Properties of Particle Production and System Evolution in BESWide Energy Range
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JiaQi Hui, ZhiJin Jiang, "A Description of Transverse Momentum Distributions in Collisions at RHIC and LHC Energies", Advances in High Energy Physics, vol. 2019, Article ID 8372416, 9 pages, 2019. https://doi.org/10.1155/2019/8372416
A Description of Transverse Momentum Distributions in Collisions at RHIC and LHC Energies
Abstract
It has long been debated whether the hydrodynamics is suitable for the smaller colliding systems such as collisions. In this paper, by assuming the existence of longitudinal collective motion and longrange interactions in the hot and dense matter created in collisions, the relativistic hydrodynamics incorporating with the nonextensive statistics is used to analyze the transverse momentum distributions of the particles. The investigations of the present paper show that the hybrid model can give a good description of the currently available experimental data obtained in collisions at RHIC and LHC energies, except for and produced in the range of GeV/c at GeV.
1. Introduction
In the past decade, the experimental results of heavy ion collisions at both RHIC and LHC energies have been extensively studied. These studies have shown that the strongly coupled quarkgluon plasma (sQGP) might be created in these collisions [1–9], which exhibits a clear collective behavior almost like a perfect fluid with very low viscosity [10–28]. Therefore, the evolution of sQGP can be described in the scope of relativistic hydrodynamics. However, unlike heavy ion collisions, collisions are a relatively smaller system with lower multiplicity, larger viscosity, and larger fluctuation [29]. The reasonableness of applying relativistic hydrodynamics in depicting the evolution of sQGP created in collisions has undergone an endless debate.
In this paper, by supposing the existence of collective flow in colliding direction, the relativistic hydrodynamics including phase transition is introduced to describe the longitudinal expansion of sQGP. Besides the collective flow, the thermal motion also exists in sQGP. The evolution of sQGP is therefore the superposition of collective flow and thermal motion. Known from the investigations of [30, 31], the longrange interactions and memory effects might appear in sQGP. This guarantees the reasonableness of nonextensive statistics in describing the thermodynamic aspects of sQGP. Hence, in this paper, we will use the nonextensive statistics instead of conventional statistics to characterize the thermal motion of the matter created in collisions.
The nonextensive statistics, i.e., Tsallis nonextensive thermostatistics, is the generalization of conventional BoltzmannGibbs statistics, which is proposed by C. Tsallis in his pioneer work of [32]. This statistical theory overcomes the inabilities of the conventional statistical mechanics by assuming the existence of longrange interactions, longrange microscopic memory, or fractal spacetime constraints in the thermodynamic system. It has a wide range of applications in cosmology [33], phase shift analyses for the pionnucleus scattering [34], dynamical linear response theory, and variational methods [35]. It has achieved a great success in solving many physical problems, such as the solar neutrino problems [36], manybody problems, the problems in astrophysical selfgravitating systems [37], and the transverse momentum spectra [38–40].
The article is organized as follows. In Section 2, a brief description is given about the employed hydrodynamics, presenting its analytical solutions. The solutions are then used in Section 3 to formulate the transverse momentum distributions of the particles produced in collisions in the light of CooperFrye prescription. The last Section 4 is about conclusions.
2. A Brief Introduction to the Hydrodynamic Model
The main content of the relativistic hydrodynamic model [15, 41] used in this paper is as follows.
The expansion of fluid obeys the continuity equationwhereis the energymomentum tensor of fluid and is the metric tensor. The fourvelocity of fluid , where is the rapidity of fluid. and in Equation (2) are the energy density and pressure of fluid, respectively, which are related by the sound speed of fluid via the equation of statewhere and are the temperature and entropy density of fluid, respectively.
The projection of Equation (1) to the direction of leads to the continuity equation for entropy conservationThe projection of Equation (1) to the direction perpendicular to gives equation which means the existence of a scalar function satisfyingBy using and Legendre transformation, Khalatnikov potential can be introduced via relationwhich changes the coordinate base of to that of where is the initial temperature of sQGP, and . In terms of , Equation (4) can be rewritten as the socalled equation of telegraphy
With the expansion of created matter, its temperature becomes lower and lower. When the temperature drops from the initial temperature to the critical temperature , phase transition occurs. This will modify the value of sound speed of fluid. In sQGP, , which is the sound speed of a massless perfect fluid, being the maximum of . In the hadronic state, . At the point of phase transition, is discontinuous.
The solutions of Equation (10) for sQGP and hadronic state are, respectively [15],where is the 0th order modified Bessel function, andwhere , , and . The in Equations (11) and (12) is a free parameter determined by fitting the theoretical results with experimental data.
3. The Transverse Momentum Distributions of the Particles Produced in Collisions
3.1. The Energy of Quantum of Produced Matter
The nonextensive statistics is based on the following two postulations [32, 36].
(a) The entropy of a statistical system possesses the form ofwhere is the probability of a given microstate among ones and is a fixed real parameter. The defined entropy has the usual properties of positivity, equiprobability, and irreversibility, and, in the limit of , it reduces to the conventional BoltzmannGibbs entropy
(b) The mean value of an observable is defined aswhere is the value of an observable in the microstate .
From the above two postulations, the average occupational number of quantum in the state with temperature can be written in a simple analytical form [42]Here, as usual, is the energy of quantum, and is its baryochemical potential. For baryons and for mesons . In the limit of , it reduces to the conventional FermiDirac or BoseEinstein distributions. Hence, the value of in the nonextensive statistics represents the degree of deviation from the conventional statistics. Known from Equation (17), the average energy of quantum in the state with temperature readswhere is the rapidity of quantum and is its transverse mass with rest mass and transverse momentum .
3.2. The Transverse Momentum Distributions of the Particles Produced in Collisions
With the expansion of hadronic matter, its temperature becomes even lower. As the temperature drops to the socalled kinetic freezeout temperature , the inelastic collisions among hadronic matter stop. The yields of produced particles remain unchanged, becoming the measured results. According to CooperFrye scheme [43], the invariant multiplicity distributions of produced particles take the form [15, 43]where is the area of overlap region of collisions, , and the integrand takes values at the moment of . The meaning of Equation (19) is evident. The part of integrand in the round brackets is proportional to the rapidity density of fluid [43]. Hence, Equation (19) is the convolution of rapidity of fluid with the energy of the particles in the state with temperature . From Equations (8) and (9)Substituting in Equation (20) by the of Equation (12) and taking the values at the moment of , it becomeswherewhere , is the 1st order modified Bessel function.
By using Equations (19) and (21)(23), we can obtain the transverse momentum distributions of produced particles as shown in Figures 1, 2, 3, and 4.
Figure 1 shows the transverse momentum spectra of , , , , , , , and produced in collisions at GeV. The solid dots, circles, and solid triangles represent the experimental data of the STAR Collaboration [44]. The solid curves are the results calculated from Equation (19). The values of free parameters , , and /NDF are listed in Table 1. It can be seen that the present model can give a good description of the transverse momentum distributions of strange particles. Since strangeness enhancement is originally proposed as a signature of sQGP produced in nuclear collisions, this proves the reasonableness of hypothesis given at the beginning of this paper that sQGP might appear in collisions.

Figure 2 presents the transverse momentum spectra of , , , , , and produced in collisions at 200 GeV. The solid dots, solid triangles, solid squares, circles, triangles, and squares represent the experimental data of the PHENIX Collaboration [45]. The solid curves are the results calculated from Equation (19). The values of free parameters , , and /NDF are summarized in Table 2. The theoretical model can give a good description of the experimental data for , , , in the whole measured transverse momentum range, and for and in the range of 3.0 GeV/c. In the range of 3.0 GeV/c, the deviation appears as shown in Figure 3, which shows the transverse momentum distributions of and in the whole measured range.

Figure 4 shows the transverse momentum spectra of , , , , , and produced in collisions at 0.9, 2.76, and 7 TeV. The solid dots, solid triangles, solid squares, circles, triangles, and squares represent the experimental data of the CMS Collaboration [46]. The solid curves are the results calculated from Equation (19). The values of free parameters , , and /NDF are summarized in Table 3.

In calculations, the sound speed in hadronic state takes the value of 0.35 [47, 48]. The critical temperature takes the value of 0.16 GeV [49]. For 200 GeV, the initial temperature takes the value of 0.35 GeV [50], the kinetic freezeout temperature takes the values of 0.12 GeV for strange particles and pions, and, for protons, 0.13 GeV from the investigation of [51], which also shows that the baryochemical potential takes the value of 0.01 GeV. For 0.9, 2.76, and 7 TeV, referring to [50], the initial temperatures are estimated to be 0.4, 0.6, and 1.5 GeV, respectively. The kinetic freezeout temperature takes the values of 0.12 GeV for pions and kaons, and, for protons, 0.13 GeV. The baryochemical potential takes the value of [51].
The parameters and have the same effects. They all affect the amplitudes of the theoretical curves. They are different from parameter which affects the slopes of the theoretical curves. From the above analysis we can see that the value of the parameter increases with the increase of the CMS beam energy. However, the values of do not seem completely consistent with the CMS and the RHIC beam energies.
4. Conclusions
By assuming the existence of longitudinal collective motion and longrange interactions in sQGP produced in collisions, the relativistic hydrodynamics including phase transition together with the nonextensive statistics is used to discuss the transverse momentum distributions of the particles produced in collisions at 0.2, 0.9, 2.76, and 7 TeV.
The theoretical model used in this paper contains rich information about the transport coefficients of fluid, such as the sound speed in sQGP, the sound speed in hadronic state, the initial temperature , the critical temperature , the kinetic freezeout temperature , and the baryochemical potential . Except for , the other five parameters take the values either from the widely accepted theoretical results or from experimental measurements. As for , there are no acknowledged values so far. In this paper, takes the values from other studies. The investigations of the present paper show the conclusions as follows.
(a) The theoretical model can give a good description of the currently available experimental data collected in collisions at RHIC and LHC energies with the only exception of and measured in the range of 3.0 GeV/c at 200 GeV, which might be caused by the hard scattering process [52]. To improve the fitting conditions, the results of perturbative QCD should be taken into account.
(b) The fitted values of are close to 1. This means that the deviation between nonextensive statistics and conventional statistics is small, while it is this small difference that plays an essential role in fitting the experimental data.
Data Availability
The experimental data used to support the findings of this study have been deposited in https://doi.org/10.1103/PhysRevC.75.064901; https://doi.org/10.1103/PhysRevC.83.064903; and https://doi.org/10.1140/epjc/s1005201221641.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by the Shanghai Key Lab of Modern Optical System.
References
 I. Arsene, I. G. Bearden, and D. Beavis, “Quarkgluon plasma and color glass condensate at RHIC? The perspective from the BRAHMS experiment,” Nuclear Physics A, vol. 757, no. 12, pp. 1–27, 2005. View at: Publisher Site  Google Scholar
 B. B. Back, M. D. Baker, and M. Ballintijn, “The PHOBOS perspective on discoveries at RHIC,” Nuclear Physics A, vol. 757, no. 12, pp. 28–101, 2005. View at: Publisher Site  Google Scholar
 J. Adams, M. M. Aggarwal, and Z. Ahammed, “Experimental and theoretical challenges in the search for the quark–gluon plasma: The STAR Collaboration's critical assessment of the evidence from RHIC collisions,” Nuclear Physics A, vol. 757, no. 12, pp. 102–183, 2005. View at: Publisher Site  Google Scholar
 K. Adcox, S. S. Adler, S. Afanasiev et al., “RHIC: Experimental evaluation by the PHENIX Collaboration,” Nuclear Physics A, vol. 757, no. 12, pp. 184–283, 2005. View at: Publisher Site  Google Scholar
 T. Alexopoulos et al., “Massidentified particle production in protonantiproton collisions at , 540, 1000, and 1800 GeV,” Physical Review D, vol. 48, Article ID 984, 1993. View at: Publisher Site  Google Scholar
 P. Lévai and B. Müller, “Transverse baryon flow as possible evidence for a quarkgluonplasma phase,” Physical Review Letters, vol. 67, no. 12, pp. 1519–1522, 1991. View at: Publisher Site  Google Scholar
 G. N. Fowler, E. M. Friedlander, R. M. Weiner, and G. Wilk, “Possible manifestation of quarkgluon plasma in multiplicity distributions from highenergy reactions,” Physical Review Letters, vol. 57, no. 17, pp. 2119–2122, 1986. View at: Publisher Site  Google Scholar
 E. M. Friedlander and R. M. Weiner, “Evidence from very large transverse momenta of a change with temperature of velocity of sound in hadronic matter,” Physical Review Letters, vol. 43, pp. 15–18, 1979. View at: Publisher Site  Google Scholar
 M. G. Albrow, S. Almehed, P. S. L. Booth et al., “Studies of protonproton collisions at the CERN ISR with an identified charged hadron of high transverse momentum at 90°: (II) On the distribution of charged particles in the central region,” Nuclear Physics B, vol. 145, no. 23, pp. 305–348, 1978. View at: Publisher Site  Google Scholar
 A. Bialas, R. A. Janik, and R. Peschanski, “Unified description of Bjorken and Landau 1+1 hydrodynamics,” Physical Review C nuclear physics, vol. 76, no. 5, Article ID 054901, 2007. View at: Publisher Site  Google Scholar
 C. Y. Wong, “Landau hydrodynamics reexamined,” Physical Review C: Nuclear Physics, vol. 78, no. 5, Article ID 054902, 2008. View at: Publisher Site  Google Scholar
 G. Beuf, R. Peschanski, and E. N. Saridakis, “Entropy flow of a perfect fluid in (1 + 1) hydrodynamics,” Physical Review C: Nuclear Physics, vol. 78, no. 6, Article ID 064909, 2008. View at: Publisher Site  Google Scholar
 Z. J. Jiang, Y. Huang, and J. Wang, “A combined model for pseudorapidity distributions in pp collisions at centerofmass energies from 23.6 to 7000 GeV,” Chinese Physics C: Nuclear Physics, vol. 40, no. 7, Article ID 074104, 2016. View at: Publisher Site  Google Scholar
 Z. W. Wang, Z. J. Jiang, and Y. S. Zhang, “The investigations of pseudorapidity distributions of final state multiplicities in Au+Au collisions at high energies,” University of Shanghai for Science and Technology, vol. 31, p. 322, 2009. View at: Google Scholar
 N. Suzuki, “Onedimensional hydrodynamical model including phase transition,” Physical Review C, vol. 81, no. 4, Article ID 044911, 2010. View at: Google Scholar
 E. K. G. Sarkisyan and A. S. Sakharov, “Relating multihadron production in hadronic and nuclear collisions,” The European Physical Journal C, vol. 70, no. 3, pp. 533–541, 2010. View at: Publisher Site  Google Scholar
 A. Bialas and R. Peschanski, “Asymmetric (1+1)dimensional hydrodynamics in highenergy collisions,” Physical Review C: Nuclear Physics, vol. 83, no. 5, Article ID 054905, 2011. View at: Publisher Site  Google Scholar
 Z.J. Jiang, J.Q. Hui, and H.P. Deng, “Unified hydrodynamics and pseudorapidity distributions of charged particles produced in heavy ion collisions at low energies at RHIC,” Chinese Physics Letters, vol. 34, no. 5, Article ID 052501, 2017. View at: Publisher Site  Google Scholar
 Z. J. Jiang, Q. G. Li, and H. L. Zhang, “Revised Landau hydrodynamic model and the pseudorapidity distributions of charged particles produced in nucleusnucleus collisions at maximum energy at the BNL Relativistic Heavy Ion Collider,” Physical Review C: Nuclear Physics, vol. 87, no. 4, Article ID 044902, 2013. View at: Publisher Site  Google Scholar
 C. Gale, S. Jeon, and B. Schenke, “Hydrodynamic modeling of heavyion collisions,” International Journal of Modern Physics A, vol. 28, no. 11, Article ID 1340011, 2013. View at: Publisher Site  Google Scholar
 U. Heinz and R. Snellings, “Collective flow and viscosity in relativistic heavyion collisions,” Annual Review of Nuclear and Particle Science, vol. 63, no. 1, pp. 123–151, 2013. View at: Publisher Site  Google Scholar
 A. N. Mishra, R. Sahoo, E. K. G. Sarkisyan, and A. S. Sakharov, “Effectiveenergy budget in multiparticle production in nuclear collisions,” The European Physical Journal C, vol. 74, Article ID 3147, 2014. View at: Publisher Site  Google Scholar
 Z. J. Jiang, Y. Zhang, H. L. Zhang, and H. P. Deng, “A description of the pseudorapidity distributions in heavy ion collisions at RHIC and LHC energies,” Nuclear Physics A, vol. 941, pp. 188–200, 2015. View at: Publisher Site  Google Scholar
 H. Niemi, K. J. Eskola, and R. Paatelainen, “Eventbyevent fluctuations in a perturbative QCD + saturation + hydrodynamics model: Determining QCD matter shear viscosity in ultrarelativistic heavyion collisions,” Physical Review C, vol. 93, Article ID 024907, 2016. View at: Publisher Site  Google Scholar
 J. NoronhaHostler, M. Luzum, and J.Y. Ollitrault, “Hydrodynamic predictions for 5.02 TeV PbPb collisions,” Physical Review C Nuclear Physics, vol. 93, no. 3, Article ID 034912, 2016. View at: Publisher Site  Google Scholar
 J. S. Moreland and R. A. Soltz, “Hydrodynamic simulations of relativistic heavyion collisions with different lattice quantum chromodynamics calculations of the equation of state,” Physical Review C, vol. 93, Article ID 044913, 2016. View at: Publisher Site  Google Scholar
 E. K. G. Sarkisyan, A. N. Mishra, R. Sahoo, and A. S. Sakharov, “Multihadron production dynamics exploring the energy balance in hadronic and nuclear collisions,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 93, Article ID 054046, 2016. View at: Publisher Site  Google Scholar
 E. K. G. Sarkisyan, A. N. Mishra, R. Sahoo, and A. S. Sakharov, “Centrality dependence of midrapidity density from GeV to TeV heavyion collisions in the effectiveenergy universality picture of hadroproduction,” Physical Review D, vol. 94, Article ID 011501, 2016. View at: Publisher Site  Google Scholar
 K. Jiang, Y. Zhu, W. Liu et al., “Onset of radial flow in collisions,” Physical Review C: Nuclear Physics, vol. 91, no. 2, Article ID 024910, 2015. View at: Publisher Site  Google Scholar
 W. Alberico, A. Lavagno, and P. Quarati, “Nonextensive statistics, fluctuations and correlations in highenergy nuclear collisions,” The European Physical Journal C, vol. 12, no. 3, pp. 499–506, 2000. View at: Publisher Site  Google Scholar
 M. Biyajima, T. Mizoguchi, N. Nakajima, N. Suzuki, and G. Wilk, “Modified Hagedorn formula including temperature fluctuation: Estimation of temperatures at RHIC experiments,” The European Physical Journal C, vol. 48, no. 2, pp. 597–603, 2006. View at: Publisher Site  Google Scholar
 C. Tsallis, “Possible generalization of BoltzmannGibbs statistics,” Journal of Statistical Physics, vol. 52, no. 12, pp. 479–487, 1988. View at: Publisher Site  Google Scholar  MathSciNet
 D. F. Torres, H. Vucetich, and A. Plastino, “Early universe test of nonextensive statistics,” Physical Review Letters, vol. 79, no. 9, pp. 1588–1590, 1997. View at: Publisher Site  Google Scholar
 D. B. Ion and M. L. Ion, “Entropic lower bound for the quantum scattering of spinless particles,” Physical Review Letters, vol. 81, no. 26, pp. 5714–5717, 1998. View at: Publisher Site  Google Scholar
 A. K. Rajagopal, R. S. Mendes, and E. K. Lenzi, “Quantum statistical mechanics for nonextensive systems: Prediction for possible experimental tests,” Physical Review Letters, vol. 80, no. 18, pp. 3907–3910, 1998. View at: Publisher Site  Google Scholar
 G. Kaniadakis, A. Lavagno, M. Lissia, and P. Quarati, “Anomalous diffusion modifies solar neutrino fluxes,” Physica A: Statistical Mechanics and its Applications, vol. 261, no. 34, pp. 359–373, 1998. View at: Publisher Site  Google Scholar
 A. R. Plastino and A. Plastino, “Information theory, approximate time dependent solutions of Boltzmann's equation and Tsallis' entropy,” Physics Letters A, vol. 193, no. 3, pp. 251–258, 1994. View at: Publisher Site  Google Scholar  MathSciNet
 U. Tirnakli, F. Bykkilic, and D. Demirhan, “Some bounds upon the nonextensivity parameter using the approximate generalized distribution functions,” Physics Letters A, vol. 245, no. 12, pp. 62–66, 1998. View at: Publisher Site  Google Scholar
 S. Grigoryan, “Using the Tsallis distribution for hadron spectra in collisions: pions and quarkonia at Gev,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 95, Article ID 056021, 2017. View at: Publisher Site  Google Scholar
 D. Thakur, S. Tripathy, P. Garg, R. Sahoo, and J. Cleymans, “Indication of a differential freezeout in protonproton and heavyion collisions at RHIC and LHC energies,” Advances in High Energy Physics, vol. 2016, Article ID 4149352, 13 pages, 2016. View at: Publisher Site  Google Scholar
 J.Q. Hui, Z.J. Jiang, and D.F. Xu, “A description of the transverse momentum distributions of charged particles produced in heavy ion collisions at RHIC and LHC energies,” Advances in High Energy Physics, vol. 2018, Article ID 7682325, 9 pages, 2018. View at: Publisher Site  Google Scholar
 J. Cleymans and D. Worku, “The Tsallis distribution in proton–proton collisions at TeV at the LHC,” Journal of Physics G: Nuclear and Particle Physics, vol. 39, no. 2, Article ID 025006, 2012. View at: Publisher Site  Google Scholar
 F. Cooper and G. Frye, “Landau's hydrodynamic model of particle production and electronpositron annihilation into hadrons,” Physical Review D, vol. 11, Article ID 192, 1975. View at: Publisher Site  Google Scholar
 B. I. Abelev, J. Adams, and M. M. Aggarwal, “Strange particle production in collisions at GeV,” Physical Review C: Nuclear Physics, vol. 75, Article ID 64901, 2007. View at: Publisher Site  Google Scholar
 A. Adare, S. Afanasiev, and C. Aidala, “Identified charged hadron production in collisions at and 62.4 GeV,” Physical Review C: Nuclear Physics, vol. 83, Article ID 064903, 2011. View at: Publisher Site  Google Scholar
 S. Chatrchyan, V. Khachatryan, and A. M. Sirunyan, “Study of the inclusive production of charged pions, kaons, and protons in pp collisions at , and 7 TeV,” The European Physical Journal C, vol. 72, Article ID 2164, 2012. View at: Publisher Site  Google Scholar
 A. Adare, S. Afanasiev, and C. Aidala, “Scaling Properties of Azimuthal Anisotropy in and Collisions at GeV,” Physical Review Letters, vol. 98, Article ID 162301, 2007. View at: Publisher Site  Google Scholar
 F.H. Liu, T. Tian, J.X. Sun, and B.C. Li, “What can we learn from (Pseudo) rapidity distribution in high energy collisions?” Advances in High Energy Physics, vol. 2014, Article ID 863863, 10 pages, 2014. View at: Publisher Site  Google Scholar
 D. Teaney, J. Lauret, and E. V. Shuryak, “Flow at the SPS and RHIC as a quarkgluon plasma signature,” Physical Review Letters, vol. 86, no. 21, pp. 4783–4786, 2001. View at: Publisher Site  Google Scholar
 M. Strickland, “Thermal and suppression at = 2.76 TeV PbPb Collisions at the LHC,” Physical Review Letters, vol. 107, Article ID 132301, 2011. View at: Publisher Site  Google Scholar
 B. I. Abelev, M. M. Aggarwal, and Z. Ahammed, “Systematic measurements of identified particle spectra in pp, d+Au, and Au+Au collisions at the STAR detector,” Physical Review C: Nuclear Physics, vol. 79, no. 3, Article ID 034909, p. 58, 2009. View at: Publisher Site  Google Scholar
 H. L. Lao, F. H. Liu, and R. A. Lacey, “Extracting kinetic freezeout temperature and radial flow velocity from an improved Tsallis distribution,” The European Physical Journal A, vol. 53, p. 44, 2017. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2019 JiaQi Hui and ZhiJin Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.