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Fluctuations and Correlations of Conserved Charges in QCD Thermodynamics Wei-jie Fu, ITP, CAS 16 Nov. 2010 AdS/CFT and Novel Approaches to Hadron and Heavy Ion Physics, week 6

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Outline Introduction Fluctuations and Correlations at Vanishing Chemical Potential Fluctuations and Correlations near the QCD critical point Summary and Discussions

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Introduction Looking for evidence of a Critical Point. Looking for evidence of a first-order phase transition. Purposes of BES at RHIC:

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How to Locate the Critical Point? Fluctuation Measures M. Cheng et al., Phys. Rev. D. 79, 074505 (2009). The characteristic signature of the existence of a critical point is an increase of fluctuations.

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Fluctuations fluctuations are challenging as there are a number of effects that can swamp the signal, for example the elliptic flow can cause a non-statistical fluctuation. fluctuations increase with the collision energy, but scaled fluctuations are independent of the energy. J. Adams et al. (STAR Collaboration), Phys. Rev. C. 72, 044902 (2005).

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K/πFluctuations Solid line---UrQMD, dashed line---HSD Comparison between the statistical hadronization model results and the experimental data. When the light quark phase space occupancy is one, corresponding to equilibrium, the calculations underestimate the experimental results at all energy. When the light quark phase space occupancy is varied to reproduce the excitation function of K+/π+ yield ratios, SH correctly predicts at higher enegier. The right figure shows the comparison of the prediction of the transport model HSD and the UrQMD models to the experimental data. C. Alt et al. (NA49), PRC 79, 044910 (2009); B.I.Abelev et al. (STAR), PRL 103, 092301 (2009).

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p/πand K/p Fluctuations C. Alt et al. (NA49), PRC 79, 044910 (2009); M.M.Aggarwal, et al. (STAR), arXiv:1007.2613; T. Schuster (NA49), PoS CPOD2009,029(2009); J.Tian (STAR), SQM 2009. The left figure shows comparison of the predictions of the UrQMD model to the experimental data for P/π. The predictions of the model is close to the experimental results. The correlation between strangeness S and baryon number B is sensitive to the state of matter. V. Koch et al., PRL 95, 182301 (2005).

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Higher Moments of Net Proton Multiplicity Distributions at RHIC M. M. Aggarwal et al. (STAR Collaboration), Phys. Rev. Lett. 105, 022302 (2010). The skewness (S) and kurtosis (k) of net proton multiplicity distributions are constant as functions of the centrality. There is no evidence for a critical point in the QCD phase diagram for baryon chemical potential below 200 MeV.

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Why are Fluctuations of Conserved Charges so important? They provide information about the degrees of freedom---confined hadrons or deconfined QGP, of strongly interacting matter. Conserved charges are conserved through the evolution of the fire ball, thus the fluctuations of conserved charges can be directly measured by heavy-ion experiments. Fluctuations are directly related to various generalized susceptibilities, and these susceptibilities can be calculated by Lattice QCD simulations and other theoretical methods. Therefore, conserved charge fluctuations provide the first direct connection between experimental observables and theoretical calculations. S. Jeon et al., PRL 85, 2076 (2000); V. Koch et al., PRL 95, 182301 (2005); B. Stokic et al., PLB 673, 192 (2009); M. Cheng et al., PRD 79, 074505 (2009) M. M. Aggarwal et al. (STAR Collaboration), arXiv:1007.2613 M. Cheng et al., PRD 79, 074505 (2009); M. M. Aggarwal et al. (STAR Collaboration), arXiv:1007.2613

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Fluctuations and Correlations Denoting the ensemble average of the conserved charge number with. The deviation of a event multiplicity from its average value is given by. Then we can relate fluctuations and correlations with the generalized susceptibilities. W. Fu et al., PRD 81, 014028 (2010); W. Fu et al., PRD 82, 074013 (2010)

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Theoretical framework---2+1 Flavor PNJL Model The Lagrangian density for the 2+1 flavor PNJL model is given as The thermodynamical potential density is C. Ratti et al., PRD 73, 014019 (2006); W. Fu et al., PRD 77, 014006 (2008)

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Fluctuations of Light Quarks and Strange Quarks The quadratic fluctuations increase monotonously with the increase of the temperature, and the fluctuations of the heavier strange quarks are suppressed relative to those of the light quarks. We find a plateau at the critical temperature in the curve of the ratio of strange and light quark quadratic fluctuations. Our calculations are consistent with the lattice calculations, except the temperature corresponding to strange quark quartic peak is relatively larger than that in lattice simulations. W. Fu, Y. Liu, and Y. Wu, PRD 81, 014028 (2010); M. Cheng et al., PRD 79, 074505 (2009)

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Ratio of the Quartic to Quadratic Fluctuations The ratio is believed to be a valuable probe of the deconfinement and chiral phase transitions. S. Ejiri et al., PLB 633, 275 (2006); S. Ejiri et al., NPA 774, 837 (2006); B. Stokic et al., PLB 673, 192 (2009). At high temperature, the system approaches the Stefan-Boltzmann limit, and the pressure can be easily obtained as At low temperature, the Polyakov-loop approaches zero, meaning the one- and two- quark states are suppressed, only permitting three-quark states. Using Boltzmann approximation, the pressure is However, the temperature –driven deconfinement transition is a continuous crossover. Assuming W. Fu, Y. Liu, and Y. Wu, PRD 81, 014028 (2010); M. Cheng et al., PRD 79, 074505 (2009)

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Fluctuations of Baryon Number at Vanishing Chemical Potentials The singular behavior of the baryon number fluctuations is expected to be controlled by the universal O(4) symmetry group at vanishing chemical potential and vanishing light quark mass. And the baryon number fluctuations are expected to scale like, Our results are consistent with the lattice calculations, except for some quantitative differences at high temperature, because of the current quark mass effect. The ratio of the quartic to quadratic baryon number fluctuations approaches 1 at low temperature. We find a pronounced cusp in the ratio at the critical temperature. The cusp is a signal of the phase transition. K. Karsch, PoS CPOD07 026 (2007); B.Stokic et al., PLB 673, 192 (2009)

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Fluctuations of Electric Charge and Strangeness W. Fu, Y. Liu, and Y. Wu, PRD 81, 014028 (2010) The fluctuations of electric charge are well consistent with the lattice results. There is a prominent cusp at the critical temperature in the ratio of the quartic to quadratic electric charge fluctuations. At low temperature, The ratio is a rising function. The sixth-order fluctuations of strangeness are qualitatively consistent with the lattice calculations. But there are some quantitative differences.

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Second-order Correlations of Conserved Charges W. Fu, Y. Liu, and Y. Wu, PRD 81, 014028 (2010); M. Cheng et al., PRD 79, 074505 (2009) The correlations of conserved charges are well consistent with the lattice results. The correlation between baryon number and electric charge decreases with the increase of the temperature, and in the high temperature limit, this correlation approaches zero, because the sum of quark electric charges is zero. The correlation between baryon number and strangeness is a useful diagnostic of strongly interacting matter. V. Koch et al., PRL 95, 182301 (2005) An interesting thing is that both our calculations and lattice simulations indicate there is a plateau at the critical temperature in the correlation between electric charge and strangeness. In summary, our effective model reproduces all the things that obtained in lattice simulations. This is expected, because what governs the critical behavior of the QCD phase transition is the universality class of the chiral symmetry, which is kept in this model.

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Fluctuations of Baryon Number in the QCD Phase Diagram W. Fu et al., PRD 82, 074013 (2010) The quadratic fluctuation of baryon number has a peak structure, and the peak becomes sharper and narrower while moving toward the QCD critical point. The cubic fluctuation changes its sign, which is argued to be used to distinguish the two sides of the QCD phase boundary. Comparing to the quadratic fluctuation, higher-order fluctuations are superior in the search for the QCD critical point. All amplitudes grow rapidly when moving toward the QCD critical point and diverge there. The chiral phase transition line in contour plots is obvious, and the region near the QCD critical point also be manifest. M. Asakawa et al., PRL 103, 262301 (2009)

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Fluctuations of Electric Charge in the QCD Phase Diagram W. Fu et al., PRD 82, 074013 (2010) The structure of the fluctuations of electric charge is similar with that of the baryon number fluctuations. The peak in the quadratic fluctuation of electric charge grows less rapidly than that in the quadratic fluctuation of baryon number. Employing the quadratic fluctuations of electric charge to search for the critical point is not easy. But the situations for the higher-order fluctuations are different. They are sensitive to the singular properties of the critical point.

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Fluctuations of Strangeness W. Fu et al., PRD 82, 074013 (2010) The structure of the fluctuations of strangeness is similar with those of the baryon number and electric charge fluctuations. But, the strangeness fluctuation is less sensitive to the singular behavior of the critical point than the baryon number and electric charge fluctuations.

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Second-order Correlations of Conserved Charges in the QCD Phase Diagram W. Fu et al., PRD 82, 074013 (2010) The common feature of the second-order correlations is that there is a peak structure during the chiral phase transition. In chiral symmetric phase, i.e. in the SB limit, the correlation between B and Q is vanishing, while other correlations have finite values. The correlation between B and Q is superior to the other correlations to be used to search the critical point.

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Third-order Correlations between Two Conserved Charge W. Fu et al., PRD 82, 074013 (2010) The common feature of the third-order correlations is that they change their signs at the chiral phase transition. Comparing with the second-order correlations, the third-order correlations are much more sensitive to the singular behavior of the critical point. The critical point is much more obvious. Among the six correlations, we find the former three correlations are better than the other ones.

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Third-order Correlations among Three Conserved Charge The correlation changes its sign from negative to positive during the chiral phase transition and diverges at the QCD critical point. Only when the system is near the chiral phase transition, the correlation has nonvanishing value. The critical point in the contour plot is very obvious, therefore, the third-order correlation among three conserved charge is an ideal probe to search for the QCD critical point. W. Fu et al., PRD 82, 074013 (2010)

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Fourth-order Correlations between Two Conserved Charge W. Fu et al., arXiv:1010.0892 The magnitudes of all the fourth-order correlations grow rapidly and oscillate drastically at the phase transition, and all the correlations diverge at the QCD critical point. The common feature of the fourth-order correlations is that there are three extrema. All the correlations except for chiBS13 and chiQS13 vanish rapidly once the system deviates from the QCD phase transition. chiBS13 and chiQS13 still have finite values at large baryon chemical potential, since the constituent mass of strange quark can not be neglected even in the chiral symmetric phase.

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Fourth-order Correlations between Two Conserved Charge W. Fu et al., arXiv:1010.0892 We observe that all the fourth-order correlations are vanishing in the chiral symmetry broken phase. The chiral phase transition line is distinct and we can easily recognize the region near around the QCD critical point. Among all the correlations, we find the former seven correlations are superior to the chiBS13 and chiQS13 to be used to search for the critical point.

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Fourth-order Correlations among Three Conserved Charge W. Fu et al., arXiv:1010.0892 There are two minima and one maximum in the curves of chiBQS121 and chiBQS211 and two maxima and one minimum in chiBQS112. The fourth-order correlations among three conserved charges approach zero rapidly once the system deviates from the chiral phase transition. The fourth-order correlations among three conserved charges are quite sensitive to the singular structure related to the critical point, and the critical point is very obvious in these contour plots, so the correlations are excellent probes to explore the critical point.

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Summary and Discussions We study the fluctuations and correlations of conserved charges, such as the baryon number, the electric charge, and the strangeness, in a effective model---2+1 PNJL model. The results are well consistent with the lattice simulations. We find the higher-order fluctuations and correlations, especially the correlations are superior to the second-order ones to be used to search for the critical point. We suggest that experimentalists at RHIC use the higher-order correlations to explore the QCD critical point. Acknowledge My collaborators: Prof. Yu-xin Liu, Prof. Yue-liang Wu Thank you for your attentions!

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