1. A new approach for locating the critical point in RHIC low energy scan experiments Introduction New approach for locating critical point Application to RHIC energy-scan Dynamics of QCD phase diagram and the locating of critical point in RHIC low energy scan Presented by Liu Lianshou SQM08, October 2008, Beijing Xu Mingmei, Yu Meiling, Liu Lianshou

2. Introduction The difficulty in understanding the mechanism of crossover Part I The Molecule-like Aggregation Model- MAM Application to the study of liquid property of sQGP Part II A new approach for locating the critical point in RHIC low energy scan experiment

3. Introduction The difficulty in understanding the mechanism of crossover

4. Commonly accepted phase diagram How is the mechanism of crossover? How is it different from 1st order phase transition? 1st order phase transition line ends at the critical point, above it is analytic crossover.

5. 1st order phase transition vs. crossover Co-existence of QGP and HG Some nucleons combine to a big bag - QGP droplet Nucleon gas Example-1 1st order phase transition in QCD Neutral atom gas E-M plasma + + + +++ + + + + + Ionization of atoms Example-2 Analytical crossover in QED Mixture of electrons, positive-ions and neutral atoms + + + Analytical crossover Mixture of two components Without phase boundary and phase separation. Boundary Co-existence of two phases 1st order phase transition

6. wQGP BEC-BCS is a phase change within deconfined phase. Example-3 Analytical crossover in QCD - 1

7. 2 quarks of opposite spin form a di-quark, leading to Bose- Einstein condensation. They might also form loose-knit Cooper pair, leading to BCS superconducting.

8. Mixed state In the intermediate stage of crossover di-quarks and Cooper pairs mixed in perturbative vacuum.

9. Partons in Quarks combine (hadronize) to hadrons Perturbative vacuum Hadrons in Hadrons decompose to quarks, A possible mechanism: Physical vacuum Example-4 Analytical crossover in QCD - 2 Crossover between HG and QGP Contradicts color confinement. In the intermediate stage there are: quarks moving in physical vacuum or hadrons moving in perturbative vacuum This case is special. Let us compare the different cases. Used in AMPTand qMD

10. No vacuum problem Color objects in perturbative vacuum. No problem Mixture of di- quarks and Cooper pairs BEC ~ BCS Contradicts confinement, The mixture of quarks (colored objects) and hadrons (color-singlets) HG ~ QGP Mixture of electrons, positive ions and neutral atoms Atomic gas ~ EM plasma + + + Causing big problem.

11. . Part I The Molecule-like Aggregation Model

12. Let us take still another example.

13. Example 5: Geometrical percolation model site A bond could be formed between two adjacent hadrons When an infinite cluster, i.e. a cluster extending from one boundary to the other, is formed, we say that the system turns to a new phase. The hadrons connected by bonds form clusters In this way the crossover from one phase to the other is realized. No contradiction with QCD Dynamical Model We borrow the concept of quark delocalization from Quark Delocalization and Color Screening Model in low energy nuclear physics. bond

14. Bond is formed by quark delocalization  When the distance of two hadrons is large, quarks are confined in each hadron by a confinement potential.  When two hadrons close enough, the infinite potential in between drops down, forming a potential barrier. Quarks can tunnel the barrier and move in a delocalized orbit. Since color can flow through bonds, hadrons in a cluster become colored objects. Only the cluster as a whole is color-singlet

15. Our basic assumption: molecule-like aggregation Usual scheme of hadron aggregation can serve as the picture for 1st order phase transition.  Form ideal gas, contradicts color confinement. Use it for crossover  Form QGP with liquid property, no contradiction with color confinement. molecule-like aggregation Inspired by the above observation we propose: Using this assumption we propose a model for the crossover between hadronic and partonic phases

16. Before crossover All hadrons are connected to an infinite cluster. End of crossover T c’ Start of crossover Begin to form infinite cluster TcTc Grape-shape QGP (gQGP) is a special form of sQGP. Grape-shape QGP (gQGP) Molecule-like Aggregation Model Clusters of various sizes At the intermediate stage of crossover, there is a mixture of gQGP (infinite cluster) and hadronic states (small clusters). Both of them are color-singlets. No contradiction with color-confinement.

17. A simple application of MAM ----- the liquid property of sQGP

18. Pair distribution function ： the probability of finding two atoms at a distance r from each other. When there is no correlation, g(r)=1. The liquid property of gQGP — Studied by pair distribution function

19. In our case, chemical distance D: D r Define new pair distribution function: : correction factor to eliminate the boundary effect.

20. Start of crossover End of crossover Middle stage The first high peak is due to intra-cell correlations among quarks; Long before crossover there is no correlation peak beside the first high one; Going nearer to crossover some shoulders appear, which develop to peaks, indicating short-range order at the start of crossover; In the process of crossover, correlation peaks appear more and more and extend farther and farther, indicating the reduction of viscosity. T=0.475T c T=0.67T c T=0.80T c T=0.93T c T=T c T=1.21T c T=1.31T c T=1.39T c Before crossover

21. Part II A new approach for locating the critical point in RHIC low energy scan experiments

22. In order to experimentally test the predicted phase diagram, RHIC has started Low-Energy Scan program LQCD predictions for critical points Chemical freeze-out Experimental freeze-out The region covered by Low-E Scan The total region covered by RHIC What to measure? What variable is effective for locating critical point?

23. Many variables have been proposed. Most of them are based on the assumption that these variables have large fluctuations at the critical point. These fluctuations are Event-by-Event, and so are contaminated by the statistical ones coming from the limited number of particles in a single event. Various attempts have been made to eliminate the statistical fluctuations, but none are decisive. It is unclear whether such a kind of fluctuation- signal for critical point could survive after the elimination of statistical fluctuations.

24. Collide A+A at various energies How to avoid the troublesome SF? Let us examine the energy scan process. first order phase transition first order phase transition second order phase transition second order phase transition Let us find a variable to chracterize the structure change while passing critical point. Since the systems have arrived equilibrium, there is no difference whether they are different systems produced in different collisions or they are the evolution of a single system A in T,μplane due to the exchange of heat and particle with an external heat- and particle- bath. During the 2nd order phase transition, the system structure will have noticeable change while passing critical point. This is a 2nd order phase transition

25. The structure change in passing CP First order phase transition Gas-like aggregation Basic elements are single particles --- partons at ABC, and hadrons at A’B’C’abc. Crossover Molecule-like aggregation Basic elements are clusters. While freezeout the hadrons get thermal + radial flow pT Besides thermal + radial flow pT, the hadrons get extra pT from zero-point vibration Thermal motion + Zero-point vibration Thermal motion When system goes along ph-trans-line, passing critical point the pT distribution will present a noticeable change This effect should be observable and could serve as a signal for locating the critical point

26. A jump at the critical point can clearly be seen. In order to get an idea about how large is the effect of we use the experimentally fitted parameters β and T fz of Au+Au collisions at 4.88 GeV and 62.3 GeV to make an interpolation. Assuming that the critical point is located in this energy region, e.g. at 20 GeV,

27. For ω = 0.15 and 0.40 GeV the relative rise of the first, second, third order moments are 5.8, 8.7, 9.9 % and 14, 23, 27 %, respectively. Such an effect should be observable in a high quality data. The sensitivity of the amount of moment-rising on the value of ω can be used to get the value of ω and wherefrom obtain the bond strength, which is a piece of useful information in the study of QCD phase structure.

28. Advantage of the new approach pT moments are averaged over whole sample instead of single event. No statistical fluctuation. For the fluctuation signal, If it is not by occasion that some energy used in the first round just locates at the vicinity of the fluctuation peak, we will see nothing in the first round and the subsequent scan has to be carried out in finer steps over the whole energy range. On the contrary, if the higher order scaled pT moments have a sudden rise while passing through the critical point, then already in the first round of energy scan we will observe a rise of these moments, and most probably the critical point is located in the region of the moment-rising.

29. Summary Applying the usual gas-like aggregation to crossover contradicts color confinement. Molecule-like aggregation is the appropriate scheme for crossover. Using Molecule-like Aggregation Model - MAM to study the pair distribution function the liquid property of sQGP is obtained. Basing on MAM a new approach for locating critical point is proposed which is free from the contamination of statistical fluctuations and is easier to be used.

30. Thank you for attention !

32. range of effective interaction potential mean free path correlation distance viscosity As the increasing of T,

33. Thank you for attention !

35. QCD has complicated phase structure. No analytical calculation is possible, due to color-confinement or “infra-red slavery”. Most reliable information comes from lattice QCD.