Bose-Einstein condensation of pions in proton-proton collisions at the Large Hadron Collider using non-extensive Tsallis statistics
Suman Deb, Dushmanta Sahu, Raghunath Sahoo, Anil Kumar Pradhan
BBose-Einstein Condensation of pions in Proton-Proton collisions at the Large HadronCollider
Suman Deb, Dushmanta Sahu, Raghunath Sahoo, ∗ and Anil Kumar Pradhan Department of Physics, Indian Institute of Technology Indore, Simrol, Indore 453552, India (Dated: January 6, 2021)The possibility of formation of Bose-Einstein Condensation (BEC) is studied in pp collisions at √ s = 7 TeV at the Large Hadron Collider. A thermodynamically consistent form of Tsallis non-extensivedistribution is used to estimate the transition temperature required to form BEC of pions, which arethe most abundant species in a multi-particle production process in hadronic and nuclear collisions.The obtained results have been contrasted with the systems produced in Pb-Pb collisions to have abetter picture. We observe an explicit dependency of BEC on the non-extensive parameter q , whichis a measure of degree of non-equilibrium – as q decreases, the transition temperature increasesand approaches to the transition temperature obtained from Bose-Einstein statistics without non-extensivity. Studies are performed on the final state multiplicity dependence of number of particlesin the pion condensates in a wide range of multiplicity covering hadronic and heavy-ion collisions,using the inputs from experimental transverse momentum spectra. PACS numbers:
I. INTRODUCTION
In 1924 with the combined efforts of Satyendra NathBose and Albert Einstein, a new phenomenon was discov-ered. In his work, Bose had treated photons as particlesof an ideal gas and then he had derived the Planck’s lawof blackbody radiation with this assumption [1]. Later,Einstein predicted that at very low temperatures thoseparticles would condense into the minimum energy levelof the system under consideration [2, 3]. This phe-nomenon is called Bose-Einstein condensation (BEC) andis applicable to particles called bosons, which have inte-gral spins and follow the Bose-Einstein (BE) statistics.BEC is called the fifth state of matter which is usu-ally formed when a gas of bosons at low densities arecooled to temperatures very close to absolute zero. Un-der such conditions, a large fraction of the particles oc-cupy the ground state, where, the wave-functions of theparticles interfere with each other and the effect is ob-served macroscopically. This is possible because of theunique property of bosons whose total wave-functions arealways symmetric. So a large number of bosons canoccupy the same state unlike the fermions which obeyPauli’s exclusion principle. The caveat here is that, BECusually occurs at very low densities and very low tem-peratures. However in high energy collisions, the tem-perature becomes extremely high. In such conditions,whether we can observe any such condensation is a ques-tion to be addressed through extensive theoretical stud-ies confronted to appropriate experimentation. The tem-perature reached in the high energy collisions is of MeVscale, which is about 10 K and is astronomically higherthan the temperature required for BEC for cold atoms.But the properties of a BEC for pions would be very dif- ∗ Electronic address: [email protected] ferent from the low temperature BEC. Firstly, the pionswould have much smaller system volume, much higherdensity and different interactions are also involved in theformation of high temperature BEC [4]. The possibilityof observing such a high-temperature BEC of pions inhadronic and heavy-ion collisions at the energy and lu-minosity frontiers in the Large Hadron Collider would beof great interest to the community.The aim of relativistic heavy-ion collisions is to un-derstand the phases of Quantum Chromo-dynamics(QCD) [5, 6]. In particular, these collisions give us an op-portunity to create and characterize a possible deconfinedstate of quarks and gluons, called Quark-Gluon Plasma(QGP) [7], which is believed to have existed after fewmicro-seconds during the infancy of the Universe throughthe much-talked about Big Bang collision. Informationabout these phases formed in such collisions is extractedfrom the distribution of the produced particles in thephase space, called (identified)particle spectra. Interpre-tations of these results are performed using different the-oretical models to draw conclusions about the propertiesof the matter formed at the extremes of temperature andenergy density. Statistical hadron gas models (thermalmodels) and hydrodynamic models [8–10] are used forthis purpose, along with many other variants to confrontto the experimental observations. Non-ideal hydrody-namics are brought in to correctly explain the anisotropyof the particles originated in the QGP [8]. Thermal mod-els mainly assume thermodynamic equilibrium to explainthe hadronic yields and extract the chemical freeze-outparameters like temperature and baryochemical poten-tial. However everything is not so smooth, mainly in ex-plaining the pion to proton ratio and mean multiplicityof proton/anti-proton. Also hydrodynamic models whichuse local thermodynamic equilibrium, are not quite goodat explaining the very low momentum part of the piontransverse momentum ( p T ) spectra [11, 12].To explain these, chemical non-equilibrium in the for- a r X i v : . [ h e p - ph ] J a n mation of the hadronic matter is assumed in some stud-ies [13]. This brings up the non-zero value of chemicalpotential which is close to the critical value of chemicalpotential needed for BEC of pions. These have been thecases studied in the heavy-ion collisions. However, anycontribution of non-equilibrium can be well preserved insmall systems like that are formed in pp collisions, be-cause of the high gluon density and net-baryon numberbeing negligibly small [14]. In such pp collisions at theLHC energies, the baryon chemical potential is close tozero . Moreover, the systems formed in pp collisions aretaken as reference to interpret the results of heavy-ioncollisions. So it is important to understand the forma-tion of BEC-like features in small systems formed in pp collisions. To this end, we investigate the possibility ofBEC in pion gas formed in pp collisions and comparethe results with heavy-ion collisions, to find a bridge be-tween the two systems. Such a study of investigatingvarious phenomena in LHC pp collisions has become anecessity in order to understand the QGP-like featuresseen in these hadronic collisions [15–17]. .It is observed that at RHIC [18, 19] and LHC [20–23] energies, the p T spectra in pp collisions deviate fromthe standard thermalized Boltzmann-Gibbs (BG) distri-bution. In such cases, Tsallis non-extensive distribution[24] describes the p T spectra very well. In view of this,we have used a thermodynamically consistent form ofTsallis distribution function [25]. The non-extensivityparameter q gives the degree of deviation from equilib-rium, where q = 1 suggests the equilibrium condition(BG scenario). For higher charged particle multiplicity,the q value tends to 1, which is an indication that in thatregime, the system has most probably attained thermalequilibrium. By fitting Tsallis distribution function tothe p T spectra of the particles, the parameters q andtemperature T are extracted [26], which are then usedto find the particle multiplicities in the condensate. Thepresent formalism is motivated by the experimental p T -spectra and we use these information to further explorethe possibility of a BEC of pions in LHC pp collisions.In this paper, we have studied the possibility of pioncondensation in high energy pp collision systems at LHCenergy of √ s = 7 TeV. We have also studied how thecritical temperature changes with the change in the non-extensive parameter q . The section II briefly gives theformulation for estimating the particle multiplicities inthe condensate and the critical temperature. In sectionIII, we discuss about our findings and finally in sectionIV, we have summarized our findings. II. FORMULATION
Before introducing the non-extensivity into the formal-ism, let’s start with a general distribution function of Bose-Einstein statistics, which is given as [27], f = 1 exp ( E − µT ) − . (1)By using the above formula, we can calculate the par-ticle multiplicities by the equation [13], N = (cid:90) ∞ d xd ph gexp (cid:18) √ p − m − µT (cid:19) − (cid:39) V (cid:90) ∞ d p (2 π ) gexp (cid:18) √ p − m − µT (cid:19) − , (2)where g is the degeneracy of the particle, p is the momen-tum, m is the mass of the particle, T is the temperatureof the system and µ is the chemical potential. The in-tegral over the space co-ordinates gives us the volume ofthe system V . In the thermodynamic limit, V → ∞ , onecan write Eq. 2 with separate terms for p = 0 and p >
0, as: N (cid:39) gexp ( m − µT ) − V (cid:90) ∞ d p (2 π ) gexp (cid:18) √ p − m − µT (cid:19) − ⇒ N total = N condensation + N excited . (3)When the chemical potential approaches to the value ofthe mass of the particle, µ → m , the N condensation be-comes dominant and becomes ∞ at µ = m . However atLHC pp collisions, the chemical potential is very smalland can be taken as zero.By taking Tsallis non-extensivity [28, 29] into account,the B-E distribution function changes to, f = 1 exp q ( E − µT ) − . (4)where, exp q ( x ) ≡ (cid:40) [1 + ( q − x ] q − if x > − q ) x ] − q if x ≤ x = ( E − µ ) /T , E is the energy of the particlegiven by E = (cid:112) p + m . It is worth noting that in thelimit, q →
1, Eq. (5) reduces to the standard exponentialfunction i.e., Maxwell-Boltzmann distribution function,lim q → exp q ( x ) → exp( x ) . The Tsallis parameter, T and q appearing in Eq. 5 areextracted from the p T spectra of the particle by usingTsallis distribution as a fitting function.Therefore, Eq.3 in the context of non-extensivity in itsthermodynamically consistent form [25] becomes, N (cid:39) g (cid:20) exp q ( m − µT ) − (cid:21) q + V (cid:90) ∞ d p (2 π ) g (cid:20) exp q (cid:18) √ p − m − µT (cid:19) − (cid:21) q . (6)The formula for the critical temperature in the ultra-relativistic limits [30] is given by, T c = 1 . × ρ / , (7)where, ρ is the pion number density of the system, givenby the formula [31] within Tsallis statistics as follows, ρ = g (cid:90) d p (2 π ) (cid:20) [1 + ( q − E − µT ] q − − (cid:21) − q . (8) III. RESULTS AND DISCUSSION
We have used Eq.6 to estimate the particle multiplic-ities in the excited states and the condensate. We havetaken certain values of temperature and have estimatedthe multiplicities at particular q values. For the sake ofsimplicity, here we have taken a constant value for thevolume of the system with the system radius of 1.2 fm,which can be approximately taken as the chemical freeze-out radius in pp collision systems. This is a reasonableassumption as the HBT radii measurement from pp col-lisions at the LHC gives the radius range of 0.8 - 1.6fm [32].In fig. 1 we have plotted the ratios of N condensation to N total (N c / N t ) and N excited to N total (N e / N t ) as afunction of temperature. We observe that at high tem-peratures, the particle multiplicities in the excited statesis dominant as compared to that in the condensate. Asthe temperature decreases, the particle multiplicities inthe condensate start to increase. After a certain criticaltemperature, the number of particles in the condensatebecomes dominant, which infers that BEC has occurred.We also clearly see that the transition temperature ishighly q dependent. For BE statistics without using non-extensivity ( q =1), we find that the transition tempera-ture is the highest at around 105 MeV. However, as weincrease the q value, the transition temperature starts todecrease. For q = 1.13, we have the lowest transitiontemperature at around 75 MeV. This means that for sys-tems which are near equilibrium, the transition temper-ature will be higher, whereas for the systems which areaway from equilibrium, the transition temperatures willbe relatively lower. This is a really interesting findingsince we know that for the highest multiplicity of π ± inthe pp collisions at √ s = 7 TeV, the kinetic freeze-out temperature obtained after fitting a thermodynamicallyconsistent Tsallis distribution function to its p T spectrais around 93 MeV [33]. This may indicate that there is apossibility that we may see BEC in pp collisions at LHCenergies. T (GeV) t N c N , t N e N t /N e N t /N c N q = 1q = 1.05q = 1.1q = 1.13
FIG. 1: (Color online) Ratio of number of particles in thecondensate with total number of particles and number of par-ticles in the excited state with total number of particles as afunction of temperature for a pion gas for different q values. In fig.2 we have plotted the critical temperature ( T c )as a function of the non-extensive parameter q . We seethat for higher values of q , which means when the systemis far away from equilibrium, the critical temperature forBEC is lower. When, the system is at equilibrium ( q = 1), the critical temperature is the highest at around105 MeV. An important thing to note here is that fora system which has acquired equilibrium, the transitiontemperature will be higher, and it may not be possiblefor the pion gas system to reach that temperature in thefreeze-out. So there is a distinct possibility that if a sys-tem is in thermal equilibrium, it may not show BEC.The relation between the critical temperature and thepion number density can be obtained by Eq.7. Figure3 shows the critical temperature ( T c ) as a function ofpion number density ( ρ ). To estimate ρ we have usedthe Eq. 8 for certain T and q values. We see that thecritical temperature is also dependent on number density ρ . The higher is the number density, higher is the criticaltemperature of the system as shown in Ref. [30].In fig. 4 we have plotted the ratios of (N c / N t ) and(N e / N t ) for pp collisions at √ s = 7 TeV using ALICEdata [34] as a function of charged particle multiplicity,which uses Eq. 6. T and q values used in Eq. 6 aretaken from Ref. [33], where the p T -spectra of producedidentified particles in a differential freeze-out scenario arefitted by Tsallis distribution function. To have a quan-titative estimation of the volume of the system at thekinetic freeze-out, we consider the chemical freeze-out q ( G e V ) c T FIG. 2: (Color online) Critical temperature as a function ofnon-extensive parameter q . ) (GeV r ( G e V ) c T q = 1q = 1.05q = 1.1q = 1.13 FIG. 3: (Color online) Critical temperature as a function ofnumber density for a pion gas. radius from Ref [35] and as the system is an expand-ing one, we add hadronic phase lifetime from our earlierstudies Ref. [36] multiplied with speed of light in vac-uum. The radii obtained by this method is quite compa-rable with that one obtains from the femtoscopy analy-sis [32]. It is worth mentioning here that this assumptionensures maximum radii of the system. We observe thatat high charged particle multiplicity which correspondsto higher kinetic freeze-out temperature, about 95% ofthe particles are in the excited states and about 5% ofthe particles occupy the ground state. With the decreasein (cid:104) dN ch /dη (cid:105) , we observe that the N excited to N total ra-tio decreases while the N condensation to N total ratio in-creases. At about (cid:104) dN ch /dη (cid:105) (cid:39)
6, which corresponds to 78 MeV temperature [33], we observe a transition. This (cid:104) dN ch /dη (cid:105) can be considered as the critical charged par-ticle multiplicity for pp collisions at √ s = 7 TeV at LHC,below which the number of particles in the condensate ishigher than the excited states. At the lowest charged par-ticle multiplicity, we observe that the particle multiplicityin the condensate is dominant over the particles in theexcited states. This is an interesting finding given thatat low charged particle multiplicity the number density,the volume and the temperature of the system are rela-tively lower as compared to the systems at high chargedparticle multiplicities. > h /d ch 6, we may observe BEC, regardlessof the collision systems. > h /d ch We have made an attempt to study the possible forma-tion of Bose-Einstein Condensates in the hadronic colli-sions at the TeV energies at the Large Hadron Collider,using the information from identified particle spectra.Motivated by the fact that a thermodynamically con-sistent form of Tsallis non-extensive distribution func-tion gives a better description of the particle transversemomentum spectra, we have extended the standard for- malism of BEC to the domain of non-extensivity. Tak-ing final state multiplicity dependence of the identifiedparticle spectra and hence the freeze-out parameters liketemperature and non-extensive parameter, q for hadronicand heavy-ion collisions, we have studied various aspectsof BEC.In summary, • In the low-multiplicity pp collisions, where the sys-tems are away from equilibrium ( q (cid:54) = 1), the prob-ability of formation of BEC is seen to be higher. • The critical temperature for the BEC to occur, de-pends on the degree of non-equilibrium in the sys-tem. This is observed from the dependency of thenumber of pions in the condensates on the non-extensive parameter q . For systems away fromequilibrium, the values of critical temperature arelower. • We observe a threshold in the final state charged-particle multiplicity, i.e. (cid:104) dN ch /dη (cid:105) (cid:39) 6, which cor-responds to an event class with freeze-out tempera-ture of around 78 MeV. 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