 1.30.1
•o . . . . . .
110
~y Fig. 4. Comparison of the ( F s ) factorial moments for the W-M model (open circles) and the model based on LPHD (full circles).
262
[ 1 ] A.H. Mueller, Nucl. Phys. B 213 (1983) 85, B 241 (1983) 141(E). [ 2 ] Yu.L. Dokshitzer and S.I. Troyan, Nonleading perturbative contributions to the dynamics of quark-gluon cascades and soft hadron spectra in e+e - annihilation, Leningrad Nuclear Physics Institute preprint No. 922 (1984). [ 3 ] B. Andersson, G. Gustafson, G. Ingelman and T. Sj6strand, Phys. Rep. 97 (1983) 31. [4] G. Marchesini and B.R. Webber, Nucl. Phys. B 238 (1984) 1. [ 5 ] R.D. Field and S. Wolfram, Nucl. Phys. B 213 ( 1983 ) 65. [6] B.R. Webber, Nucl. Phys. B 238 (1984) 492. [7] G. Marchesini and B.R. Webber, Nucl. Phys. B 310 (1988) 461.
Volume 228, number 2
PHYSICS LETTERS B
[8 ] T.D. Gottschalk and D.A. Morris, Nucl. Phys. B 288 ( 1987 ) 729. [9] D. Amati and G. Veneziano, Phys. Lett. B 83 (1979) 87. [10]Ya.l. Azimov, Yu.L. Dokshitzer, V.A. Khoze and S.I. Troyan, Z. Phys. C 27 ( 1985 ) 65; QCD portrait of hadronic jets, Leningrad Nuclear Physics Institute preprint No. 1230 (1987); Yu.L. Dokshitzer, V.A. Khoze, S.I. Troyan and A.H. Mueller, Rev. Mod. Phys. 60 (1988) 373. [ 11 ] A. Biatas and R. Peschanski, Nucl. Phys. B 273 (1986) 703; B 308 (1988) 857; Phys. Lett. B 207 ( 1988 ) 59.
14 September 1989
[ 12 ] A. Bia[as, K. Fiatkowski and R. Peschanski, Europhys. Lett. 7 (1988) 125; W. Ochs and J. Wosiek, Phys. Lett. B 214 (1988) 617. [ 13 ] JACEE Collab., Phys. Rev. Lett. 50 ( 1983 ) 2062; R. Ho|ytiski et al., Phys. Rev. Lett. 62 (1989) 733; Ames-Bologna-CERN-Dortmund-Warsaw Collab., Nijmegen preprint HEN-309 (1989). [ 14] K. Fiatkowski, B. Wosiek and J. Wosiek, Cracow preprint TPJU-6/89 (March 1989). [ 15 ] B. Buschbeck, P. Lipa and R. Peschanski, Phys. Lett. B 215 ( 1988 ) 788.
263
FACTORIAL
PHYSICSLETTERSB
MOMENTS
-
14 September 1989
A NEW TEST OF H A D R O N I Z A T I O N M O D E L S ~
Maciej JI~DRZEJCZAK
Institute of Physics, WarsawUniversity,Biatystok Branch, Lipowa 41, Biatystok, Poland Received 8 February 1989;revised manuscript received 4 July 1989
Westudythe dependenceof the factorialmomentsof the rapiditydistribution on the rapidityresolutionin the Webber-Marchcsini model and the model based on local parton-hadron duality. Saturation in the first case and intermittent behaviour in the second case is obtained,
During the last few years many efforts have been directed to find new tests of perturbative QCD. From this point of view, the most interesting observables are the ones that are not sensitive to hadronization processes. New results in perturbative QCD - the summation of leading infrared logarithms as well as some next-to-leading corrections [ 1,2 ] - allow for a more detailed comparison between the partonic and hadronic level. Such a comparison can shed new light on the problem of confinement. To study this particular problem, observables that are most sensitive to the nonperturbative part of the process should be considered. There are several coexisting hadronization models. The most important difference between them is the mass scale Qo at which the QCD evolution is broken and nonperturbative effects turned on. In earlier versions of the string model [3 ] the parameter Qo is, for e+e - annihilation as an example, of the order of Q the virtual photon mass. On the other hand, in the class of models [4,5-8] based on the "preconfinement" idea [9 ] this critical scale is reduced to the value at which perturbation calculus ceases to be valid. Finally, it is possible to go even further, reducing Qo down to AQCI~,i.e., beyond the values allowed by perturbation theory [ 10 ]. Experimental e+e - annihilation data indicate that the one-particle inclusive distribution ofhadrons rather closely follows the parton distribution in such a formal limit. This observation, called local parton-hadron duality ~r This research has been supported in part by the RPBRRRI.14. 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(LPHD), reduces nonperturbative effects to a constant factor that relates partonic and hadronic levels. On the basis of inclusive hadron distributions alone it is quite difficult to discriminate between various hadronization models. More fine characteristics are needed. Recently Bialas and Peschanski [ 11,12 ] have proposed to study multiplicity distributions in small rapidity bins and have predicted a power-like growth of factorial moments with resolution in rapidity. Subsequently, such a dependence has been found in experimental data [ 13 ]. These new observables are sensitive to fluctuations of different scales and we show in this letter that it may be used to distinguish possible hadronization mechanisms. The plan of the letter is as follows. In the first part we present the analysis of the factorial moments of the rapidity distribution calculated in the WebberMarchesini ( W - M ) model [6,7], i.e., the model with hadronization based on "preconfinement". We show, that the rapidity resolution dependence of the moments indeed gives some insight into the branching process. Next, we check that the partonic level of the W - M model in the limit Qo=2Aoco is finite and consistent with the LPHD hypothesis. In the last part of the letter we compare the factorial moments calculated for the hadronic final state generated in the W - M model and the partonic state in the LPHD limit. Extended to the case of inclusive distributions, the ith factorial moment for the division of rapidity range to M b i n s is defined [ 11 ] by
~ 1 km(km__l)...(km__i+l)) '
Volume 228, n u m b e r 2
PHYSICS LETTERS B
where km is the number of particles in the rnth bin and ( ) denotes the average over a chosen class of events with an average multiplicity n. Let us identify the structures characteristic for the W-M model in the M dependence of ( F i ) . The evolution variable of perturbative QCD branching in this model is (=pl'p2/oglO)2, where Pl.2((.01,2) are fourmomenta (energies) ofpartons 1 and 2 from the same branching. The effects of soft gluon interference is taken into account (to leading order) by a kinematical cut-off imposed on (. In each branching its value is lower than in the previous one. The ( ordering with infrared cut-off Qo implies a shrinkage of the energy fraction phase space in successive branchings [4 ]. The maximum rapidity separation of the daughter partons for given ( reads
14 September 1989
1.6 Qo= 1.0 GeV 1.5
A V o2
1.4
o
Q0=0,65 GeV O o
o
o
• • O o o
"ll 1.3
Q= 1 TeV
$ 1.2
$ . . . . . . .
110
6y
Ay ~ In (wx/~/Qo - 1 ), where o9 is the energy of the parent parton. We define the rapidity as y = ½1n[ (E+p)/(E-p) ] instead of the usual definition of the momentum parallel to the jet axis. It is an entirely inclusive quantity, independent of the overall jet characteristics [ 10]. As long as the rapidity resolution ~y exceeds the separation of the partons Ay, the branching products usually fall into the same bin and the corresponding fluctuations do not contribute to the observed distributions. The finer the resolution is, the smaller rapidity separations contribute and the larger is the part of the branching process that is visible. In particular, changing 8y one can follow the QCD evolution and look for the sign of fluctuations of nonperturbative origin. In fig. 1 the dependence of the fifth factorial moment on the rapidity resolution in the W - M model with Qo = 1.0 GeV and 0.65 GeV is shown. The resuits presented are for charged and neutral pions produced in e+e - annihilation at a virtual photon mass of 1 TeV. We have used here two samples of 10 000 events with multiplicities exceeding 80. An interesting feature of the results is the sharp step for the cut-off Qo = 1 GeV. Its origin is the following. Perturbative branching in the W - M model breaks when the parton virtualities fall to a value of the order of Qo. Then for all final gluons the virtual mass is set exactly to the cut-off value and nonperturbative splitting to a light qcl pair is enforced. It simplifies 260
Fig. 1. The dependence of the factorial m o m e n t ( F s ) on the rapidity resolution in the W - M model (program BIGWIG v.4.3 without weak decays and with correlations due to soft gluon interference taken into account [Azimuthal correlations due to gluon interference have been considered in ref. [ 7 ]. Here we have taken into account only the first two terms of the eikonalized amplitude Fourier expansion in the azimuthal angle 0. The probability distributions of ~ for the soft gluon emitted by parton i under angel O~ in the vicinity of parton j reads (when Oi<
considerably the identification of colour singlet groups of partons (clusters). The splitting is assumed to have an isotropic distribution in the gluon rest-frame. The average rapidity separation of the quarks than reads D(v)=-I [(l+v) ln(l+v)+(1-v)ln(1-v)], v
(1)
where v = x / l - (2mJQo) 2 is the quark velocity in the gluon rest-frame and mq is the quark mass. The D(v) values for both Qo values are marked in fig. 1 by arrows. It is seen that nonperturbative splitting gives its contribution to ( F s ) only for the resolution ~y<~D(v). The step is due to the fact that all final gluons have identical masses and split in the same
Volume 228, number 2
PHYSICS LETTERS B
way. Therefore the presence of this "structure" has particularly strong influence on the fluctuations of the corresponding scale. For the best fit (Qo=0.65 GeV, AQCD=0.2 GeV) proposed in ref. [ 7 ] the cut-off value is very close to the limit <20= 2mq. In this limit the energy separation of quarks tends to zero, and the splitting cannot be resolved, even for finest binning. After gluon splitting, the hadronic phase of the evolution begins. This phase does not contribute to the fluctuations on a scale less than 6y~- 1. Now let us consider the L P H D problem originally formulated [ I0 ] in the framework of the modified leading logarithmic approximation [2 ] which includes all leading logarithms and next-to-leading corrections to infrared sensitive quantities. It has been conjectured that the W - M Monte Carlo algorithm for the QCD evolution also takes all these corrections properly into account. If we take q 2 = z 2 ( 1 - z ) 2k2, as an argument of as, than in the formal limit Qo-~ 2AQCDthe Sudakov form factor is finite, although at the integration limits z--, ½, k---,2Qo the coupling constant diverges. The comparison of the parton rapidity distribution (for Q o = 2 A Q c D = 0 . 3 2 GeV) with the charged pion distribution from e+e - annihilation at 29 GeV calculated in the W - M model is shown in fig. 2. We have adopted the simplest one-parameter fit, entirely ignoring the conversion factor K~ between partonic and hadronic distributions (in ref. [10] K~=I.1 was assumed). From our analysis it follows that instead of breaking the perturbative QCD evolution at some scale and supplementing it by a hadronic phase, as in the W M model, one can prolong the QCD evolution to the "nonperturbative" limit Qo = 2AQcD. The one-particle inclusive distributions obtained in both cases are very similar. Now, let us proceed to the main point of this letter and calculate the factorial moments for the two cases considered above. We have generated in each case a sample of 70 000 events with multiplicities exceeding 12 at Q = 2 9 GeV. In fig. 3a the moments calcu,ted for pure QCD evolution in the L P H D limit with = 0.32 GeV are shown. The slopes of the linear fits mall rapidity bins are presented in fig. 3b. An imately linear growth of the slope with facto,ent rank is observed. The fifth moments for
5.0
14 September 1989
.
.
.
.
t
.
.
.
.
+
'
~
,
,
i
.
.
.
.
i
.
.
.
.
Q=29 GeV 4.0
+ •
3.0
o. Cll
o r~
2.0
1.0 o 1.0
2+0
3.0
4.0
5.0
Fig. 2. Rapidity distribution for pure QCD cascading in the Qo = 2AQcD limit (full circles) compared with W-M model predictions for charged pions (open circles).
both cases at Q = 1 TeV are compared in fig. 4. In the rapidity intervals considered (from ~y--~0.1 to ~y,-~ 1.0 ) the model based on the LPHD manifests an intermittent [ 12 ] type ofbehaviour: for finer binning additional fluctuations of finer scale contribute. The W - M model behaves in a quite different manner. After a short rise for larger values of fly, the ( F 5 ) moment saturates with better and better resolution. (Similar results concerning the W - M model have been also presented in a recent paper [ 14 ]. ) The saturation can be related to the transition from the partonic level, with small scale fluctuations, to the hadronic level, which produces fluctuations of larger scales only. The general behaviour of the moments is approximately energy independent. Thus we arrive at the conclusion that the factorial moment dependence on the resolution can serve as a sensitive test of the multiparticle production models. In particular, it can help in judging whether the LPHD hypothesis refers only to the one-particle inclusive characteristics of the final state or whether it is a deeper feature of confinement. The existing data suggest [ 15 ] that the second possibility is more favorable, but extensive studies are needed to find the final answer.
261
Volume 228, number 2
PHYSICS LETTERS B
.40
I
|
I
I
I
I
14 September 1989
.'q
I
,
I
I 'i
,
!
I
1=5 •
1=4 .30
.2
&
t=3
V
-1--2 ~
.~0
0 • oo
.I
q=29 GeV
~->~=o.a~ e~v a)
All partons .|0
I
I
I
t
I
t
,1
b) t
.0
8.
d~y
*
i
,
l
,
3.
,
,
i
l
l
i
*
4,
,
6.
R a n k of t h e m o m e n t .
Fig. 3. (a) Dependence of factorial moments on the rapidity resolution in the LPHD case. The moments are rescaled to the value of (F2) for ~y~-0.7. (b) Slopes of the log (FD-log ~y dependence versus factorial moment rank.
1.50
•
We are grateful to G. Biatkowski, M. Krawczyk and J. Kalinowski for useful and stimulating discussions. Q= 1 TeV
1.45
I
References Ato v
1.40
o
0 o
(3
• o O
•© 1.35 o 1.30.1
•o . . . . . .
110
~y Fig. 4. Comparison of the ( F s ) factorial moments for the W-M model (open circles) and the model based on LPHD (full circles).
262
[ 1 ] A.H. Mueller, Nucl. Phys. B 213 (1983) 85, B 241 (1983) 141(E). [ 2 ] Yu.L. Dokshitzer and S.I. Troyan, Nonleading perturbative contributions to the dynamics of quark-gluon cascades and soft hadron spectra in e+e - annihilation, Leningrad Nuclear Physics Institute preprint No. 922 (1984). [ 3 ] B. Andersson, G. Gustafson, G. Ingelman and T. Sj6strand, Phys. Rep. 97 (1983) 31. [4] G. Marchesini and B.R. Webber, Nucl. Phys. B 238 (1984) 1. [ 5 ] R.D. Field and S. Wolfram, Nucl. Phys. B 213 ( 1983 ) 65. [6] B.R. Webber, Nucl. Phys. B 238 (1984) 492. [7] G. Marchesini and B.R. Webber, Nucl. Phys. B 310 (1988) 461.
Volume 228, number 2
PHYSICS LETTERS B
[8 ] T.D. Gottschalk and D.A. Morris, Nucl. Phys. B 288 ( 1987 ) 729. [9] D. Amati and G. Veneziano, Phys. Lett. B 83 (1979) 87. [10]Ya.l. Azimov, Yu.L. Dokshitzer, V.A. Khoze and S.I. Troyan, Z. Phys. C 27 ( 1985 ) 65; QCD portrait of hadronic jets, Leningrad Nuclear Physics Institute preprint No. 1230 (1987); Yu.L. Dokshitzer, V.A. Khoze, S.I. Troyan and A.H. Mueller, Rev. Mod. Phys. 60 (1988) 373. [ 11 ] A. Biatas and R. Peschanski, Nucl. Phys. B 273 (1986) 703; B 308 (1988) 857; Phys. Lett. B 207 ( 1988 ) 59.
14 September 1989
[ 12 ] A. Bia[as, K. Fiatkowski and R. Peschanski, Europhys. Lett. 7 (1988) 125; W. Ochs and J. Wosiek, Phys. Lett. B 214 (1988) 617. [ 13 ] JACEE Collab., Phys. Rev. Lett. 50 ( 1983 ) 2062; R. Ho|ytiski et al., Phys. Rev. Lett. 62 (1989) 733; Ames-Bologna-CERN-Dortmund-Warsaw Collab., Nijmegen preprint HEN-309 (1989). [ 14] K. Fiatkowski, B. Wosiek and J. Wosiek, Cracow preprint TPJU-6/89 (March 1989). [ 15 ] B. Buschbeck, P. Lipa and R. Peschanski, Phys. Lett. B 215 ( 1988 ) 788.
263