QCD critical point and observables M. Stephanov U. of Illinois at Chicago QCD critical point and observables – p. 1/15
QCD phase diagram (a sketch) T , GeV QGP crossover critical point 0.1 hadron gas quark(yonic) matter phases: c.s.c., nuclear crystals, ..? CFL matter vacuum 0 µ B , GeV 1 Models (and lattice) suggest the transition becomes 1st order at some µ B . Can we observe the critical point in heavy ion collisions, and how? QCD critical point and observables – p. 2/15
Critical point(s) in known liquids Most liquids have a critical point (seen, e.g., by critical opalescence). Water: Does QCD “perfect liquid” have one? QCD critical point and observables – p. 3/15
What do we need to discover the critical point? 200 LTE04 LTE03 LTE08 T , LR01 130 LR04 MeV 19 150 11 R H I C 7.7 s c a n 5 100 50 2 0 0 200 400 600 800 µ B , MeV Experiments: RHIC, NA61/SPS, FAIR/GSI, NICA. Better lattice predictions, with controllable systematics. Sensitive experimental signatures. QCD critical point and observables – p. 4/15
Critical fluctuations: theory Consider an observable such as, Ω( σ V ) 1 V σ , where σ ∼ ¯ � e.g., σ V = ψψ . µ < µ CP Einstein, 1910: V � ∼ (Ω ′′ ) − 1 � σ 2 P ( σ V ) ∼ number σ V of states with that σ V i.e., e S , or e − Ω /T 2 (Ω ′′ ) − 1 → ∞ µ = µ CP large equilibrium fluctuations T 1 3 2 3 µ > µ CP µ Why does CP defy the central limit theorem? Because, correlation length ξ → ∞ . This is a collective phenomenon. The magnitude of fluctuations � σ 2 V � ∼ ξ 2 . QCD critical point and observables – p. 5/15
Fluctuation signatures Experiments measure multiplicities N π , N p , . . . , 6 10 Au+Au 200 GeV 0-5% mean p T , etc. Number of Events 0.4<p <0.8 (GeV/c) 5 30-40% 10 T |y|<0.5 70-80% 4 10 These quantities fluctuate event-by-event. 3 10 Fluctuation magnitude is quantified by e.g., 2 10 � ( δN ) 2 � , � ( δp T ) 2 � . 10 1 -20 -10 0 10 20 What is the magnitude of these fluctuations ∆ Net Proton ( N ) p near the QCD C.P .? (Rajagopal-Shuryak-MS, 1998) Universality tells us how it grows at the critical point: � ( δN ) 2 � ∼ ξ 2 . Magnitude of ξ is limited < O ( 2–3 fm ) (Berdnikov-Rajagopal) . “Shape” of the fluctuations can be measured: non-Gaussian moments. As ξ → ∞ fluctuations become less Gaussian. Higher cumulants show even stronger dependence on ξ (PRL 102:032301,2009) : � ( δN ) 4 � − 3 � ( δN ) 2 � 2 ∼ ξ 7 � ( δN ) 3 � ∼ ξ 4 . 5 , which makes them more sensitive signatures of the critical point. QCD critical point and observables – p. 6/15
Higher moments (cumulants) and ξ Consider probability distribution for the order-parameter field: P [ σ ] ∼ exp {− Ω[ σ ] /T } , 2( ∇ σ ) 2 + m 2 Z » 1 – 2 σ 2 + λ 3 3 σ 3 + λ 4 4 σ 4 + . . . d 3 x ξ = m − 1 σ Ω = . ⇒ σ d 3 x σ ( x ) : R Moments (connected) of q = 0 mode σ V ≡ V � = V T ξ 2 ; V � = 2 V T 2 λ 3 ξ 6 ; κ 2 = � σ 2 κ 3 = � σ 3 V � 2 = 6 V T 3 [ 2( λ 3 ξ ) 2 − λ 4 ] ξ 8 . κ 4 = � σ 4 V � c ≡ � σ 4 V � − 3 � σ 2 Tree graphs. Each propagator gives ξ 2 . + λ 3 T ( Tξ ) − 3 / 2 and λ 4 = ˜ Scaling requires “running”: λ 3 = ˜ λ 4 ( Tξ ) − 1 , i.e., V � = 2 V T 3 / 2 ˜ λ 3 ξ 4 . 5 ; κ 4 = 6 V T 2 [ 2(˜ λ 3 ) 2 − ˜ λ 4 ] ξ 7 . κ 3 = � σ 3 QCD critical point and observables – p. 7/15
Moments of observables Example: Fluctuation of multiplicity is the fluctuation of occup. numbers, X δN = p δn p . Any moment of the multiplicity distribution is related to a correlator of δn p : d 3 p κ 3 π = � ( δN ) 3 � = X X X p 3 � δn p 1 δn p 2 δn p 3 � , (2 π ) 3 . R where P p = V p 1 p 2 n p fluctuates around ¯ n p ( m ) , which also fluctuates: δm = gδσ , i.e., p + ∂ ¯ n p δn p = δn 0 ∂m g δσ . „ g « 3 v 2 v 2 v 2 � δn p 1 δn p 2 δn p 3 � σ = 2 λ 3 p 1 p 2 p 3 V 2 T m 2 γ p 1 γ p 2 γ p 3 σ v 2 γ p = ( dE p /dm ) − 1 p = ¯ n p (1 ± ¯ n p ) , Similarly for � ( δN ) 4 � c . Since � ( δN ) 3 � scales as V 1 we suggest ω 3 ( N ) ≡ � ( δN ) 3 � which is V 0 . ¯ N QCD critical point and observables – p. 8/15
Energy scan and fluctuation signatures: notes crossover ( ˜ λ 3 = 0 ) T 1st order critical point freeze-out point with max ξ contours of equal ξ freeze-out points vs √ s µ B Higher moments provide more sensitive signatures. As usual, value comes at a price: Harder to predict – more theoretical uncertainties. Signal/noise is worse for higher moments. But one can, e.g., combine various higher moments to optimize or eliminate uncertainties. QCD critical point and observables – p. 9/15
Using ratios and mixed moments Athanasiou, Rajagopal, MS (2010) The dominant dependence on µ B (i.e., on √ s ) is from two sources ξ and n p , e.g., κ 3 p ∼ ˜ λ 3 g 3 p ξ 4 . 5 n 3 p . ξ ( µ B ) has a peak at µ B = µ critical ; B /T determines the height of the peak; n B ∼ e µ critical B p and ˜ other factors: g 3 λ 3 depend on µ B weaker. Leading dependence on µ critical can be cancelled in ratios. E.g., B « 2 „ N π κ 3 p ∼ ˜ λ 3 g 3 p ξ 4 . 5 N p N p Unknown/poorly known coupling parameters g p or g π can be also cancelled in ratios. E.g., no uncertainties in these ratios κ 3 κ 2 κ 4 κ 4 p 4 p 2 π 3 π κ 4 π , or . κ 2 κ 4 κ 3 2 p 3 p 4 π when critical fluctuations dominate. They are 1. Mixed moments allow more possibilities. E.g., κ 2 2 p 2 π κ 4 p κ 4 π . Mixed moments have no trivial Poisson contribution. QCD critical point and observables – p. 10/15
Using ratios and mixed moments Athanasiou, Rajagopal, MS (2010) The dominant dependence on µ B (i.e., on √ s ) is from two sources ξ and n p , e.g., κ 3 p ∼ ˜ λ 3 g 3 p ξ 4 . 5 n 3 p . ξ ( µ B ) has a peak at µ B = µ critical ; B /T determines the height of the peak; n B ∼ e µ critical B p and ˜ other factors: g 3 λ 3 depend on µ B weaker. Leading dependence on µ critical can be cancelled in ratios. E.g., B « 2 „ N π κ 3 p ∼ ˜ λ 3 g 3 p ξ 4 . 5 N p N p Unknown/poorly known coupling parameters g p or g π can be also cancelled in ratios. E.g., no uncertainties in these ratios κ 3 κ 2 κ 4 κ 4 p 4 p 2 π 3 π κ 4 π , or . κ 2 κ 4 κ 3 2 p 3 p 4 π when critical fluctuations dominate. They are 1. Mixed moments allow more possibilities. E.g., κ 2 2 p 2 π κ 4 p κ 4 π . Mixed moments have no trivial Poisson contribution. QCD critical point and observables – p. 10/15
Experiment (pre-QM) 200 LTE04 LTE03 LTE08 T , LR01 MeV µ LR04 (MeV) B 150 210 54 720 420 20 10 R H 3 I C STAR Data Au+Au Collisions Lattice QCD s c a n AMPT 0.4<p <0.8 (GeV/c) AMPT (SM) T |y|<0.5 Hijing 100 UrQMD 2 Therminator 2 σ κ 50 1 STAR Preliminary Critical Point Search 0 0 4 5 20 100 200 10 0 200 400 600 800 s (GeV) µ B , MeV NN ( κσ 2 = κ 4 /κ 2 ≈ ω 4 if κ 2 ≈ N ). No critical signatures seen at those values of µ B . Consistent with expectations that µ critical > 200 MeV. B What is happening at √ s = 19 . 6 GeV? Low statistics. Large positive contribution to Poisson is excluded, but large negative — is not. QCD critical point and observables – p. 11/15
Negative kurtosis? Could the critical contribution to kurtosis be negative? (MS, arxiv:1104.1627) „ g « 4 v 2 Z � ( δN ) 4 � c = � N � + � σ 4 p V � c + . . . , T γ p p V � c = 6 V T 2 [ 2˜ λ 4 ] ξ 7 . 3 − ˜ � σ 4 λ 2 On the crossover line ˜ λ 3 = 0 by symmetry, while ˜ λ 4 ≈ 4 . > 0 . P ( σ V ) : → Thus � σ 4 V � c < 0 and ω 4 ( N ) < 1 on the crossover line. And around it. Universal Ising eq. of state: M = R β θ , t = R (1 − θ 2 ) , H = R βδ h ( θ ) here κ 4 is κ 4 ( M ) ≡ � M 4 � c 120 100 80 60 Κ 4 40 20 0 � 20 � 0.4 � 0.2 0.0 0.2 0.4 0.6 t QCD critical point and observables – p. 12/15
Implications for the energy scan , GeV T QGP critical point freezeout 0.1 curve CFL+ hadron gas nuclear matter 0 µ B , GeV 1 QCD critical point and observables – p. 13/15
Implications for the energy scan , GeV T QGP critical point t H 0.1 CFL+ hadron gas nuclear matter 0 µ B , GeV 1 QCD critical point and observables – p. 13/15
Implications for the energy scan , GeV T QGP critical point t 19 H freezeout 0.1 curve CFL+ hadron gas nuclear matter 0 µ B , GeV 1 QCD critical point and observables – p. 13/15
Implications for the energy scan , GeV T QGP critical point t 19 ? 11 H freezeout 0.1 curve CFL+ hadron gas nuclear matter 0 µ B , GeV 1 If the kurtosis stays significantly below Poisson value in 19 GeV data, the logical place to take a closer look is between 19 and 11 GeV. QCD critical point and observables – p. 13/15
Implications for the energy scan ω 4 baseline √ s 11 19 If the kurtosis stays significantly below Poisson value in 19 GeV data, the logical place to take a closer look is between 19 and 11 GeV. QCD critical point and observables – p. 13/15
QM notes Potential sources of baseline shift (from Poisson) at high baryon density: Fermi statistics: ω 4 ≈ 1 − 7 � n p � p (small effect, but grows with µ B ). O(4) critical line (Friman-Karsch-Redlich-Skokov). Baryon number conservation? QCD critical point and observables – p. 14/15
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