# Event Shape and Multiplicity Dependence of Freeze-Out Scenario and System Thermodynamics in Proton+Proton Collisions ats=13TeV Using PYTHIA8

Authors: Sushanta Tripathy; Ashish Bisht; Raghunath Sahoo; Arvind Khuntia; Malavika Panikkassery Salvan
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

Recent observations of QGP-like conditions in high-multiplicity pp collisions from ALICE experiment at the LHC warrant an introspection whether to use pp collisions as a baseline measurement to characterize heavy-ion collisions for the possible formation of a Quark-Gluon Plasma. A double differential study of the particle spectra and thermodynamics of the produced system as a function of charged-particle multiplicity and transverse spherocity in pp collisions would shed light on the underlying event dynamics. Transverse spherocity, one of the event shape observables, allows to separate the events in terms of jetty and isotropic events. We analyse the identified particle transverse momentum (pT) spectra as a function of charged-particle multiplicity and transverse spherocity using Tsallis nonextensive statistics and Boltzmann-Gibbs Blast-Wave (BGBW) model in pp collisions at s=13TeV using PYTHIA8 event generator. The extracted parameters such as temperature (T), radial flow (β), and nonextensive parameter (q) are shown as a function of charged-particle multiplicity for different spherocity classes. We observe that the isotropic events approach thermal equilibrium while the jetty ones remain far from equilibrium. We argue that, while studying the QGP-like conditions in small systems, one should separate the isotropic events from the spherocity-integrated events, as the production dynamics are different.


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## Body

## 1. Introduction

Although it was envisaged long back that central heavy-ion collisions at ultrarelativistic energies could produce a deconfined state of partons called Quark-Gluon Plasma (QGP) [1, 2], the unprecedented collision energies available at the Large Hadron Collider (LHC) at CERN, Switzerland, has brought up new challenges in characterizing the proton+proton (pp) collisions to understand a possible formation of QGP droplets in these hadronic collisions. There are various signatures of QGP, which are already observed in pp collisions at the LHC. These include strangeness enhancement [3], hardening of pT-spectra [4, 5], and the thermal effective temperature being comparable to that observed in heavy-ion collisions [3], degree of collectivity [6], etc. In view of these observations in pp collisions, it has become more challenging to understand the system formed in pp collisions, although pp has been considered as a baseline measurement to understand nuclear effects like RAA and suppression of J/ψ. The new measurements at the LHC keeping in mind that the final state multiplicity drives the particle production (excellent scaling observed) necessiate a closer look into the underlying physics mechanisms of particle production in pp collisions. The phenomena like color reconnection, multipartonic interactions, rope hadronization, and string fragmentation have done a wonderful job in explaining various new heavy-ion-like observations in pp collisions.Event shape engineering has given a new direction to underlying events in pp collisions to have a differential study taking various observables. The transverse spherocity successfully separates jetty events from isotropic ones in pp collisions. As is clearly understandable, the particle production mechanism in jetty events is different from isotropic ones. When the former one involves high-pT phenomena, the latter is soft-physics dominated. In view of this, recently, we have carried out a double differential study of particle ratios in pp collisions at the LHC energies, taking transverse spherocity, transverse momentum, and multiplicity [7]. A natural question which pops up is whether the thermodynamics of jetty events are different from the isotropic ones. To quantify this, in the present study, we have taken pQCD-inspired PYTHIA8 event generator which includes multiparton interactions (MPI) along with color reconnection (CR), to study the event shape and multiplicity dependence of freeze-out scenario and system thermodynamics in pp collisions at s=13TeV. It has been reported that the MPI scenario is crucial to explain the underlying events, multiplicity distributions, and flow-like patterns in terms of color reconnection [8]. Thus, it is a preferable tune to study the possible thermodynamics in small systems, as experimental data are not available yet. It should be worth noting that PYTHIA8 does not have inbuilt thermalization. However, as reported in Ref. [8], the color reconnection (CR) mechanism along with the multipartonic interactions (MPI) in PYTHIA8 produces the properties which arise from the thermalization of a system such as radial flow and mass dependent rise of mean transverse momentum. In the PYTHIA model, a single string connecting two partons follows the movement of the partonic endpoints, and this movement gives a common boost to the string fragments (final state hadrons). With CR along with MPI, two partons from independent hard scatterings can reconnect and they increase the transverse boost. This microscopic treatment of final state particle production is quite successful in explaining the similar features which arise from a macroscopic picture via hydrodynamical description of high-energy collisions. Thus, it is apparent to say that the PYTHIA8 model with MPI and CR has a plausible ability to produce the features of thermalization. The current results, which use PYTHIA8 with MPI and CR to obtain event shape and multiplicity dependence of freeze-out scenario and system thermodynamics, will help to compare with the upcoming experimental data. Such a study has also been done for heavy flavor particles like J/ψ in Ref. [9]. This paper is intended solely for presenting a noble and unique study, which would give an outlook on similarities/differences between jetty and isotropic events in LHC pp events and their multiplicity dependence. This will help in making a proper bridge in understanding the particle production from hadronic to heavy-ion collisions.Furthermore, the spacetime evolution of hadronic and heavy-ion collisions at the LHC energies could be thought of following a cosmological expansion of the produced fireball. In this scenario, as the fireball expands and cools down, it leaves a temperature profile with time. Different identified particles decouple from the fireball giving the signature of a mass-dependent particle freeze-out—higher mass particles decoupling from the system earlier in time. In this work, we have considered such a scenario and have performed a differential study taking final state event multiplicity and event topology.The paper is organized as follows. After the introduction and identification of the problem under consideration, we discuss the methodology of event generation and data analysis in Section2. In Section 3, we discuss the identified pT-spectra in pp collisions to extract the thermodynamic parameters. Finally, we summarize the work in Section 4 with important findings, which could be tested when experimental data become available.


## 2. Event Generation and Analysis Methodology

PYTHIA, one of the popular and most useful event generators in the LHC era, is used to simulate ultrarelativistic collision events among the elementary particles likee±, p, and p¯. It is incorporated with many known physics mechanisms like hard and soft interactions, parton distributions, initial- and final-state parton showers, multipartonic interactions, string fragmentation, color reconnection, and resonance decays [10].In our present study, we have used PYTHIA 8.235 to generate pp collisions ats=13TeV with Monash 2013 Tune (Tune:14) [11]. PYTHIA 8.235 is an advanced version of PYTHIA 6 which includes the multipartonic interaction (MPI) scenario as one of the key improvements. The detailed physics processes in PYTHIA 8.235 can be found in Ref. [12]. We have implemented the inelastic, nondiffractive component of the total cross-section for all soft QCD processes with the switch SoftQCD: all = on. This analysis is carried out with around 250 million minimum bias events at s=13TeV, and we have chosen MPI-based scheme of default color reconnection mode (ColorReconnection:mode(0)). Here, the minimum bias events are those events where no selection on charged-particle multiplicity and/or spherocity is applied. For the generated events, we let all the resonances to decay except the ones used in our study with the switch HadronLevel:Decay = on. Throughout the analysis, the event selection criteria are such that only those events were chosen which have at least 5 charged particles. To match with experimental conditions, charged particle multiplicities (Nch) have been chosen in the acceptance of V0 detector in ALICE at the LHC with pseudorapidity coverage of V0A (2.8<η<5.1) and V0C (−3.7<η<−1.7) [13]. The generated events are categorized in ten V0 multiplicity (V0M) bins, each with 10% of the total number of events. The number of charged particle multiplicities in an event in different V0 multiplicity classes is listed in Table 1.Table 1
V0 multiplicity classes and the corresponding charged particle multiplicities.
V0M classIIIIIIIVVVIVIIVIIIIXXNch50-14042-4936-4131-3527-3023-2619-2215-1810-140-9For an event, transverse spherocity is defined for a unit vectorn^nT,0 which minimizes the ratio [14–16]:(1)S0=π24Σip→Ti×n^ΣipTi2.By restricting it to the transverse plane, transverse spherocity becomes infrared and collinear safe [17], and by construction, the extreme limits of transverse spherocity are related to specific configurations of events in the transverse plane. The value of transverse spherocity ranges from 0 to 1. Transverse spherocity becoming 0 means that the events are pencil-like (back-to-back structure) while 1 would mean the events are isotropic as shown in Figure 1. The pencil-like events are usually the hard events, while the isotropic events are the result of soft processes. Here onwards, for the sake of simplicity, the transverse spherocity is referred to as spherocity. To disentangle the jetty and isotropic events from the average-shaped events, we have applied spherocity cuts on our generated events. In this analysis, the spherocity distributions are selected in the pseudorapidity range of η<0.8 with a minimum constraint of 5 charged particles with pT>0.15 GeV/c. The jetty events are those having 0≤S0<0.29 with lowest 20 percent (≃50 M events), and the isotropic events are those having 0.64<S0≤1 with highest 20 percent (≃50 M events) of the total events [18].Figure 1
(Color online) Figure showing jetty and isotropic events in the transverse plane.To assure the quality of the generated events, we show in Figure2, the correlation between the spherocity with charged-particle multiplicity. As expected, the high multiplicity pp collisions are dominated by isotropic events, while the low multiplicity events are dominated by the jetty ones. From our earlier event shape analysis [7], it is evident that spherocity along with the charged particle multiplicity (which is correlated with nMPI) should be preferred for a better selectivity of events.Figure 2
(Color online) Spherocity distribution for different multiplicity classes in pp collisions ats=13TeV using PYTHIA8. Different colors and line styles are for different multiplicity classes.


## 3. Transverse Momentum Spectra of Identified Particles

To check the compatibility of PYTHIA8 simulated data with the experimental data, we have compared the charged particlepT spectra for pp collisions at s=13TeV from ALICE data [19]. The comparison is shown in Figure 3. The lower panel shows the ratio of the predictions from PYTHIA8 to experimental data. In order to see the agreement of spectral shapes, we have used an arbitrary scaling factor (1.35) to scale the simulated data. The used scaling factor is to check the matching of the spectral shape, and it bears no physical significance. We found that the scaled simulated data agree with the spectral shape from experimental data within (10-20)% at low-pT and consistent to unity for intermediate and high-pT.Figure 3
(Color online) Upper panel: comparison of charged particlepT spectra in pp collisions at s=13TeV between ALICE data [19] and PYTHIA8 simulation, which is used for this analysis. Lower panel: the ratio between scaled simulated data and experimental data.For the first time, we combine spherocity with event multiplicity and study the freeze-out scenario and thermodynamics of the system formed in pp collisions ats=13TeV. We use experimentally motivated thermodynamically consistent Tsallis nonextensive distribution function [20] for analysing the complete range of the pT-spectra, whereas to extract the kinetic freeze-out temperature and the possible collective radial flow we use the Boltzmann-Gibbs Blast-Wave model [21, 22] taking pT≤2 GeV/c. We begin with the fitting and analysis procedure with a short description on Tsallis nonextensive statistics and Boltzmann-Gibbs Blast-Wave model. Here onwards, π++π−,K++K−,p+p¯,K∗0+K∗0¯,andΛ0+Λ0¯ are denoted as pion (π), kaon (K), proton (p), K∗0, and Λ, respectively.


### 3.1. Experimentally Motivated Tsallis Nonextensive Statistics

ThepT-spectra of produced particles in high-energy collisions have been proposed to follow a thermalised Boltzmann type of distribution given as [23],(2)Ed3σd3p≃Cexp−pTTkin.Here,C is the normalisation constant, and Tkin is the kinetic freeze-out temperature. Due to possible QCD contributions at high-pT, the identified particle spectra at RHIC and LHC do not follow the above distribution, while the low-pT-region can be explained by incorporating the radial flow (β) into the Boltzmann-Gibbs distribution function, which is known as Boltzmann-Gibbs Blast-Wave (BGBW) model [22]. One can extract Tkin and radial flow (β) by fitting the identified particle transverse momentum spectra at low-pT. The detailed description along with the fitting of the Boltzmann-Gibbs Blast-Wave model to the identified particle spectra is discussed in the next subsection.To describe the completepT-spectra, one has to account for the power-law contribution at high-pT [24–26], which empirically takes care of the possible QCD contributions. A combination of both low and high-pT aspects has been proposed by Hagedorn, which describes the experimental data over a wide pT-range [27]. The distribution proposed by Hagedorn is given by(3)Ed3σd3p=C1+pTp0−n⟶exp−npTp0forpT⟶0,p0pTnforpT⟶∞.Here,C, p0, and n are fitting parameters. The above expression acts as an exponential and a power-law function for low and high-pT, respectively. However, deviations are observed by experiments at RHIC [28, 29] and LHC [30–33] while describing the pT-spectra of identified particles using a Boltzmann-Gibbs distribution function, even if the domain of temperature of the produced systems are high enough. On the other hand, Tsallis statistics with its nonextensivity features can be regarded as a generalization of Boltzmann-Gibbs statistics, and it gives a better description of systems, which have not yet reached equilibration. Its low-pT exponential and high-pT power-law behavior gives a complete spectral description of identified secondaries produced in pp collisions. In addition, a nonextensive entropic q-parameter shows the extent of nonequilibration of any particle in a thermal bath. There are few different versions of Tsallis distribution, which are being used by experimentalists and theoreticians. However, we use a thermodynamically consistent Tsallis nonextensive distribution function as shown in Ref. [20]. By saying thermodynamically consistent, we mean that the used distribution function satisfies all the standard thermodynamic relations for entropy, temperature, energy, pressure, and number density. The Tsallis distribution function at midrapidity is given by,(4)1pTd2NdpTdyy=0=gVmT2π21+q−1mTT−q/q−1,where g is the degeneracy factor, V is the system volume, mT=pT2+m2 is the transverse mass, and q is the nonextensive parameter. In the limit of q⟶1, Tsallis distribution (Eq. (4)) reduces to the standard Boltzmann-Gibbs distribution (Eq. (2)). It should be noted here that the use of the Tsallis nonextensive distribution function is purely motivated by its excellent description of experimental particle spectra. It has been widely used to explain the particle spectra in high-energy collisions [34–38] starting from elementary e++e− and hadronic to heavy-ion collisions [28, 29, 39–52]. Recently, few comprehensive studies have been carried out using Tsallis distribution for pions and quarkonium spectra in pp collisions [53, 54]. In this subsection, we employ the Tsallis nonextensive distribution function as shown in Eq. (4) to describe the pT-spectra. It should be noted here that the used Tsallis distribution function of Eq. (4) [55] has an extra power q, compared to the original distribution function proposed by Tsallis [56]. However, this form of the distribution function is thermodynamically consistent, which makes no major change in the observables as the values of q in hadronic collisions lie between, 1≤q≤1.22 [57]. While Tsallis nonextensive statistics makes a connection between entropy and thermodynamics of a system, the dynamics of the system in terms of long-range correlations and fluctuations (q−1 being the strength of the fluctuation [58]) are encoded in the entropic parameter, q.Figures4–6 show the fitting of pT-spectra of pions, kaons, protons, K∗0, ϕ, and Λ as a function of charged-particle multiplicity using Tsallis distribution function (Eq. (4)) for spherocity-integrated, isotropic, and jetty events, respectively. We observe that the Tsallis distribution fits the generated data till pT≃10 GeV/c. Figure 7 shows the quality of fitting in terms of the reduced-χ2, χ2/NDF as a function of multiplicity for different spherocity classes. The values of χ2/NDF show that the quality of fitting is reasonably good for all the multiplicity and spherocity classes.Figure 4
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using Tsallis distribution for spherocity integrated events in various multiplicity classes as shown in Table 1.Figure 5
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using Tsallis distribution for isotropic events in various multiplicity classes as shown in Table 1.Figure 6
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using Tsallis distribution for jetty events in various multiplicity classes as shown in Table 1.Figure 7
(Color online)χ2/NDF for the fitting of generated pT-spectra of identified hadrons using Tsallis distribution in different spherocity and multiplicity classes.Figures8 and 9 show the extracted parameters from the fitting of Tsallis distribution using Eq. (4) as a function of charged-particle multiplicity for different spherocity classes. Figure 8 shows that the temperature parameter increases with charged-particle multiplicity for spherocity-integrated and isotropic events. However, the jetty events seem to show a reverse trend for pions, kaons, and protons. For K∗0, ϕ, and Λ, the temperature parameter shows an increase with multiplicity for jetty events. For all cases, the temperature for jetty events is lower compared to the other spherocity classes. We also observe that the temperature for lighter particles does not change significantly with multiplicity, while with the increase in mass, the temperature increases steeply as a function of multiplicity. Figure 9 shows that for isotropic events, the nonextensive parameter, q values remain lower compared to the spherocity-integrated events, which suggests that isotropic events have got a higher degree of equilibration compared to spherocity-integrated events. This indicates that while studying the QGP-like conditions in small systems, one should separate the isotropic events from the spherocity-integrated events, as the production dynamics are different. On the contrary, the q-values for jetty are always higher compared to spherocity-integrated events, indicating that the jetty events remain far away from equilibrium. The present study is very useful in understanding the microscopic features of degrees of equilibration and their dependencies on the number of particles in the system and on the geometrical shape of an event. It would be interesting to study the particle mass dependence of these thermodynamic parameters. In order to do that, we have taken the events with the highest multiplicity class and done the same spherocity analysis, taking different particles as discussed here, which is shown in Figure 10. For the isotropic and spherocity-integrated events in high multiplicity pp collisions, the temperature remains higher for particles with higher masses, which supports a differential freeze-out scenario. This suggests that massive particles freeze-out early from the system. However, the jetty events show a reverse trend.Figure 8
(Color online) Multiplicity dependence ofT in different spherocity classes from the fitting of Tsallis distribution using Eq. (4).Figure 9
(Color online) Multiplicity dependence ofq in different spherocity classes from the fitting of Tsallis distribution using Eq. (4).Figure 10
(Color online) Mass dependence ofT and q in different spherocity classes for the highest multiplicity class.To explore the flow-like features in small systems, one needs to focus on the low-pT of the particle spectra with Boltzmann-Gibbs Blast-Wave (BGBW) model, which is discussed in the next subsection. As we saw an indication of a differential freeze-out scenario, in the following section, we consider making individual spectral analysis using BGBW, instead of a simultaneous fitting, which is usually necessitated by a single freeze-out scenario.


### 3.2. Boltzmann-Gibbs Blast-Wave Model

The expression for invariant yield in Boltzmann-Gibbs Blast-Wave (BGBW) model is given by [21, 22](5)Ed3Ndp3=D∫d3σμpμexp−pμuμT.Here, the four-velocity denoting flow velocities in space-time is given byuμ=coshρcoshη,tanhρcosϕr,tanhρsinϕr,sinhη, and the particle four-momentum is pμ=mTcoshy,pTcosϕ,pTsinϕ,mTsinhy, while the kinetic freeze-out surface is given by d3σμ=coshη,0,0,−sinhητrdrdηdϕr. Here, η is the space-time rapidity and assuming Bjorken correlation in rapidity, i.e., y=η [59], Eq. (5) can be expressed as(6)d2NdpTdyy=0=D∫0R0rdrK1mTcoshρTI0pTsinhρT.Here,D=gVmT/2π2, where g is the degeneracy factor, V is the system volume, and mT=pT2+m2 is the transverse mass. Here, I0pTsinhρ/T and K1mTcoshρ/T are the modified Bessel’s functions. They are given by(7)K1mTcoshρT=∫0∞coshyexp−mTcoshycoshρTdy,(8)I0pTsinhρT=12π∫02πexppTsinhρcosϕTdϕ.Here,ρ is a parameter given by ρ=tanh−1β. β=βmaxξn [22, 60–62] is the radial flow, where βmax is the maximum surface velocity and ξ=r/R0 with r as the radial distance. In the BGBW model, the particles closer to the center of the fireball move slower than the ones at the edges and the average of the transverse velocity can be evaluated as [63](9)β=∫βmaxξnξdξ∫ξdξ=22+nβmax.In our calculation, for the sake of simplicity, we use a linear velocity profile, i.e.,n=1, and R0 is the maximum radius of the expanding source at freeze-out (0<ξ<1).Figures11–13 show the fitting of pT-spectra of pions, kaons, protons, K∗0, ϕ, and Λ as a function of charged-particle multiplicity using BGBW distribution using Eq. (6) for spherocity-integrated, isotropic, and jetty events, respectively. The BGBW distribution fits the spectra for identified hadrons till pT≃2 GeV/c. Figure 14 shows the χ2/NDF for the fitting of generated pT-spectra of identified hadrons using BGBW model in different spherocity and multiplicity classes. For pions, the χ2/NDF is relatively lower compared to kaons and protons, indicating pions are better described by the BGBW model. The decrease of χ2/NDF for pions with increasing charged-particle multiplicity is due to the fact that the number of particles is less in the lower multiplicity classes, which makes the fitting worse. As expected, the fitting for jetty events is worse compared to isotropic and spherocity-integrated events, indicating that the jetty events remain far from equilibrium and a BGBW description, hence, becomes less significant.Figure 11
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using BGBW model for spherocity integrated events in various multiplicity classes as shown in Table 1.Figure 12
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using the BGBW model for isotropic events in various multiplicity classes as shown in Table 1.Figure 13
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using the BGBW model for jetty events in various multiplicity classes as shown in Table 1.Figure 14
(Color online)χ2/NDF for the fitting of generated pT-spectra of identified hadrons using the BGBW model in different spherocity and multiplicity classes.It is interesting to note that since BGBW analysis is in the soft sector of particle production, as expected, we do not see any difference between jetty, isotropic, and spherocity-integrated events so far the multiplicity dependence of kinetic freeze-out temperature and the radial flow velocity are concerned, except pions. This is depicted in Figures15 and 16. For all the discussed particles except the pions, the kinetic freeze-out temperature shows a linear increase with final state multiplicity. The radial flow velocity also shows a monotonic increase with multiplicity class for all the particles. Taking the highest multiplicity class, let us now look into the particle mass dependence of the freeze-out parameters. Figure 17 shows that the temperature increases with mass for the highest multiplicity pp collisions, indicating a differential freeze-out scenario. As seen in the previous subsection, the temperature from the BGBW model also suggests that the particles with heavier mass freeze-out early in time. The radial flow velocity is seen to decrease with particle mass, which supports a hydrodynamical behavior. We observe β≃0.62 for pions, whereas it is 0.31 for Λ.Figure 15
(Color online) Multiplicity dependence ofT in different spherocity classes from the fitting of the BGBW model.Figure 16
(Color online) Multiplicity dependence ofβ in different spherocity classes from the fitting of the BGBW model.Figure 17
(Color online) Mass dependence ofT and β in different spherocity classes for the highest multiplicity class using BGBW fit up to pT~2 GeV/c.


## 3.1. Experimentally Motivated Tsallis Nonextensive Statistics

ThepT-spectra of produced particles in high-energy collisions have been proposed to follow a thermalised Boltzmann type of distribution given as [23],(2)Ed3σd3p≃Cexp−pTTkin.Here,C is the normalisation constant, and Tkin is the kinetic freeze-out temperature. Due to possible QCD contributions at high-pT, the identified particle spectra at RHIC and LHC do not follow the above distribution, while the low-pT-region can be explained by incorporating the radial flow (β) into the Boltzmann-Gibbs distribution function, which is known as Boltzmann-Gibbs Blast-Wave (BGBW) model [22]. One can extract Tkin and radial flow (β) by fitting the identified particle transverse momentum spectra at low-pT. The detailed description along with the fitting of the Boltzmann-Gibbs Blast-Wave model to the identified particle spectra is discussed in the next subsection.To describe the completepT-spectra, one has to account for the power-law contribution at high-pT [24–26], which empirically takes care of the possible QCD contributions. A combination of both low and high-pT aspects has been proposed by Hagedorn, which describes the experimental data over a wide pT-range [27]. The distribution proposed by Hagedorn is given by(3)Ed3σd3p=C1+pTp0−n⟶exp−npTp0forpT⟶0,p0pTnforpT⟶∞.Here,C, p0, and n are fitting parameters. The above expression acts as an exponential and a power-law function for low and high-pT, respectively. However, deviations are observed by experiments at RHIC [28, 29] and LHC [30–33] while describing the pT-spectra of identified particles using a Boltzmann-Gibbs distribution function, even if the domain of temperature of the produced systems are high enough. On the other hand, Tsallis statistics with its nonextensivity features can be regarded as a generalization of Boltzmann-Gibbs statistics, and it gives a better description of systems, which have not yet reached equilibration. Its low-pT exponential and high-pT power-law behavior gives a complete spectral description of identified secondaries produced in pp collisions. In addition, a nonextensive entropic q-parameter shows the extent of nonequilibration of any particle in a thermal bath. There are few different versions of Tsallis distribution, which are being used by experimentalists and theoreticians. However, we use a thermodynamically consistent Tsallis nonextensive distribution function as shown in Ref. [20]. By saying thermodynamically consistent, we mean that the used distribution function satisfies all the standard thermodynamic relations for entropy, temperature, energy, pressure, and number density. The Tsallis distribution function at midrapidity is given by,(4)1pTd2NdpTdyy=0=gVmT2π21+q−1mTT−q/q−1,where g is the degeneracy factor, V is the system volume, mT=pT2+m2 is the transverse mass, and q is the nonextensive parameter. In the limit of q⟶1, Tsallis distribution (Eq. (4)) reduces to the standard Boltzmann-Gibbs distribution (Eq. (2)). It should be noted here that the use of the Tsallis nonextensive distribution function is purely motivated by its excellent description of experimental particle spectra. It has been widely used to explain the particle spectra in high-energy collisions [34–38] starting from elementary e++e− and hadronic to heavy-ion collisions [28, 29, 39–52]. Recently, few comprehensive studies have been carried out using Tsallis distribution for pions and quarkonium spectra in pp collisions [53, 54]. In this subsection, we employ the Tsallis nonextensive distribution function as shown in Eq. (4) to describe the pT-spectra. It should be noted here that the used Tsallis distribution function of Eq. (4) [55] has an extra power q, compared to the original distribution function proposed by Tsallis [56]. However, this form of the distribution function is thermodynamically consistent, which makes no major change in the observables as the values of q in hadronic collisions lie between, 1≤q≤1.22 [57]. While Tsallis nonextensive statistics makes a connection between entropy and thermodynamics of a system, the dynamics of the system in terms of long-range correlations and fluctuations (q−1 being the strength of the fluctuation [58]) are encoded in the entropic parameter, q.Figures4–6 show the fitting of pT-spectra of pions, kaons, protons, K∗0, ϕ, and Λ as a function of charged-particle multiplicity using Tsallis distribution function (Eq. (4)) for spherocity-integrated, isotropic, and jetty events, respectively. We observe that the Tsallis distribution fits the generated data till pT≃10 GeV/c. Figure 7 shows the quality of fitting in terms of the reduced-χ2, χ2/NDF as a function of multiplicity for different spherocity classes. The values of χ2/NDF show that the quality of fitting is reasonably good for all the multiplicity and spherocity classes.Figure 4
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using Tsallis distribution for spherocity integrated events in various multiplicity classes as shown in Table 1.Figure 5
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using Tsallis distribution for isotropic events in various multiplicity classes as shown in Table 1.Figure 6
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using Tsallis distribution for jetty events in various multiplicity classes as shown in Table 1.Figure 7
(Color online)χ2/NDF for the fitting of generated pT-spectra of identified hadrons using Tsallis distribution in different spherocity and multiplicity classes.Figures8 and 9 show the extracted parameters from the fitting of Tsallis distribution using Eq. (4) as a function of charged-particle multiplicity for different spherocity classes. Figure 8 shows that the temperature parameter increases with charged-particle multiplicity for spherocity-integrated and isotropic events. However, the jetty events seem to show a reverse trend for pions, kaons, and protons. For K∗0, ϕ, and Λ, the temperature parameter shows an increase with multiplicity for jetty events. For all cases, the temperature for jetty events is lower compared to the other spherocity classes. We also observe that the temperature for lighter particles does not change significantly with multiplicity, while with the increase in mass, the temperature increases steeply as a function of multiplicity. Figure 9 shows that for isotropic events, the nonextensive parameter, q values remain lower compared to the spherocity-integrated events, which suggests that isotropic events have got a higher degree of equilibration compared to spherocity-integrated events. This indicates that while studying the QGP-like conditions in small systems, one should separate the isotropic events from the spherocity-integrated events, as the production dynamics are different. On the contrary, the q-values for jetty are always higher compared to spherocity-integrated events, indicating that the jetty events remain far away from equilibrium. The present study is very useful in understanding the microscopic features of degrees of equilibration and their dependencies on the number of particles in the system and on the geometrical shape of an event. It would be interesting to study the particle mass dependence of these thermodynamic parameters. In order to do that, we have taken the events with the highest multiplicity class and done the same spherocity analysis, taking different particles as discussed here, which is shown in Figure 10. For the isotropic and spherocity-integrated events in high multiplicity pp collisions, the temperature remains higher for particles with higher masses, which supports a differential freeze-out scenario. This suggests that massive particles freeze-out early from the system. However, the jetty events show a reverse trend.Figure 8
(Color online) Multiplicity dependence ofT in different spherocity classes from the fitting of Tsallis distribution using Eq. (4).Figure 9
(Color online) Multiplicity dependence ofq in different spherocity classes from the fitting of Tsallis distribution using Eq. (4).Figure 10
(Color online) Mass dependence ofT and q in different spherocity classes for the highest multiplicity class.To explore the flow-like features in small systems, one needs to focus on the low-pT of the particle spectra with Boltzmann-Gibbs Blast-Wave (BGBW) model, which is discussed in the next subsection. As we saw an indication of a differential freeze-out scenario, in the following section, we consider making individual spectral analysis using BGBW, instead of a simultaneous fitting, which is usually necessitated by a single freeze-out scenario.


## 3.2. Boltzmann-Gibbs Blast-Wave Model

The expression for invariant yield in Boltzmann-Gibbs Blast-Wave (BGBW) model is given by [21, 22](5)Ed3Ndp3=D∫d3σμpμexp−pμuμT.Here, the four-velocity denoting flow velocities in space-time is given byuμ=coshρcoshη,tanhρcosϕr,tanhρsinϕr,sinhη, and the particle four-momentum is pμ=mTcoshy,pTcosϕ,pTsinϕ,mTsinhy, while the kinetic freeze-out surface is given by d3σμ=coshη,0,0,−sinhητrdrdηdϕr. Here, η is the space-time rapidity and assuming Bjorken correlation in rapidity, i.e., y=η [59], Eq. (5) can be expressed as(6)d2NdpTdyy=0=D∫0R0rdrK1mTcoshρTI0pTsinhρT.Here,D=gVmT/2π2, where g is the degeneracy factor, V is the system volume, and mT=pT2+m2 is the transverse mass. Here, I0pTsinhρ/T and K1mTcoshρ/T are the modified Bessel’s functions. They are given by(7)K1mTcoshρT=∫0∞coshyexp−mTcoshycoshρTdy,(8)I0pTsinhρT=12π∫02πexppTsinhρcosϕTdϕ.Here,ρ is a parameter given by ρ=tanh−1β. β=βmaxξn [22, 60–62] is the radial flow, where βmax is the maximum surface velocity and ξ=r/R0 with r as the radial distance. In the BGBW model, the particles closer to the center of the fireball move slower than the ones at the edges and the average of the transverse velocity can be evaluated as [63](9)β=∫βmaxξnξdξ∫ξdξ=22+nβmax.In our calculation, for the sake of simplicity, we use a linear velocity profile, i.e.,n=1, and R0 is the maximum radius of the expanding source at freeze-out (0<ξ<1).Figures11–13 show the fitting of pT-spectra of pions, kaons, protons, K∗0, ϕ, and Λ as a function of charged-particle multiplicity using BGBW distribution using Eq. (6) for spherocity-integrated, isotropic, and jetty events, respectively. The BGBW distribution fits the spectra for identified hadrons till pT≃2 GeV/c. Figure 14 shows the χ2/NDF for the fitting of generated pT-spectra of identified hadrons using BGBW model in different spherocity and multiplicity classes. For pions, the χ2/NDF is relatively lower compared to kaons and protons, indicating pions are better described by the BGBW model. The decrease of χ2/NDF for pions with increasing charged-particle multiplicity is due to the fact that the number of particles is less in the lower multiplicity classes, which makes the fitting worse. As expected, the fitting for jetty events is worse compared to isotropic and spherocity-integrated events, indicating that the jetty events remain far from equilibrium and a BGBW description, hence, becomes less significant.Figure 11
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using BGBW model for spherocity integrated events in various multiplicity classes as shown in Table 1.Figure 12
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using the BGBW model for isotropic events in various multiplicity classes as shown in Table 1.Figure 13
(Color online) Fitting of generatedpT-spectra of identified hadrons from PYTHIA8 using the BGBW model for jetty events in various multiplicity classes as shown in Table 1.Figure 14
(Color online)χ2/NDF for the fitting of generated pT-spectra of identified hadrons using the BGBW model in different spherocity and multiplicity classes.It is interesting to note that since BGBW analysis is in the soft sector of particle production, as expected, we do not see any difference between jetty, isotropic, and spherocity-integrated events so far the multiplicity dependence of kinetic freeze-out temperature and the radial flow velocity are concerned, except pions. This is depicted in Figures15 and 16. For all the discussed particles except the pions, the kinetic freeze-out temperature shows a linear increase with final state multiplicity. The radial flow velocity also shows a monotonic increase with multiplicity class for all the particles. Taking the highest multiplicity class, let us now look into the particle mass dependence of the freeze-out parameters. Figure 17 shows that the temperature increases with mass for the highest multiplicity pp collisions, indicating a differential freeze-out scenario. As seen in the previous subsection, the temperature from the BGBW model also suggests that the particles with heavier mass freeze-out early in time. The radial flow velocity is seen to decrease with particle mass, which supports a hydrodynamical behavior. We observe β≃0.62 for pions, whereas it is 0.31 for Λ.Figure 15
(Color online) Multiplicity dependence ofT in different spherocity classes from the fitting of the BGBW model.Figure 16
(Color online) Multiplicity dependence ofβ in different spherocity classes from the fitting of the BGBW model.Figure 17
(Color online) Mass dependence ofT and β in different spherocity classes for the highest multiplicity class using BGBW fit up to pT~2 GeV/c.


## 4. Summary and Conclusion

We perform a double differential study of the identified particle spectra and the system thermodynamics as a function of charged-particle multiplicity and transverse spherocity in pp collisions ats=13TeV using PYTHIA8. In order to understand the production dynamics of particles in high-multiplicity pp collisions, an event shape-dependent study becomes inevitable. Furthermore, to study the event topology dependence of the kinetic freeze-out properties, we have taken a cosmological expansion scenario of the produced fireball with a differential particle freeze-out. This work would shed light into the underlying event dynamics and help in understanding the possible differences and/or similarities in freeze-out parameters, when the hadronic collisions are compared with heavy-ion collisions. For the analysis of the identified particle pT-spectra as a function of charged-particle multiplicity and transverse spherocity, we use the thermodynamically consistent and experimentally motivated Tsallis nonextensive distribution function. In the soft sector of particle production, which corresponds to low-pT, Boltzmann-Gibbs Blast-Wave (BGBW) model is used to extract the kinetic freeze-out temperature and the radial flow velocity to study the particle mass and event multiplicity dependence. The important findings of this work are summarized below:(i)
We observe that the temperature parameter obtained by fitting the full range of thepT-spectra using Tsallis distribution function is dependent on spherocity class, and it increases with multiplicity for isotropic events, showing a steeper increase for higher mass particles(ii)
The entropic parameterq is found to be spherocity and multiplicity dependent. The jetty events have a tendency of staying away from equilibrium. For isotropic events, the q values remain lower compared to the spherocity-integrated events, which suggests that isotropic events approach more towards equilibrium compared to spherocity-integrated events. This hints for separating isotropic events from spherocity-integrated ones while studying the QGP-like conditions in small systems. This is because the production dynamics are different. In addition, while taking pp collisions as the baseline measurement to study any possible system formation in heavy-ion collisions at the LHC energies, the technique of transverse spherocity would be very useful(iii)
From BGBW analysis, it is observed that the higher mass particles show higher freeze-out temperature, which is an indication of a differential freeze-out scenario(iv)
The radial flow velocity is found to be mass dependent: higher for lighter mass particles—an indication of a hydrodynamic behavior in small systems. The obtained average flow velocities indicate a substantial collectivity in small systems in high multiplicity pp events at the LHC energies(v)
The kinetic freeze-out temperature and the radial flow velocity obtained in the BGBW framework are observed to be independent of spherocity class, except for pionsThe present study is very useful in understanding the microscopic features of degrees of equilibration and their dependencies on the number of particles in the system and on the geometrical shape of an event. In the absence of spherocity-dependent experimental data, the present study should give an outlook on similarities/differences between jetty and isotropic events in LHC pp events and their multiplicity dependence. This will help in making a proper bridge in understanding the particle production from hadronic to heavy-ion collisions.



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