Limiting fragmentation in high-energy nuclear collisions at the CERN Large Hadron Collider
Pragati Sahoo, Pooja Pareek, Swatantra Kumar Tiwari, Raghunath Sahoo
LLimiting fragmentation in high-energy nuclear collisions at the CERN Large HadronCollider
Pragati Sahoo, Pooja Pareek, Swatantra Kumar Tiwari, and Raghunath Sahoo ∗ Discipline of Physics, School of Basic Sciences,Indian Institute of Technology Indore, Simrol,Khandwa Road, Indore- 453552, INDIA (Dated: April 11, 2019)The hypothesis of limiting fragmentation (LF) or it is called otherwise recently, as extended longi-tudinal scaling, is an interesting phenomena in high energy multiparticle production process. Thispaper discusses about different regions of phase space and their importance in hadron production,giving special emphasis on the fragmentation region. Although it was conjectured as a universalphenomenon in high energy physics, with the advent of higher center-of-mass energies, it has becomeprudent to analyse and understand the validity of such hypothesis in view of the increasing inelasticnucleon-nucleon cross-section ( σ in ). In this work, we revisit the phenomenon of limiting fragmenta-tion for nucleus-nucleus (A+A) collisions in the pseudorapidity distribution of charged particles atvarious energies. We use energy dependent σ in to transform the charged particle pseudorapidity dis-tributions ( dN AA ch /dη ) into differential cross-section per unit pseudorapidity ( dσ AA /dη ) of chargedparticles and study the phenomenon of LF. We find that in dσ AA /dη LF seems to be violated atLHC energies while considering the energy dependent σ in . We also perform a similar study usingA Multi-Phase Transport (AMPT) Model with string melting scenario and also find that LF isviolated at LHC energies. PACS numbers: 12.38.Mh, 12.38.Gc, 25.75.Nq, 24.10.Pa
I. INTRODUCTION
Understanding the particle productions in high energynuclear collisions is always fascinating. The particle pro-duction in high energy collisions happens from three dif-ferent regions: the projectile, the target and the centralregion. Particles emitted from the outer region are calledprojectile/target fragments. There are various nuclearfragmentation mechanisms discussed in literature [1, 2].The most important are: a sudden fragmentation by ex-plosive mechanisms, such as shock waves [1] and a slowfragmentation by the “fission” of the spectator regions,mainly because of the interactions with the particles orfragments emitted from the participant region at trans-verse angles in the center-of-momentum system [1]. Thelatter is a purely low-energy nuclear phenomenon, whereas the former is more applicable to relativistic domain ofenergies. During the late 1960s, the hypothesis of limit-ing fragmentation became important to understand theparticle production [2, 3]. According to this hypothesisthe produced particles, in the rest frame of one of theprojectiles become independent of centre-of-mass ener-gies, thus following a possible scaling (as a function of η (cid:48) = η ± y beam ), known as limiting fragmentation (LF).As (pseudo)rapidity is a longitudinal variable it is alsocalled longitudinal scaling. Here y beam = ln( √ s NN /m p ),is beam rapidity and m p is the mass of proton. Therehave been several attempts to understand the nature ofhadronic interactions which lead to limiting fragmenta- ∗ Electronic address: [email protected] tion and the deviations from it [4–6].It is expected that a central plateau develops at higherenergies, which clearly separates the central rapidity fromthe fragmentation region. However, as such, there is noseparating boundary between the central rapidity and thefragmentation region. The width of the fragmentation re-gion is around 2-units in rapidity [7]. The fragmentationregion thus, is expected to be well separated from thecentral region only in very high energies, as the kinemat-ically available rapidity region is much wider than 4-unitsin rapidity. The particle production in fragmentation re-gion is attributable to the valence quarks participatingin hadronization, whereas in central rapidity region, it isdominated by the mid-rapidity gluonic sources at highenergies [8, 9]. The central rapidity region is called Pio-nization region [7] and is shown in the Fig. 1.There have been several experimental efforts to under-stand the particle production in both mid and forwardrapidities [10–17]. As LF is the thrust area of this pa-per we focus on the particle production in the forwardrapidity region. The experimental observation of LF wasfirst reported by the PHOBOS experiment at RHIC withcharged particles [17], later STAR experiment also con-firmed the hypothesis with inclusive photons in the for-ward rapidity [13]. The Limiting fragmentation was ob-served by UA5 experiment at CERN for pp and p¯p col-lisions from 53 GeV to 900 GeV [18]. However, ALICEexperiment at the LHC has reported a violation of LF hy-pothesis for inclusive photons in pp collisions with limitedforward rapidity coverage [19].Various theoretical works [5, 6, 20–25] have reportedthe observation of limiting fragmentation phenomenon inheavy-ion collisions. Recently, limiting fragmentation in a r X i v : . [ h e p - ph ] A p r d N / d y ( η ) Region of PionizationFragmentation Region
FIG. 1: A schematic of (pseudo)rapidity distribution showingthe pionization and fragmentation regions. the era of RHIC and LHC has got a special mention witha new concept called the hypothesis of “energy-balancedlimiting fragmentation” [26, 27]. In Ref. [5], it is claimedthat the cross-section plays an important role in frag-mentation regions. Marian [6] has shown that the LFphenomenon is observed in the differential cross-sectionper unit pseudorapidity in proton+nucleus collisions atRHIC energies.Our main aim in this work is to study the phenom-ena of LF for A+A collisions in view of increasing in-elastic particle production cross-section from RHIC toLHC energies. The hypothesis of limiting fragmentationcan be tested for both the observables, namely the par-ticle multiplicity density and also the differential cross-section. As LF is least explored in the case of differentialcross-section, this work focuses on the later observablewith a detailed discussion on multiplicity as well, fora clear comparison of the expected results at the LHCenergies. The total hadronic cross-section does not re-main constant from lower RHIC energies to the highestLHC energy but is a slowly increasing function of √ s [28].The particle production in heavy-ion collisions dependson the hadronic cross-section. Thus, a detailed studyof the longitudinal scaling behaviour in terms of cross-section could be a prudent attempt. The longitudinalvariables are expected to be sensitive to the available en-ergy and the multiplicity of the produced secondaries. Inthis context, the study of possible longitudinal scalingof the final state multiplicity as a function of collisionenergy becomes judicious, in view of increasing inelasticparticle production cross-sections at LHC energies. Thepaper is organised as follows: in Sec. II, we recapitulatethe basics of Landau hydrodynamics and its connectionwith the limiting fragmentation hypothesis. In Sec. III, we present the methodology to calculate the differentialcross-section per unit pseudorapidity and discuss the re-sults obtained using experimental data and AMPT. Fi-nally, we summarise our findings in Sec. IV. II. LANDAU HYDRODYNAMICS ANDLIMITING FRAGMENTATION HYPOTHESIS
The angular distribution of the particles produced inhigh-energy collisions is described by the famous Landaumodel with relativistic hydrodynamics given by the con-servation of energy momentum tensor, ∂ µ T µν = 0 witha blackbody equation of state, p = (cid:15)/ p is the pressureand (cid:15) is the energy density [29, 30]. Landau hydrody-namical model assumes complete thermalization of thetotal energy in the Lorentz contracted volume of the fire-ball, which makes the initial energy density to grow withcollision energy [31]. The formulation given in [31] givesrise to the initial entropy of the system, which is pro-duced in the thermalization process of the quanta of thesystem, to follow a Gaussian distribution in the rapidityspace. The width of the rapidity distribution is deter-mined by the Lorentz contraction factor and is related tothe speed of sound [32]. The multiplicity distribution inthe rapidity space, thus becomes [29, 30, 33] dNdy = Ks / √ πL exp (cid:18) − y L (cid:19) , (1)where L = σ y = (1 /
2) ln( s/m p ) = ln( γ ). Eq. 1 can berewritten as dNdy = Ks / √ πy beam exp (cid:18) − y y beam (cid:19) . (2)The conclusion from Ref. [31] shows that the hypothe-sis of limiting fragmentation comes naturally in Landau’smodel of multiparticle production. Following the LF hy-pothesis, when the rapidity distribution is seen from oneof the projectiles’ rest frame, i.e. by transforming to y (cid:48) = y − y beam , the above expression for rapidity distri-bution becomes ( dN/dy = dN/dy (cid:48) ) [31], dNdy (cid:48) = Ks / √ πy beam exp (cid:18) − ( y (cid:48) + y beam ) y beam (cid:19) , = Ks / √ πy beam exp − (cid:18) y (cid:48) y beam + y (cid:48) (cid:19) exp (cid:18) − y beam (cid:19) , = 1 √ y beam exp (cid:18) − y (cid:48) y beam − y (cid:48) (cid:19) . (3)For y (cid:48) = 0, the distribution only depends on the Lorentzcontraction factor, which is a function of collision energy.When we make the transformation, y (cid:48) = y − y beam , thefragmentation region shifts by a factor y beam , a valuewhich increases with the collision energies, making theregion to overlap with each other. (GeV)s10 ( m b ) i ne l s ATLAS (MBTS) ATLAS (ALFA) pp (non ALICE)ALICE LHCbAugerpp a )s) + C (sHybrid: A + B ln()s( n A + B ln )sA + B ln(
FIG. 2: The inelastic cross-section as a function of √ s . Thesymbols are experimental data [34–37] and the fitted lines arephenomenologically motivated functions. III. LIMITING FRAGMENTATION AT THELHC
In this section, we study the limiting fragmentationphenomenon in the pseudorapidity distributions of dif-ferential cross-section of charged particles ( dσ/dη ) forA+A collisions at various center-of-mass energies startingfrom 19.6 GeV to 5.02 TeV. Due to lack of experimen-tal data of dσ AA /dη , we take the experimentally mea-sured dN AA ch /dη at various collision energies. We trans-form dN AA ch /dη into dσ AA /dη using nucleon-nucleon in-elastic cross-sections ( σ in ) for different energies applyingthe method discussed below. A very detailed study isneeded to make the connection possible. Recent stud-ies [6] shows that the longitudinal scaling of the differ-ential cross-section per unit pseudorapidity is observedin the experimental data for higher RHIC energies. Therationale behind our work is to bring in the direct center-of-mass energy dependence of σ in , which has a differ-ent low-energy behaviour up to the top RHIC energy incomparison to the LHC energies. This is also observedfrom the experimentally measured values of σ in [34–37],which are shown in the Fig. 2. In this figure, we showthe variation of σ in with collision energy. It is clearlyseen that there is a very slow rise of σ in at lower colli-sion energies up to the top RHIC energy. We have fittedthe experimental data with various phenomenologicallymotivated functions in order to understand the energy-dependent behaviour of σ in . A logarithmic function, A+ B ln( √ s ), with A and B as free fitting parameters ex-plains the data only up to RHIC energies. This seemsto deviate completely after the top RHIC energy. The σ in data beyond the top RHIC energy do not follow alogarithmic behaviour. To study the complete energy-dependent behaviour, we have used a hybrid function, A+ B ln( √ s ) + C( √ s ) α , which combines logarithmic anda power-law to fit the data. Here A, B, C and α are free parameters. This hybrid function explains the datafrom lower to higher energies. We have also fitted thedata with a function A + B ln n ( √ s ), where A and B arefree parameters. A more detailed discussions could befound in Ref. [28]. This seems to describe the data verywell. These findings suggest that the logarithmic func-tion alone cannot explain the data for higher energies,while the power of logarithmic function and the hybridfunction mentioned above could explain from lower tohigher energies shown in the figure. The σ in at LHC en-ergies showing a different functional behaviour than thelower energies necessitates a relook into the hypothesisof limiting fragmentation.Considering the crude approximation to the physi-cal situation in the framework of Landau hydrodynam-ical model of particle production, the relationship be-tween the differential cross-section per unit pseudora-pidity ( dσ pp /dη ) and the pseudorapidity distribution( dN ppch /dη ) of charged particles for pp collisions is givenas [38], dσ pp dη = σ in (cid:18) dN ppch dη (cid:19) . (4)Now, the relation of charged particle pseudorapidity dis-tribution in A+A collisions with the charged particlepseudorapidity distribution in pp collisions using a two-component model, where the contributions from soft andhard processes in the particle production are taken sep-arately, is given as [39, 40], dN AAch dη = dN ppch dη (cid:18) (1 − x ) < N part > x < N coll > (cid:19) . (5)Here, x and (1 − x ) are the fractions of contributionto the particle production from hard and soft processes,respectively.Using Eq. 5 in Eq. 4, we get a relation between thedifferential cross-section per unit pseudorapidity in ppcollisions and the charged particle pseudorapidity distri-bution in heavy-ion collisions as follows: dσ pp dη = σ in (cid:16) dN AAch dη (cid:17)(cid:16) (1 − x )
FIG. 3: The number of participant pair normalized pseudora-pidity distribution of charged particles ( dN AAch /dη ) in heavy-ion collisions versus η − y beam for various energies. The sym-bols are experimental data [35, 43–45] and the lines are thedouble Gaussian fits. collide in a central way and the pseudorapidity spectrumtransforms as [42], dσ AA dη = A (cid:18) dσ pp dη (cid:19) . (7)Using Eqs. 6 and 7, we write the differential cross-section per unit pseudorapidity in terms of charged par-ticle pseudorapidity distribution for the heavy-ion colli-sions as, dσ AA dη = A σ in (cid:16) dN AAch dη (cid:17)(cid:16) (1 − x )
FIG. 4: The differential cross-section per unit pseudorapidity( dσ AA /dη ) as a function of η − y beam for various collisionenergies. The symbols are experimental points and the linesare double Gaussian fits. lows, f ( η ) = A e − η σ − A e − η σ , (9)where A , A are the amplitudes and σ , σ are widthsof the double Gaussian function. This function describesthe experimental data very well at LHC energies withinuncertainties [44, 45]. The fitting parameters are givenin the table I for √ s NN = 2.76 and 5.02 TeV. We ob-serve that the limiting fragmentation phenomenon seemsto be violated at √ s NN = 5.02 TeV, while it is observedat energies from √ s NN = 19.6 GeV to 2.76 TeV. De-spite this, at √ s NN = 5.02 TeV, the extrapolation of thecharged particle pseudorapidity density scaled with av-erage number of participant does not show a similar be-haviour in the fragmentation region as observed at lowerenergies. The lack of data around the beam rapidity re-gion and the asymmetric values around η = 0 refrain usto draw any solid conclusion on the behaviour observedat highest LHC energies. It should also be noted herethat a Gaussian extrapolation to the fragmentation re-gion is assumption-based and its validity is subjected toa check against the experimental data.Now, we evaluate dσ AA /dη using Eq. 8 for √ s NN =19.6 to 5.02 TeV taking the x parameters from Ref. [40],which is almost energy independent from RHIC to LHCenergies. The inelastic cross-sections for various energiesare taken from Ref. [34–37]. The Monte Carlo Glaubermodel [46] is used to calculate number of participants( N part ) and number of binary collisions ( N coll ) at dif-ferent energies. The differential cross-section per unitpseudorapidity for various center-of-mass energies start-ing from √ s NN = 19.6 to 5.02 TeV are shown in Fig. 4with respect to η − y beam . We notice that the limitingfragmentation hypothesis appears to be violated at LHC TABLE I: The values of parameters obtained from the fitting of experimental data of dN ch /dη with the double Gaussianfunction given by Eq. 9 Parameters √ s NN = 2.76 TeV √ s NN = 5.02 TeV A ± ± A ± ± σ ± ± σ ± ± beam - y h - - - - h / d c h AA d N PHOBOS 19.6 GeVPHOBOS 62.4 GeVPHOBOS 130 GeVPHOBOS 200 GeVALICE 2.76 TeVALICE 5.02 TeV AMPT 19.6 GeVAMPT 62.4 GeVAMPT 130 GeVAMPT 200 GeVAMPT 2.76 TeVAMPT 5.02 TeV
FIG. 5: The comparison of AMPT model predictions withexperimental data on dN AAch /dη versus η − y beam for variousenergies. energies, i.e. at √ s NN = 2.76 and 5.02 TeV. These find-ings suggest that, it is very important to consider theenergy dependent σ in in order to study LF phenomenonparticularly at LHC energies.The experimental data for pseudorapidity distributionsof charged particles in the full phase space are not avail-able at the LHC energies. In addition, a double Gaus-sian extrapolation of dN ch /dη to the y beam at a givenenergy, seem to introduce an artefact in the spectra,which forbids one to look into the hypothesis of limit-ing fragmentation. To circumvent this problem, we takeAMPT model in string melting scenario [47] as tunedin Ref. [48] for the most central bin 0-6% and 0-5% forRHIC and LHC energies, respectively. We have thencompared the measured experimental data for pseudora-pidity distribution of charged particles [35, 43–45] withthe results obtained in AMPT model. The comparisonof experimental data with the AMPT model predictionis shown in Fig. 5. AMPT predictions reproduce themid-rapidity and the fragmentation region very well butcannot reproduce around the peak region ( η ∼
0) atRHIC energies. For LHC energy at √ s NN = 2.76 TeV,the AMPT predictions are in good agreement with theexperimental data except for the mid-rapidity region,where the predictions slightly underestimate the mea-sured data. Similarly, for √ s NN = 5.02 TeV, the predic- beam - y h - - - - - - - ( m b ) h / d AA s d · AMPT 19.6 GeVAMPT 62.4 GeVAMPT 130 GeVAMPT 200 GeVAMPT 2.76 TeVAMPT 5.02 TeV
FIG. 6: dσ AA /dη versus η − y beam using AMPT results. tions from AMPT model slightly overestimate the datameasured for 0-5% centrality bin. In this figure, we seethat the phenomenon of longitudinal scaling is observedat RHIC and LHC energies. Theses findings are also de-scribed in the Ref. [49], where various transport modelslike AMPT and the Ultra-relativistic Quantum Molec-ular Dynamics (UrQMD) model are used to study thisphenomenon. They observed that AMPT (both defaultand string melting versions) and UrQMD with defaultversion show the longitudinal scaling in pseudorapiditydistributions of charged particles at RHIC and LHC en-ergies.We convert the AMPT results of dN AAch /dη into dσ AA /dη using Eq. 8. In Fig. 6, we have shown dσ AA /dη versus η − y beam to see the longitudinal scaling phenom-ena in the fragmentation region for different energies from19.6 GeV to 5.02 TeV. Again, we have found a similarobservation for the AMPT model as observed in the ex-perimental data i.e. LF is observed up to RHIC energiesin dσ AA /dη and seems to be violated for LHC energies.Theses findings are very important while discussing thelongitudinal scaling hypothesis at LHC energies. IV. CONCLUSIONS AND OUTLOOK
In this work, we have revisited the phenomenon of lim-iting fragmentation in the pseudorapidity distributions ofdifferential cross-sections of the charged particles usingthe energy dependent inelastic cross-section. The find-ings of this analysis are: • We have observed the limiting fragmentation phe-nomenon in the experimental data of dN AAch /dη from √ s NN = 19.6 GeV to 2.76 TeV and it is vio-lated at √ s NN = 5.02 TeV. Here, the double Gaus-sian function is used to extrapolate the experimen-tal data in the fragmentation region. However, onthe basis of extrapolation method, one can not inferany exact physics conclusions. • We have transformed experimental data of dN AAch /dη to dσ AA /dη for various energies from √ s NN = 19.6 GeV to 5.02 TeV and see the dis-tributions in the rest frame of one of the nuclei.We have found that the LF hypothesis seems tobe violated at both the energies i.e. at √ s NN =2.76 and 5.02 TeV, when one considers the energydependent inelastic cross-section. • We have also studied the phenomenon of longitu-dinal scaling using AMPT model and employingthe same procedure as used for the experimentaldata. Our studies suggest that, AMPT seems toshow a possible violation of limiting fragmentationphenomenon for dσ AA /dη at LHC energies. • The hypothesis of LF comes as a natural outcomewhen the particle production follows the Landauhydrodynamics, with a Gaussian pseudorapidityprofile. • LF works fine, when the hadronic cross-section isassumed to be almost independent of energy, whichis not the case and hence it is expected to be vio-lated at higher energies. We find that the limitingfragmentation appears to be violated at LHC ener-gies while using the energy dependent cross-section. • The thermal model with Landau extrapolation toLHC for charged particles, predicts a violation ofLF at LHC [50]. What about photons in this frame-work? It has been observed that for pions in ther-mal model with longitudinal flow, the LF is violatedat the LHC energies [51]. What about photons witha longitudinal flow? These need further investiga-tions. • It is expected that at higher energies, Landauhydrodynamics should fail and we should expectBjorken boost invariant hydrodynamics to workout, with the observation of a mid-rapidity plateau.If LF is a natural outcome of Landau model, thenLF should be violated at LHC for two reasons:i) failure to see a Gaussian pseudorapidity distri-bution and ii) cross-sections vary substantially to-wards higher collision energies. • At lower collision energies, baryon stopping at themid-rapidity is expected and the dN ch /dη ( y ) is ex-pected to follow a Gaussian-like behaviour, whichcould be described by the particle production inLandau hydrodynamic model. Hence, at these en-ergies, the observation of a limiting fragmentationhypothesis in particle production is expected. Butat higher energies, where Landau hydrodynamicsfails due to the absence of Gaussian rapidity distri-bution, LF is found to be violated. • Going from the top RHIC energy to the LHC en-ergies, there is an order of magnitude increase inthe collision energy. Considering at least two unitsof (pseudo)rapidity overlap for the LF to be valid,the observed y beam at √ s NN = 200 GeV and 5.02TeV makes hardly any overlap in (pseudo)rapidity.While looking into the possible observation of lim-iting fragmentation, one looks at spectral overlapin the fragmentation region, which may not be ex-pected as mentioned. Hence, RHIC can’t be com-bined with LHC while looking for the hypothesis ofLimiting Fragmentation. • Theoretical models are mostly assumption depen-dent. In order to validate a model, one needsto confront a model to experimental data. Weneed forward charged particle and photon detec-tors at the LHC energies in order to validate theLF hypothesis. In the absence of this, extrapo-lation of any theoretical findings from mid-rapidityto extreme forward rapidity would be a speculationsometimes or a mere coincidence, as the physics ofparticle production is highly rapidity dependent. Inview of this, in the present work we have taken theinelastic cross-section with the collision geometryto study the LF hypothesis. This is the novelty ofthe present work.
ACKNOWLEDGEMENTS
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