Experimental equation of state in proton-proton and proton-antiproton collisions and phase transition to quark gluon plasma
aa r X i v : . [ h e p - ph ] J un Experimental equation of state in pp and p¯p collisionsand phase transition to quark gluon plasma
Renato Campanini a,b, ∗ , Gianluca Ferri a a Universit`a di Bologna, Dipartimento di Fisica, viale C. Berti Pichat 6/2, I-40127,Bologna, Italy b INFN, Sezione di Bologna, viale C. Berti Pichat 6/2, I-40127, Bologna, Italy
Abstract
We deduce approximate equations of state from experimental measurementsin pp and p¯p collisions. Thermodynamic quantities are estimated combiningthe measure of average transverse momentum h p T i vs pseudorapidity density dN ch dη with the estimation of the interaction region size from measures of BoseEinstein correlation, or from a theoretical model which relates dN ch dη to theimpact parameter. The results are very similar to theory predictions in case ofcrossover from hadron gas to quark gluon plasma. According to our analysis,the possible crossover should start at dN ch dη ≃ dN ch dη ≃ Keywords: quark gluon plasma, average transverse momentum vs pseudorapiditydensity, equation of state, Bose Einstein correlation, hadron gas, soundvelocity ∗ Corresponding Author: Tel. +39 051 2095078, Fax +39 051 2095047,Mobile +39 3485925020
Email addresses:
[email protected] (Renato Campanini), [email protected] (Gianluca Ferri)
Preprint submitted to Physics Letters B September 10, 2018 . Introduction Some of the most important questions about the transition to the quark gluon plasma (QGP), a new state of matter with partonic degrees freedom, are not yet fully answered. Among them the location of phase boundaries between hadronic gas and the QGP. The results of lattice QCD simulations concerning the order of phase transition depend strongly on the number of quark flavors and on the quark masses. For vanishing baryon chemical potential µ b = 0, the nature of transition can be a genuine phase transition (first order or continuous), or just a rapid change (crossover) over a small temperature range [1]. Estimates of energy densities which can be achieved in ultra-relativistic pp or p¯p collisions with high multiplicities suggest values sufficiently high for experimental formation of the QGP [2]. However it may be that, unlike what happens in heavy ion interactions, in pp and p¯p the central blob of created matter never thermalizes [3], although there are different opinions [2, 4–7] which predict that thermodynamics con- cepts may be applied in pp or p¯p high multiplicity events. Probes of equation of state are among possible signatures of phase tran- sition or crossover. The basic idea behind this class of signatures is the identification of modifications in the dependence of energy density ǫ , pres- sure P and entropy density σ of hadronic matter on temperature T . One wants to search for a rapid rise in the effective number of degrees of freedom, as expressed by the ratio ǫ/T or σ/T , over a small temperature range. One can expect a step-like rise as predicted by lattice simulations (Fig. 1), more or less steep depending from the presence of transition or crossover, and from the order of the transition in the former case. Finite volume effects may ǫ/T and σ/T : the latent heat and the jump in the entropy density are considerably reduced for small systems [8]. Besides that, the critical temperature may shift to higher temperatures and the width of the transition may broaden for smaller volumes and there may be a smoothening of singularities due to the finite size of the system [8–10]. In 1982 it has been suggested by Van Hove [12] that an anomalous be- havior of average transverse momentum h p T i as function of the multiplicity could be a signal for the occurrence of a phase transition in hadronic matter. His conjecture is based on the idea that the h p T i distribution of secondaries reflects the temperature of the system and its evolution in the transverse direction, while the multiplicity per unit rapidity provides a measure of en- tropy [13, 14]. In a recent paper [15] one of us showed that from 22 to h p T i vs pseudorapidity density dN ch dη curves there is a slope change at dN ch dη = 5 . ± .
2. Signals related to these slope changes may indi- cate transition to a new mechanism of particle production. Many years ago, in [16], we pointed out that pp at ISR and p¯p data at CERN collider showed a kind of jump at dN ch dη = 6 and that it had to be investigated as a possible phase transition signal [17]. In 2002, Alexopoulos et al. [18] assumed that the system produced in p¯p at √ s = 1800 GeV for dN ch dη > .
75 was above the deconfinement transition to explain their experimental results. In present article, taking into account experimental results in pp and p¯p at high energies [19–32], we show how measured physical quantities satisfy relations which, given proper approximations and correspondences, can give a representation of the equations of state (EOS) that describe the created system in the central region in pseudorapidity in high energy pp and p¯p a) (b)(c) Figure 1: Results of the lattice simulations of QCD for
T > T c (critical tem-perature) and from ideal hadron-gas model for T < T c .1a: entropy density σ scaled by T calculated in the hadron-gas model andby lattice simulations of QCD shown as function of temperature. The verti-cal line indicates the critical temperature.1b: sound velocity c shown as function of the energy density ǫ .1c: temperature dependence of the square of the sound velocity at zerobaryon density as function of T . In this case the critical temperature T is equal to 170 MeV.From [11]. 4ollisions. Starting from h p T i vs dN ch dη experimental results together with the estimation of the size S of the interaction area, which is obtained from the measurements of the radii of emission in function of multiplicities [23–26], or from a model which relates multiplicity to impact parameter [33], we obtain relations among h p T i and particle density σ S , which seem to resemble EOS curves predicted for hadronic matter with crossover to QGP. The h p T i and S vs dN ch dη relations contain the relevant information, which translates in h p T i vs σ S correlations. According to our knowledge, this is the first attempt to obtain an esti- mation of the complete EOS for hadronic matter using experimental data only.
2. Methods The experimental results and the approximations made in this work are the following. h p T i vs dN ch dη As we mentioned before, h p T i vs dN ch dη correlation at about dN ch dη = 6 shows a slope change in all the experiments. In Van Hove scheme h p T i reflects tem- perature and the system evolution. On the other hand, the biggest part of emitted particles is constituted by pions and the pion h p T i is rather insensitive to flow [34]. Thus, not identified charged particles h p T i may be considered as an estimation of the system temperature because it’s not influenced very much by a possible transverse expansion. Furthermore, transverse radius R side vs pair transverse momentum k T in pp Bose Einstein correlation mea- sures [24] show that, at least until dN ch dη ≃ .
4, results are consistent with the h p T i as an identifier of temperature, because it’s little affected by the expansion. Since a substantial number of pions is the product of resonance decay and the particles originating from the resonance decays populate the low p T region [11], in this work we consider mainly h p T i vs dN ch dη correlations with a p T min cut ( >
400 MeV/c in CDF experiments Run I and Run II , > MeV/c in ALICE, and two different cuts, >
500 MeV/c and > in ATLAS experiment) in order to work with h p T i values less influenced by this effect. Furthermore, diffractive events are substantially reduced for dN ch dη & h p T i vs dN ch dη plots with p T min ≥
400 MeV/c [21, 22]. We will show anyway also some results for h p T i computed with p T min cut >
100 MeV/c. The structure of the relations we are going to show is still present in these measures. In this work, h p T i computed for different p T cuts will be plotted without the application of corrections due to the cut in the used p T range, apart from the case of events energy density estimation, in which we will use a corrected h p T i . Regarding dN ch dη , it is computed from the number of particle in a given region of pseudorapidity η and p T , dividing by the amplitude of the η range and properly correcting for p T cuts. In order to perform this last correction, we considered dN ch dη curves for the different experiments, measured with and without p T cuts, and multiplied by the ratio between correspondent values of dN ch dη in the central region. All data are obtained from minimum bias experiments. For CDF run II 1960 GeV, high multiplicity trigger data are added to minimum bias data, for charged particle multiplicity N ch ≥ ( | η | < p T >
400 MeV/c corresponding to dN ch dη corrected ≥
22) [15, 28]. out experimental inefficiencies corrections) because the h p T i vs dN ch dη plot and its derived plots are much sensible to experimental losses and, on the other hand, the application of corrections may involve some “smearing” of data which could highly modify the analyzed effects [19, 35]. For reasons of space we don’t show behaviors for raw data in this paper, because results are very similar to those for corrected data. The initial energy density in the rest system of a head-on collision hasbeen argued to be [6]: ǫ ≃ dN ch dη · h p T i VV denotes the volume into which the energy is deposited. Similarly the initialentropy density is [2]: σ ≃ dN ch dη · V As a result, ǫ is equal to σ · h p T i . The volume V may be estimated as V = S · ct , where S is the interaction area and ct is a longitudinal dimension we can traditionally consider to be about 1 fm long. In order to study our system, we will use the quantity σ S = dN ch dη · S as an estimation of entropy density. In models like color glass condensate and percolation, the system physics depends on σ S [36–38]. For the estimation of the area of interaction S , we proceed in different ways, our target being the obtainment of results which are robust respect to the definition of the area.
7n the other hand, we are more interested in relations between variables than in their absolute values.
Using Bose Einstein correlation among emitted particles, measurements of particle emission regions in many pp and p¯p experiments have been done [23–
26, 31, 32, 39–41] In [23, 24], as already mentioned, the measurement of R side in function of k T in pp, shows that the transverse radius doesn’t depend on k T for low dN ch dη values ( < pansion of the particle emission source, at least at these dN ch dη values. For dN ch dη values greater than 7, there is a dependence of R side on k T , so that probably a source expansion is possible at least from this dN ch dη value. It is thus possi- ble that a new phenomenon is started in events with dN ch dη between 3.4 and
7. The hypothesis of no expansion for low dN ch dη values lets us approximate the initial interaction section radius to be coincident to the final emission radius. We take into account that for dN ch dη values greater than about 7.5, this approximation is more uncertain. Furthermore, resonance effects are present, but, once more, we are not interested in the absolute values of the interaction section, but in its behavior in function of dN ch dη . Not taking into account these effects yields a systematic error on the value of the radius, which we consider invariant for different values of dN ch dη . Given the similarity in both the behavior and the absolute values of R side and invariant radius R inv versus multiplicity, we use R inv as an estimation of the interaction region radius, mainly because R inv data where measured for a larger dN ch dη range than R side ones [23–26, 31, 32, 39, 40]. In Fig. 2, R inv is shown as a function of pseudorapidity density. In the left we only show data for CMS (preliminary) h k T i ≃ .
35, while in the right side we show the same results along data from other experiments (UA1, ABCDHW ISR,
STAR). We fitted the data of Fig. 2a with two functional relations between R inv and dN ch dη : the first is linear in the cube root of dN ch dη [23, 24, 42] and the second is linear in cube root of dN ch dη for dN ch dη > .
5, matched with a 5 th de- gree polynomial fit for smaller dN ch dη values. The first fit gives a χ NDoF = 0.84 with p -value: 0.67, while the second gives a χ NDoF = 0.45 with p -value: 0.95. Considering these results, we opted to use the second fit for the following analysis.
It has been stated that the behavior of radii in function of dN ch dη doesn’t depend on the experiment energy [41]. Data in Fig. 2 seem to confirm this statement, and justify our choice of a single relation for R inv vs dN ch dη for all energies. In order to estimate the interaction region, we used the following alter- natives:
1. An area obtained using R inv from the combined (polynomial + linear in cube root of dN ch dη ) fit from Fig 2a. This choice may overestimate the interaction region in case of system expansion, being R inv a measure of the emission region;
2. following ALICE results in R side vs k T , we make the hypothesis that no expansion is present in events with sufficiently low dN ch dη . So we use R inv from the left (polynomial) part of the combined fit in Fig 2a, then we use a constant radius for dN ch dη > . dimensions of the initial region before the possible expansion, making the assumption that at dN ch dη ≃ . dNd (cid:0) R i n v [ f m ] fit: linear in dN ch d (cid:1) (cid:2) NDoF : 0.84 p-value: 0.67fit: combined: poly5 + linear in dN ch d (cid:3) (cid:4) NDoF : 0.45 p-value: 0.95ALICE 0.9 TeV (Preliminary)ALICE 7 TeV (Preliminary)CMS 7 TeV (Preliminary)CMS 0.9 TeV (Preliminary) (a) dNd (cid:5) R i n v [ f m ] ISR_SFM 63 GeV elec. trig.ISR_SFM 63 GeV min. biasISR SFM 44 GeV min. biasUA1 630GeVSTARALICE 0.9 TeV (Preliminary)ALICE 7 TeV (Preliminary)CMS 0.9 TeV (Preliminary)CMS 7 TeV (Preliminary) (b)
Figure 2: R inv vs dN ch dη for different experiments [23–26, 31, 32, 40]. Leftplot shows data from CMS (preliminary) and ALICE(preliminary) both at h k T i ≃ .
35, along with linear fit in dN ch dη and a combined fit: 5 th degreepolynomial for dN ch dη < . dN ch dη for dN ch dη > .
5. Rightplot shows data from the first plot along with data from UA1, ABCDHWISR, and STAR experiments. 10aximum; at dN ch dη = 7 . inv value is 1 .
08 fm;
3. an area obtained from a model which relates the impact parameter to the multiplicity of events [33].
From dN ch dη values and from interaction areas, estimated as described above, we obtained the values of density of particles for transverse area. We considered the simplified case where the central blob volume V is the same in all collisions for a given dN ch dη [12]. We estimate an average σ from the ratio between dN ch dη and the estimated average V . σ S / h p T i vs h p T i Using the estimated σ S , the relation h p T i vs σ S can be studied. A slope change in h p T i vs σ S plots is found at σ S between 2.5 and 3 fm − , depending on the method used for the estimation of area S and corresponds directly to the slope change seen in h p T i vs dN ch dη at dN ch dη ≃ Starting from σ S and h p T i , we plotted σ S / h p T i vs h p T i curves, as an experimental approximation of σ/T vs T curves. See Figs. 3 and 4. We obtained very similar σ S / h p T i vs h p T i curves from other pp and p¯p experi- ments [27, 29–31] (not shown). In figures, we put labels with corresponding dN ch dη values for interesting points, in order to relate these points to the characteristic values in dN ch dη . In the different plots, different regions are recognizable. In particular, in all plots we see that from the σ S / h p T i value corresponding to dN ch dη ≃
2, up to a value correspondent to dN ch dη ≃
6, the curve is almost flat, then rises very quickly. This behavior is similar to the one in σ/T curve, in presence of crossover, starting from a state of matter, identified by σ/T nearly constant (cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:7) (cid:5)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8)(cid:9)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8) (a) (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:7) (cid:5)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8)(cid:9)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8) (b) (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:7)(cid:5)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8)(cid:9)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8) (c) (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6) (cid:5)(cid:7)(cid:5)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8)(cid:9)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:10) (d) Figure 3: σ S / h p T i vs h p T i plots. Area S from “5 th degree polynomial +constant after dN ch dη > .
5” fit.3a: ALICE at √ s = 0.9 TeV, 0 . < p T < | η | < .
8, Minimum Bias.3b: ATLAS at √ s = 0.9 TeV, p T > . | η | < .
5, Minimum Bias.3c: ATLAS at √ s = 7 TeV, p T > . | η | < .
5, Minimum Bias.3d: ATLAS at √ s = 7 TeV, p T > . | η | < .
5, Minimum Bias.12 (cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:7)(cid:5)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8)(cid:9)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8) (a) (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:7)(cid:5)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8)(cid:9)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8) (b) (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:7) (cid:5)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8)(cid:9)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8) (c) (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:7)(cid:5)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8)(cid:9)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:8) (d)
Figure 4: σ S / h p T i vs h p T i plots. Area S from “5 th degree polynomial +constant after dN ch dη > .
5” fit.4a: CDF Run I at √ s = 1.8 TeV, p T > . | η | < .
0, Minimum Bias.4b: CDF Run II at √ s = 1.96 TeV, p T > . | η | < .
0, MinimumBias + High multiplicity trigger.4c: ATLAS at √ s = 7 TeV, p T > . | η | < .
5, Minimum Bias.4d: CMS at √ s = 7 TeV, p T > | η | < .
4. Minimum Bias.13region 2 . dN ch dη . dN ch dη ≃ in plots with many points at high dN ch dη values (ATLAS with p T > MeV/c, and CDF Run II 1960 GeV with p T >
400 MeV), we observe a strong slope change around corresponding dN ch dη values of about 24 or higher. It’s worth noting that what seems to be a different behavior in the left side for ATLAS with p T > points correspond to dN ch dη &
7, apart from the first point, which correspond to dN ch dη ≃ .
4. The ratio between σ S / h p T i values corresponding to dN ch dη ≥ and those corresponding to dN ch dη ≤ area calculation method used for the estimation of σ S . This ratio in the case of EOS would correspond to the ratio between the number of the degrees of freedom of the state before and after the transition or the crossover. We note that for small size systems as it would be in the pp case, the jump in entropy density is considerably reduced [8–10] in comparison to the theoretical infinite volume case. In plots with p T >
100 MeV/c (ATLAS 7 TeV) or p T > h p T i with varying dN ch dη , which leads to an initial steep rise. After that, the curves assume the same behavior of previously seen plots. c One of the physical quantities used to characterize the state of a system is its squared sound velocity, defined as c = σT · dTdσ , for constant V [11]. In our study, we approximate it with c = σ S h p T i · d h p T i dσ S . It is really interesting that if h p T i is proportional to T and if σ S is proportional to the entropy density, then the c value obtained in this approximation is equal to the right value
14f c = σT · dTdσ , because proportionality constants cancel out. In order to obtain our c estimation, from h p T i vs dN ch dη curves and from σ S values, we compute the curve h p T i vs σ S , to which we apply numerical derivation. We cope with the statistical fluctuation in data points using a combination of Gaussian and Savitzky-Golay filters [43]. Examples of c vs h p T i curves are shown in Figs. 5a and 5c. The so obtained c estimation resembles the typical shape of a phase transition or a crossover: a descent, a minimum region and a following rise, as it’s also obtained analytically from EOSs which present a phase transition or a crossover. The minimum value reached by the estimation of c in the different experimental curves varies from 0.08 to 0.18 and could correspond to what it’s called the EOS softest point [17]. Recently Refs. [44–46] estimate c minimum value for realistic EOS to be around 0.14. From ǫ S ≃ h p T i · σ S we compute c vs ǫ S curves, that are approximations of c vs energy density. We report these curves in Figs. 5b and 5d. In this case, ǫ S values are calculated using h p T i values from h p T i vs dN ch dη curves with no p T min cut at corresponding energies, estimated in correspon- dence with the different dN ch dη values. As Figs. 5 show, the numerical estima- tion of c vs ǫ S , is characterized by a maximum at low energy density fol- lowed by a minimum region, which is obtained for ǫ S values in range 1 . − . GeV/fm , and a subsequent rise. We note that ǫ S as computed here is an estimation of the energy density for pseudorapidity unit and unit of transverse area. In order to estimate the volume energy density this should be divided by ct. .70 0.75 0.80 0.85 0.90 0.95 1.00 1.05
[GeV/c] (cid:6) S < p T > d < p T > d (cid:7) S ALICE Corrected 0.9 TeV 0.5
d < p T > d (cid:11) S ALICE Corrected 0.9 TeV 0.5
[GeV/c] (cid:12) S < p T > d < p T > d (cid:13) S CDF - Run II p t >0.4GeV/c (c) (cid:14) S [GeVfm (cid:15) ] (cid:16) S < p T > d < p T > d (cid:17) S CDF - Run II p t >0.4GeV/c (d) Figure 5: c = σ S h p T i · d h p T i dσ S vs h p T i or ǫ S , using two different fits for area S estimation.ALICE at √ s = 0.9 TeV, 0 . < p T < | η | < .
8, Minimum Bias;area S from “5 th degree polynomial + linear in dN ch dη / after dN ch dη > .
5” fit.5a: c vs h p T i , 5b: c vs ǫ S .CDF Run II at √ s = 1.96 TeV, p T > . | η | < .
0, Minimum Bias+ High multiplicity trigger; area S from “5 th degree polynomial + constantafter dN ch dη > .
5” fit.5c: c vs h p T i , 5d: c vs ǫ S . 16 . Discussion The shape of the σ S / h p T i approximation to the EOS is very similar, using both R inv from the fit on all dN ch dη space and R inv fitted up to dN ch dη = from the impact parameter model, but the slope change at σ S / h p T i values corresponding to dN ch dη around 6 is still present, as well as the change at σ S / h p T i values corresponding to dN ch dη about 24. In order to avoid possible systematics due to calculation involved in the area definition, we plotted directly dN ch dη / h p T i vs h p T i : this is equivalent to obtain σ S / h p T i curves considering a transverse section which is constant for all dN ch dη values. For space reason we don’t show these plots in this paper. In this case the shape doesn’t resemble an EOS shape anymore, but the slope changes at dN ch dη ≃ dN ch dη ≃
24 are still present, because they are contained in the h p T i vs dN ch dη correlation. The shape of the curves obtained from experimental data ( σ S / h p T i vs h p T i , c vs h p T i and c vs energy) depends on experimental h p T i vs dN ch dη curves and from the value of the area used to obtain density sigmas. Systematic errors in h p T i , dN ch dη , and R inv measurements don’t lead to appreciable variations in h p T i vs σ S behavior, which is what we are interested on. It seems to us that the main result of this work is that putting together experimental data of h p T i vs dN ch dη and R inv vs dN ch dη , curves are obtained which are the reproduction of theoretical EOS curves. Regarding model comparison, we obtained σ S / h p T i vs h p T i plots starting from Montecarlo curves ( Pythia ATLAS AMBT1 and
Pythia8 for ATLAS and CMS experiment respectively), which are shown in Fig. 6. .70 0.75 0.80 0.85 0.90 0.95 1.00 1.05
[GeV/c] (cid:18) S < p T > [ f m (cid:19) ( G e V / c ) (cid:20) ] Pythia ATLAS AMBT1ATLAS 0.9 TeV p t >500MeV/c (a)
[GeV/c] (cid:21) S < p T > [ f m (cid:22) ( G e V / c ) (cid:23) ] Pythia ATLAS AMBT1ATLAS 7 TeV p t >500MeV/c (b)
[GeV/c] (cid:24) S < p T > [ f m (cid:25) ( G e V / c ) (cid:26) ] Pythia ATLAS AMBT1ATLAS 7 TeV p t >100MeV/c (c)
[GeV/c] (cid:27) S < p T > [ f m (cid:28) ( G e V / c ) (cid:29) ] Pythia8CMS 7 TeV (d)
[GeV/c] (cid:30) S < p T > [ f m (cid:31) ( G e V / c ) ] Pythia ATLAS AMBT1ATLAS 7 TeV p t >2.5GeV/c (e) Figure 6: σ S / h p T i vs h p T i . Comparison with models.6a: ATLAS at √ s = 0.9 TeV, p T > . | η | < .
5, Minimum Bias and
Pythia ATLAS AMBT1 .6b: ATLAS at √ s = 7 TeV, p T > . | η | < .
5, Minimum Bias and
Pythia ATLAS AMBT1 .6c: ATLAS at √ s = 7 TeV, p T > . | η | < .
5, Minimum Bias and
Pythia ATLAS AMBT1 .6d: CMS at √ s = 7 TeV, p T > | η | < .
4, Minimum Bias and
Pythia8 .6e: ATLAS at √ s = 7 TeV, p T > . | η | < .
5, Minimum Bias and
Pythia ATLAS AMBT1 . 18ome models on which tuning has been done, for example with CDF Run
II data at 1960 GeV for p T >
400 MeV/c, well reproduce the h p T i vs dN ch dη curve at higher (7 TeV) or lower (0.9 TeV) energies with p T > is clear that in these cases, starting from the h p T i vs dN ch dη curves of models and using Bose Einstein correlation or the impact parameter–multiplicity relation for σ S estimation, curves similar to the experimental ones are obtained. On the other hand, models don’t predict well h p T i vs dN ch dη curves with low p T min , and consequently σ S / h p T i vs h p T i curves as shown for the comparison of models at 7 TeV for CMS and ATLAS data, respectively with p T > and p T >
100 MeV/c [20–22, 47].
The interpretation of curve shapes as experimental “estimation” of EOS depends on how much likely are the correspondences between h p T i and T , and between measured σ S and entropy.
4. Conclusion
The result we consider to be the most important is the following: in many experiments [19–22, 27–31] from 31 GeV to 7000 GeV, starting from h p T i vs dN ch dη and using results from measures of radii with Bose Einstein correlation or from a model that relates impact parameter and multiplicity, we obtained that σ S / h p T i vs h p T i and c = σ S h p T i · d h p T i dσ S reproduce the shape of hadronic matter EOSs and squared sound velocity respectively, in presence of crossover or phase transition. From the plots, a neat change around dN ch dη around 6, where the crossover or the phase transition seems to start, and another possible change at dN ch dη around 24 are observed. The curve c vs ǫ S has a minimum around a “transverse” energy density of about 1.5 GeV/fm .
19n order to understand if these behaviors have a real physical meaning or are just casual, results of measures in the following regions should be compared: 2 . dN ch dη . dN ch dη &
6, 6 . dN ch dη .
24 and dN ch dη & References
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