DFTT 52/93
October 1, 1993





THE GENERALIZED SIMPLIFIED

PARTON SHOWER MODEL



R. UGOCCIONI, A. GIOVANNINI, S. LUPIA

Dipartimento di Fisica Teorica, Universit`a di Torino and
INFN, Sezione di Torino, via Giuria 1, 10125 Torino, Italy

ABSTRACT
We explore the consequences of considering clans real physical objects in
the framework of a generalized version of the Simplified Parton Shower
model for a single jet. We predict that the average number of clans at
fixed energy grows linearly in rapidity and slowly decreases with energy
in a fixed rapidity interval.


Talk presented by R. Ugoccioni at the
XXIII International Symposium on Multiparticle Dynamics
Aspen, Colorado, USA, 1217 September 1993.
To be published in the Proceedings.



 2 Nov 1993





Work supported in part by M.U.R.S.T. (Italy) under Grant 1992.


THE GENERALIZED SIMPLIFIED
PARTON SHOWER MODEL





R. UGOCCIONI*, A. GIOVANNINI and S. LUPIA
Dipartimento di Fisica Teorica, Universit`a di Torino and
INFN, Sezione di Torino, via Giuria 1, 10125 Torino, Italy




ABSTRACT



We explore the consequences of considering clans real physical objects in the framework of
a generalized version of the Simplified Parton Shower model for a single jet. We predict that
the average number of clans at fixed energy grows linearly in rapidity and slowly decreases
with energy in a fixed rapidity interval.




1. The Generalized SPS model


Clan analysis has been a very useful tool for interpreting experimental data in high
energy collisions; through its use, two regularities were discovered: first, the average
number of clans is approximately constant with energy in a fixed rapidity interval, and
second, the average number of clans grows linearly with the rapidity interval at fixed energy.
These results have not been explained so far by any theoretical calculation, although they are
well reproduced by Monte Carlo simulations. In an attempt to remedy this lack of analytical
understanding of experimental facts, we propose to generalize an existing model 1, based on
essentials of QCD, by assuming clans as real objects. This assumption should be contrasted
with the statistical clan concept used in the original model.
In the original Simplified Parton Shower (SPS) model 1, we consider an initial parton

of maximum allowed virtuality which splits at virtuality into two partons of virtuality






0 and 1. We require 0 + 1 and min = 1 GeV. We define the probability for
      






a parton of virtuality to split at , ( ), which is normalized by a Sudakov form
   





factor. The joint probability density ( 0 1 ) for a parton of virtuality to split into two
     





partons of virtuality 0 and 1 is defined by ( 0 1 ) = ( 0 ) ( 1 ) ( ), where
                





( ) is a normalization factor. This general scheme is valid for any splitting function
 






( ) which is factorizable in terms of its variables and . However, for numerical
   





simulations we choose:









(log ) 1 log 
  
( ) = = (1)

     (log )    log











Work supported in part by M.U.R.S.T. (Italy) under Grant 1992
* Talk presented by R. Ugoccioni


1





being the only free parameter of the model. This form of ( ) was motivated in 1
  





by our request of simplicity in the structure of the model. Notice that eq. (1) is exactly the
virtuality dependence of the standard QCD kernel controlling gluon emission by a parton.
For describing the rapidity structure of the model, we use the singular part of the QCD
kernel controlling gluon branching:







1 1

( ) + (2)
         
1
    











Here is the energy fraction of the produced parton ( = 0 1).
     





We propose now to incorporate in the original SPS model which we have just
sketched the concept of clan as follows: we decide to pay attention for each event to the
ancestor which, splitting times, gives rise to subprocesses (one at each splitting) and
 





we identify them with clans. Therefore in this model for a single event the concept of
clan is no more as it has been in 2, i.e., only a statistical one. In the present picture the
number of clans in each event coincides with the number of splittings of the ancestor, i.e.,
with the number of steps in the cascade. It will be shown that this simple consideration is
enough to predict that the distribution of clans in full phase space is Poissonian; we stress
that this result can be obtained a priori in the generalized version of the model, differently
from what has been done in 2 where the independent production of clans was an "ansatz"
introduced a posteriori in order to explain the occurrence of NB regularity.
The generalization of SPS to a genuine Markov process proceeds by assuming
that the splitting function of the first step is the same at any step. This fact implies
that our kinematical limits are extended from a triangle ( 0 + 1 ) to a square
  

















( 0 1 ), so that subsequent steps are not correlated. We call this new version
   











of the SPS model Generalized Simplified Parton Shower (GSPS) model. It follows that the
generating function of the number of clans in full phase space (fps) is given by:

fps
(fps )( 1)
( ) = (3)
clan
 





i.e., a shifted Poisson distribution. Accordingly, the average number of clans can easily be
calculated by using eq. (1):







"






 log









(fps ) = (fps ) + 1 ( ) + 1 = log + 1 (4)

!       
2 log 2

In order to explain the clan scaling in rapidity, we extend our study to clan distri-






butions in symmetric rapidity intervals = [ cut cut ]. We can relate the probability to
#  # #











$ % &
have clans in the symmetric rapidity interval , ( cut ), to the corresponding
# #











%
probability defined in full phase space, (fps ), by the relation


(











$



% & %
(
cut ) = ( cut fps) (fps ) (5)
# ' # 





&

% %





=


2








$ $
where ( cut fps) is the conditional probability to have clans in when one
#  #






has clans in full phase space. This new function contains all dynamical information
on the rapidity structure of the production process. Since clans are not correlated in full
phase space and the only allowed correlations are among particles within the same clan,






$

the probability ( cut fps) results to be a positive binomial distribution
# 











& &





% % %






$






( cut fps) = (1 ) (6)






#  




$












where ( cut ) is the probability to produce a single clan in the rapidity interval from
# #






the maximum virtuality . Accordingly, it can be shown that the probability generating
function for clans in symmetric rapidity intervals turns out to be

y
( )( 1)
( ) = [ ( ) + 1 ( )] cut
 (7)
clan cut cut

 #   #












with the average number given by ( cut ) = ( cut ) (fps ).

# #





Equation (7), being the sum of two Poissonian distributions (the first a shifted one),
has a nice physical meaning: the two terms correspond to the probability of having the
ancestor within or outside the given rapidity interval. In addition, it should be noticed that







the void probability 3 in a given rapidity interval, 0( cut ), is different from zero and is
#





given by







( cut ) (fps )

0( cut ) = [1 ( cut )] (8)






#  # 





This fact is far rich in consequences, which we propose to explore in a forthcoming paper.







We also notice that, when cut is very small and therefore ( cut ) tends to zero, the

# #





unshifted Poissonian dominates. Thus, it can be stated that in the smallest rapidity intervals
the clan multiplicity is Poissonian to a good approximation. When cut tends to full
#












phase space and therefore ( cut ) tends to 1, we approach the exact, full phase space,
#












shifted Poisson distribution. We find also that, when ( cut ) is sufficiently large, the
#






shifted Poissonian dominates at large (the tail of the distribution) while the unshifted one

dominates at small (the head of the distribution). This fact might have some consequences

in interpreting the anomalies found in NB behavior for small . Furthermore, the average
number of clans in a symmetric rapidity interval, cut , can be calculated from eq. (7)
 #  #











and is given by the following formula:








( cut ) = ( cut )  (fps ) (9)

# # 












The next step is to calculate the explicit form of ( cut ), i.e.,
#





"


cut













( cut ) = ~( clan ) clan (10)
#  #  #





cut



















where ~( clan ) is the probability to find a clan of rapidity clan generated at any step of
 # #






the cascade initiated by an ancestor of maximum allowed virtuality . The just mentioned


3



Figure 1. Average number of clans  ( cut ) Figure 2. Average number of clans  ( cut ) at
 





   





as a function of rapidity interval cut at different partonic level as a function of rapidity interval cut
 





ancestor parton energies in the GSPS model. at different jet energies for lowest energy jets
 











from a sample of 100000 3-jet events generated
with JETSET 7.2 Parton Shower at = 1000
 





GeV. Jet selection has been performed using the
LUCLUS algorithm; partonic level has been ob-
tained via GLPHD.


probability can be expressed, within the GSPS, in terms of probability distributions calcu-







lable from eq. (1) and (2). It results that the explicit form of ( cut ), valid in the central
#





rapidity region, is given by







cut
#
( cut ) = 

# 1 log






(11)







(log 2 cut ) (log 2 +
 cut ) 
 #  #





+ ;



2 ( 1)(log )






 






0 cut log log 2
# 





By inserting eq. (11) and (4) into eq. (9), we determine the behavior of the average
number of clans as a function of the rapidity interval and of the energy of the ancestor


parton, which is shown for = 2 in Fig. 1. A similar behavior occurs for other values of
 

0 and our choice of = 2 should be considered only indicative. As can be seen by
inspection, the average number of clans grows linearly with rapidity at fixed energy and is
a smoothly decreasing function of energy in a fixed rapidity interval. We point out that no
one model to our knowledge has a so sharp analytical prediction on clan behavior in single
jets. It is clear that our approximation consists in treating produced partons in a jet as pure
gluons: our parton shower therefore should be considered a gluon shower. We notice that a
similar behavior was seen by us 5 at hadronic level in single lowest energy jets disentangled
by using the algorithm from a sample of 3-jet events generated with 7.2






Parton Shower 6. We remind that the lowest energy jets in 3-jet events are usually identified
with gluon jets. In order to test our predictions in a region where data are not available, we


4


propose to reconstruct the parton level from the above mentioned sample of lowest energy
jets in spite of the intrinsic limitations of Monte Carlo models and jet-finding algorithms.
In view of NB regularity occurring in this sample 5, we proceed by applying Generalized
Local Parton Hadron Duality (GLPHD) prescription 7. Results at the same jet energies
seen in Fig. 1 are shown in Fig. 2: the predicted trend of the GSPS model is confirmed.
The above predictions can be tested in the real world at LEP I and LEP II once a definite
jet finding algorithm is universally accepted. Were our analytical predictions confirmed
in experimental data, this fact would raise the question of the real existence of clans, i.e.,
groups of particles of common ancestor, in the production process and of their intimate
physical nature.



2. Acknowledgements

One of us (R.U.) would like to thank prof. M. Block and prof. A. White, and the
Local Organizing Committee, for the fruitful and stimulating atmosphere that was created
at this symposium.



3. References

1. R. Ugoccioni and A. Giovannini, Z. Phys. C53 (1992) 239;
A. Giovannini and R. Ugoccioni, invited talk, in: Fluctuations and Fractal Struc-
ture, Proc. of the Ringberg Workshop on Multiparticle Production (Ringberg Castle,
Germany, 1991), eds. R.C. Hwa, W. Ochs and N. Schmitz, World Scientific, Singa-
pore, 1992, p. 264.
2. L. Van Hove and A. Giovannini, in Proc. of XVII International Symposium on Mul-
tiparticle Dynamics, (Seewinkel, 1985), eds. M. Markitan et al., World Scientific,
Singapore 1987, p. 561;
A. Giovannini and L. Van Hove, Z. Phys. C30 (1986) 391.
3. S. Lupia, A. Giovannini and R. Ugoccioni, Z. Phys. C59 (1993) 427;
R. Ugoccioni, A. Giovannini and S. Lupia, "Void Analysis and Hierarchical Struc-
ture for Single Jets", DFTT 41/93, to be published in the Proceedings of the Work-
shop "Soft Physics and Fluctuations", Cracow, Poland, 27 May 1993.
4. S. Lupia, A. Giovannini and R. Ugoccioni, in preparation.
5. F. Bianchi, A. Giovannini, S. Lupia and R. Ugoccioni, Z. Phys. C58 (1993) 71;
F. Bianchi, A. Giovannini, S. Lupia and R. Ugoccioni, "Single Jet Multiplicity
Distributions in + Annihilation at High Energies", DFTT 49/92, to be published







in the Proceedings of the XXII International Symposium on Multiparticle Dynamics,
Santiago de Compostela, Spain, 1317 July 1992.
6. M. Bengstsson and T. Sjostrand, Nucl. Phys. B289 (1987) 810;
T. Sjostrand and M. Bengstsson, Computer Physics Commun. 43 (1987) 367.
7. L. Van Hove and A. Giovannini, Acta Phys. Pol. B19 (1988) 917.



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