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Open Journal of Microphysics, 2012, 2, 1-11
http://dx.doi.org/10.4236/ojm.2012.21001 Published Online February 2012 (http://www.SciRP.org/journal/ojm)
Characteristics of Strange Hadron Production in Some
High Energy Collisions and the Role of Power Laws
Sunil Kumar Biswas1, Goutam Sau2, Amar Chandra Das Ghosh3, Subrata Bhattacharyya4*
1West Kodalia Adarsha Siksha Sadan, New Barrackpore, Kolkata, India
2Beramara RamChandrapur High School, Kolkata, India
3Department of Microbiology, Surendranath College, Kolkata, India
4Physics and Applied Mathematics Unit (PAMU), Indian Statistical Institute, Kolkata, India
Email: {sunil_biswas2004, sau_goutam}@yahoo.com, [email protected],
*[email protected]
Received November 8, 2011; revised December 28, 2011; accepted January 10, 2012
ABSTRACT
Studies on “strange” particle production have always occupied a very important space in the domain of Particle Physics.
This was and is so, just because of some conjectures about specially abundant or excess production of “strange” parti-
cles, at certain stages and under certain conditions arising out of what goes by the name of “Standard” model in Particle
Physics. With the help of Hagedornian power laws we have attempted to understand and interpret here the nature of the
-spectra for the strange particle production in a few high energy nuclear collisions, some interesting ratio-behaviors
and the characteristics of the nuclear modification factors that are measured in laboratory experiments. After obtaining
and analysing the final results we do not confront any peculiarities or oddities or extraneous excesses in the properties
of the relevant observables with no left-over problems or puzzles. The model(s) used by us work(s) quite well for ex-
plaining the measured data.
T
p
Keywords: Hadron-Nucleus Collisions; Inclusive Production; Scaling Phenomena; Power Laws
1. Introduction
Studying the nature of particle production in proton-
proton collisions is important and interesting in itself, as
it might shed light on the basic mechanism for pro-
duction of particles. Besides, it could also serve as a
necessary benchmark for the physics developments in
ultra-relativistic heavy ion collisions [1,2]. This is
specially important at the large hadron collider (LHC)
where the heavy ion programme had started by Nove-
mber 2010 delivering some preliminary results on some
aspects of strange hadrons produced as the final product
in high energy nuclear collisions and these strange
secondaries are supposed to provide valuable insights
into the properties of the “controversial” system newly
formed. One of the main motivation for measuring
strange particles in heavy nucleus-nucleus reactions at
LHC is the expectation that their production-rates for
participating nucleon should be enhanced with respect to
basic nucleon-nucleon interactions. Strangeness enhance-
ment has consistently been proposed as one of the strong
diagnostics for a Quark Gluon Plasma (QGP) state [3,4].
The enhancement factor (E) is defined as rapidity-density
of multiplicity (yield) par mean number of nucleon parti-
cipants [
p
art
N] in heavy ion collisions, divided by the
respective value in p + p collisions. The requisite
information about
p
art
N etc are to be obtained from
Abelev et al. [5,6].
As the strange hadrons are not at all present in the
initial system (A), the question rises very sharply: how
do they make their appearance in the final products. So
there must be some specific reflections on the constituent
pictures of the particles and specifically the nucleons.
Besides, enhancement of strangeness productions was/is
one of the powerful diagnostics for the formation of
quark gluon plasma (QGP) in relativistic heavy ion col-
lisions and the colliders (RHIC). The observations of an
increase of strange baryon production relative to p + p
collision in SPS data, confirmed later at the RHIC studies,
has brought excitement in this area.Besides, the increase
of πp ratio (B) in such collisions in the non-strange
domain had its parallel in the strangeness sector with the
observation of slow rise of the 0
k values.
The organization of this work is as follows. In Section
2 we give an outline of the model to be applied. In the
next section (Section 3) we deliver the results by figures
and tables with a short discussion on the results obtained.
*Corres
p
ondin
g
autho
r
.
C
opyright © 2012 SciRes. OJM
S. K. BISWAS ET AL.
2
In the last chapter, we precisely point out the conclusions
to be arrived at.
2. The Background in Some Detail and the
Working Formulae
With gradual attainments of larger transverse momenta
(T) of the secondaries in high-T (hard) interactions,
the problems of deviations from exponential nature of
fits on invariant spectra began to crop up steadily.
Gazdzicki and Gorenstein [7] observed rightly that for
GeV/c, the data sharply deviate from the
exponential nature, for which Darriulat[8] proposed a
power law distribution of certain forms for both -
spectra and particle multiplicity. Indeed, for both T-
spectra and multiplicity such power law forms have
become now the most dominant tools in dealing with the
transverse momentum spectra of all hadrons. Gazdzicki
and Gorenstein showed that the normalised multiplicities
and (T
m) spectra of neutral mesons obey the
T-scaling which has had an approximately power law
structure of the form , where is called
p
T
p
m
p
>2
T
p
p
T
p
n
T
mT
m
transverse mass and is defined by 2
mp2
TT
m. This
scaling behaviour was analogous to that expected in
statistical mechanics, the parameter n plays the role of
temperature and any normalization constant to be used
resembles the system volume.Thus the basic modi-
fication of the statistical approach needed to reproduce
the experimental results on some hadron production
process in

pp p interaction in the large
TT
mp
domain is to change the shape of the distribution func-
tions
*
exp E
T

had to be altered to the power law
form as given by
*n
E



with some changed parame-
ters, viz, a scale parameter and an exponent n, both
are assumed to be common for all hadrons.
Let us now dwell, in brief, on the clues to the possible
origin of power laws. One of the basic features of the
hadronuclear collisions is: irrespective of the initial
state,agitations caused by the impinging projectile (be it a
parton or particle/nucleon) generate system effects of
producing avalanches of new kind of partons (called glu-
ons) which form an open dissipative system.And these
production processes are not at all gradual; rather they
are very sudden,drastic and complex. And such complex
properties and processes in nature do generally subscribe
to the power-law behaviours.In the recent times, it is
being propounded consistently that the power law
behaviours put into use here are “manifestations of the
dynamics of complex systems whose striking feature is
of showing universal laws characterized by exponents in
scale invariant distributions that happen to be basically
independent of the details in the microscopic dynamics”
[9]. The avalanches caused by production of excessive
number of some new variety of parton called ‘gluons’
(the process called “gluonisation”) give rise to the
jettiness of particle production and of cascadisation of
the particle production processes leading to the fractality
as is shown by Sarcevic [10]. These cascades are
self-organizing, self-similar and do just have the fractal
behaviour. Driven by the physical impacts of these
well-established factors, in the high energy collision
processes do crop up the several power-laws.And how
such power laws do evolve from exponential origins or
roots is now-a-days being taken care of by the induction
of Tsallis entropy [11] and a generalisation of Gibbs-
Boltzmann statistics for long-range and multifractal pro-
cesses.
In what follows we are going to choose a specific form
of power law which was previously applied by us in the
case of hadron-nucleus collisions.With a view to accom-
modating some observed facts for strange particle pro-
duction, it is tempting to try to fit the whole distribution
for the inclusive T-spectra with one single expression
in the form of power law as was done by G. Arnison et al.
[12] and Hagedorn [13].
p

3
3
dd
d=. =
2π
d
n
TT T
dNyq
E constA
pdpp q
p


(1)
where the letters and expressions have their contextual
significance.This parametrization describe the data well
over the entire range of .
T
Indeed for we have,
p
0,
T
p
1
exp for 0
and
for
n
T
T
TT
n
T
T
qn
p
pq q
npp
q
qp
p














(2)
Thus along with impressive fit, which now includes
the large T domain, the estimate of pT
p assumes
with the help of expression (1):


2d2
==
3
d
n
TTT
Tn
TTT
qp qppq
pn
qp qpp
(3)
So, in clearer terms, let us put the final working for-
mulae here as follows with substitution of T (trans-
verse momentum) as
p
x
in the power-law model [14,15]
respectively
Copyright © 2012 SciRes. OJM
S. K. BISWAS ET AL.
Copyright © 2012 SciRes. OJM
3

=1 n
f
xA xq
(4)
There is yet another very important observable called
nuclear modification factor (NMF), denoted here by
CP , which for production of any hadron is defined by
[16]
R
 



2
2
2πddd
=
2πddd
Cenrral
TT bin
c
CP T
P
eripheral
TT bin
c
dNp pyN
Rp dNp pyN
(5)
number of degrees of freedom is too limited for many
cases. The quality of fits to the data indicated in the
tables by 2ndf
terms in the columns is attempted to
be kept at a modestly satisfactory value (tending as
nearly as possible to unity). And the figures are drawn by
Wgnuplot, wherein there are some inbuilt statistical
procedures and techniques.
Quite observably, the results are depicted here in
graphical plots. And the used values of the corresponding
parameters for fits are shown in separate tables. The
Figures 1(a) and (b) show the production of secondaries
0,,kkk
,
and
in proton-proton collisions at
s
= 200 GeV at the rapidity (Table 1). The
figures in Figures 2(a) and (b) depict the results for the
<0.5y
,,,
3. Results
In obtaining the results presented here, no serious sta-
tistical calculational procedure was adopted. The graphs
are drawn more as fitological-cum-phenomenological
exercises with mainly statistical errors in considerations.
The experimental data do not provide, in the most cases,
any systematic errors. Data points for the heavy, high
strangeness-valued particles are too scarce; for which the

 particle production for the same collision
at the same energy (Table 2).The cases of k
and k
production in gold-gold reaction at the same energy and
at different centralities are reproduced by power laws in
Figures 3(a) and (b) (Table 3). In Figures 4(a) and (b)
(a) (b)
Figure 1. Transverse momentum spectra for production of keon (0,,KKK
), lamda (
), lamdabar (), cascade minus
(), cascade plus bar () particles in pp collisions at
bar
s = 200 GeV. The ex perimental data are taken from Ref. [5].
The solid curves are fits for power-law model.
Table 1. Numerical values of the fit parameters of power law equation for keon and lamda production in p-p collisions at
NN
s= 200 GeV, = 0 to 5 GeV/c.
T
p
Sesondaries
A
q n
2
ndf
0
s
K
0.563 0.023 3.108 0.025
15.005 0.032
20.707/17
K
1.067 0.032 1.581 0.055
10.000 0.209
10.943/9
K
0.066 0.008 2.895 0.026
15.116 0.405
0.584/6
0.273 0.036 3.092 0.068
15.007 0.074
18.393/15
bar 0.029 0.001 3.010 0.068
15.016 0.068
15.673/15
S. K. BISWAS ET AL.
4
(a) (b)
Figure 2. Transverse momentum spectra for production of cascade (
,), lamda and lamdabar particles in pp
collisions at
bar
s= 200 GeV. The experimental data are taken from Ref. [17]. The solid curves are fits for power-law model.
(a) (b)
Figure 3. Transverse momentum spectra for production of kaon (,KK
) at different centrality at s = 200GeV in Au-Au
collisions. The experimental data are taken from Ref. [18]. The solid curves are fits for power-law model.
Table 2. Numerical values of the fit parameters of power law equation for lamda and cascade particle production in p-p
collisions at NN
s= 200 GeV, T
p
= 0 to 5 GeV/c.
Sesondaries
A
q n
2
ndf
1.8480.094 1.3060.061
10.4170.244
26.883/12
bar 0.541 0.026 1.9100.068
12.0050.061
36.675/13
0.022 0.001 1.8040.023
10.0330.119
21.4/8
bar
0.063 0.002 2.3010.015
9.9510.466
24.246/9
Copyright © 2012 SciRes. OJM
S. K. BISWAS ET AL.
Copyright © 2012 SciRes. OJM
5
(a) (b)
Figure 4. Transverse momentum spectra for production of lamda (
) and lamda bar (
) particles at different centrality in
Au-Au collisions. The experimental data are taken from Ref. [18] The solid curves are fits for power - law model.
Table 3. Numerical values of the fit parameters of Power Law equation for keon production () in Au-Au collisions at
,kk
NN
s = 200 GeV at different Centrality, T
p
= 0 to 2 GeV/c.
Secondaries Centrality (%)
A
q n
2
ndf
0 - 5% 120.1148.341
2.5080.057
10.3760.172
4.555/14
20% - 30% 47.371 2.28
1.9980.029
9.9890.112
2.477/15
40% - 50% 16.727 2.703
1.8430.082
10.0060.111 1.294/12
k
60% - 70% 3468.89253.7
2.0300.033
11.9810.154
3.022/13
0 - 5% 435.675 15.85
2.2860.076
12.000.264
1.032/15
20% - 30% 68.462 6.505
2.3790.061
12.0030.07 0.246/12
40% - 50% 25.454 0.901
2.2390.084
12.0520.297
1.566/14
k
60% - 70% 5.058 0.283
2.0160.101
11.8190.382
3.356/14
the production of and
are shown in the same
collision at the same energy(Table 4). In Figures 5(a),
(b) and (c) the production of cascade, cascade-bar and
omega particles production are shown at different cen-
tralities and at the same energy (Table 5). The cases of
production of neutral kaons and lamda particles in
Copper-Copper collision at
s
= 200 GeV are plotted
in Figures 6(a) and (b) with reckoning of the para-
meters presented in Tables 6 and 7.
In Table 8 the values of average transverse momenta
(T) for different produced secondaries in proton-
proton and gold-gold collisions have been computed. All
these values tally with the similar ranges arrived at by
experimental measurements. This helps us to obtain for
us a consistency check-up of the parameter values used
for getting fits to the data on
p
T-spectra. In Figures
7(a) and (b) we see, the lamda-bar to lamda and cascade-
minus to cascade-plus particle production cross-section
ratio as a function of transverse momentum respectively
and the ratio gradually fall off with increasing values of
T. In Figures 8(a) and (b) the nuclear modification
factors (CP ) are plotted against transverse momentum
for the production of neutral meson and lamda particles in
copper-copper collision. With the increasing T, the
CP -values fall off. In addition, the data show the CP
for baryons exhibits a lower fall-off compared with that
of mesons in intermediate transverse momentum region.
The experimental data show that the baryonmeson
difference of disappears at higher .
p
p
R
R
pR
CP T
he data on production of strange particles described
R p
T
S. K. BISWAS ET AL.
6
Table 4. Numerical values of the fit parameters in Power Law form for (
and
) production in Au-Au collisions at
NN
s= 200 GeV at T
p
= 1 to 5 GeV/c ranges and for various Centrality-values as given below.
Secondaries Centrality (%)
A
q n
2
ndf
0 - 5% 63.570 4.841
3.5420.105
16.1240.140
10.064/18
10% - 20% 772.323 48.46
2.0000.087
13.8700.259
5.778/14
20% - 40% 141.42 11.33
2.0020.023
12.8150.099
7.512/12
40% - 60% 34.334 2.317
2.0250.109
12.8200.308
7.238/12
60% - 80% 19.811 0.968
2.0000.014
11.8690.235
14.913/12
0 - 5% 198.2330.269
2.0000.014
10.7650.269
7.588/10
10% - 20% 29.161 1.727
2.0020.121
12.0030.072
17.298/10
20% - 40% 122.194 4.406
2.8990.014
15.0010.054
3.589/8
40% - 60% 25.1830.646
2.5000.008
12.8880.028
9.902/12
bar
60% - 80% 10.994 0.788
2.0010.022
11.9950.088
31.900/13
Table 5. Numerical values of the fit parameters in Power Law equation for cascade-minus (
), cascade-plus bar ()
and particle production in Au-Au collisions at
bar

NN
s 200 GeV at different T
=
p
-values = 1 to 5 GeV/c for various
Centrality values.
Secondaries Centrality (%)
A
q n
2
ndf
0 - 5% 213.632 16.06
1.3440.015
10.0420.271 14.593/8
10% - 20% 1019.24 81.73
1.0010.010
10.0170.058
20.800/6
20% - 40% 2978.87 155.4
1.0000.006
11.0030.001 6.138/6
40% - 60% 16.505 0.288
1.5000.003
10.0190.055
0.675/7
60% - 80% 3.065 0.339
1.5030.027
10.0010.111
16.968/5
0 - 5% 1531.25144
1.0010.012
9.9960.070
18.845/7
10% - 20% 5248.25 468.3
0.8010.008
10.0110.053
22.498/6
20% - 40% 735.176 31.68
0.9990.005
9.9960.031
24.258/8
40% - 60% 233.981 15.39
0.9990.008
9.9970.047
16.173/7
bar
60% - 80% 2.993 0.188
1.4960.014
10.0060.057
4.049/4
0 - 5% 1.248 0.094
3.0180.055
9.9880.139
2.073/3
20% - 40% 2.970 0.229
1.9970.028
9.9780.100
1.304/2


40% - 60% 0.342 0.037
2.0010.043
9.2350.131
4.755/3
here pertain, in the main, to the “hard” sector of high
energy interactions. And it is well-known that Hage-
dornian power law forms which have their roots in the
physics of quantum chromodynamics (QCD) describe
hard particle production in a modestly successful manner
for, at least, the light hadrons of which strange K-mesons
constitute a part.But some of the strange particles have
moderately high masses, for which our objective here
was primarily to check whether this generalized power
law form could address the issues of invariant T-
spectra and some other related observables in a satis-
factory manner for all the strange hadrons. And the
outcome is: this study is strongly affirmative by all
indications and yardsticks of actual performances.
p
The data on strange particles are in general quite
sparse. The errors in measurements are also in most cases
Copyright © 2012 SciRes. OJM
S. K. BISWAS ET AL. 7
(a) (b)
(c)
Figure 5. Transverse momentum spectra for production of cascade minus (
),cascade plus bar () and omega
particles () at different centrality in Au-Au collisions at
bar

s = 200 GeV. The experimental data are taken fr om Ref.
[19]. The solid curves are fits for power-law model.
Table 6. Numerical values of the fit parameters of Power Law equation for keon production () in Cu-Cu collisions at
0
s
k
NN
s = 200GeV at different T
p
-values = 1 to 9 GeV/c and for several Centrality domains.
Secondaries Centrality (%)
A
q n
2
ndf
0 - 10% 112.079 4.169
2.0200.035
12.2290.094
6.675/21
10% - 20% 67.284 2.154
1.9210.031
11.7120.477
6.068/21
20% - 30% 24.860 0.568
2.8790.030
14.1550.262
3.667/21
30% - 40% 17.099 0.286
2.5650.070
13.1810.0.245
6.230/21
40% - 50% 18.504 0.598
1.9850.036
11.5830.476
7.738/21
0
s
k
50% - 60% 12.159 0.536
2.0000.006
11.5510.054
7.842/21
Copyright © 2012 SciRes. OJM
S. K. BISWAS ET AL.
8
(a) (b)
Figure 6. Transverse momentum spectra for production of neutral keons () and lamda (
0
s
k
) particles at different centrality
in Au-Au collisions at s = 200 GeV. The experimental data are taken from Ref. [18]. The solid curves are fits for
power-law model.
(a) (b)
Figure 7. Transverse momentum-dependence spectra of
bar/
and
/ for pp collision at bar
s = 200 GeV. The
ata type-points are taken from Ref. [5]. The solid curves or lines are drawn on the basis of Power Law Model. d
Table 7. Numerical values of the fit parameters of Power Law equation for lamda particle () production in Cu-Cu
collisions at NN
s = 200 GeV at different Centrality, T
p
= 1 to 7 GeV/c.
Secondaries Centrality (%)
A
q n
2
ndf
0 - 10% 1091.23 47.1
2.0000.009
14.4280.166
2.635/12
10% - 20% 139.7312.91
2.0340.103
13.0330.275
15.067/14
20% - 30% 105.009 5.974
2.0240.073
12.9960.193
16.425/17
30% - 40% 46.450 1.523
2.0260.081
12.6290.216
24.731/17
40% - 50% 27.007 1.798
2.0010.018
12.3400.215
16.803/15
50% - 60% 24.819 2.077
2.0290.023
13.0420.097
17.437/14
Copyright © 2012 SciRes. OJM
S. K. BISWAS ET AL.
Copyright © 2012 SciRes. OJM
9
(a) (b)
Figure 8. Plots of the nuclear modification factor (cp
R
) versus T
p
spectra for Cu-Cu collisions at s = 200 GeV. The
experimental data are taken from Ref. [18]. The solid curves indicate the power-law-based description of the data.
Table 8. Calcul ated values of average transverse m omen tum
T
p
for at <0.5
cm
yNN
s = 200 GeV.
Nature of Collisions SecondariesValue of T
p
in GeV/c
0
s
k 0.52
k 0.45
k 0.48
0.52
0.50
0.51
p-p Collisions
bar
0.86
Au-Au Collisions k 0.68
k 0.50
0.54
0.52
0.38
bar
0.28
(Central)

 0.86
quite considerable. If these limitations are taken into
account, the importance of this comprehensive work,
though entirely phenomenological, assumes some degree
of importance. Barring these generalized comments on
an overall basis, we have some specific observations as
well, which are also quite well-merited and are being
presented hereafter: 1) The power indices for all the
varieties of strange hadrons lie in the most cases,in the
range of 10 - 12. This is in accord with the prescription
on the limit set by Brodsky [20], with the q-values
bordering on the values, 2 - 3. 2) The 2
/ndf values for
production of cascade and omega particles have suffered
quantitatively due to very small values of the number of
degrees of freedom. 3) Thirdly, the q and n values do not
exhibit any clear centrality-dependence or the mass-de-
pendence of the observed heavy secondaries. 4) However,
we cannot ascertain at this moment the properties of
these parameters with regard to the nature of their energy-
dependence(s), if any. 5) The average momentum values
of these measured heavy strange baryons are found to be
quantitatively compatible with other non-strange light
hadrons, though the expression for the average transverse
momentum is not very rigorously derived, for which
reliability of expression (3) is certainly limited. 6) Some
ratio-values shown by Figures 7(a) and (b) are mo-
destly well-described. 7) However, the values of the nu-
clear modification factor, denoted by CP , are not
reproduced satisfactorily, especially on the lower-side of
the T-values. But this is no wonder, as the used power
laws are suited to high-T values as was remarked
above very concretely. 7) Still, with one of the simplest
approaches, that we have succeeded in explaining the
characteristics of a large bulk of data on these rare
hadrons is certainly quite stimulating to and encouraging
for us.
R
pp
4. Concluding Remarks
Let us now sum up;
1) The used power laws explain quite well the mea-
S. K. BISWAS ET AL.
10
sured data on the observables like, T-spectra, some
ratio-behaviours and the nuclear modification factors; so
none of the questions related to suppression or enhance-
ment is consequential. 2) The centrality-dependences of
the T-spectra of strange hadrons too are well-repro-
duced. 3) The essential physical content of the power-law
models is the modest observance of theT
p-scaling (as
is reflected in the
p
p
0Tp term). And in terms of the
functional efficacy, this model seems, so far, to be at par,
if not better, with all other numerous existing approaches
within the frameworks of either the Hadronic transport
models or the statistical models [21-25]. 4) The model
obviously bears no relationship with the concept of the
quark-gluon plasma (QGP), which is, still, just a con-
jecture with so far no clear and concrete experimental
support. 5) In a conclusive vein, this has to be asserted
that we observe nothing too strange about “strangeness”
production in high energy interactions.
p
The latter two points in the preceding paragraph high-
light, in the main, the novelty of this study from a global
viewpoint against the background of the current trends
and streaming ideas in the present day Particle Physics.
5. Acknowledgements
The authors express their thankful gratitude to the hon-
ourable referee for making some valuable comments and
suggestions.
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