Download Relativistic nucleus-nucleus collisions, Transverse mass, Effective

Journal of Nuclear and Particle Physics 2015, 5(2): 38-43
DOI: 10.5923/j.jnpp.20150502.03
Transverse Mass Spectra for Charged Pions in
Nucleus–Nucleus Collisions in Terms of Relativistic
Quantum Molecular Dynamics Model
A. M. Abdalla
Department of Mathematical and Physical Engineering, Faculty of Engineering in Shoubra Benha University, Cairo, Egypt
Abstract Transverse mass spectrum is one of the experimental parameter used to describe relativistic kinematic and
mechanisms for hadron-hadron collisions and extended to nucleus-nucleus collisions. Charged pions emitted from Au+Au
and Pb+Pb collisions at central rapidity (|𝑦𝑦| < 0.5) and impact parameter 𝑏𝑏 ≤ 3.4 𝑓𝑓𝑓𝑓 at different values of lab energies
are investigated. The transverse mass distributions of secondaries depend on the energy of the interacting system. The
experimental results compared with predictions by Quantum Molecular Dynamic model QMD and it ultra-relativistic
implementation Ur-QMD using art version Ur-QMD-2.3. This model can give suitable distribution for transverse mass and
describe experimental data for collision energy below 40 GeV. Above this energy, the hadron productions are not systematic
which may due to the creation of different nuclear states of non-thermal equilibrium and high energy densities. The inverse
slop parameter represents the effective temperature of the mechanism responsible for hadron productions. The temperature
increases in regular tool with collision energy up to 40 GeV.
Keywords Relativistic nucleus-nucleus collisions, Transverse mass, Effective temperature, Charged pions production
1. Introductions
One of the major goals of research on high-energy
heavy-ion collisions is to explore properties of strongly
interacting matter, particularly its phase structure [1].
Nucleus-nucleus collisions such as, Au+Au collisions at
√𝑠𝑠𝑁𝑁𝑁𝑁 = 200 GeV at Relativistic Heavy Ion Collider (RHIC),
allow to study strongly interacting nuclear matter at extreme
temperatures and densities. At this condition, nuclear matter
will reach to new phase quark gluon plasma (QGP) [2].
Several signatures of the formation of transient QGP state
during the early stage of a nucleus-nucleus collision at high
energies were proposed in the past [3]. Quantum
chromodynamics QCD, the main theory of strong
interactions, believed that a gas of hadrons when its
temperature exceeds a critical value [4] undergo a transition
to state of quasi-free quarks and gluons [1]. In this state of
matter, the normal confinement of quarks and gluons in
hadrons removed and the partons can exist as quasi-free
particles in an extended region of space-time. In the above
given example for high-energy nucleus-nucleus collisions,
could give many signals point to the formation of
quark-gluon plasma (QGP) [5-9]. One of the first signatures
proposed was an enhancement of strange particle production
* Corresponding author:
[email protected] (A. M. Abdalla)
Published online at http://journal.sapub.org/jnpp
Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved
in A+A with respect to p+p collisions [10].
Theoretical analysis of these data is still in progress. RHIC
continues to study the detailed properties of the strongly
interacting matter using p+p, d+Au and Au+Au systems at
various colliding energies from 7.7 GeV up to 200 GeV.
Measurements of transverse momentum spectra for particles
emerging from p+p collisions used as a baseline to which
similar measurements from heavy ion collisions are
compared to study collective effects. Explanation of
transverse-mass spectra of hadrons in collisions of heavy
nuclei turned out to be one of the most difficult tasks [11-13].
Experimental data on transverse-mass spectra of pions
produced in central Au+Au [14] and Pb+Pb [15, 16] collisions
reveal peculiar dependence on the incident energy. The
inverse-slope parameter (called effective temperature T) of
these spectra at mid-rapidity increases with incident energy
in the energy domain of BNL Alternating Gradient
Synchrotron (AGS) and then saturates at energies of CERN
Super Proton Synchrotron (SPS). At these values of collision
energies, nuclear materials reach to a state of nuclear
saturation. It was assumed that this saturation is associated
with the de-confinement phase transition [17, 18].
Many theoretical models are partly able to give new
concepts on nuclear materials under these conditions and
special formulations for treatments of the assumed processes
that are responsible for the creations and production of
experimentally observed secondary particles. One of the
important of these models is the Quantum Molecular
Journal of Nuclear and Particle Physics 2015, 5(2): 38-43
Dynamic model QMD and it ultra-relativistic
implementation Ur-QMD and recent version is the art
version Ur-QMD-2.3 [19, 20]. This model is a microscopic
many body approach to pp, pA, and AA interactions at
relativistic energies. It can be applied for studying
hadron-hadron, hadron-nucleus and heavy ion collisions
from Elab = 100 A MeV to √𝑠𝑠𝑁𝑁𝑁𝑁 = 200 GeV. Its microscopic
transport approach is based on the covariant propagation of
color strings, constituent quarks and di-quarks (as string ends)
accompanied by mesonic and baryonic degrees of freedom.
It simulates multiple interactions of ingoing and newly
produced particles, the excitation and fragmentation of color
strings and the formation and decay of hadronic resonances.
At higher energies, the treatment of sub-hadronic degrees of
freedom is of major importance. In the present model, these
degrees of freedom enter via the introduction of a formation
time for hadrons produced in the fragmentation of strings
[21-23]. The major aspects and formulation of Ur-QMD are
discussed in Refs. [24, 25]. Details on transverse mass
distributions will discussed in section 2. A comparison and
discussions for transverse mass distributions of charged
pions produced from central and mid rapidity from Pb-Pb
and Au-Au collisions at relativistic energies and results
suggested by Ur-QMD-2.3 model will be in section 3.
Transverse mass 𝑚𝑚 𝑇𝑇 (momentum 𝑝𝑝𝑇𝑇 ) is one of the
variable parameter used to describe the relativistic kinematic
for nucleon-nucleon collisions and extended to
hadron-nucleus and nucleus-nucleus collisions. At hadron
colliders, a significant and unknown proportion of the energy
of the incoming hadrons in each event escapes down the
beam-pipe. Consequently, if invisible particles are created in
the final state, their net momentum can only be constrained
in the plane transverse to the beam direction. Defining the
z-axis as the beam direction, this net momentum is equal to
the missing transverse energy vector
𝐸𝐸𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = − ∑𝑖𝑖 𝑝𝑝𝑇𝑇(𝑖𝑖)
quantity 𝑚𝑚 𝑇𝑇 defined by
𝑚𝑚2𝑇𝑇 ≡ [𝐸𝐸𝑇𝑇 (1) + 𝐸𝐸𝑇𝑇 (2)]2 − [𝑝𝑝𝑇𝑇 (1) + 𝑝𝑝𝑇𝑇 (2)]2
where
= 𝑚𝑚12 + 𝑚𝑚22 + 2[𝐸𝐸𝑇𝑇 (1)𝐸𝐸𝑇𝑇 (2) − 𝑝𝑝𝑇𝑇 (1) ∙ 𝑝𝑝𝑇𝑇 (2)] (2)
𝑝𝑝𝑇𝑇 (1) = 𝐸𝐸𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
(3)
𝑚𝑚2𝑇𝑇 = 2|𝑝𝑝𝑇𝑇 (1)||𝑝𝑝𝑇𝑇 (2)|(1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝜑𝜑12 )
(4)
This quantity 𝑚𝑚 𝑇𝑇 is called the ‘transverse mass’ by
hadron collider experimentalists. The distribution of event
= 𝑚𝑚 . If
𝑚𝑚 𝑇𝑇 values possesses an end-point at 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
𝑇𝑇
𝑚𝑚1 = 𝑚𝑚2 = 0 then
where, 𝜑𝜑𝑖𝑖𝑖𝑖 is defined as the angle between particles i, and j,
in the transverse plane. The transverse momentum 𝑝𝑝𝑇𝑇 is the
projection of the particle's 3-momentum onto the plane that
is transverse to the collision axis z: 𝑝𝑝𝑇𝑇 = 𝑝𝑝 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠, where 𝜃𝜃 is
the initial polar angle of the particle with respect to the event
vertex position along the collision axis z. The transverse
energy of the particle with rest mass 𝑚𝑚0 is
𝑚𝑚 𝑇𝑇 = �𝑝𝑝𝑇𝑇2 + 𝑚𝑚02
(1)
where the sum runs over the transverse momenta of all
visible final state particles.
(5)
Along the beam axis z, a quantity called rapidity y defines
the longitudinal motion scale for a particle with a known
mass 𝑚𝑚0 :
𝑦𝑦 =
2. Theoretical Treatments
39
1
2
𝑙𝑙𝑙𝑙𝑙𝑙 �
𝐸𝐸+𝑝𝑝 𝑧𝑧
𝐸𝐸−𝑝𝑝 𝑧𝑧
�
(6)
where 𝑝𝑝𝑧𝑧 = 𝑚𝑚 𝑇𝑇 sinh(𝑦𝑦). The four-momentum of a particle
can be described in terms of its transverse momentum 𝑝𝑝𝑇𝑇 , its
rapidity 𝑦𝑦, and transverse energy 𝑚𝑚 𝑇𝑇 and
𝑝𝑝𝜇𝜇 = (𝑚𝑚 𝑇𝑇 cosh 𝑦𝑦, 𝑝𝑝𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝜑𝜑0 , 𝑝𝑝𝑇𝑇 sin 𝜑𝜑0 , 𝑚𝑚 𝑇𝑇 𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝑦𝑦) (7)
The transport models Ur-QMD that employ hadronic and
string degrees of freedom, [21-23] are used for our study.
They take into account the formation and multiple
re-scattering of hadrons and thus dynamically describe the
generation of pressure in the hadronic expansion phase. The
Ur-QMD transport approach [21-23] includes all baryonic
resonances up to masses of 2 GeV as well as mesonic
resonances up to 1.9 GeV. For hadronic continuum
excitations, a string model used with hadron formation times
in the order of 1-2 fm/c depending on the momentum and
energy of the created hadron. The transport approach is
matching to reproduce the nucleon-nucleon, meson nucleon
and meson-meson cross section data in a wide kinematic
range [26].
3. Comparison with Experimental
Results and Discussions
Figure 1. Two body decay
Consider a single heavy particle of mass m produced in
association with visible particles which decays as in Fig.1 to
two particles, of which one (labeled particle 1) is invisible.
The mass of the parent particle can be constrained with the
The pion is one of the particles that mediate the interaction
between a pair of nucleons. This interaction is attractive pulls
the nucleons together. Pions are mesons with zero spin, and
they are composed of first-generation quarks. In the quark
model, an up quark and an anti-down quark make up a π+,
whereas a down quark and an anti-up quark make up the π−,
and these are the antiparticles of one another. The neutral
pion π0 is a combination of an up quark with an anti-up quark
40
A. M. Abdalla: Transverse Mass Spectra for Charged Pions in Nucleus–Nucleus
Collisions in Terms of Relativistic Quantum Molecular Dynamics Model
or a down quark with an anti-down quark. The two
combinations have identical quantum numbers, and hence
they are only found in superposition. The lowest-energy
superposition of these is the π0, which is its own antiparticle.
Together, the pions form a triplet of isospin. Each pion has
isospin (I = 1) and third-component isospin equal to its
charge (Iz = +1,0 or −1) [27]. Figure 2 shows the transverse
mass spectra for positive pions at central rapidity
(|𝑦𝑦| < 0.5) and impact parameter 𝑏𝑏 ≤ 3.4 𝑓𝑓𝑓𝑓 for Pb+Pb
collisions at lab energies 4, 6, 8 and 11 GeV. The solid lines
represent the corresponding calculations using QMD model
with new art version Ur-QMD-2.3. The corresponding data
from different experiments [28-33] are depicted with symbols.
The same comparison applied for negative pions from
Pb+Pb/Au+Au at energies 30, 40, 80, 160 and 200 GeV
shown in Fig. 3. The UrQMD calculations give valuable
representation for 𝑚𝑚 𝑇𝑇 spectra especially for low values of
transverse mass. The Ur-QMD transport model successfully
used to predict and interpret experimental data at various
energies and for a multitude of observables and reaction
systems, e.g. hadron yields, transverse spectra [34-36]. It
treats the initial nucleon-nucleon interactions within a
string-hadronic framework. In addition, this model includes
effects such as string-string interactions and hadronic
re-scattering, which are expected to be relevant in A+A
collisions.
Figure 2. Transverse mass distribution for secondary positive pions emitted from Au-Au collisions at 4, 6, 8 and 11 lab energies. The solid line the
corresponding predictions by UrQMD version 2.3
Journal of Nuclear and Particle Physics 2015, 5(2): 38-43
41
Figure 3. Transverse mass distribution for secondary negative pions emitted from Pb-Pb collisions at 30, 40, 80, 160 and 200 lab energies. The solid line
the corresponding predictions by UrQMD version 2.3
42
A. M. Abdalla: Transverse Mass Spectra for Charged Pions in Nucleus–Nucleus
Collisions in Terms of Relativistic Quantum Molecular Dynamics Model
Table 1. The temperature of positive pion production system at lab collision energies. The magnitudes in parentheses are the corresponding predicted by
Ur-QMD
𝑇𝑇(𝜋𝜋 +) MeV
4 GeV
6 GeV
8 GeV
11 GeV
326.57±4.19
(372.81±5,29)
341.20±6.19
(393.11±3.40)
379.84±9.27
(407.07±3.11)
369.79±8.82
(415.11±4.16)
Table 2. The temperature of negative pion production system at lab collision energies. The magnitudes in parentheses are the corresponding predicted by
Ur-QMD
𝑇𝑇(𝜋𝜋 − ) MeV
30 GeV
40 GeV
80 GeV
160 GeV
200 GeV
375.90±12.98
(426.07±3.59)
409.83±14.41
(428.38±4.20)
321.82±31.88
(455.16±27.29)
397.90±16.87
(431.53±3.63)
514.32±9.10
(474.67±3.50)
the two-pion correlation function [38]. This may be due to
decrease of collision formation time and suitable conditions
of high energy density. The processes become more difficult
and confirm of creation states of non-thermal equilibrium of
nuclear matter and non-regular mechanisms of hadron
productions.
4. Conclusions
Figure 4. Temperature of the charged pion production system as a
function of the energies for Au-Au and Pb-Pb collisions
The 𝑚𝑚 𝑇𝑇 spectra for hadrons like 𝜋𝜋 + and 𝜋𝜋 − production
described by a power law relation in the full phase space
accessible to the experiment with
1
𝑑𝑑𝑑𝑑
𝑚𝑚 𝑇𝑇 𝑑𝑑𝑚𝑚 𝑇𝑇 𝑑𝑑𝑑𝑑
= 𝐶𝐶 𝑒𝑒𝑒𝑒𝑒𝑒 �−
𝑚𝑚 𝑇𝑇
𝑇𝑇
�
(8)
In Eq. 8 the inverse slope parameter T and the
normalization parameter C are free parameters and their
values are extracted from the least square fits to the
experimental spectra. The magnitude of T is the effective
temperature of the system responsible for the emissions of
secondary hadrons. The values of T, for system responsible
for production of positive pions at above energies are given
in table 1. The same values obtained for negative pion given
in table 2. The values in parentheses are the corresponding
theoretical values. The behavior of this dependence, as a
function of collision energies in the lab system are shown in
Fig. 4. The solid line represents the linear fitting of the
experimental data. For energies below 40 GeV the
temperature of the system responsible for charged pion
production is uniformly changes with collision energies from
Pb+Pb (Au+Au) collisions. Above this energies there is a
suddenly changes in T and UrQMD assumption is however
difficult to justify with the dynamical models of the collision
process [20, 37] and the results on the energy dependence of
Transverse mass is one of the experimental parameter
used to study kinematic of nucleus-nucleus collision at
relativistic energies. The ultra-relativistic Quantum
Molecular Dynamic Model using version Ur-QMD-2.3 give
suitable transverse mass distribution for charged pions
production compared with experimental data for collision
energies below 40 GeV and above the hadrons production is
not systematic which may due to the creation of nuclear
states of non-thermal equilibrium and states of high energy
density. The inverse slop parameter represents the effective
temperature of the mechanism responsible for hadrons
production. The effective temperatures are regular increase
with collision energy up to 40 GeV.
ACKNOWLEDGEMENTS
I would like to thank RQMD group for answering the
questions and their suitable advice to make the best use of the
model. I am grateful to the group for their supply by the
program of calculations.
REFERENCES
[1]
R. Stock, J. Phys. G 30, 633 (2004).
[2]
J. C. Collins and M. J. Perry, Phys. Rev. Lett. 34, 1353 (1975),
E. V. Shuryak, Phys. Rep. 61, 71 (1980) and 115, 151(1984).
[3]
J. Rafelski and B. M¨uller, Phys. Rev. Lett. 48, 1066 (1982),
T. Matsui and H. Satz, Phys. Lett.B 178, 416 (1986).
[4]
Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo, Phys. Lett. B
643, 46 (2006), M. Cheng et al., Phys. Rev. D 74,
054507(2006).
Journal of Nuclear and Particle Physics 2015, 5(2): 38-43
[5]
I. Arseneet. al. (BRAHMS), Nucl. Phys. A757, 1 (2005); B. B.
Back et. al. (PHOBOS), Nucl., Phys. A757, 28 (2005); J.
Adams et. al. (STAR), Nucl. Phys.A757, 10 (2005); K. Adcox
et.al. (PHENIX), Nucl. Phys. A757, 184 (2005).
43
[21] B. Andersson, G. Gustafson and B. Nilsson-Almqvist, Nucl.
Phys. B 281 (1987) 289.
[22] B. Nilsson-Almqvist and E. Stenlund, Comput. Phys.
Commun. 43 (1987) 387.
[6]
S. S. Adler et al. (PHENIX), Phys. Rev. Lett. 94, 082302
(2005).
[7]
A. Adare et al. (PHENIX), Phys. Rev. Lett. 104, 132301
(2010).
[24] M. Belkacem et al., Phys. Rev. C 58 (1998) 1727
[arXiv:nucl-th/9804058].
[8]
A. Adare et al. (PHENIX), Phys. Rev. C81, 034911 (2010).
[9]
J. Rafelski and B. Muller, Phys. Rev. Lett. 48, 1066(1982).
[25] L. V. Bravina et al., Phys. Rev. C 60 (1999) 024904
[arXiv:hep-ph/9906548].
[10] J. Bartke et al., Z. Phys. C48, 191 (1990), T. Alber et al., Z.
Phys. C64, 195 (1994).
[11] E.L. Bratkovskaya, S. Soff, H. Stoecker, M. van Leeuwen,
and W. Cassing, Phys. Rev. Lett. 92, 032302 (2004);E.L.
Bratkovskaya, M. Bleicher, M. Reiter, S. Soff, H. Stoecker,
M. van Leeuwen, S. Bass, and W. Cassing, Phys. Rev. C 69,
054907 (2004).
[12] M. Wagner, A.B. Larionov, and U. Mosel, Phys. Rev. C71,
034910 (2005).
[13] M. Gazdzicki, M.I. Gorenstein, F. Grassi, Y. Hama,
T.Kodama, and O. SocolowskiJr, Braz. J. Phys. 34,
322(2004).
[14] L. Ahle, et al. (E866 and E917 Collaborations), Phys.Lett.
B476, 1 (2000).
[15] M. Gazdzicki, et al. (NA49 Collab.), J. Phys. G30,
S701(2004).
[23] T. Sjostrand, Comput. Phys. Commun. 82 (1994) 74.
[26] A.M.Abdalla, International Journal of High Energy Physics,
2014; 1(1): 6-12
[27] C. Amsler et al. (Particle Data Group), PL B667, 1 (2008)
(URL: http://pdg.lbl.gov)
[28] J. L. Klay et al. [E-0895 Collaboration], Phys. Rev. C 68
(2003) 054905[arXiv:nucl-ex/0306033].
[29] C. Alt et al. [NA49 Collaboration], Phys. Rev. C 77 (2008)
024903 [arXiv:0710.0118 [nuclex]].
[30] S. V. Afanasiev et al. [The NA49 Collaboration], Phys. Rev.
C 66 (2002) 054902[arXiv:nucl-ex/0205002].
[31] J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 92
(2004) 112301[arXiv:nucl-ex/0310004].
[32] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. C 69
(2004) 034909[arXiv:nucl-ex/0307022].
[33] I. Arsene et al. [BRAHMS Collaboration], Phys. Rev. C 72
(2005) 014908[arXiv:nucl-ex/0503010].
[16] S. V. Afanasiev, et al. (NA49 Collab.), Phys. Rev.
C66,054902 (2002); C. Alt, et al. (NA49 Collab.), J.
Phys.G30, S119 (2004).
[34] S. A. Bass et al., Nucl. Phys. A 661 (1999) 205
[arXiv:nucl-th/9907090].
[17] M.I. Gorenstein, M. Gazdzicki, and K. Bugaev, Phys. Lett.
B567, 175 (2003).
[35] M. J. Bleicher et al., Phys. Rev. C 62 (2000) 024904
[arXiv:hep-ph/9911420].
[18] B. Mohanty, J. Alam, S. Sarkar, T.K. Nayak, B.K. Nandi,
Phys. Rev. C 68, 021901 (2003).
[36] E. L. Bratkovskaya et al., Phys. Rev. C 69 (2004) 054907
[arXiv:nucl-th/0402026].
[19] M. Bleicher et al., J. Phys. G 25 (1999) 1859
[arXiv:hep-ph/9909407].
[37] W. Cassing, E. L. Bratkovskaya and S. Juchem, Nucl. Phys. A
674, 249 (2000).
[20] S. A. Bass et al., Prog. Part. Nucl. Phys. 41 (1998) 255 [Prog.
Part. Nucl. Phys. 41 (1998)225] [arXiv:nucl-th/9803035].
[38] S. Kniege et al. [NA49 Collaboration], J. Phys. G 30, S1073
(2004).