1. BB Hadronic matter Quark-Gluon Plasma Chiral symmetry broken x Exploring QCD Phase Diagram in Heavy Ion Collisions Krzysztof Redlich University of Wroclaw and EMMI/GSI QCD phase boundary and freezeout in HIC Cumulants and probability distributions of conserved charges as Probe for the Chiral phase transition: theoretical expectations and recent STAR data at RHIC -CEP ? ? AA collisions

2. 2 – probing the response of a thermal medium to an external field, i.e. variation of one of its external control parameters: (generalized) response functions == (generalized) susceptibilities pressure: thermal fluctuations density fluctuations condensate fluctuations generalized susceptibilities: energy density net charge number order parameter Bulk thermodynamics and critical behaviour Mean particle yields

3. 3 Susceptibilities of net charge number – The generalized susceptibilities probing fluctuations of net -charge number in a system and its critical properties pressure: particle number density quark number susceptibility 4 th order cumulant net-charge q susceptibilities expressed by and central moment

4. 4 Only 3-parameters needed to fix all particle yields Tests of equlibration of 1 st “moments”: particle yields resonance dominance: Rolf Hagedorn partition funct ion Breit-Wigner res. particle yield thermal density BR thermal density of resonances

5. Chemical freezeout and the QCD chiral crossover A. Andronic et al., Nucl.Phys.A837:65-86,2010. O(4) universality HRG model Chiral crossover Thermal origin of particle production: with respect to HRG partition function Chiral crosover Temperature from LGT HotQCD Coll. (QM’12)

6. Chemical freezeout and the QCD chiral crossover A. Andronic et al., Nucl.Phys.A837:65-86,2010. O(4) universality HRG model Chiral crossover Is there a memory that the system has passed through a region of QCD chiral transition ? What is the nature of this transition? Chiral crosover Temperature from LGT HotQCD Coll. (QM’12)

7. QCD phase diagram and the O(4) criticality In QCD the quark masses are finite: the diagram has to be modified Expected phase diagram in the chiral limit, for massless u and d quarks: Pisarki & Wilczek conjecture TCP: Rajagopal, Shuryak, Stephanov Y. Hatta & Y. Ikeda TCP

8. The phase diagram at finite quark masses The u,d quark masses are small Is there a remnant of the O(4) criticality at the QCD crossover line? CP Asakawa-Yazaki Stephanov et al., Hatta & Ikeda At the CP: Divergence of Fluctuations, Correlation Length and Specific Heat

9. The phase diagram at finite quark masses Can the QCD crossover line appear in the O(4) critical region? It has been confirmed in LQCD calculations TCP CP LQCD results: BNL-Bielefeld Critical region Phys. Rev. D83, 014504 (2011 ) Phys. Rev. D80, 094505 (2009)

10. 10 singular critical behavior controlled by two relevant fields: t, h Close to the chiral limit, thermodynamics in the vicinity of the QCD transition(s) is controlled by a universal scaling function K. G. Wilson, Nobel prize, 1982 Bulk Thermodynamics and Critical Behavior non-universal scales control parameter for amount of chiral symmetry breaking regular

11. O(4) scaling and magnetic equation of state Phase transition encoded in the magnetic equation of state pseudo-critical line F. Karsch et al universal scaling function common for all models belonging to the O(4) universality class: known from spin models J. Engels & F. Karsch (2012) QCD chiral crossover transition in the critical region of the O(4) 2 nd order

12. 12 Find a HIC observable which is sensitive to the O(4) criticality Consider generalized susceptibilities of net-quark number Search for deviations from the HRG results, which for quantifies the regular part Quark fluctuations and O(4) universality class To probe O(4) crossover consider fluctuations of net- baryon and electric charge: particularly their higher order cumulants with F. Karsch & K. R. Phys.Lett. B695 (2011) 136 B. Friman, V. Skokov et al, P. Braun- Munzinger et al. Phys.Lett. B708 (2012) 179 Nucl.Phys. A880 (2012) 48 or compare HIC data directly to the LGT results, S. Mukheriee QM^12 for BNL lattice group

13. Effective chiral models Renormalisation Group Approach coupling with meson fileds PQM chiral model FRG thermodynamics of PQM model: Nambu-Jona-Lasinio model PNJL chiral model the SU(2)xSU(2) invariant quark interactions described through: K. Fukushima ; C. Ratti & W. Weise; B. Friman, C. Sasaki., …. B.-J. Schaefer, J.M. Pawlowski & J. Wambach; B. Friman, V. Skokov,... the invariant Polyakov loop potential (Get potential from YM theory, C. Sasaki &K.R. Phys.Rev. D86, (2012); Parametrized LGT data: Pok Man Lo, B. Friman, O. Kaczmarek &K.R. ) B. Friman, V. Skokov, B. Stokic & K.R. fields

14. Including quantum fluctuations: FRG approach FRG flow equation (C. Wetterich 93) J. Berges, D. Litim, B. Friman, J. Pawlowski, B. J. Schafer, J. Wambach, …. start at classical action and include quantum fluctuations successively by lowering k R egulator function suppresses particle propagation with momentum Lower than k k-dependent full propagator B. Stokic, V. Skokov, B. Friman, K.R.

15. FRG for quark-meson model LO derivative expansion (J. Berges, D. Jungnicket, C. Wetterich) ( η small) Optimized regulators ( D. Litim, J.P. Blaizot et al., B. Stokic, V. Skokov et al.) Thermodynamic potential: B.J. Schaefer, J. Wambach, B. Friman et al. Non-linearity through self-consistent determination of disp. rel. with andwith

16. Employed Taylor expansion around minim Get Potential Ignore flow of mesonic field get Mean Field result Essential to include fermionic vacuum fluctuations: Solving the flow equation with approximations: E. Nakano et al.

17. Deviations of the ratios of odd and even order cumulants from their asymptotic, low T-value: are increasing with and the cumulant order Properties essential in HIC to discriminate the phase change by measuring baryon number fluctuations ! Ratios of cumulants at finite density: PQM +FRG HRG B. Friman, F. Karsch, V. Skokov &K.R. Eur.Phys.J. C71 (2011) 1694 HRG

18. STAR data on the first four moments of net baryon number Deviations from the HRG Data qualitatively consistent with the change of these ratios due to the contribution of the O(4) singular part to the free energy HRG

19. Kurtosis saturates near the O(4) phase boundary The energy dependence of measured kurtosis consistent with expectations due to contribution of the O(4) criticality. Can that be also seen in the higher moments? B. Friman, et al. EPJC 71, (2011)

20. Deviations of the ratios from their asymptotic, low T-value, are increasing with the order of the cumulant Ratio of higher order cumulants in PQM model B. Friman, V. Skokov &K.R. Phys.Rev. C83 (2011) 054904 Negative ratio!

21. STAR DATA Presented at QM’12 Lizhu Chen for STAR Coll. V. Skokov, B. Friman & K.R., F. Karsch et al. The HRG reference predicts: HRG O(4) singular part contribution: strong deviations from HRG: negative structure already at vanishing baryon density

22. Moments of the net conserved charges Obtained as susceptibilities from Pressure or since they are expressed as polynomials in the central moment

23. Moments obtained from probability distributions Moments obtained from probability distribution Probability quantified by all cumulants In statistical physics Cumulants generating function:

24. Probability distribution of the net baryon number For the net baryon number P(N) is described as Skellam distribution P(N) for net baryon number N entirely given by measured mean number of baryons and antibaryons In Skellam distribution all cummulants expressed by the net mean and variance P. Braun-Munzinger, B. Friman, F. Karsch, V Skokov &K.R. Phys.Rev. C84 (2011) 064911 Nucl. Phys. A880 (2012) 48)

25. Probability distribution of net proton number STAR Coll. data at RHIC STAR data Do we also see the O(4) critical structure in these probability distributions ? Thanks to Nu Xu and Xiofeng Luo

26. Influence of O(4) criticality on P(N) Consider Landau model: Scaling properties: Mean Field O(4) scaling

27. Contribution of a sigular part to P(N) Get numerically from: For MF broadening of P(N) For O(4) narrower P(N)

28. Take the ratio of which contains O(4) dynamics to Skellam distribution with the same Mean and Variance at different Ratios less than unity near the chiral critical point, indicating the contribution of the O(4) singular part to the thermodynamic pressure K. Morita, B. Friman et al. The influence of O(4) criticality on P(N) for

29. Take the ratio of which contains O(4) dynamics to Skellam distribution with the same Mean and Variance near Asymmetric P(N) Near the ratios less than unity for For sufficiently large the for K. Morita, B. Friman et al.

30. The influence of O(4) criticality on P(N) for K. Morita, B. Friman & K.R. In central collisions the probability behaves as being influenced by the chiral transition

31. Centrality dependence of probability ratio O(4) critical Non- critical behavior For less central collisions, the freezeout appears away the pseudocritical line, resulting in an absence of the O(4) critical structure in the probability ratio. STAR analysis of freezeoutK. Morita et al. Cleymans & Redlich Andronic, Braun- Munzinger & Stachel

32. Energy dependence for different centralities Ratios at central collisions show properties expected near O(4) chiral pseudocritical line For less central collisions the critical structure is lost

33. Conclusions: Hadron resonance gas provides reference for O(4) critical behavior in HIC and LGT results Probability distributions and higher order cumulants are excellent probes of O(4) criticality in HIC Observed deviations of the and by STAR from the HRG qualitatively expected due to the O(4) criticality Deviations of the P(N) from the HRG Skellam distribution follows expectations of the O(4) criticality Present STAR data are consistent with expectations, that in central collisions the chemical freezeout appears near the O(4) pseudocritical line in QCD phase diagram