Diagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions
Arghya Chatterjee, Sandeep Chatterjee, Tapan K. Nayak, Nihar Ranjan Sahoo
aa r X i v : . [ nu c l - e x ] O c t Diagonal and off-diagonal susceptibilities ofconserved quantities in relativistic heavy-ioncollisions
Arghya Chatterjee , Sandeep Chatterjee , , Tapan K. Nayak and Nihar Ranjan Sahoo Variable Energy Cyclotron Centre, HBNI, Kolkata 700064, India School of Physical Sciences, National Institute of Science Education and Research,Jatni, 752050, India Texas A&M University, College Station, Texas 77843, USAE-mail: [email protected]
Abstract.
Susceptibilities of conserved quantities, such as baryon number, strangeness andelectric charge are sensitive to the onset of quantum chromodynamics (QCD) phasetransition, and are expected to provide information on the matter produced in heavy-ion collision experiments. A comprehensive study of the second order diagonalsusceptibilities and cross correlations has been made within a thermal model approachof the hadron resonance gas (HRG) model as well as with a hadronic transport model,UrQMD. We perform a detailed analysis of the effect of detector acceptances andchoice of particle species in the experimental measurements of the susceptibilities forheavy-ion collisions corresponding to √ s NN = 4 GeV to 200 GeV. The transversemomentum cutoff dependence of suitably normalised susceptibilities are proposed asuseful observables to probe the properties of the medium at freezeout.PACS numbers: 25.75.-q,25.75.Gz,25.75.Nq,12.38.Mh
1. Introduction
Heavy-ion collisions at relativistic energies create matter at extreme conditions oftemperature and energy density, consisting of deconfined quarks and gluons. The majormotivations of facilities at the Relativistic Heavy Ion Collider (RHIC) at BrookhavenNational Laboratory, the Large Hadron Collider (LHC) at CERN and FAIR at GSI areto study the formation of this new form of matter, called quark-gluon plasma (QGP) andstudy its basic properties. In the Quantum Chromodynamics (QCD) phase diagram,location of the QCD critical point could be one of the compelling discovery in the heavy-ion collisions. The end point of the first order phase transition between hadronic matterto QGP phase is known as QCD critical point, after which there is no genuine phasetransition but a cross over from hadronic to quark-gluon degrees of freedom [1, 2]. While iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions √ s NN > . N B ) is as good as the proton ( p ),is not measured. Λ-baryon is the lightest strange baryon and hence contributes mostsignificantly to the baryon-strangeness correlation. Although Λ-baryon is measured overan ensemble through its charged daughter particles, its measurement on an event-by-event level poses a daunting task. Thus, only the moments of net charge ( N Q ) aremeasured faithfully. For the other charges we have to rely on proxies, e.g. net protonserves as a good proxy for net baryon and net kaons serve as a proxy for net strangeness.We have studied the dependence on the observed particle sets for all the secondorder susceptibilities. We have made a comprehensive study of the effect due tofinite number of the conserved quantities. Ideally, in order to observe grand canonicalfluctuations of the conserved charges, the fraction R of the conserved charge carriedby the system to that of the total available charge carried by the bath as well as thesystem should be much smaller than half ( R << . N B equals to twice the mass number of the nucleus and N Q equals to twicethe atomic number of the nucleus that are distributed in the momentum rapidity ( y )direction within | y max | ∼ log (cid:0) √ s NN /m p (cid:1) . Thus, the final distribution of this N B and iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions N Q into the system and bath depends on three factors: (a) the available experimentalacceptance, (b) √ s NN , and (c) the baryon stopping phenomenon which decides the initialrapidity distribution of the conserved charges. It is important to note that the strong √ s NN dependence of baryon stopping rubs off on to R , complicating the comparison ofmeasured moments at different √ s NN .In this article, we have used the Ultra-relativistic Quantum Molecular Dynamics(UrQMD v3.3) as well as the hadron resonance gas (HRG) model to analyze the diagonaland off-diagonal susceptibilities of the conserved charges. Model descriptions forUrQMD and HRG are presented in Section II. In Section III, we discuss the observablesused for the present study. In Section IV, we present the collision energy dependence ofall the susceptibilities. We discuss the species dependence on the observables and effectof detector acceptances. We present the results from both the models. The article issummarised in Section V. An appendix at the end discusses the estimation of statisticalerrors associated with the observables.
2. Model considerations
UrQMD is a microscopic transport model [32, 33]. In this model, the space-timeevolution of the fireball is studied in terms of excitation of color strings which fragmentfurther into hadrons, the covariant propagation of hadrons and resonances whichundergo scatterings and finally the decay of all the resonances. This model setup hasbeen quite successful and widely applied in heavy-ion phenomenology [32, 33]. It hasalso been previously used to compute several susceptibilities [34, 35, 36, 37, 38]. Theacceptance window plays an important role in such studies. The initial distributionof N B and N Q in y is a consequence of the baryon stopping phenomenon whichhas a strong √ s NN dependence- as a result at higher √ s NN the mid-rapidity regionis almost free of N B and N Q while at lower √ s NN almost all the N B and N Q aredeposited in the mid-rapidity region. This is also expected to have significant effecton the fluctuations of conserved quantities. This √ s NN dependent baryon stoppingphenomenon is dynamically included in the UrQMD approach. Here, we have generatedaround a million events per beam energy from √ s NN = 4 −
200 GeV.We have compared the UrQMD results with those from a thermal approach, byusing the HRG model. Here it is important to note that while the HRG results areexpected to reflect the susceptibilities at the time of chemical freezeout where inelasticcollisions cease, the UrQMD results reveal the above quantities at the kinetic freezeoutsurface where the elastic collisions cease. Thus, while the latter are more realisiticin the context of heavy ion collisions, the HRG results here serve as useful guide fora qualitative understanding of the results. The HRG model consists of all hadronsand resonances as listed in the Particle Data Book [39] within the framework of amultiple species non-interacting ideal gas in complete thermal and chemical equilibrium.The only parameters are the temperature T , chemical potentials µ B , µ Q and µ S corresponding to the conserved quantities of baryon number B , electric charge Q and iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions √ s NN (GeV) 10 V (MeV − ) T (MeV) µ B (MeV) µ Q (MeV) µ S (MeV)6.27 1.4 130.8 482.4 12.7 106.57.62 1.3 139.2 424.6 11.7 96.17.7 1.3 139.6 421.6 11.6 95.58.76 1.1 144.2 385.7 10.9 88.811.5 1.5 151.6 316.0 9.5 75.017.3 2.0 158.6 228.6 7.4 56.539 1.7 164.2 112.3 4.1 29.462.4 1.8 165.3 72.3 2.8 19.4130 2.1 165.8 35.8 1.4 9.8200 2.5 165.9 23.5 1.0 6.4 Table 1.
The chemical freezeout parameters extracted from mid-rapidity data atdifferent √ s NN [20, 40]. strangeness S , and the volume V of the fireball which are obtained by fits to data. TheHRG model has been found to provide a very good description of the mean hadronyields using a few thermodynamic parameters at freeze-out (for a recent compilationof the freeze-out parameters, see Ref. [40]). In this work, we have used the sameparametrisation for the √ s NN dependence of T , µ B , µ Q and µ S as given in Ref. [20].The chemical freezeout volumes, V (cid:0) √ s NN (cid:1) are taken from Ref. [40]. These parametersare listed in Table 1.Lately, susceptibilities have also been employed to study the freeze-out conditionsof the fireball within the HRG model [27]. All the quantities of interest can be computedfrom the partition function Z ( V, T, µ B , µ Q , µ S )ln Z = X i ln Z i = X i aV g i (2 π ) d p ln (cid:16) ae ( − ( p + m i ) + µ i ) /T (cid:17) (1)= V T X i g i π (cid:16) m i T (cid:17) ∞ X l =1 ( − a ) l +1 l − K ( lm i /T )exp[ l ( B i µ B + Q i µ Q + S i µ S ) /T ] , (2)where a = − g i , m i , B i , Q i , S i and µ i = B i µ B + Q i µ Q + S i µ S (3)refer to the degeneracy factor, mass, baryon number, electric charge, strangeness andhadron chemical potential respectively of the i th hadron species, V is the part of thefireball volume under study that can also be called the system and K is the modifiedBessel function of the second kind. From ln Z , all thermodynamic quantities could becomputed. iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions
3. Observables and methods
The susceptibilities of the conserved quantities of the strongly interacting matter inthermal and chemical equilibrium can be computed within the grand canonical ensemble(GCE) from partial derivatives of the pressure ( P ) with respect to the chemicalpotentials χ ijkBQS = ∂ i + j + k ( P/T ) ∂ i ( µ B /T ) ∂ j ( µ Q /T ) ∂ k ( µ S /T ) , (4)where the P is obtained from ln Z as follows P = TV ln Z (5)From the experimental point of view it is straightforward to compute the centralmoments, M, M ijkBQS = h ( B − h B i ) i ( Q − h Q i ) j ( S − h S i ) k i . (6)Using the fact that the generating function for the cumulants is given by the logarithmof that of the moments, one can express one in terms of the other. Up to the secondorder, this relationship is one-to-one, such as, χ XY = 1 V T M . (7)Using this relation, all the diagonal and non-diagonal susceptibilities of second ordercan be expressed in terms of second order central moments ( σ ): σ Q σ , QB σ , QS σ , BQ σ B σ , BS σ , SQ σ , SB σ S The ratios of of χ ijXY to χ Y can be suitably constructed to cancel the volume effect.In the quasiparticle picture of quarks and gluons, the ratios χ BS /χ S and χ QS /χ S become-1/3 and 1/3 respectively. It is not possible to find such simple factors for other ratioslike χ QB /χ B that receive contribution from both light and strange quarks. Thus thefollowing the ratios of off-diagonal and diagonal susceptibilities have been constructed: C BS = − χ BS χ S , C SB = − χ BS χ B , (8) C QS = 3 χ QS χ S , C SQ = χ QS χ Q , (9) C QB = χ QB χ B , C BQ = χ QB χ Q . (10)We have studied the susceptibilities and these ratios as a function of collision energy anddetector acceptances in terms of pseudorapidity ( η ) and transverse momentum ( p T ). iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions BS C UrQMDHRG Q S C (GeV) NN S10 Q B C Figure 1. (Color online) Beam energy dependence of ratios of off-diagonal to diagonalsusceptibilities for central (0-5%) Au+Au collisions using UrQMD and HRG models.
4. Beam energy dependence
In Figure 1, we present the ratios, C BS , C QS , and C QB as a function of √ s NN for topcentral (0-5% of total cross section) Au+Au collisions using both UrQMD and HRGmodels. In this construction, the ratios are calculated by considering all the charges,baryons and strange particles within | η | < . . < p T < . C BS decrease and remain constantafter √ s NN = 27GeV, whereas the values of C QS increase and then remain constantafter √ s NN = 20GeV. The values of C QB show decreasing trend with increasing √ s NN .These trends can be understood on the basis of the thermal model framework ofthe HRG. C BS gets dominant contribution from Λ (being the lightest strange baryon)in the numerator while in the denominator, it gets contribution mainly from the kaons(being the lightest strange mesons). With decreasing √ s NN , µ B increases which inturn enhances the relative contribution from Λ compared to kaons resulting in theincreasing trend of C BS . In case of C QS , it receives dominant contribution from thekaons both in the numerator and denominator. However the contribution from thelightest strange baryon Λ to χ S keeps growing with decreasing √ s NN . This results inthe monotonic increasing trend for C QS with √ s NN . Finally, the weak rising trend of C QB with decreasing √ s NN can be traced to the contribution from the multiply charged∆ to the numerator. Susceptibilities and their ratios have a strong species dependence. In the experiments,only charged hadrons are measured. At present, it is possible to perform an event-by- iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions Allp,k, π,Λ p,k, π
10 100√ S NN
01 10 100 √ S NN √ S NN (GeV) (GeV) (GeV) σ σ σ B Q S σ σ σ BS QS QB - C BS C QS C QB Figure 2. (Color online) Particle species dependence on the diagonal and off-diagonalsusceptibilities and their ratios in central (0-5%) Au+Au collisions using the UrQMDmodel calculations. event analysis of only π + , K + , p and their anti-particles. Neutral hadrons, like K , n and Λ, which contribute significantly to the strangeness fluctuation, baryon fluctuationand baryon-strangeness correlation, respectively, are not measured on an event-by-eventbasis. We have estimated the effect of such missing contributions by computing all theratios for three different hadron sets: (i) considering all hadrons as in Fig. 1, (ii) with π ± , K ± , p , ¯ p and Λ and ¯Λ, and (iii) with only π ± , K ± , p and ¯ p . The results of the studyusing UrQMD model are shown in Fig. 2 as a function of collision energy. Ratios like C BS , C QS and C QB are constructed using variances (like σ B , σ S and σ Q ), and covariances(like σ , BQ , σ , SB and σ , QB ) of different conserved quantities. Large differences have beenobserved for the three cases. In all cases, it is clear that using only π ± , K ± , p , ¯ p asevent-by-event measurements, the sensitiveness of the ratios as a function of collisionenergy reduces to a large extent. For C BS , there is practically no difference between thecases with all particles and those with π ± , K ± , p , ¯ p and Λ and ¯Λ. But the trend foronly π ± , K ± , p , ¯ p is very different. It is necessary to analyse at least Λ and ¯Λ includedin the measurements. Here we note that an additional hurdle in experiments will beto distinguish Σ from Λ. However, our results here with all the particles as well asonly π ± , K ± , p , ¯ p and Λ and ¯Λ are almost identical. Thus the role of Σ is probablysub-dominant. It is well known that in UrQMD, the yields of multi-strange baryons are iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions S Q C all particles Λ ,K,P, π ,K,P π (GeV) NN S B Q C Figure 3. (Color online) The measurements for C SQ and C BQ are robust under thedifferent choices of particle set for central (0-5%) Au+Au collisions .highly underestimated [41, 42]. It might be possible that similar reason also leads to theabove observation of marginal influence of the multi-strange baryons towards strangesusceptibilities as well as baryon-strange correlations. C QB on the other hand is almostconstant across all √ s NN - the case with only π ± , K ± , p and ¯ p is roughly twice of thecase when we include all hadrons due to the missing neutrons in the denominator in theformer.In order to reduce the dependence on the choice of hadron set we also look at theother ratios, C SQ and C BQ , where only charged hadrons contribute. We show the resultsof these two quantities from UrQMD in Fig. 3. For both of these quantities, the leadingcontributors to numerator as well as denominator come from the measured hadron sets π ± , K ± , p and ¯ p . Thus, we find that these results are quite stable to further inclusionof other hadrons and hence can be estimated in experiments with the limited particleidentification capability. Limited event-by-event particle identification, realistic efficiency corrections in theexperiments and finite kinematic acceptances in p T and η - all of these contribute todilute the signal for susceptibilities in the experiments [43]. Some of these effectshave been already discussed earlier [44, 45, 22, 46, 47, 48]. In the previous sectionwe discussed the effect of limited particle identification on susceptibilities. We will nowdiscuss specifically the effect of the window of detector acceptances in terms of η and p T windows for measuring the susceptibilities.Ideally, grand canonical fluctuations trivially scale with system volume when incontact with an infinite bath. On the other hand, in heavy-ion collisions for large enough iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions > ba t h < X > sys t e m < X R = − ≤ | η | 1.0 ≤ | η | 1.5 ≤ | η | BX = B - (GeV) NN S > ba t h < X > sys t e m < X R = − BX = B +
Figure 4. (Color online) The ratios of net baryon (top panel) and total baryon(bottom panel) numbers within the system to the bath + system for central Au+Aucollisions are plotted as a function of collision energy using the UrQMD model. Theresults are presented for three different η windows. acceptance it is possible that the system size becomes comparable with that of the bathand even larger. Global charge conservation in such cases cause suppression of thermalfluctuation resulting in non-thermal fluctuations [49, 50]. Thus, the interpretationof conserved charge fluctuations in terms of thermal and critical fluctuations is notstraightforward in such cases. Another factor that adds to the above complicationis the fact that baryon stopping is not constant across √ s NN , resulting in completelydifferent distributions of conserved charges in η for different √ s NN .The rapidity dependence of baryons affects the beam energy dependence of the ratio R of the number of baryons carried by the system to the total carried by the system aswell as bath. Ideally, in order to observe grand canonical fluctuations R << .
5. Westudy this ratio for net baryons and total number of baryons as a function of collisionenergy for central Au+Au collisions using UrQMD model. The ratios are plotted forthree different η windows (0.5, 1.0 and 1.5) in Fig. 4. We find that at high √ s NN , R is sufficiently less than 0 .
5. However, at lower √ s NN , the | η | < . .
25 while for even larger acceptance windows, R > . N B fluctuations. Thus, a fixed η window across allbeam energies does not correspond to same system to bath effective volume ratio forall √ s NN [51]. This suggests that for low √ s NN , | η | < . η . We have studied in detail the dependence of susceptibilitieson the acceptance window in p T and η . Let us first analyse the acceptance dependence of the susceptibilities of conservedquantities using the HRG model. From Eq. 2 and 4, we can write the susceptibilities iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions χ - χ χ (GeV) maxT p χ HRG χ χ Figure 5. (Color online) The p T max dependence of the second order susceptibilitiesin the HRG model for central Au+Au collisions at three colliding energies. BS1,1 χ - S2 χ QS1,1 χ | max η | Q2 χ HRG
QB1,1 χ B2 χ Figure 6. (Color online) The η max dependence of the second order susceptibilities inthe HRG model for central Au+Au collisions at three colliding energies. due to the h -th hadron as, χ hijkBQS = g h (2 π ) ∞ X l =1 e lµ i /T ( − a ) l +1 l ( i + j + k ) − B i Q j S k Z y rmax − y rmax dy r Cosh ( y r ) Z y max y min dy y e − y Cosh( y r ) (11)where y min = lT p p T min + m h and similarly for y max . For ease of understanding, we havewritten down the explicit cutoff dependence on the momentum rapidity y r instead of thepseudorapidity η which is more relevant experimentally. As noted from the integrand,the Boltzmann factor in terms of y r and transverse momentum p T is e − l √ p T + m /T Cosh( y r ) . iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions BS1,1 σ -3 S2 σ QS1,1 σ (GeV) maxT p σ UrQMD QB1,1 σ B2 σ Figure 7. (Color online) The p T max dependence of the second order susceptibilitiesin the UrQMD model for central Au+Au collisions at three colliding energies. − BS1,1 σ -3 − S2 σ QS1,1 σ | max η | Q2 σ UrQMD QB1,1 σ
110 B2 σ Figure 8. (Color online) The η max dependence of the second order susceptibilities inthe UrQMD model for central Au+Au collisions at three colliding energies. The Cosh ( y r ) dependence ensures that thermal production of particles is more stronglysuppressed in y r as compared to p T /T .The susceptibilities have a strong dependence on the maximum value of transversemomentum ( p T max ) value and maximum η -window ( η max ). These dependences arestudied for central Au+Au collisions for three collision energies and plotted in Figs. 5and 6 as a function of ( p T max ) and η max , respectively. All the susceptibilities show similartrends with both p T max and η max . Small values of susceptibilities are observed for small p T max and η max which steadily grow with increasing p T max and η max as more phase spaceis included, finally saturating off to a constant value as the Boltzmann factor suppressesany further contribution from high p T and η . iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions Q2 σ B2 σ S2 σ BS1,1 σ QS1,1 σ QB1,1 σ = 7.7 GeV NN S UrQMD (GeV) maxT p1 20.51 Q2 χ B2 χ S2 χ BS1,1 χ QS1,1 χ QB1,1 χ = 200 GeV NN S HRG
Figure 9. (Color online) The p T max dependence of all second order susceptibilitiesnormalised by their values for p T max = 2 GeV and η max = 0 . √ s NN = 7.7 GeV (upper panel) and 200 GeV (lower panel) by using theHRG and UrQMD models. Figures 7 and 8 show the different second order susceptibilities with acceptances interms of p T max and η max as obtained using the UrQMD model. The p T max dependence ofsusceptibilities turns out to be similar to those obtained from the HRG model. However,the η max dependence exhibits quite different behaviour, primarily because of globalconservation. There is an initial increase for small η max and attain maximum valueat intermediate rapidity window within η max ∼ η ∼ The p T max and η max dependence of the susceptibilities within the HRG and UrQMDmodels presented in the earlier sections can be compared and nicely summarised afterthey are suitably normalised. Here, we have normalised by their values at p T max = 2GeV and η max = 0 .
5. We denote these normalised susceptibilities as ˆ χ and ˆ σ . Thusby construction, for p T max = 2 GeV and η max = 0 .
5, ˆ χ and ˆ σ are unity. Figure 9shows p T max dependence of the variance and covariance of conserved quantities, fromboth UrQMD and HRG models, in central Au+Au collisions for two collision energies.For small p T max , the susceptibilities in both the models approach zero as the systemphase space volume approaches zero. In all the cases the fluctuations grow with p T max before saturating to a constant value. It is interesting to observe a clear conserved iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions (GeV) maxT p s i ng l e2 χ π m π π
10 m
Figure 10. (Color online) The p T max dependence of the second order susceptibilitynormalised by the value for p T max = 2 GeV and η max = 0 . charge ordering in these normalised susceptibilities with the increase of p T max . This isobserved in both HRG and UrQMD model calculations. ˆ χ Q that receives contributionmainly from net pion reaches its saturation value fastest while ˆ χ B that gets contributionfrom net proton saturates at larger values of p T max . ˆ χ S which mainly gets contributionfrom kaons saturates at intermediate p T max , closer to that of ˆ χ Q .In the HRG setup, it is easy to understand that such conserved charge orderingarises from the ordering of the masses of the hadrons that contribute dominantly tothe different susceptibilities. This can be clearly understood within the framework ofa single particle ideal gas. We have plotted the second order normalised susceptibilityof this single particle ideal gas for three different masses of the particle. The resultsof these calculations are shown in Fig. 10. As the mass of the particle increases, thesaturation in ˆ χ single kicks in at a higher p T max . As we go from HRG to UrQMD, we findthe effect of the mass ordering gets even more pronounced as seen in Fig. 9.We have plotted the η max dependence of all the normalised susceptibilities with p T max = 2 GeV in Fig. 11. They are all normalised by their values for η max = 0 . p T max dependence, in this case all the plots collapse on each other and wedon’t see any mass ordering in the HRG results. The UrQMD results show a similarcollapse for η max ≤ η max ∼ η max . The location of the peaks of the baryonic moments are clearlydictated by the baryon stopping effect and closely follow the peak in the η distributionof baryons. The mesonic moment peaks are much less sensitive to √ s NN . iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions σ σ σ σ σ σ χ χ χ χ χ χ = 7.7 GeV NN S UrQMD
HRG | max η |2 4 6024 = 200 GeV NN S Figure 11. (Color online) The η max dependence of all second order susceptibilitiesnormalised by their values for p T max = 2 GeV and η max = 0 .
5. Summary
The moments of conserved charges are important observables to probe the QCD phasediagram in heavy ion collisions. These provide glimpses of the thermal conditionsprevalent in the fireball. They quantities are also expected to carry the signaturesof non-thermal behaviour, like those close to the QCD critical point. Although thestudy of these observables is very well motivated, there are several experimental issuesthat need to be understood in order to interpret the data and draw physics conclusions.Using UrQMD and HRG models, we have studied some of the issues with respect toparticle identification as well as detector acceptances in the experiments with regard toall the second order susceptibilities. The main results of our study are as follows:(i) χ Q , χ BQ and χ QS can be measured accurately with the event-by-eventmeasurements of the limited particle set: π ± , K ± , p and ¯ p , while the measured χ B from net proton roughly scale as half of that expected from the complete particleset due to the missing neutrons,(ii) We should have | η max | ≤ . √ s NN >
10 GeV,(iii) Suitably normalised susceptibilities show a conserved charge ordering in the p T acceptance in HRG as well as in UrQMD. The net charge susceptibilities saturateto their maximum value at smaller p T max value followed by net strangeness andnet baryon. For a thermal medium, the p T max dependence arises from the differentmasses of the relevant degrees of freedom that contribute to these conserved charge iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions
6. Acknowledgement
We acknowledge many helpful discussions on susceptibilities with Sourendu Gupta andPrithwish Tribedy. SC acknowledges XII th plan project no. 12-R&D-NIS-5.11-0300 andCNT project PIC XII-R&D-VECC-5.02.0500 for support. NRS is supported by theUS Department of Energy under Grant No. DE-FG02-07ER41485. This research usedresources of the LHC grid computing center at the Variable Energy Cyclotron Center,Kolkata, India.
7. Appendix: Statistical error estimation
Let us consider the observable, which is the ratio of off-diagonal ( c , ) to diagonal ( c , )cumulants of conserved charged distributions, C XY = α c , c , , (12)where α is a constant, and X and Y are the net charge ( Q ), net baryon ( B ) or netstrangeness ( S ). C X,Y can be expressed as C X,Y = φ ( c , , c , ) . (13)Now using the error propagation formula, one can find the variance of φ ( X i , X j )as, V ( φ ) = n X i =1 ,j =1 ∂φ∂X i ∂φ∂X j Cov ( X i , Xj )= n X i =1 ( ∂φ∂X i ) V ( X i )+ 1 N n X i,j =1 ,i = j ∂φ∂X i ∂φ∂X j Cov ( X i , Xj ) (14)Using X = c , and X = c , , the variance of φ ( c , , c , ) can be expressed in termsof the following: V ( φ ) = ( ∂φ∂c , ) V ( c , ) + ( ∂φ∂c , ) V ( c , )+ 2 ∂φ∂c , ∂φ∂c , Cov ( c , , c , ) . (15) iagonal and off-diagonal susceptibilities of conserved quantities in relativistic heavy-ion collisions c m,n and c k,l ) [52] is Cov ( c m,n , c k,l ) = 1 N ( c m + k,n + l − c m,n c k,l ) . (16)Where N is the total number of events in an ensemble. Using above two equations oneobtains, V ( C X,Y ) = α N [( 1 c , )( c , − c , ) + ( c , c , )( c , − c , )+ 2( − c , c , )( c , − c , c , )] . (17)The error in C X,Y is finally written aserror = p V ( C XY ) . (18)Similarly, one can derive error of any observable (the higher order off-diagonal anddiagonal cumulants of different multiplicity distributions). References [1] L. McLerran et al., Phys.Rev.
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