1. Quantum Fluctuation and scaling in the Polyakov loop -Quark-Meson model Chiral models: predictions under mean field dynamics Role of Quantum and Thermal fluctuations: Functional Renormalization Group (FRG) Approach FRG in QM model at work: O(4) scaling of an order parameter FRG in PQM model Fluctuations of net quark number density in the beyond MF Work done with: B. Friman, E. Nakano, C. Sasaki, V. Skokov, B. Stokic & B.-J. Schaefer Krzysztof Redlich, University of Wroclaw & CERN

2. Effective QCD-like models Polyakov loop K. Fukushim a, C. Ratti & W. Weise, B. Friman & C. Sasaki,.,..... B.-J. Schaefer, J.M. Pawlowski & J. Wambach; B. Friman et al.

3. Inverse compressibility and 1 st order transtion at any spinodal points: Singularity at CEP are the remnant of that along the spinodals CEP spinodals C. Sasaki, B. Friman & K.R., Phys.Rev.Lett.99:232301,2007.

4. Experimental Evidence for 1 st order transition Specific heat for constant pressure: Low energy nuclear collisions

5. Including quantum fluctuations: FRG approach FRG flow equation (C. Wetterich 93) 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

6. 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

7. Two independent methods employed  Grid method – exact solution  Taylor expansion around minimum: Solving the flow equation Flow eqns. for coefficients (truncated at N=3): Follows minimum

8. 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

9. FRG-Scaling of an order parameter in QM model The order parameter shows scaling. From the one slope 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 3th order when solving FRG flow equation: see D. Litim analysis for O(4) field Lagrangian

10. 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:

11. Two type of susceptibility related with order parameter 1. longitudinal 2. transverse Fluctuations & susceptibilities Scaling properties at t=0 and

12. Extracting delta from chiral susceptibilities Within the scaling region and at t=0 the ratio is independent on h FRG in QM model consitent with expected O(4) scaling

13. Lines of constant s/n Grid Taylor Two indep. calculations FRG at work –global observables FRG - E. Nakano et al. MF results: see also K. Fukushima

14. Focusing of Isentrops and their signature Idea: ratio sensitive to μ B Isentropic trajectories dependent on EoS In Equilibrium: momentum-dep. of ratio reflects history Caveats: Critical slowing down Focusing non-unsiversal!! Asakawa, Bass, Müller, Nonaka large baryons emitted early, small later

15. Renormalization Group equations in PQM model Quark densities modify by the background gluon fields Flow equation for the thermodynamic potential density in the PQM model with Quarks Coupled to the Background Gluonic Fields The FRG flow equation has to be solved together with

16. 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 smother Mean Field dynamics FRG results QM PQM QM PQM

17. Fluctuations of an order parameter Mean Field dynamics FRG results Deconfinement and chiral transition approximately same Within FRG broadening of fluctuations and their strength: essential modifications compare with MF

18. Thermodynamics of PQM model in the presence of mesonic fluctuations within FRG approach Quantitative modification of the phase diagram due to quantum and thermal fluctuations. Shift of CP to the lower density and higher temperature

19. Net quark number density fluctuations Coupling to Polykov loops suppresses fluctuations in broken phase Large influence of quantum fluctuations. Problem with cut-off effects at high T in FRG calulations. ! Probes of chiral trans. MF-results FRG results PQM QM

20. Removing cut-off dependence in FRG Matching of flow equations We integrate the flow equation bellow from and switch at to discussed previously

21. 4 th order quark number density fluctuations Peak structure might appear due to chiral dynamics. In GL-theory PQM QM Kink-like structure Dicontinuity MF results FRG results QM model PQM model

22. Kurtosis as excellent probe of deconfinement HRG factorization of pressure: consequently: in HRG In QGP, Kurtosis=Ratio of cumulants excellent probe of deconfinement F. Karsch, Ch. Schmidt et al., S. Ejiri et al. Kurtosis Observed quark mass dependence of kurtosis, remnant of chiral O(4) dynamics?

23. Kurtosis of net quark number density in PQM model Strong dependence on pion mass, remnant of O(4) dynamics!? MF results FRG results V. Skokov, B. Friman &K.R. Smooth change with a rather weak dependen- ce on the pion mass

24. Conclusions The FRG method is very efficient to include quantum and thermal fluctuations in thermodynamic potential in QM and PQM models The FRG provide correct scaling of physical obesrvables 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 calculations indicate the remnant of the O(4) dynamics at finite pion mass, however much weaker than that expected in the mean field approach