T BB Hadronic matter Quark-Gluon Plasma Chiral symmetry broken Chiral symmetry restored LHC Modelling QCD phase diagram Polyakov loop extended quark-meson model Including quantum and thermal fluctuations: FRG-approach Probing phase diagram with fluctuations of net baryon number theory & experiment A-A collisions fixed x 1 st principle calculations: perturbation theory pQCD LGT Renormalization Group and Quark number fluctuations near chiral phase transition Krzysztof Redlich University of Wroclaw Based on recent work with: B. Friman, V. Skokov; & F. Karsch

Modelling QCD phase diagram Preserve chiral symmetry with condensate as an order parameter K. Fukushima (2004) C. Ratti & W. Weise (07) Preserve center symmetry with Polyakov loop as an order parameter Polyakov loop dynamics Confinement Nambu & Jona-Lasinio Spontaneous Chiral symmetry Breaking Synthesis PNJL Model R. Pisarski (2000)

Sketch of effective chiral models coupled to Polyakov loop coupling with meson fileds PQM chiral model Nambu-Jona-Lasinio model PNJL chiral model the chiral 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

Polyakov loop potential fixed from pure glue Lattice Thermodynamics First order deconfinement phase transition at fixed to reproduce pure SU(3) lattice results C. Ratti & W. Weise 07

Thermodynamics of PQM model under MF approximation Fermion contribution to thermodynamic potential Entanglement of deconfinement and chiral symmetry Suppresion of thermodynamics due to „statistical confinement” Suppresion suppresion SB limit

The existence and position of CP and transition is model and parameter dependent Introducing di-quarks and their interactions with quark condensate results in CSC phase and dependently on the strength of interactions to new CP’s 6 Generic Phase diagram from effective chiral Lagrangians 1st order Zhang et al, Kitazawa et al., Hatta, Ikeda; Fukushima et al., Ratti et al., Sasaki et al., Blaschke et al., Hell et al., Roessner et al.,.. (Pisarski-Wilczek) O(4)/O(2) univ.; see LGT, Eijri et al 09 crossover Asakawa-Yazaki CP 2 nd order, Z(2) (Stephanov et al.) Hatsuda et al. Alford et al. Shuryak et al. Rajagopal et al.

7 Probing CEP with charge fluctuations Net quark-number fluctuations where The CP ( ) and TCP ( ) are the only points where in an equilibrium medium the diverge ( M. Stephanov et al.) A non-monotonic behavior of charge fluctuations is an excellent probe of the CP CP

Phase diagram in chiral models with CP at any spinodal points: Singularity at CEP is the remnant of that along spinodals CP spinodals C. Sasaki, B. Friman & K.R. V. Koch et al., I. Mishustin et al., ….

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

Renormalization Group equations in PQM model Quark densities modified by the background gluon fields Flow equation for the thermodynamic potential density in the PQM model with Quarks Coupled to the Background Gluonic Fields with fixed such that to minimise quantum potential, highly non-linear equation due to V. Skokov, B. Friman &K.R.

FRG at work –O(4) scaling: Near critical properties obtained from the singular part of the free energy density Resulting in the well known scaling behavior of external field Phase transition encoded in the “equation of state” coexistence line pseudo-critical point

FRG-Scaling of an order parameter in QM model The order parameter shows scaling. From the slopes one gets However we have neglected field-dependent wave function renormal. Consequently and. The 3% difference can be attributed to truncation of the Taylor expansion at 3rd order when solving FRG flow equation: see D. Litim analysis for O(4) field Lagrangian

Effective critical exponents Approaching from the side of the symmetric phase, t >0, with small but finite h : from Widom- Griffiths form of the equation of state For and, thus Define:

Two types of susceptibility related with order parameter 1. longitudinal 2. transverse Fluctuations of the order parameter Scaling properties at t=0 and

The order parameter in PQM model in FRG approach For a physical pion mass, model has crossover transition Essential modification due to coupling to Polyakov loop The quantum fluctuations makes transition smoother Mean Field dynamics FRG results QM PQM QM PQM

Fluctuations of order parameters Mean Field dynamics FRG results Deconfinement and chiral transition approximately same Within FRG broadening of fluctuations and their strength: essential modifications compared with MF

The Phase Diagram and EQS Mean-field approximation Function Renormalization Group Mesonic fluctuations shift the CEP to higher temperature The transition is smoother No focusing of isentropes ( see E. Nakano et al. (09) ) (c.f., C. Nonaka & M. Asakawa &. B. Muller) CEP

Probing phase diagram with moments of net quark number fluctuations Smooth change of and peak-like structure in as in O(4) Moments probes of the chiral transition MF-results PQM QM FRG-results QM PQM QM model PQM model PQM model QM model SB limit SB

Kurtosis of net quark number density in PQM model V. Skokov, B. Friman &K.R. For the assymptotic value due to „ confinement” properties Smooth change with a rather weak dependen- ce on the pion mass For

Kurtosis as an excellent probe of deconfinement HRG factorization of pressure: consequently: in HRG In QGP, Kurtosis=Ratio of cumulants excellent probe of deconfinement S. Ejiri, F. Karsch & K.R. Kurtosis F. Karsch, Ch. Schmidt The measures the quark content of particles carrying baryon number

Quark number fluctuations at finite density Strong increase of fluctuations with baryon-chemical potential In the chiral limit the and daverge at the O(4) critical line at finite chemical potential

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 ! Ratio of cumulants at finite density

QCD phase boundary & Heavy Ion Data QCD phase boundary appears near freezeout line Particle yields and their ratio, well described by the Hadron Resonance Gas

QCD phase boundary & Heavy Ion Data Excellent description of LGT EQS by HRG A. Majumder&B. Muller LGT by Z. Fodor et al.. R. Hagedorn

QCD phase boundary & Heavy Ion Data Is there a memory that the system has passed through the region of the QCD phase transition ? Consider the net-quark number fluctuations and their higher moments

STAR DATA ON MOMENTS of FLUCTUATIONS Mean Variance Skewness Kurtosis Phys. Rev. Lett. 105, (2010)

Properties of fluctuations in HRG Calculate generalized susceptibilities: from Hadron Resonance Gas (HRG) partition function: then, and resulting in: Compare this HRG model predictions with STAR data at RHIC:

Comparison of the Hadron Resonance Gas Model with STAR data Frithjof Karsch & K.R. K.R. RHIC data follow generic properties expected within HRG model for different ratios of the first four moments of baryon number fluctuations Can we also quantify the energy dependence of each moment separately using thermal parameters along the chemical freezeout curve?

Mean, variance, skewness and kurtosis obtained by STAR and rescaled HRG STAR Au-Au these data, due to restricted phase space: Account effectively for the above in the HRG model by rescaling the volume parameter by the factor 1.8/8.5

30 LGT and phenomenological HRG model C. Allton et al., Smooth change of and peak in at expected from O(4) universality argument and HRG S. Ejiri, F. Karsch & K.R. For fluctuations as expected in the Hadron Resonance Gas

31 To see deviations from HRG results due to deconfinement and chiral transtion one needs to measure higher order fluctuations: Lattice QCD results model calculations : C. Schmidt

Conclusions The FRG method is very efficient to include quantum and thermal fluctuations in thermodynamic potential in QM and PQM model The FRG provides correct scaling of physical observables expected in the O(4) universality class The quantum fluctuations modified the mean field results leading to smearing of the chiral cross over transition The RHIC data on the first four moments of net- proton fluctuations consitent with expectations from HRG: particle indeed produced from thermal source To observe large fluctuations related with O(4) cross-over, measure higher order fluctuations, N>6