International Journal of Scientific & Engineering Research, Volume 4, Issue 5, May‐2013 1
ISSN 2229‐5518
A Genetic programming for modeling Hadron- nucleus Interactions at 200 GeV/c
*Mahmoud Y. El-Bakry
University of Tabuk, Faculty of Science, Department of Physics, Tabuk, KSA.
** El-Sayed A. El-Dahshan, *** A. Radi
Ain Shams University, Faculty of Science, Department of Physics, Abbassia, Cairo, Egypt.
** Egyptian E-Learning University- 33 El-mesah St., El-Dokki- Giza- Postal code 12611.
*** The British University in Egypt (BUE).
M. Tantawy
*Ain Shams University, Faculty of Education, Department of Physics, Roxi, Cairo, Egypt.
*Moaaz A. Moussa
Buraydah Colleges, Al-Qassim, Buraydah, King Abdulazziz Road, East Qassim University, P.O.Box 31717, KSA.
moaaz2030@Yahoo.com
—————————— ——————————
igh energies experimental data on hadron-nucleus (h-A) interactions are required for understanding high energy interactions. These data provide a useful link between hadron-hadron (h-h) interactions and the complex phenomena of nucleus-nucleus (A-A) interactions. These types of interac- tions investigate space time picture and highlight on phenome- non which doesn't exist in (h-h) such as gray particles, cascade, multi-collisions, etc. There are various models for (h-A) interac- tion like diffractive excitation model[1], collective tube model [2], quark model [3], energy flux cascade model [4], interanuc- lear cascade model [5], hydrodynamical model [6], multiple
scattering model [7] and many others .
Conventional models like parton two fireball model (PTFM), treate nucleons as composite objects of loosely bound states of the spatially separated constituents (quarks) which in turn are
some problems in high energy physics [30–34]. The effort to understand the interactions of fundamental particles requires complex data analysis for which machine learning (ML) algo- rithms are vital. Machine learning (ML) algorithms are becom- ing useful as alternate approaches to conventional techniques [35]. The complex behavior of the h-A interactions due to the nonlinear relationship between the interaction parameters and the output often becomes complicated. In this sense, ML tech- niques such as artificial neural network [36], genetic algorithm [37], PYTHIA [38] and PHOJET [39] Monte Carlo models. The PHOJET model combines the ideas based on a dual parton model [40] on soft process of particle production and uses low- est-order perturbative QCD for hard process. PYTHIA on the other hand uses string fragmentation as a process of hadroniza- tion and tends to use the perturbative parton-parton scattering
composed of point-like particles (partons) [8]. This may allow
for low to high PT
particle production, and genetic program-
one to consider the nucleons as consisting of a fixed number of
partons. This nucleon structure has been used in different mod- els [8-10] along with other assumptions to describe h-A interac- tions. PTFM, which is proposed by Hagedorn [11] has been used to explain the high energy interactions of hadrons and nuclei [12-18]. All these studies showed qualitative predictions of the measured parameters [19-23]. Extremely high energy collisions are required to get the fundamental particles close enough to study and understand the interactions between them [24–29].
Artificial intelligence techniques (or the machine learning)
such as genetic programming (GP) are applicable for solving
ming [41] can be used as alternative tool for the simulation of these interactions [30–34, 42–47].
The motivation of using a GP approach is its ability to de- velop a model based entirely on prior data without the need of making underlying assumptions. Another motivation for ap- plying such machine learning approach (e.g. GP) is simply the lack of knowledge (in most cases) about the mathematical de- pendence of the quantity of interest on the relevant measured variables [48].
In the present work, we illustrate the GP technique to model the multiplicity distribution of charged and negative pions for different beams at 200 GeV/c in hadron-nucleus col-
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lisions. The rest of the paper is organized as follows; Sec. 2 presents parton two fireball model PTFM at high energies for the multiparticle production in hadron-nucleus collisions Sec- tion 3 gives a review to the basics of the GP technique. Finally, the results and conclusion are provided.
According to references [13, 15-18], the charged multip- licity distribution will be given by,
nch
P (n ch ) (n)(nch n) (1)
n1
computer programs) is created, the fitness of individuals is evaluated and then the ‘parents’ are selected out of these indi- viduals. The parents are then made to yield ‘offspring’s’ by following the process of reproduction, mutation and crossov- er. The creation of offspring’s continues (in an iterative man- ner) until a specified number of offspring’s in a generation are produced and further until another specified number of gen- erations are created. The resulting offspring at the end of all this process is the solution of the problem. The GP thus trans- forms one population of individuals into another one in an iterative manner by following the natural genetic operations like reproduction, mutation and crossover. Each individual contributes with its own genetic information to the building of new ones (offspring) adapted to the environment with higher chances of surviving. This is the basis of genetic algorithms and programming. The representation of a solution for the
;n ch 2,4,6,........Q,
/
problem provided by the GP algorithm is a tree (Fig. 1).
Where,
(n)
=
n
( ) P
2
( n o )
, ( )
2
0
is the Poisson distribution of the form,
n
N 2
( ) =
2
p n 2
e - NP
(2)
n 2 !
Where, N: is the number of pairs of created particles from one
fireball ( N 
n 0 ),
2
n 2 the number of pairs of charged
pions, n2
charged.
n 1
2
, P the probability that the pair of pions is
The number of negative particles from one fireball equals the
half of new created charged pions n 
n ch
2
“Figure 1. Tree representation of the program
The probability distribution of negative particles
P(n ) is
-
( X / Y ) ( 5 Z * 13 ) ”
the same as the probability distribution of charged particles
P (n ) P (n
- ch
2n )
-
(3)
The GP is implemented using the experimental data to si- mulate multiplicity distribution of charged and negative pions
; n
0 ,1, 2 ,3,........,
Q / 2
for p Ar
, p Xe
, p Au
and
p He
collisions
40
131
197 4
Genetic programming is an extension to Genetic Algo- rithms (GA). GA is an optimization and search technique based on the principles of genetics and natural selection. A GA allows a population composed of many individuals (chromo- some) to evolve under specified selection rules to a state that maximizes the “fitness” (i.e. minimizes the cost function). The GP is similar to genetic algorithms but unlike the latter its so- lution is a computer program or an equation as against a set of numbers in the GA. A good explanation of various concepts related to GP can be found in Koza (1992) [41, 49].
In GP a random population of individuals (equations or
at 200 GeV/c. The GP model was constructed with training
sets and the accuracy was verified by the test sets. In order to
generate the GP model we have implemented the GP steps
(Fitness evaluation, reproduction, crossover and mutation)
that were mentioned in Section 3. Table 1 lists the values of the
control parameters and the set of function genes that are used
in modeling the multiplicity distribution. The fitness function
evaluates how accurate the mathematical model.
This discovered function has been used to predict the mul-
tiplicity distribution of pions for h-A interactions.
Simulation results based on GP model, for modeling the
multiplicity distribution of charged pions for h-A interactions
at 200 GeV/c (the training cases) are given in Fig. 2 (b, c, d,e)
and Fig. 3 (b, c, d, e, f) for negative pions. While Figs. 2, 3 (a)
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describe the predicted results for
p Ar40 interaction at 200
GeV/c, we notice that the curves (for training and prediction cases) obtained by the trained GP model show a best fitting to the experimental data in all cases. Then, the GP model is able to exactly model the multiplicity distribution at 200 GeV/c for different beams in h-A collisions. If the large dataset is used in training, the best GP model is obtained.
TABLE 1
VALUES OF THE CONTROL PARAMETERS USED IN MULTIPLICI- TY DISTRIBUTION
“Figure 2. Comparison between the experimental data, PTFM and simulated multiplicity distribution of charged pions P (nch ) for h-A collisions at 200
GeV/c: (—) GP model, (……) PTFM, ( ) experimen-
tal data”
Negative and charged pions multiplicity distributions, Eq. (1, 3), are calculated by PTFM for p Ar40 , p Xe131 , p Au 197 and p He 4 assuming is given by , an b
, Where, a = 0.04, b= 0.35 as in references [17, 18]. The results
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of these calculations are represented in figure 2 (a, b, c, d, e)
and figure 3 (a, b, c, d, e, f) along with experimental data [50,
51] which show fair agreement with the corresponding expe-
rimental data. It can be seen from figs. (2, 3) that charged and
negative pions multiplicity distributions are not in accordance
with the experimental data for heavy nuclei although the situ-
ation becomes better for the light ones. The emission of sec-
ondary particles is assumed to follow a Poisson distribution.
As mass number increases the multiplicity distribution is not
broaden but its peak is shifted to high numbers.
“Figure 3. Experimental Data, PTFM, trained and predicted simulated multiplicity distribution of pions P ( n ) for h-A collisions at 200 GeV/c: (—) GP model, (……) PTFM, ( ) experimental data”
Genetic programming, with its advantage of discovering mathematical equations (see APPENDIX), has been shown to be an efficient method for modeling the h-A interactions par- ticularly above the pion production threshold. This paper presents an efficient approach for calculating the multiplicity distribution of charged and negative pions at 200 GeV/c
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International Journal of Scientific & Engineering Research, Volume 4, Issue 5, May‐2013 5
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through the obtained discovered functions as shown in ap- pendix.
The discovered function shows an excellent match to the
v B
k u
experimental data. Moreover, the discovered function is capa- ble of predicting the experimental data that are not used in the
u t / cos(log
2 ( X 2 ))
training set. The present study has shown that the GP ap- proach can be employed successfully to model the h-A interac-
t sin(sin(cos(log2 (X1 s))))
r
tions at high energies. Finally, we conclude that GP has be- come one of the important research areas in the field of ha-
s B
/ X3
dron-nucleus collisions
APPENDIX
r (n o) q
q sin(cos( X 3 )) / cos(log 2 (X 2 ) )
Our discovered function (for charged pion multiplicity dis- tribution) is generated using the obtained control GP parame-
ters as follows,
p sin( 10 X 3 sin( X
3 ))
P(n
) (((X3 A)1.732021B * D E ) / F ) / G
o sin(cos(log 2 (X1 ) * (X 2 X 3 ))) X1
m
ch
Where,
n log 2 ( l)
X 1
A ( X 1 - X 3 ) /(10 X 3
)10 x1
l ( 2 X 3
0 .56464
X 2 0 .46078 ) / X 1
B cos( X 1 /(log((cos
(X 2
* (0 .58503
X 3 )
(log 2 ( X 2 ) * ( X 3 X 2 ))))
)))
m sin(0.700228 sin(X2 ))
j
b log 2 ( X 3 ) 0 .1X 1
k (log 2 ( X 2 )
X 1 ) X 1 (0.076463)
D cos((sin(
0.510701
a bb ) 10 ))
j sin( 0.519666 sin(sin( i)))
bb ((log2 ( X 3 ) 0.1X1 )
i (sin((
X ) h )) h
a cos(sin((0.510701 (0.60512 /(0.1X3 )
h sin(log 2 ( X 2 ) sin( X 3 )
D -0.83907 / cos(log 2 ( X 2 ))
* cos(log
2 ( 0 .29895
/ X 3 ) 0 .26714
))) 10 ))
E ((log
2 ( X 3 ) 0 .1X 1 )
The actual parameters are, X 1 number of charged particles
nch , n , X 2 , lab momentum ( P L ) and X 3 , mass number (A).
F cos(log 2
G cos(log 2
(log 2
(log 2
(cos(log 2
(cos(log 2
(0.29895 X
(0.29895 X
1 ))))) 1
1 ))))) 1
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