1. Third Moments of Conserved Charges as Probes of QCD Phase Structure Masakiyo Kitazawa (Osaka Univ.) M. Asakawa, S. Ejiri and MK, PRL103, 262301 (2009). xQCD, Bad Honnef, June 22, 2010

2. Phase Diagram of QCD Phase Diagram of QCD T 0 Hadrons Color SC  Quark-Gluon Plasma QCD critical point How can we map these components of phase diagram in heavy-ion collision experiments? Third moments of conserved charges (including skewness) would smartly do this!

3. Fluctuations at QCD Critical Point Fluctuations at QCD Critical Point 2 nd order phase transition at the CP. divergences of fluctuations of p T distribution freezeout T baryon number, proton, chage, … Stephanov, Rajagopal, Shuryak ’98,’99 baryon # susceptibility However, Region with large fluctuations may be narrow. Fluctuations may not be formed well due to critical slowing down. Fluctuations will be blurred by final state interaction.

4. (Net-)Charge Fluctuations (Net-)Charge Fluctuations D-measure: Asakawa, Heinz, Muller, ’00 Jeon, Koch, ’00 When is experimentally measured D formed? Conserved charges can remember fluctuations at early stage, if diffusions are sufficiently slow. NQNQ N Q : net charge # / N ch : total # yy hadrons:quark-gluon: D ~ 3-4 D ~ 1 large  small values of D:

5. Experimental Results for D-measure Experimental Results for D-measure Failure of QGP formation? Is the diffusion so fast? NO! The result does not contradict these statements. Large uncertainty in N ch. Bialas(’02), Nonaka, et al.(’05) RHIC results: D ~ 3 PHENIX ’02, STAR ’03 hadron gas: D ~ 3-4 free quark-gluon gas: D ~ 1 STAR, ’10

6. Higher Order Moments Higher Order Moments Ratios between higher order moments (cumulants) Higher order moments increase much faster near the CP. Ejiri, Karsch, Redlich, ’05 Gupta, ’09 Stephanov, ’09 We want much clearer signals to map the phase diagram, such as changing signs. We want much clearer signals to map the phase diagram, such as changing signs. RBC-Bielefeld ’09 Hadrons:1 Quarks:1/3 2 4th/2nd at  =0 reflects the charge of quasi-particles

7. Take a Derivative of  B Take a Derivative of  B  B has an edge along the phase boundary changes the sign at QCD phase boundary! m 3 (BBB) can be measured by event-by-event analysis if N B in  y is determined for each event. Note: : third moment of fluctuations yy NBNB

8. Impact of Negative Third Moments Impact of Negative Third Moments Once negative m 3 (BBB) is established, it is evidences that (1)  B has a peak structure in the QCD phase diagram. (2) Hot matter beyond the peak is created in the collisions. No dependence on any specific models. Just the sign! No normalization (such as by N ch ).

9. Third Moment of Electric Charge Third Moment of Electric Charge net baryon # in  y : difficult to measure net charge # in  y : measurable! Experimentally,  Q : chemical potential associated to N Q

10. Under isospin symmetry, singular @CEP isospin susceptibility (nonsingular) Hatta, Stephanov ’02 Third Moment of Electric Charge Third Moment of Electric Charge net baryon # in  y : difficult to measure net charge # in  y : measurable! Experimentally,  Q : chemical potential associated to N Q  B  I /9

11. The Ridge of Susceptibility The Ridge of Susceptibility = 0 at  B =0 (C-symmetry) m 3 (BBB) is positive for small  B (from Lattice QCD) Region with m 3 (BBB)<0 is limited near the critical point: ~  B at  B >>  QCD (since  ~  B 4 for free Fermi gas) T 

12. The Ridge of Susceptibility The Ridge of Susceptibility Analysis in NJL model: = 0 at  B =0 (C-symmetry) m 3 (BBB) is positive for small  B (from Lattice QCD) Region with m 3 (BBB)<0 is limited near the critical point: ~  B at  B >>  QCD (since  ~  B 4 for free Fermi gas) T  m 3 (BBB)<0 m 3 (QQQ)<0

13. Proton # Skewness @STAR Proton # Skewness @STAR STAR, 1004.4959 Measurement of the skewness of proton number @STAR shows that for 19.6-200GeV.

14. Proton # Skewness @STAR Proton # Skewness @STAR Remark: Proton number, N P, is not a conserved charge. No geometrical connection b/w 2nd & 3rd moments. Measurement of the skewness of proton number @STAR shows that for 19.6-200GeV. STAR, 1004.4959

15. Derivative along T Direction Derivative along T Direction Signs of m 3 (BBE) and m 3 (QQE) change at the critical point, too. T  E : total energy in a subvolume measurable experimentally

16. More Third Moments More Third Moments T  “specific heat” at constant diverges at critical point edge along phase boundary

17. More Third Moments More Third Moments Signs of these three moments change, too! T  “specific heat” at constant diverges at critical point edge along phase boundary

18. Model Analysis Model Analysis Regions with m 3 (*EE)<0 exist even on T-axis.  This behavior can be checked 2-flavor NJL; G=5.5GeV -2, m q =5.5MeV,  =631MeV on the lattice at RHIC and LHC energies

19. Trails to the Edge of Mountains Trails to the Edge of Mountains m 3 (EEE) on the T-axis Experimentally: RHIC and LHC On the lattice:

20. Trails to the Edge of Mountains Trails to the Edge of Mountains m 3 (EEE) on the T-axis Experimentally: RHIC and LHC On the lattice: Experimentally: energy scan at RHIC On the lattice: ex.) Taylor expansion Cheng, et al. ‘08 c4c4 c6c6 m 3 (QQQ), etc. at   >0

21. Summary 1 Summary 1 Seven third moments all change signs at QCD phase boundary near the critical point. To create a contour map of the third moments on the QCD phase diagram should be an interesting theoretical subject. m 3 (BBB), m 3 (BBE), m 3 (BEE), m 3 (EEE), m 3 (QQQ), m 3 (QQE), and m 3 (QEE) Negative moments would be measured and confirmed both in heavy-ion collisions and on the lattice. In particular, (1) m 3 (EEE) at RHIC and LHC energies, (2) m 3 (QQQ)=0 at energy scan, are interesting!

22. Summary 2 Summary 2 Let’s go see the scenery over the ridge! But, do not forget to first draw a map for a safe expedition. Critial Point Loreley, photo by MK, 2005

23. Derivative along T direction Derivative along T direction simple T-derivative: E : total energy in a subvolume measurable experimentally Problem: T and  can not be determined experimentally. mixed 3 rd moments:

24. Further Possibility Further Possibility If measured moments originate from a narrow region in the T-  plane, and if experimental resolution is sufficiently fine, exp. lattice This formula is used to determine  /T experimentally. Moreover, third moments provide the divergence vector of  and C . These information may enable us to pin down the initial state of fireballs.

25. Loreley