to Look for the Critical Point and Phase Transition New Ways to Look for the Critical Point and Phase Transition Masayuki Asakawa Department of Physics, Osaka University in collaboration with C. Nonaka, B. Müller, and S.A. Bass S. Ejiri and M. Kitazawa

CEP(critical end point) QCD Phase Diagram LHC T RHIC QGP (quark-gluon plasma) CEP(critical end point) 160-190 MeV crossover 1st order Hadron Phase order ? chiral symmetry breaking confinement CSC (color superconductivity) 5-10r0 mB

Punchlines: Quantities to Look at Quantities not subject to Critical Slowing Down and Final State Interactions Quantities sensitive to CEP or Phase Transition Conserved Quantities unlike correlation lengths, (usual) fluctuations...etc. What to see? How to see? Question

CEP = 2nd order phase transition, but... Divergence of Fluctuation Correlation Length Specific Heat ? Critical Opalescence? CEP = 2nd Order Phase Transition Point 2SC CFL even if the system goes right through the critical end point… Subject to Final State Interactions If expansion is adiabatic AND no final state int.

Critical Slowing Down and Final State Int. Furthermore, critical slowing down limits the size of fluctuation, correlation length ! Time Evolution along given isentropic trajectories (nB/s : fixed) x and xeq big difference almost no enhancement 1-dim Bjorken Flow Nonaka and M.A. (2004) Model H (Hohenberg and Halperin RMP49(77)435)

Locally Conserved Charges Fluctuations of Locally Conserved Quantities Net Baryon Number Net Electric Charge Net Strangeness Energy ... Net Electric Charge Hadron Gas Phase ~2/3 of hadrons carry electric charge QGP Phase only ~ 1/2 of d.o.f., i.e. quarks and anti-quarks carry electric charge Heinz, Müller, and M.A., PRL (2000) Also, Jeon and Koch, PRL (2000)

Charge Fluctuation at RHIC D-measure Predictions QGP phase Hadron phase ( : hadron resonance gas) Experimental Value (STAR) (PHENIX) Białas, PLB(2002) Nonaka, Müller, Bass, M.A., PRC (2005) Well-Explained by Quark Recombination

Odd Power Fluctuation Moments We are in need of observables that are not subject to final state interactions Fluctuation of Conserved Charges: not subject to final state interactions Usually fluctuations such as have been considered even power One of exceptions: Stephanov (2008) in conjunction with CEP Usual Fluctuations such as : positive definite Absolute values carry information of states Asakawa, Heinz, Müller, Jeon, Koch On the other hand, Odd power fluctuations : NOT positive definite In general, do not vanish (exception, ) Sign also carry information of states

Physical Meaning of 3rd Fluc. Moment cB : Baryon number susceptibility in general, has a peak along phase transition changes the sign at QCD phase boundary ! In the Language of fluctuation moments: more information than usual fluctuation

(Hopefully) More Easily Measured Moments Third Moment of Electric Charge Fluctuation cB cI/9 singular @CEP iso-vector susceptibility nonsingular when Isospin-symm. Hatta and Stephanov 2002

Mixed Moments T m Energy is also a conserved charge and mesurable ! E : total energy in a subvolume Regions with Negative fluctuation moments Result with the standard NJL parameters (Nf=2)

More and More 3rd Moments These quantities are expressed concisely as derivatives of “specific heat” at constant T m “specific heat” at constant diverges at critical end point peaks on phase transition line

Comparison of Various Moments 2-flavor NJL with standard parameters Far Side G=5.5GeV-2 mq=5.5MeV L=631MeV Near Side Different moments have different regions with negative moments By comparing the signs of various moments, possible to pin down the origin of moments Negative m3(EEE) region extends to T-axis (in this particular model) Sign of m3(EEE) may be used to estimate heat conductivity

Principles to Look for Other Observables We are in need of observables that are not subject to final state interactions After Freezeout, no effect of final state interactions Chemical Freezeout usually assumed momentum independent but this is not right chemical freezeout time: pT (or bT) dependent Larger pT (or bT), earlier ch. freezeout Principle I

Emission Time Distribution Larger bT, earlier emission No CEP effect (UrQMD)

Principle II (T,mB) (r,h) Universality: QCD CEP belongs to the same universality class as 3d Ising Model (T,mB) (r,h) QCD Critical End Point 3d Ising Model End Point same universality class h : external magnetic field Further Assumptions Size of Critical Region No general universality Lattice calculation: not yet V limit Effective Model Results ? need to be treated as an input, at the moment

Singular Part + Non-singular Part Matching between Hadronic and QGP EOS Entropy Density consists of Singular and Non-Singular Parts Only Singular Part shows universal behavior Requirement: reproduce both the singular behavior and known asymptotic limits Matched Entropy Density Hadron Phase (excluded volume model) QGP phase crit Bcrit Dimensionless Quantity: Sc D: related to extent of critical region

Isentropic Trajectories In each volume element, Entropy (S) and Baryon Number (NB) are conserved, as long as entropy production can be ignored (= when viscosities are small) Isentropic Trajectories (nB/s = const.) An Example CEP Near CEP s and nB change rapidly isentropic trajectories show non-trivial behavior Bag Model EOS case

Consequence FO, CO QCP FO, CO QCP ratio For a given chemical freezeout point, prepare three isentropic trajectories: w/ and w/o CEP Along isentropic trajectory: FO, CO QCP Principle I As a function of pT(bT): FO, CO QCP Bag Model EOS (w/o CEP, usual hydro input) ratio : near CEP steeper

Evolution along Isentropic Trajectory As a function of pT(bT): FO, CO QCP with CEP steeper spectra at high PT

Final State Interaction and Critical Slowing Down Summary Final State Interaction and Critical Slowing Down Usual observables such as large fluctuation, critical opalescence, ...etc. do not work Conserved Charges and Higher Moments: i) Third Fluctuation Moments of Conserved Charges take negative values in regions on the FAR SIDE of Phase Transition 　　　(more information!) ii) Different Moments have different regions with negative moments By comparing different moments, possible to pin down the origin of hot matter, get an idea about heat conductivity...etc. Ratio : Two Principles i) Chemical Freezeout is pT(bT) dependent ii) Isentropic Trajectory behaves non-trivially near CEP (focusing) ratio behaves non-monotonously near CEP Information on the QCD critical point: such as location, size of critical region, existence...