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Renormalization Group and Quark number fluctuations near chiral phase transition. Krzysztof Redlich University of Wrocl aw. Modelling QCD phase diagram Polyakov loop extended quark-meson model Including q uantum and t hermal fluctuations : FRG-approach

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

  2. Modelling QCD phase diagram • Preserve chiral symmetry with condensate as an order parameter • Preserve center symmetrywithPolyakovloop as an order parameter R. Pisarski (2000) PolyakovloopdynamicsConfinement K. Fukushima (2004) C. Ratti & W. Weise (07) PNJL Model Synthesis Spontaneous Chiralsymmetry Nambu & Jona-Lasinio Breaking

  3. Sketch of effective chiral models coupled to Polyakov loop • coupling with meson fileds PQM chiral model • Nambu-Jona-Lasinio model PNJL chiral model the invariant Polyakov loop potential 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, ...

  4. 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

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

  6. GenericPhase diagram from effective chiral Lagrangians (Pisarski-Wilczek) O(4)/O(2) univ.; see LGT , Eijri et al 09 • 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 crossover 2nd order, Z(2) (Stephanov et al.) CP Asakawa-Yazaki 1st order Alford et al. Shuryak et al. Rajagopal et al. Hatsuda et al. 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., ..

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

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

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

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

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

  12. 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

  13. 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:

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

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

  16. 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

  17. The Phase Diagram and EQS Function Renormalization Group Mean-field approximation CEP CEP • Mesonicfluctuationsshiftthe CEP to highertemperature • Thetransitionissmoother • No focusing of isentropes (see E. Nakano et al. (09)) (c.f., C. Nonaka & M. Asakawa &. B. Muller)

  18. 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 SB FRG-results PQM model QM PQM QM QM model PQM model SB limit QM model PQM PQM MF-results

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

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

  21. 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

  22. Ratio of cumulantsatfinitedensity Deviations of theratios of odd and even order cumulantsfromtheirasymptotic, lowT-value, areincreasingwith and thecumulant order Propertiesessentialin HIC to discriminatethephasechange by measuringbaryonnumberfluctuations !

  23. 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

  24. 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

  25. QCD phase boundary & Heavy Ion Data • Isthere a memorythatthe system haspassedthroughthe region of the QCD phasetransition ? • Considerthenet-quarknumberfluctuations and theirhighermoments

  26. STAR DATA ON MOMENTS of FLUCTUATIONS Phys. Rev. Lett. 105, 022302 (2010) • Mean • Variance • Skewness • Kurtosis

  27. Properties of fluctuationsin 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:

  28. 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?

  29. Mean, variance, skewness and kurtosisobtained by STAR and rescaled HRG • STAR Au-Au • 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., S. Ejiri, F. Karsch & K.R. • Smooth change of and peak in at expected from O(4) universality argument and HRG • 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

  32. 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

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