OOverview of experimental critical point search
Tobiasz Czopowicz , Jan Kochanowski University, Kielce, Poland Warsaw University of Technology, Warsaw, Poland
Abstract.
The existence and location of the QCD critical point is anobject of vivid experimental and theoretical studies. Rich and beautifuldata recorded by experiments at SPS and RHIC allow for a systematicsearch for the critical point – the search for a non-monotonic dependenceof various correlation and fluctuation observables on collision energy andsize of colliding nuclei.
Keywords:
Quark-Gluon Plasma, critical point, fluctuations T μ μ T cc Quark-Gluon Plasmahadronicmatter criticalpointcross-over f i r s t o r d e r ph a s e t r a n s i t i o n Fig. 1.
A sketch of the phase diagram ofstrongly-interacting matter.
A sketch of the most popular phasediagram of strongly-interacting mat-ter is shown in Fig. 1. At low tem-peratures and baryon chemical po-tential, the system consists of quarksand gluons confined inside hadrons.At higher temperature and/or baryonchemical potential, quarks and glu-ons may act like quasi-free particles,forming a different state of matter– the Quark-Gluon Plasma. Betweenthe two phases, a first-order transitionis expected at high µ . Critical point(CP) is a hypothetical end point ofthis first-order phase transition linethat has properties of a second-orderphase transition [1,2].It is commonly expected that theQCD critical point should lead to an anomaly in fluctuations in a narrow domainof the phase diagram. However predictions on the CP existence, its location andwhat and how should fluctuate are model dependent [3].The experimental search for the critical point requires a two-dimensional scanin freeze-out parameters (T, µ ) by changing collision parameters controlled inlaboratory, i.e. energy and size of the colliding nuclei (or collision centrality). a r X i v : . [ nu c l - e x ] J a n Tobiasz Czopowicz
An extensive quantity is a quantity that is proportional to the number of WoundedNucleons (W) in the Wounded Nucleon Model [4] (WNM) or to the volume (V)in the Ideal Boltzmann Grand Canonical Ensemble (IB-GCE). The most popu-lar are particle number (multiplicity) distribution P ( N ) cumulants: κ = (cid:104) N (cid:105) , κ = (cid:10) ( δN ) (cid:11) = σ , κ = (cid:10) ( δN ) (cid:11) = Sσ , κ = (cid:10) ( δN ) (cid:11) − (cid:10) ( δN ) (cid:11) = κσ . Ratio of any two extensive quantities is independent of W (WNM) or V (IB-GCE)for an event sample with fixed W (or V) – it is an intensive quantity. For example: (cid:104) A (cid:105) / (cid:104) B (cid:105) = W · (cid:104) a (cid:105) /W · (cid:104) b (cid:105) = (cid:104) a (cid:105) / (cid:104) b (cid:105) , where A and B are any extensive event quantities, i.e. (cid:104) A (cid:105) ∼ W , (cid:104) B (cid:105) ∼ W and (cid:104) a (cid:105) = (cid:104) A (cid:105) and (cid:104) b (cid:105) = (cid:104) B (cid:105) for W = 1. Popular examples are: κ /κ = ω [ N ] = σ [ N ] (cid:104) N (cid:105) = W · σ [ n ] W ·(cid:104) n (cid:105) = ω [ n ] (scaled variance), κ /κ = Sσ , κ /κ = κσ . For an event sample with varying W (or V), cumulants are not extensive quan-tities any more. For example: κ = σ [ N ] = σ [ n ] (cid:104) W (cid:105) + (cid:104) n (cid:105) σ [ W ] . However, having two extensive event quantities, one can construct quantities thatare independent of the fluctuations of W (or V). Popular examples include [5,6]: (cid:104) K (cid:105) / (cid:104) π (cid:105) , ∆ [ N, P T ] = ( ω [ N ] (cid:104) P T (cid:105) − ω [ P T ] (cid:104) N (cid:105) ) /c , Σ [ N, P T ] = ( ω [ N ] (cid:104) P T (cid:105) + ω [ B ] (cid:104) N (cid:105) − (cid:104) N P T (cid:105) − (cid:104) P T (cid:105) (cid:104) N (cid:105) ) /c ,where P T = N (cid:80) i =1 p T,i and C is any extensive quantity (e.g. (cid:104) N (cid:105) ). Quantum statistics leads to short-range correlations in momentum space, whichare sensitive to particle correlations in configuration space (e.g. of CP origin). verview of experimental CP search 3
Popular measures include momentum difference in Longitudinal ComovingSystem (LCMS), q , that is decomposed into three components: q long – denotingmomentum difference along the beam, q out – parallel to the pair transverse-momentum vector ( k t = ( p T, + p T, ) /
2) and q side – perpendicular to q out and q long . The two-particle correlation function C ( q ) is often approximated by athree-dimensional Gauss function: C ( q ) ∼ = 1 + λ · exp (cid:0) − R long q long − R out q out − R side q side (cid:1) , where λ describes the correlation strength and R out , R side , R long denote Gaus-sian HBT radii.A more parametrization of the correlation function is possible via introducingL´evy-shaped source (1-D) [7]: C ( q ) ∼ = 1 + λ · e ( − qR ) α , where q = | p − p | LCMS , λ describes correlation length, R determines the lengthof homogenity and L´evy exponent α determines source shape: α = 2: Gaussian, predicted from a simple hydro, α <
2: anomalous diffusion,generalized central limit theorem, α = 0 .
5: conjectured value at the critical point.
When a system crosses the second-order phase transition, it becomes scale invari-ant, which leads to power-law form of correlation function. The second factorialmoment is calculated as a function of the momentum cell size (or bin number M ): F ( M ) ≡ (cid:28) M M (cid:88) i =1 n i ( n i − (cid:29)(cid:44)(cid:28) M M (cid:88) i =1 n i (cid:29) , where n i is particle multiplicity in cell i .At the second-order phase transition the system is a simple fractal and thefactorial moment exhibits a power-law dependence on M [8,9,10,11]: F ( M ) ∼ ( M ) ϕ . In case the system freezes-out in the vicinity of the critical point, ϕ = 5 / F ( M ) dependence on the single-particle inclusive momentumdistribution, one needs a uniform distribution of particles in bins or subtractionof the F ( M ) values for mixed events: ∆F ( M ) = F data ( M ) − F mixed ( M ) . Based on coalescence model, particle ratios of light nuclei are sensitive to thenucleon density fluctuations at kinetic freeze-out and thus to CP. In the vicin-ity of the critical point or the first-order phase transition, density fluctuationbecomes larger [12,13].
Tobiasz Czopowicz
Nucleon density fluctuation can be expressed by proton, triton and deuteronyields as: ∆n = (cid:10) ( δn ) (cid:11) (cid:104) n (cid:105) ≈ √ N p · N t N d − . [GeV] NN s [h] pp ω [h] BeBe 0-1% ω [h] ArSc 0-0.2% ω EPOS ppEPOS BeBe 0-1%EPOS ArSc 0-0.2% ] - [ h ω beam Sc, 0-5% Ar+ Be, 0-5% Be+ p+p [GeV] NN s6 8 10 12 14 16 18 , N ] T [ P ∆ - h NA61/SHINE preliminary Sc, 0-5% Ar+ Be, 0-5% Be+ p+p Fig. 2. Results on multiplicity [14] ( left ) and multiplicity-transverse momen-tum [15] ( center, right ) fluctuations for all negatively charged particles recorded byNA61/SHINE. Results on energy dependence of multiplicity fluctuations by NA61/SHINE [14]quantified by the scaled variance are presented in Fig. 2 ( left ). No prominentstructures that could be related to the critical point are observed. Results on energy dependence of multiplicity-transverse momentum fluctuationsby NA61/SHINE [15] expressed in ∆ and Σ strongly intensive quantities are pre-sented in Fig. 2 ( center, right ). No prominent structures that could be attributedto the critical point are observed. Figure 3 ( left ) presents energy dependence of fourth-order net-proton fluctuationin 5% most central Au+Au collisions recorded by STAR [16]. The observed non-monotonic dependence is consistent with theoretical predictions [17] and mightsuggest a critical point around √ s NN ≈ verview of experimental CP search 5 Fig. 3. Results on κσ of net-proton [16] ( left ) as well as net-kaon and net-charge [18,19] ( right ) distributions measured by STAR. The STAR Collaboration has also studied net-kaon and net-charge distributionsin central Au+Au collisions [18,19]. However, the results, presented in Fig. 3( right ), show no (within errors) energy dependence. Fig. 4 presents compilations of Au+Au ( √ s NN = 7.7–200GeV) data from STAR [21] and Pb+Pb ( √ s NN = 2 . 76 TeV) data from AL-ICE [22]. The Gaussian emission source radii ( R out − R side ) [20] show clearnon-monotonic energy dependence with a maximum at √ s NN ≈ . T = 165 MeV and µ = 95 MeV. Transverse-mass dependence of L´evy exponent Transverse-mass depen-dence of L´evy exponent α have been studied both at SPS and RHIC. Figure 5presents the results for Be+Be at 17 GeV by NA61/SHINE [23] and for Au+Auat 200 GeV by PHENIX [24]. Both studies revealed similar results, i.e. α ≈ . NA49 and NA61/SHINE have studied the second factorial moment, ∆F , formid-rapidity protons at 17 GeV. Tobiasz Czopowicz (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:7)(cid:7)(cid:9) (cid:7)(cid:8)(cid:7)(cid:9) (cid:7)(cid:8)(cid:9) (cid:9) (cid:9)(cid:7) (cid:10) (cid:11) (cid:12)(cid:13) (cid:14) (cid:1) (cid:15) (cid:1) (cid:10) (cid:11) (cid:16) (cid:17) (cid:18)(cid:4) (cid:1) (cid:2) (cid:19) (cid:20) (cid:11) (cid:6) (cid:7)(cid:21)(cid:9)(cid:7) (cid:7)(cid:7)(cid:15)(cid:7)(cid:21)(cid:22)(cid:7)(cid:21)(cid:15)(cid:9)(cid:7)(cid:22)(cid:9)(cid:7)(cid:15)(cid:11)(cid:7)(cid:22)(cid:11)(cid:7)(cid:15)(cid:23)(cid:7)(cid:22)(cid:23)(cid:7)(cid:15)(cid:24)(cid:7)(cid:22)(cid:24)(cid:7)(cid:15)(cid:21)(cid:7)(cid:22)(cid:21)(cid:7)(cid:15)(cid:25)(cid:7)(cid:22) NN s (cid:2)(cid:3) (cid:2)(cid:3)(cid:4) (cid:4) (cid:5) (cid:6) (cid:1)(cid:2)(cid:7)(cid:8) (cid:2)(cid:3)(cid:4) (cid:4)(cid:9)(cid:10)(cid:11)(cid:12) (cid:9)(cid:10)(cid:11)(cid:13) (cid:9)(cid:10)(cid:11)(cid:14) (cid:10)(cid:11)(cid:10) (cid:2) (cid:3) (cid:9) (cid:5) (cid:3) (cid:4) (cid:4)(cid:2) (cid:3) (cid:14) (cid:15)(cid:16) (cid:5) (cid:1) (cid:9) (cid:1) (cid:3) (cid:14) (cid:17) (cid:18) (cid:19)(cid:20) (cid:4) (cid:1)(cid:1) (cid:2) (cid:7) (cid:8) (cid:14) (cid:9) (cid:5) (cid:3) (cid:4) (cid:6) (cid:21)(cid:14)(cid:22)(cid:13) (cid:2)(cid:3) (cid:2)(cid:3)(cid:4) (cid:4)(cid:5) (cid:7) (cid:23) (cid:1)(cid:2)(cid:7)(cid:8) (cid:2)(cid:3)(cid:4) (cid:4)(cid:10) (cid:13) (cid:24) (cid:21)(cid:14)(cid:21)(cid:14)(cid:22)(cid:13) (cid:10)(cid:10)(cid:9)(cid:10)(cid:25)(cid:26)(cid:10)(cid:25)(cid:9)(cid:21)(cid:10)(cid:26)(cid:21)(cid:10)(cid:9)(cid:14)(cid:10)(cid:26)(cid:14)(cid:10)(cid:9)(cid:22)(cid:10)(cid:26)(cid:22)(cid:10)(cid:9)(cid:13)(cid:10)(cid:26)(cid:13)(cid:10)(cid:9)(cid:25)(cid:10)(cid:26) (cid:2)(cid:27)(cid:4) (cid:2)(cid:28)(cid:4) Fig. 4. Compilations of Au+Au ( √ s NN = 7.7–200 GeV, STAR [21]) and Pb+Pb( √ s NN = 2 . 76 TeV, ALICE [22]) data: energy dependence of Gaussian emission sourceradii [20] ( left ) and one of the result for initial Finite-Size Scaling analysis [20] ( right ). (GeV) T m α (R fixed) + π + π + - π - π (all pars. free) + π + π + - π - π NA61/SHINE NN PRELIMINARY ] [GeV/c T m α = 200 GeV NN sPHENIX 0-30% Au+Au / NDF = 208/61 , CL < 0.1 % χ = 1.207 , α - π - π + π + π Fig. 5. Transverse mass dependence of the L´evy exponent α for 20% most centralBe+Be collisions at 17 GeV by NA61/SHINE [23] ( left ) and for 30% Au+Au at 200GeV by PHENIX [24] ( right ). Although in central Be+Be, C+C, Ar+Sc and Pb+Pb no signal has beenobserved, a deviation of ∆F from zero seems apparent in central Si+Si andmid-central Ar+Sc as shown in Fig. 6. The nucleon density fluctuations, ∆n , for central Pb+Pb by NA49 [28] andcentral Au+Au by STAR [29,30] show a non-monotonic dependence on collisionenergy with a peak for √ s NN ≈ 20 GeV [31] as presented in Fig. 7. verview of experimental CP search 7 -0.500.51 0 5000 10000 15000 20000 Δ F ( M ) M datapower-law fitNA49 ’’Si’’+Si @ 158A GeV/c -0.2500.250.50.751 10 100 1000 10000 Δ F ( M ) Mdatapower law NA61/SHINE preliminary Ar+Sc NA61,cent.5-10%, pur > 90% -0.2500.250.50.751 10 100 1000 10000 Δ F ( M ) Mdatapower law NA61/SHINE preliminary Ar+Sc NA61,cent.10-15%, pur > 90% Fig. 6. Second factorial moment, ∆F , for mid-rapidity protons at 17 GeV in Si+Siby NA49 [26] ( left ) and in 5–10% and 10–15% Ar+Sc by NA61/SINE [27] ( center,right ). Fig. 7. Nucleon density fluctuation, ∆n , for central Pb+Pb [28] and Au+Au [29,30]collisions. The experimental search for the critical point is ongoing. There are four indi-cations of anomalies in fluctuations in heavy-ion collisions at different collisionenergies ( √ s NN ≈ , , , 47 GeV). Interpreting them as due to CP allows oneto estimate four hypothetical CP locations depicted in Fig. 8.Fortunately, there are high-quality, beautiful new data coming soon bothfrom SPS (NA61/SHINE) and RHIC (STAR Beam energy Scan II). References 1. M. Asakawa and K. Yazaki, Nucl. Phys. A , 668 (1989).2. A. Barducci, R. Casalbuoni, S. De Curtis, R. Gatto and G. Pettini, Phys. Lett. B , 463 (1989).3. M. A. Stephanov, PoS LAT , 024 (2006).4. A. Bialas, M. Bleszynski and W. Czyz, Nucl. Phys. B , 461 (1976).5. M. I. Gorenstein and M. Gazdzicki, Phys. Rev. C , 014904 (2011).6. M. Gazdzicki, M. I. Gorenstein and M. Mackowiak-Pawlowska, Phys. Rev. C ,no. 2, 024907 (2013). 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