# Transverse momentum distribution and elliptic flow of charged hadrons in \(U+U\) collisions at \(\sqrt{s_{NN}}=193\) GeV using HYDJET++

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## Abstract

Recent experimental observations of the charged hadron properties in \(U+U\) collisions at 193 GeV contradict many of the theoretical models of particle production including two-component Monte Carlo Glauber model. The experimental results show a small correlation between the charged hadron properties and the initial geometrical configurations (e.g. body–body, tip–tip etc.) of \(U+U\) collisions. In this article, we have modified the Monte Carlo HYDJET++ model to study the charged hadron production in \(U+U\) collisions at 193 GeV center-of-mass energy in tip–tip and body–body initial configurations. We have modified the hard as well as soft production processes to make this model suitable for \(U+U\) collisions. We have calculated the pseudorapidity distribution, transverse momentum distribution and elliptic flow distribution of charged hadrons with different control parameters in various geometrical configurations possible for \(U+U\) collision. We find that HYDJET++ model supports a small correlation between the various properties of charged hadrons and the initial geometrical configurations of \(U+U\) collision. Further, the results obtained in modified HYDJET++ model regarding \(dn_{ch}/d\eta \) and elliptic flow (\(v_{2}\)) suitably matches with the experimental data of \(U+U\) collisions in minimum bias configuration.

## 1 Introduction

The basic motivation of heavy ion collision experiments is to understand the properties and behaviour of quantum chromodynamics (QCD) at very high temperature and chemical potentials via analysing the data on multi-particle production and by matching experimental measurements to the simulation models for the entire evolution of the fireball. There are existing computational models which use the theoretical or phenomenological foundation of strong interactions to mimic the space-time evolution of collision experiments. One can broadly classify these models in two types: dynamical models [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and semi dynamical models [11, 12, 13]. Dynamical models are those which consider the pre-equilibrium evolution as well as post equilibrium hydrodynamic evolution like IP-Glasma model etc [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. However, most of the models are semi dynamical models which use a static initial condition at proper thermalization time and then evolve the system using viscous or ideal hydrodynamics like AMPT, MC-Glauber etc [11, 12, 13]. The particle production mechanism of both types of model are quite different. In dynamical models, the parton saturation is a viable mechanism for particle production e.g., IP-Glasma model is based on the ab-initio color glass condensate framework which combines the impact parameter dependent saturation model for parton distributions with an event-by-event classical Yang-Mills description of early-tile glasma fields [1]. Similarly EKRT model is based on the assumption of final state gluon saturation and thus the initial energy density and produced number of partons scales with atomic number and beam energy [2, 3]. In KLN model, the inclusive production of partons is driven by the parton saturation in strong gluon fields [4, 5, 6, 7]. In saturation regime, the multiplicity of produced partons should be proportional to atomic number [4, 5, 6, 7]. On the other hand the particle production mechanism in semi-classical models are implemented via some phenomenological parameterization or using Monte Carlo event generator e.g., in MC-Glauber model, the particle production is based on static initial conditions and two-component parameterization in which first term is proportional to mean number of participants and second term is proportional to mean number of collisions [11]. In AMPT model initial conditions are obtained from HIJING event generator then ZPC for parton scatterings. After that Lund string model for hadronization and ART model to treat the hadronic scatterings [12]. UrQMD model describes the particle production at low and intermediate energies in terms of scatterings amongst hadrons and their resonances. At higher energies, the excitation of colour strings and their subsequent fragmentation is the particle production mechanism in this model [13].

Most of the simulation models are successful in providing the multiplicity of charged hadrons produced in various heavy ion collision experiments. Vast experimental data on multi-particle production and distributions with collision control parameters like centrality, rapidity and/or transverse momentum etc., put a stringent constraint on these models so that we can understand the production mechanism more deeply and make our models more realistic. To strengthen our understanding about quantum chromodynamics (QCD), these collider experiments collide various nuclei at different colliding energies. Recently RHIC experiment has collided uranium (*U*) nuclei at the center-of-mass energy \(\sqrt{s_{NN}}= 193\) GeV [14]. As we know that uranium is a deformed nuclei (prolate in shape) so various kind of initial configurations are possible in \(U+U\) collision e.g., body–body, tip–tip, body-tip etc. The various computational models previously predicted a large difference in multiplicity and elliptic flow between body–body and tip–tip configurations of \(U+U\) collisions [15, 16]. However, the experimental data of multi-particle production in \(U+U\) collisions regarding multiplicity and elliptic flow (\(v_{2}\)) contradicts the earlier expectations of most of these computational and theoretical models and shows a small correlation between multiplicity (and/or \(v_{2}\)) and initial configurations of \(U+U\) collision [14]. This contradiction may have two possible reasons. Either the simulation models have something missing or experimentally we are not quite able to disentangle the events with different geometrical orientations. Thus we have to work on both the aspects since \(U+U\) collision in its various orientations is quite useful to understand wide range of physics. Quark gluon plasma (QGP) phase which is characterized by the observables like elliptic flow, jet quenching, charmonia suppression and multiplicity can be better understood in the collision of deformed uranium nuclei due to its initial geometry and specific orientation [16, 17, 18, 19, 20, 21]. Further \(U+U\) collisions can provide a reliable tool to subtract the background elliptic flow effect from the signal so that one can detect the chiral magnetic effect (CME) [16]. In spherical nuclei, it is difficult to disentangle both these effect since the strength of both the signals generated from elliptic flow and CME is of similar strength in peripheral collisions. However, in \(U+U\) central collisions, the different geometrical orientations can provide a way to subtract the background signal from CME signal due to a measurable difference in their strength. Thus central collisions of \(U+U\) nuclei in tip–tip configuration can possibly be a good tool to characterize the signal of CME [15, 22].

Very recently different methods have been proposed to modify some of the models to incorporate the experimental \(U+U\) observations in that particular simulation models [23, 24, 25]. The constituent quark model is also proposed to describe the experimental observation on \(v_{2}\) in \(U+U\) collisions [26, 27]. In this article we want to study the \(U+U\) collision at \(\sqrt{s_{NN}} = 193\) GeV in body–body and tip–tip configurations by modifying HYDJET++ model which uses PYTHIA type initial condition for hard part and Glauber type initial condition for soft part. Further most of the existing models either consist of high \(p_{T}\) particle production from jet fragmentation or involve low \(p_{T}\) hadron production using thermal statistical processes. However, HYDJET++ model [28] consistently includes production of hard as well as soft \(p_{T}\) hadrons, to calculate the charged hadron production in \(U+U\) collisions at center-of-mass energy \(\sqrt{s_{NN}}=193\) GeV. We study the pseudorapidity distribution, transverse momentum (\(p_{T}\)) distribution of charged hadrons. Moreover we calculate the elliptic flow of these produced particles in body–body and tip–tip configurations of \(U+U\) collisions. Rest of the article is organised as follows: In Sect. 2, we have provided a brief detail of modified HYDJET++ model and described its various physical parts under different sections. Further we have written down the equation to calculate the elliptic flow of charged hadrons. In Sect. 3, we have provided the results and discussions under two sections: (A) pseudorapidity distributions and, (B) transverse momentum distribution and elliptic flow. At last we have summarized our current work.

## 2 Model formalism

The heavy ion event generator HYDJET++ simulates relativistic heavy ion collisions as a superposition of the soft, hydro-type state and the hard state resulting from multi-parton fragmentation. The soft and hard components are treated independently in HYDJET++. The details on physics model and simulation procedure of HYDJET++ can be found in the corresponding manual [28, 29]. The main features of HYDJET++ model are listed very briefly in this section.

### 2.1 Hard multi-jet production

The model for the hard multi-parton production of HYDJET++ event is based on PYQUEN partonic loss model [30, 31, 32]. In brief the hard part of hadron production in HYDJET++ uses PYQUEN [30] which includes generation of initial parton spectra according to PYTHIA and production vertices is measured at a given impact parameter. After that rescattering of partons is incorporated using an algorithm of the parton path in a dense medium along with their radiative and collisional energy loss. Finally hadronization takes place according to the Lund string model [33] for hard partons and in-medium emitted gluons. An important cold nuclear matter effect which is shadowing of parton’s distribution function is included using Glauber–Gribov theory [34]. As a simplification to the model, the collisional energy loss due to scattering [35, 36] with low momentum transfer is not considered because its contribution to the total collisional energy loss is very less in comparison with high momentum scattering. The medium where partonic rescattering occurs is treated as a boost invariant longitudinally expanding quark-gluon fluid, and partons are produced on a hypersurface of equal proper times \(\tau \) [37]. Since we use Bjorken hydrodynamics thus the results in this model have limited applicability at larger rapidities where one should use Landau hydrodynamics for the proper description of medium expansion.

*r*is basically \(\rho \) of cylindrical polar coordinate system and not spherical polar coordinate

*r*. We follow this representation so that readers do not get confused it with nuclear density function (\(\rho \)). The values and range of \(\psi \) remains equal to \(\phi \) during this coordinate conversion as far these two configurations are concerned. It is quite difficult to make conversion mapping between these two coordinate systems to incorporate random values of theta from its whole range i.e. 0 to \(\pi \). Thus we reserve this topic for our future research work. To show the validity of our modification in deformed Woods-Saxon function and make the readers visualize, the nuclear density profiles (in cylindrical coordinate system) for non-deformed gold nucleus along with tip and body configuration of uranium nucleus are shown in Figs. 1, 2 and 3, respectively. Now the two quantities, nuclear thickness function (\(T_{A}\)) and nuclear overlap function (\(T_{AA}\)) can be calculated using this modified and deformed Woods-Saxon nuclear density profile function in cylindrical coordinates \(\rho (r,~z,~\psi )\) by following expressions [41] (Please see Fig. 4a, b):

where \(r_{1,2}(b,r,\psi )\) are the distances between the centers of colliding nuclei and the jet production vertex \(V(r\cos \psi , r\sin \psi )\), *r* is the distance from the nuclear collision axis to *V*, \(R_{eff}(b,\psi )\) is the transverse distance from the nuclear collision axis to the effective boundary of nuclear overlapping area in the given azimuthal direction \(\psi \).

*b*) with maximum possible radius of uranium nucleus (\(R_{A}\)). This \(b/R_{A}\) actually represents the centrality of the event. We have also shown the maximum possible value of \(N_{part}(0)\) for both the configurations which must not depend on centrality.

### 2.2 Soft ‘thermal’ hadron production

The soft part of HYDJET++ is the thermal hadronic state generated on the chemical and thermal freeze-out hypersurface obtained from the parameterization of relativistic hydrodynamics with a given freeze-out condition [42, 43]. The first and foremost modification which we have done in soft part is to change the nuclear density profile function for deformed uranium nucleus as discussed in above section. After that we have to modify the freeze-out hypersurface to properly include the effect of nuclear deformation via change in number of participants.

*b*[41] and is calculated as:

*a*,

*b*,

*c*,

*d*, and

*e*have been determined from the best fit of the particle ratios at various collision energies: \(a =1.290\pm 0.113\) GeV, \(b=0.28\pm 0.046\) GeV\(^{-1}\), \(c= 0.170\pm 0.1\) GeV, \(d=0.169\pm 0.02\) GeV\(^{-1}\), and \(e=0.015\pm 0.01\) GeV\(^{-3}\). The temperature for other i.e., semi-central, semi-peripheral and peripheral, events is calculated by using the following relation so that one can convert the fixed freeze-out hypersurface into a centrality(or \(N_{part}\)) dependent hypersurface which is much needed modification in soft particle production in HYDJET++:

### 2.3 Elliptic flow

*x*–

*y*at the freezeout can be approximated by an ellipse in non-central collision. Radii \(R_{x}\) and \(R_{y}\) of the ellipse at a given impact parameter

*b*can be parameterized [52, 53, 54, 55] in terms of spatial anisotropy at freezeout \(\epsilon _{2}(b) = (R_{y}^{2}-R_{x}^{2})/(R_{x}^{2}+R_{y}^{2})\) and the scale factor \(R_{f}(b) = [(R_{x}^{2}+R_{y}^{2})/2]^{1/2}\) as:

## 3 Results and discussions

### 3.1 Pseudorapidity distributions

We have generated one million events for each centrality class for each of the configuration (tip–tip and body–body) separately using HYDJET++. Probability distribution curves for body–body and tip–tip events are shown in Fig. 7. We start our analysis with pseudorapidity distribution of charged hadrons. Pseudorapidity distribution of charged hadrons is a useful observable which can help us to understand various properties of the fireball formed and the particle production process.

In Fig. 11, \(dn_{ch}/d\eta \) with respect to \(\eta \) is shown for most central tip–tip collision. Further we have presented the jet (hard) part and hydro (soft) part separately to show their relative contribution in the total multiplicity. From Fig. 11, one can see that the hard part has relatively low contribution than the soft part and hydro part is almost 3 times larger than the jet part. One can also see that the jet part is almost flat in central rapidity region and the dip at \(\eta = 0\) is mainly due to soft part of particle production. Further we have compared these most central tip–tip results with the most central body–body results. One can see that the combined multiplicity (soft plus hard) is larger in most central tip–tip configuration than the most central body–body configuration. Jet part has also the same behaviour. However, soft part shows an opposite behaviour. Here the body–body soft multiplicity is larger than tip–tip results. Similarly Fig. 12 presents the variation of \(dn_{ch}/d\eta \) with respect to \(\eta \) for most peripheral tip–tip configuration along with separate soft and jet part. Further we have compared these results with most peripheral body–body configuration. Here we found that the combined multiplicity is larger in body–body configuration than corresponding tip–tip result. Furthermore both jet as well as hydro part is larger in comparison to tip–tip configuration. Even the hydro part in body–body configuration is larger than the overall multiplicity in tip–tip configuration in most peripheral events.

### 3.2 Transverse momentum distribution and elliptic flow

In Fig. 20, we have demonstrated the variation of elliptic flow distribution of charged hadrons with respect to transverse momentum in central \(Au+Au\) and of charged pions in central \(Pb+Pb\) collisions at 200 GeV and 2.76 TeV and compare HYDJET++ results with the experimental data. We have observed a suitable match between data and the model results. Figure 21 demonstrates the variation of elliptic flow with respect to transverse momentum (\(p_{T}\)) for various centrality class in body–body configuration of \(U+U\) collisions. We have shown these results for charged hadrons with \(|\eta |< 0.5\). From this plot, one can observe that the elliptic flow increases with \(p_{T}\) upto \(p_{T}\approx 3\) GeV and then starts to decrease with further increase in \(p_{T}\) for each centrality class. Further, it is clearly shown that for any given \(p_{T}\) upto 3 GeV, the elliptic flow increases as the collision becomes more and more peripheral. It is quite obvious since the initial geometrical anisotropy is very small for central collisions which actually reflects in low \(v_{2}\) value for central collision. At higher \(p_{T}\) the elliptic flow in each centrality class overlaps on each other. Further we have shown the elliptic flow of charged hadrons in central \(Au+Au\) collision [61] for comparison. We observed that \(v_{2}\) in most central \(Au+Au\) collision is less than \(v_{2}\) in most central body–body \(U+U\) collisions over the entire \(p_{T}\) range considered here. Similarly, Fig. 22 presents the variation of \(v_{2}\) with \(p_{T}\) in different centrality class for tip–tip configurations. The qualitative behaviour of elliptic flow is quite similar to the body–body configuration. However when we see the comparison of \(v_{2}\) in central \(Au+Au\) data with the \(v_{2}\) in tip–tip \(U+U\) most central collision then one can see that \(v_{2}\) of charged hadrons in \(Au+Au\) collision is less than \(v_{2}\) of charged hadrons in tip–tip configuration of \(U+U\) most central collision.

In Fig. 25, we have shown the effect of centrality on mean elliptic flow for body–body and compared them with the corresponding results of tip–tip configurations. We have integrated over \(p_{T}\) from 0.001 to 5 GeV. From this result one can see that \(\langle v_{2}\rangle \) increases in going from central to peripheral which is actually due to a increase in eccentricity going from central to peripheral collisions. However from here it is clear that in central collisions the difference in magnitude between body–body and tip–tip collisions is small. However, in semi-peripheral as well as in peripheral collisions, one can distinguish between body–body and tip–tip events by observing the \(\langle v_{2}\rangle \) magnitude of charged hadrons. We have also shown the results obtained in Ref. [38] using AMPT model in two different modes (string melting mode and default mode). We found that the qualitative behaviour of variation of \(\langle v_{2}\rangle \) with centrality in HYDJET++ is quite opposite to AMPT model and shows a small difference in \(v_{2}\) for tip–tip and body–body configuration in central events and a large difference in peripheral events. On the other side AMPT has shown opposite behaviour. We have also plotted the STAR experimental data [14] of \(v_{2}\) as a function of centrality for minimum bias events. We found that the experiment results are nearly in between the HYDJET++ model results for tip–tip and body–body configurations. However, the experimental data is between the AMPT-Default mode results for tip–tip and body–body configurations in central and mid-central events but not in peripheral events. In AMPT-SM mode, the experimental data is very close to tip–tip configuration results.

In summary, we have calculated and shown the pseudorapidity density and transverse momentum distributions of charged hadrons produced in \(U+U\) collisions at \(\sqrt{s_{NN}}=193\) GeV in various initial geometrical configurations. In present study, it has been shown that the correlation between multiplicity and initial geometrical configurations of \(U+U\) collisions is small which is in accordance with the recent experimental observation. However, the experimental results are quite preliminary due to complexity in disentangling the tip–tip and body–body events. We have shown the midrapidity charged-particle multiplicity distribution from HYDJET++ model which is in good agreement with the experimental results for minimum bias events. Further, we have shown the evolution of elliptic flow with \(p_{T}\) and centrality in different configurations of \(U+U\) collisions. It has been observed that elliptic flow generated in body–body collisions is larger than tip–tip collisions but the difference in magnitude of \(v_{2}\) is small in central collisions and large in peripheral events. Further, we found that our tip–tip results of elliptic flow matches with STAR experiment result of 0–0.5% centrality class when \(p_{T}<1\). At last, we have observed that the experimental results of \(v_{2}\) as a function of centrality for minimum bias events are nearly in between the tip–tip and body–body configuration results of our model. However, this is not the case for AMPT results. Finally we may conclude that our present study will shed some light on the particle production mechanism and the evolution of the fireball created in various geometrical configurations of \(U+U\) collisions specially the entanglement of hard (jet) and soft (hydro) part in body–body and tip–tip configurations.

## Notes

### Acknowledgements

Authors gratefully acknowledge personal communications and the useful comments/suggestions on the present manuscript by Prof. I.P. Lokhtin and his group. PKS is grateful to IIT Ropar, India for providing an institute postdoctoral research grant. PKS is also thankful for the pleasant stay at Department of Physics, BHU where work was done. OSKC would like to thank Council of Scientific and Industrial Research (CSIR), New Delhi for providing a research fellowship. AS acknowledges the financial support obtained from UGC under research fellowship scheme in central universities. This research work is supported partially by DST FIST, DST PURSE and UGC-CAS programmes.

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