Chiral models: predictions under mean field dynamics

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 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 K. Fukushima, C. Ratti & W. Weise, B. Friman & C. Sasaki , ., ..... B.-J. Schaefer, J.M. Pawlowski & J. Wambach; B. Friman et al. Polyakov loop

3. GenericPhase diagram from effective chiral Lagrangians then (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 Hatsuda et Sasaki et al. Alford et al. Shuryak et al. Rajagopal 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., ..

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

5. 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) Regulator function suppresses particle propagation with momentum Lower than k

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 and with

7. 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

8. 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

9. 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:

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

11. 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

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

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

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

15. 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

16. 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

17. Net quark number density fluctuations FRG results • 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. QM PQM QM MF-results PQM

18. 4th order quark number density fluctuations MF results • Peak structure might appear due to chiral dynamics. In GL-theory FRG results QM PQM model PQM PQM model QM model QM model Kink-like structure Dicontinuity

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

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

21. 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

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

23. 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 smother PQM PQM QM <L> <L> QM

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

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