Abstract

Elliptic flow of hadrons observed at relativistic heavy ion collision experiments at relativistic heavy ion collider (RHIC) and large hadron collider (LHC) provides us an important signature of possible deconfinement transition from the hadronic phase to partonic phase. However, hadronization processes of deconfined partons back into final hadrons are found to play a vital role in the observed hadronic flow. In the present work, we use a coalescence mechanism also known as recombination (ReCo) to combine quarks into hadrons. To get there, we have used the Boltzmann transport equation in relaxation time approximation to transport the quarks into equilibration and finally to freeze-out the surface, before coalescence takes place. A Boltzmann-Gibbs blast wave (BGBW) function is taken as an equilibrium function to get the final distribution and a power-like function to describe the initial distributions of partons produced in heavy ion collisions. In the present work, we try to estimate the elliptic flow of identified hadrons such as , , and , produced in Pb+Pb collisions at at the LHC for different centralities. The elliptic flow () of identified hadrons seems to be described quite well in the available range. After the evolution of quarks until freeze-out time has been calculated using BTE-RTA, the approach used in this paper consists of combining two or more quarks to explain the produced hadrons at intermediate momenta regions. The formalism is found to describe the elliptic flow of hadrons produced in Pb+Pb collisions to a large extent.

1. Introduction

The study of collective phenomena, among many contemporary signatures of quark-gluon plasma (QGP), continues to remain in the forefront of scientific investigations [1, 2]. While on the experimental front [3–9] analysis of final hadrons’ data from RHIC and LHC experiments has enabled us to look back in time and reconstruct the flow phenomena, the phenomenological models using theoretical and numerical techniques have been able to simulate the events, starting from the point of collision of heavy ions to the freeze-out. The theoretical results have been successful in explaining experimental data to a large extent. With the advent of new techniques, the time is however ripe to be able to resolve differences in theories and experiments and precisely determine various observables of QGP.

Earlier, attempts were made through extensive theoretical modelling and analysis of data, to reconstruct azimuthal anisotropy, (also known as elliptic flow) , of hadrons in a transverse momentum plane [10–13]. However, it is believed that azimuthal anisotropy would develop at the early phase of heavy ion collision, when bulk of the deconfined quarks and gluons from noncentral collision between two heavy ions goes into local thermalization or the quark-gluon plasma (QGP) state. The geometrical asymmetry of the spatial overlap zone is transformed into momentum anisotropy of the produced particles. With the onset of the local themalization of the bulk partonic matter, the azimuthal anisotropy (mathematically, the second coefficient of the Fourier expansion of particle transverse momentum spectrum), is exhibited strongly in the collective behaviour of the quark-gluon plasma. This information on initial anisotropy is carried till freeze-out and finally reflected in the hadron spectra. However, the rapid expansion of the medium towards isotropization may smear this information to some extent. But on the other hand, hadronic medium effects may add to the partonic flow until kinetic freeze-out sets in. Thus, it is necessary to develop robust calculation to be able to discern factors emanating from various phases of heavy ion collision, which may affect the particles’ flow. While phenomenologies of hadronic matter try to reconstruct the from the final hadronic spectra, initial anisotropy in the partons’ configuration space on the other hand affects the formation of flow and is calculated using phenomenological models such as the Glauber mechanism along with perturbative QCD-based calculations. However, the two phases of initial anisotropy and hadrons’ interaction remain separated by the QGP phase. As mentioned earlier, the QGP phase contributes to the evolution of the particle flow to a great extent. Hence, it is up to the transport models which may properly bring in the QGP effects and bridge the initial anisotropy and effects of the hadronic phase in the observed [14]. The transport models help us in studying collision centrality dependency of QGP properties. They not only shed light on properties of hot and dense matter viz. average temperature and momentum reached by equilibrated system and their dependency on collision centrality but also provide us with vast information on transport properties such as radial flow coefficients, momentum broadening, drag and diffusion coefficients, and electrical and thermal conductivity of QGP matter [15–19].

The available transport calculations are based on either the hydrodynamical equation, Langevin equation, or Boltzmann transport equation. In this paper, we have used the Boltzmann transport equation (BTE) in relaxation time approximation (RTA). BTE-RTA would transport the entire parton distribution to equilibration and then to freeze-out surface whereafter kinetic or chemical interaction among particles ceases completely. Using the BTE-RTA approach in the present work, we have attempted to study parameters associated with particle production such as the radial flow and relaxation time of the final-state particles. Neglecting the effects of the hadron medium on the particles’ flow as approximation, the current work focuses on the interplay of the various parameters and mechanisms on the partonic states. We will discuss our approach in detail in subsequent sections. We have also assumed that the final quarks would hadronize into mesons and baryons using the partonic coalescence mechanism at the hadronization hypersurface. We will discuss this formalism in one of the following sections. Finally, the elliptic flow, , for various hadrons is calculated and presented in the results and discussion section. We have also presented figures on our study of the parameters and their interdependencies. We have then a concluding section in our paper, which is followed by bibliography.

Let us now discuss BTE-RTA formalism briefly.

2. Relaxation Time Approximation (RTA) of Boltzmann Transport Equation (BTE)

As mentioned in the introductory section, the evolution of quarks within the medium towards the freeze-out surface has major effects on the observed final particle spectra. The transport calculations such as hydrodynamics and BTE are commonly used as the evolution mechanisms and provide description of hadron spectra in both qualitative manner and quantitative manner [20–26]. We know that various dynamical features ranging from multiparton interaction, in-medium energy loss, thermal and chemical equilibrations, to dynamics at freeze-out surfaces contribute extensively to the particle flow and can be studied using BTE. We also know that partons evolving through space and time undergo several collisions and thermalize. Furthermore, they continue to evolve and expand until freeze-out even after hadronization. Any of these features can be studied using BTE. The BTE in general can be written as where is the distribution of particles which depends on position, momentum, and time. is the velocity, and is the external force. and are the partial derivatives with respect to position and momentum, respectively. is the collision term which depicts the interaction of the particles with the medium or among themselves. Earlier, BTE has also been used in RTA to study the time evolution of temperature fluctuation in a nonequilibrated system [27] and also for studying the and of various light and heavy flavours at RHIC and LHC energies [28, 29].

We have considered the evolution of particle momentum distribution with time. We have taken assuming particle distribution to be homogeneous in space and the configuration space distribution or spatial distribution has been parametrized accordingly. There are no external forces acting on the system 0. Hence, the second and third terms of equation (1) become zero and it reduces to

The full kernel of the collision term contains microscopic interaction cross-sections of particles. For any transport models such as AMPT and UrQMD containing microscopic Boltzmann equation, the full interaction kernel along with space and time evolution of the system becomes important. In our calculation owing to assumed homogeneous spatial distribution, the spatial variables have been parametrized.

In BTE-RTA [30], which is an effective model, the collision term is however expressed as where is Boltzmann local equilibrium distribution characterized by a freeze-out temperature . is the relaxation time, the time taken by a nonequilibrium system to reach equilibrium. Using equation (3), equation (2) becomes

Solving the above equation with the initial conditions, i.e., at and at , in general, we get final distribution for any quark flavour as where is the freeze-out time parameter. The initial distribution at is taken as power-like distribution. We will come back to this later. BTE-RTA computes to give the final distribution as a function of parameter . If the system is given enough time or is large compared to , might converge to .

We use equation (5) in the definition of the elliptic flow () at midrapidity, which is expressed as

Equation (6) gives azimuthal anisotropy after incorporating RTA in BTE. The Boltzmann-Gibbs blast wave (BGBW) function has been taken as the equilibrium distribution function, , as where the particle four-momentum is and the four-velocity denoting flow velocities in space time is given by , while the kinetic freeze-out surface is given by . Here, is the space-time rapidity. Assuming boost-invariant scenario where we have taken Bjorken correlation in rapidity, i.e., [31] along the longitudinal or beam axis. Thus, equation (7) can be expressed as where . Here, is the quark degeneracy factor, is the particle emission time, and is the transverse mass.

in the integrand is a transverse rapidity variable which is given by , with as a function of impact parameter, , and gives the anisotropy dependence in the flow. [32–35] is the radial flow, where is the maximum surface velocity and , with as the radial distance from the center of the fireball. In the blast wave model, the particles closer to the center of the fireball move slower than the ones at the edges. The average of the transverse velocity can be evaluated as [36]

While the anisotropic parameter, is written as

In our calculation, we use a linear velocity profile, (), is the maximum radius of the expanding source at freeze-out , and is the radius of colliding nucleus. is the impact parameter to include the centrality dependence of anisotropy. In this paper, we have parametrized the initial distribution given by particle production using perturbative QCD leading order (pQCD LO) calculations for collision,

Here, and are momentum fractions carried by interacting partons from their respective colliding protons and are given by

A cut of 2 GeV/c is taken for the jet production following other event generators like PYTHIA and HIJING [37, 38]. The parton density functions, , are taken to be CTEQ5M [39]. The partonic differential scattering cross-sections, , is calculated from the LO processes, and . To incorporate NLO processes, we have taken a factor, , of value 2.5, and finally, nuclear overlap function, and EKS98 parametrization for shadowing effects are taken into account to convert particle production cross-section from collision into , particle spectra. Equation (11) is parametrized using a function with a power-like structure (Juttner distr.), and we fixed the parameters of the preequilibrated partons shown in Table 1. However, it is worthwhile to mention that other types of functions can be utilized to obtain the initial quark distributions.

Here too, we have assumed Bjorken correlation in rapidity, i.e., . Using the Glauber model, is calculated to be for 0-5% centrality and for 50-60% centrality of the colliding nuclei at . Using equations (8) and (13), the final distribution can be expressed as in equation (5). This gives the final distribution for quarks. The quark masses for the initial distributions are taken to be , , , and . After the transport quark coalescence, formalism has been used to combine the quarks into hadrons. This will be discussed next.

3. Quark Coalescence

The quark coalescence model (ReCo) is used to recombine quarks into hadrons and is found to be one of the prominent hadronization mechanisms beside parton fragmentation [40, 41]. In Refs. [42, 43], the authors have used a two-component behaviour of hadronic spectra. For low- (), they have used the recombination mechanism for thermalized partons, while for a power law-like distribution with fragmentation has been used. In our work, we have concentrated our transport approach to low and intermediate (5 GeV/c), where instead of adopting thermalized distribution directly, we have allowed jet distribution of partons to relax or thermalize and then proceed to hadronization. The idea was to use the BTE-RTA equation to study the applicability of interpolation of jet distribution with the blast wave equation at the intermediate region. The coalescence or recombination of partons into hadrons has been able to explain experimentally observed hadron spectra in the intermediate and perhaps at the low-momentum regions, while parton fragmentation processes are aptly suitable in explaining hadrons with high momenta. And thus, the ReCo mechanism has been used for final transported quark distributions in the present work. The ReCo mechanism also highlights the major contribution of partonic degrees of freedom in the observed hadron flow. The process such as has been neglected as gluon contributions are mostly at low , while high gluons contribute to hadrons via a fragmentation mechanism. In the intermediate-momentum region, constituent quark counting becomes important for recombination process. At the hadronization surface, only constituent quarks behave as effective degrees of freedom with mass. However, it must be noted that for net entropy and energy density calculation, gluon contribution is most vital [42–46]. In the present work, we have not included gluon contribution to the hadron production which is one of the main differences from the earlier works and it is most visible at the low momenta where gluon contribution is important. Unlike earlier works, we have not included the fragmentation mechanism for high particles and only focused our observations to the intermediate . Another difference in the current work is the absence of the hadron decay mechanism which is important at a low-momentum region. However, these are out of the scope of the current work and could constitute a better prospective for future research.

The coalescence model can be applied to the quarks at the hadronization surface when two (three) quarks recombine to form mesons (baryons) [47–49]. The model can be further utilized in describing observed spectra of light nuclei such as deuteron which contains a neutron and a proton [50].

The coalescence model combines two or more quark distributions using convoluting functions also known as Wigner functions. The basic equation showing the number of mesons from two combining quarks can be broadly written as where and are the spatial and transverse momentum coordinates of the combining quarks and antiquarks and are the quark distribution functions. is the Wigner function convoluting two partonic distributions. in the front of equation (14) is meson degeneracy factor. We have also assumed Bjorken correlation in rapidities, and , throughout. We also assumed . This ensures a close phase space for quarks in both momentum and configuration spaces.

We have assumed the delta functions correlation, , and . We have defined the partons in the spatial and momentum coordinates in the C.M. frame of meson, such as so that we can derive

Here, we have also assumed that is small compared to and thus neglected in the quark distributions, . Thus, we have

We have now, where meson transverse mass factor is given by, .

As for the Wigner function, , we can use the following relation,

Therefore, equation (18) is transformed as

To simplify our equations, we convert our momentum variable into light-cone coordinates, of the interacting quarks in the momentum space of the hadron as follows: so that, and

We also assume that the partons recombining into hadrons have their momenta almost parallel to the final hadron. So can be considered to be very small compared to and its dependency in the quark distribution, has also been neglected. It can be shown following equation (22) that the parton momentum, , where is the momentum fraction of the final hadron’s momentum, carried by its constituent quarks during recombination [51, 52]. Putting the above conditions into the equation, and assuming the normalization,

Finally, we can write

We have along the unit normal direction, at the freeze-out hypersurface. Similarly, for the baryons, one can derive to show

To illustrate on our calculations, one may use equation (8) as an example and show that

Thus, one may calculate to show

Similarly for the baryons, one may write

We have replaced transverse mass, , by expressions from equations (26) and (28), throughout our calculations. We have also assumed a general Gaussian distribution as Wigner functions, (for mesons), (for baryons), which are given by

Here, is the width of the Gaussian function, and its small values would give us the narrow Wigner function closer to being a delta function or on other hand, its larger values would give us a broad convoluting function instead. The values can be chosen according to the best fit with the particle spectra. We will resume its discussion in the results section.

Thus, using equation (4), in equations (24) and (25), we calculate of the final hadrons at midrapidity as

4. Results and Discussions

We would like to reiterate that in the current work, using BTE in RTA, we have transported quarks of various flavors ( and ) produced promptly (assumed to be out of equilibrium) from initial gluon fusion to thermalization and freeze-out time and recombined them into hadrons using the coalescence mechanism. We have neglected the decay contributions to final hadron spectra as well as effects due to hadronic interactions. We have tried to extract the correlation between parameters such as the radial flow, , and the ratio of freeze-out time and relaxation/thermalization time, , We have presented the results on the elliptic flow () of various identified hadrons like pions, kaons, protons, D meson, and lambda for different centralities of Pb+Pb collision at . While analysing the data, we have kept the freeze-out temperature () for the hadrons at 0.095 GeV for the most central collisions (0-5)% and 0.11 GeV for most peripheral collisions (50-60)% [53]. We assume that the value of is smaller for the central collisions in comparison to the peripheral collisions. The above assumption on the freeze-out temperature is based on the fact that the freeze-out in peripheral collisions occurs quicker than in the most central collisions [53]. The dependence of identified hadrons’ is studied by varying two parameters, and , using equation (29). Based on the closest explanation of the data, we have kept the width of the Wigner function fixed at 0.0009 for mesons and 0.04 for baryons in our calculations. As discussed earlier that the constituent quarks for recombination process occupy a close phase space, we needed a narrow Gaussian function and not a delta function so as to avoid the collinear divergences as well as to satisfy the above condition. We have not included the flow from hadronic medium as they are most visible for particles below [54, 55] where our results focus on .

In Figure 1, we have shown spectra of various charged hadrons in the most central collisions of Pb+Pb at . The coalescence method is employed to form hadrons from quarks at the freeze-out surface. The resulting transverse momentum distributions are then drawn and compared with the experimental data from ALICE@CERN [57–59]. It is found that the discussed model in the above section explains the experimental data in the moderate region.

Figure 1 

The transverse momentum spectra of (), (), (), and mesons versus for most central Pb+Pb collisions at . Symbols are experimental data points [56, 57], and lines are the model results from equations (24) and (25).

In Figure 2, we have shown of (). The left plot shows the variation of with for different surface velocity parameters, , while the right plot shows for different . Three different values of keeping fixed are taken and vice versa. Generally speaking, our theoretical results match with the experimental data within errors for the discussed particles, from the mid- region to the max shown. However, the model fails to explain the data for . The reason may be due to the absence of pions from decays of resonances [60]. Pions also stand out as an example that shows coalescence picture should work mostly in the mid- region.

Figure 2 

The elliptic flow () of () versus at constant and for peripheral collisions (50-60)% at . Symbols are experimental data points [58], and lines are the model results.

In Figure 3, the elliptic flow of pions () is presented as a function of for various centralities at for Pb+Pb collisions. Symbols are the experimental data, and lines are model results. Here, freeze-out temperatures () are taken smaller for most central collision than to peripheral collisions. The model results are found to explain the data qualitatively beyond for all the centralities within the error bar. However, the quark coalescence mechanism is not able to explain the data below . Experimentally, it is found that for (50-60)% appears to be inverse in order compared to (40-50)% due to statistical fluctuations. However, the model follows the expected trend of higher for higher centralities.

Figure 3 

The elliptic flow () of () versus for various centralities for Pb+Pb collisions at . Symbols are experimental data points [58], and lines are the model results.

In the left panel of Figure 4, we have shown the elliptic flow or azimuthal anisotropy and spatial anisotropy of the pions versus is generally defined in terms of spatial coordinates of participants’ nucleons in the transverse plane. It can be written as

(a)

(b)

(a)
(b)

Figure 4 

(a) The elliptic flow () and spatial anisotropy () of () versus for Pb+Pb collisions at . (b) The ratio of and of pions versus .

In this paper, Glauber-MC formalism [61] has been employed to calculate . Both and decrease with , which is expected. In the right panel of Figure 4, we show the ratio of and vs. or centrality. We find that the ratio tends to increase towards central collisions but drops suddenly for most central. This ratio approximately shows the strength of anisotropy developed as we move towards central collisions and may indicate the extent of collectivity undertaken by the bulk of the partons within quark gluon plasma. However, the sudden drop in this ratio at the most central will be investigated further in our future reports.

In Figure 5, we have presented the variations of of Kaons, () with for 50-60% centrality. The left panel is versus for different at constant , while the right panel shows versus for different at constant . Three different values of keeping fixed are taken and vice versa. The theoretical curves tend to overestimate the data although it gives a consistent explanation as to the nature of the shape of Kaons shown by the experimental data. Also, the plot on the left side shows that the theoretical lines cross each other for the different values of , which shows greater sensitivity of on the surface velocity of the fireball. The theoretical line is quite close to experimental points at low which shows that large mass should have less contribution from resonance decays.

Figure 5 

The elliptic flow () of () versus at constant and for peripheral collisions (50-60)% at . Symbols are experimental data points [58], and lines are the model results.

In Figure 6, we have shown of K-short (). The left plot shows the variation of with taking various values of . The right plot of the figure represents the variation of with taking different values of . Three different values of keeping fixed are taken and vice versa. is a little heavier than Kaons, which is why the and values are almost similar in both the cases. Similarly, the theoretical curve tends to overestimate the data up to . However, the theoretical curve shows a gradually increasing trend and slopes down smoothly at high .