1. Non-equilibrium critical phenomena in the chiral phase transition 1.Introduction 2.Review : Dynamic critical phenomena 3.Propagating mode in the O(N) model 4.Over-damping near the critical point 5.Conclusion Kazuaki Ohnishi (NTU) K.O., Fukushima & Ohta : NPA 748 (2005) 260 K.O. & Kunihiro : PLB 632 (2006) 252

2. Strong interaction between hadrons (proton, neutron, pion, ρ-meson) QCD (quark & gluon) Chiral symmetry in the u-, d-quark sector 1. Introduction

3. Ferromagnet O(3) symmetry is spontaneously broken NG mode = spin wave Spontaneous Breaking of Chiral symmetry pion is the massless Nambu-Goldstone particle 1. Introduction

4. Static (Equilibrium) critical phenomena Dynamic (Non-equilibrium) critical phenomena Heavy Ion Collision, Early universe Quark-Gluon-Plasma phase Color-Superconducting phase Hadron phase Early universe Heavy Ion Collision (RHIC,LHC) 1st TCP 2nd 1. Introduction Lattice simulation, Effective theory, Universality argument, etc. Real world

5. Anomalous dynamic critical phenomena Critical slowing down Softening of propagating modes Divergence of transport coefficients... Long relaxation time Slow motion of long wavelength fluctuations of Slow variables 2. Review : Dynamic Critical Phenomena Non-equilibrium, time-dependent Non-equilibrium state Equilibrium state Relax

6. 2. Review : Dynamic Critical Phenomena 2 kinds of slow variables 1. Order parameter 2. Conserved quantity Flat potential Continuity Eq. Slow variables (Order parameter & Conserved quantities) are the fundamental degrees of freedom in the critical slow dynamics

7. 2 types of Slow modes for slow variables 1. Diffusive (Relaxational) mode 2. Propagating (Oscillatory) mode (Spin wave, Sound wave, Phonon mode, etc) t t Propagating mode (Damped Oscillatory mode) Diffusive mode (Damping mode) 2. Review : Dynamic Critical Phenomena

8. Spectral func. for slow variables ( : fixed) ( Dynamic critical exponent) Critical slowing down Softening Propagating mode pole with Real and Imaginary parts Diffusive mode pole with only Imaginary part Dynamic scaling hypothesis 2. Review : Dynamic Critical Phenomena

9. Static universality class critical behavior (critical exponents) is identical if symmetry and (spatial) dimension are same. Ferromagnet and anti-ferromagnet belong to the O(3) universality class Chiral transition belongs to the same universality class as ferromagnet and anti-ferromagnet Pisarski & Wilczek:PRD29(1984)338 2. Review : Dynamic Critical Phenomena Universality class

10. 1.Whether the order parameter is conserved or not 2.What kinds of conserved quantities in the system Whole critical points in condensed matter physics (Ferromagnet, Anti-Ferromagnet, λtransition, Liquid-Gas, etc) have been classified into model A, B, C,.... 2. Review : Dynamic Critical Phenomena Classification scheme Dynamic universality class Slow variables Hohenberg & Halperin: Rev.Mod.Phys.49 (1977) 435

11. Dynamic universality class of chiral transition Slow variables for Chiral phase transition Meson field Chiral charge Energy Momentum Order parameter (Non-conserved) Conserved quantities Slow variables for Anti-Ferromagnet Staggered Magnetization Magnetization Energy Momentum Order parameter (Non-conserved) Conserved quantities Rajagopal & Wilczek: NPB 399 (1993) 395 Meson mode is a diffusive mode 2. Review : Dynamic Critical Phenomena Chiral transition belongs to anti-ferromagnet

12. Hatsuda & Kunihiro: PRL 55 (1985) 158 Meson (particle) is an oscillatory mode of field Diffusive mode Rajagopal & Wilczek Propagating mode Hatsuda & Kunihiro 2. Review : Dynamic Critical Phenomena Meson mode is a propagating mode ?

13. 3. Propagating mode in the O(N) model Langevin Eq. Brownian particle Zwanzig J.Stat.Phys.9(1973)215 O(N) Ginzburg-Landau potential Meson mode (Propagating mode) (K.O., Fukushima & Ohta: NPA 748 (2005) 260) (Koide & Maruyama: NPA 742 (2004) 95) Square of propagating velocity Damping constant Canonical momentum conjugate to order parameter Neither Order parameter nor Conserved quantity!

14. Renormalization Group (RG) analysis of the order parameter fluctuation with canonical momentum K.O. & Kunihiro: PLB 632 (2006) 252 4. Over-damping near the critical point Langevin Eq.

15. Large damping constant limit of the propagating mode If we impose the large damping condition, then the propagating mode is over-damped. For, we can integrate out explicitly the faster degree of freedom to obtain (Ma: “Modern theory of critical phenomena” (1976)) is the faster degree of freedom is the slower degree of freedom t t Oscillatory (propagating) mode Over-damped (diffusive) mode 4. Over-damping near the critical point Langevin eq. for a diffusive mode

16. RG analysis of the Langevin Eq. for the propagating mode RG transformation ● Integration of short-wavelength fluctuations ● Scale transformation : Recursion relation : 4. Over-damping near the critical point

17. ε-expansion Green func. Green func. for diffusive mode Self-energy Full Green func. New parameters ・・・ 4. Over-damping near the critical point

18. Recursion Relation We can find fixed points in the space Usual recursion for the static G-L theory Dynamic parameters Gaussian & Wilson-Fisher (WF) fixed points (Hohenberg & Halperin: Rev.Mod.Phys. 49 (1977) 435) 4. Over-damping near the critical point

19. Two fixed points with respect to Wilson-Fisher fixed point Crossover between the two fixed points Propagating mode becomes over-damped near the critical point 4. Over-damping near the critical point Gaussian WF z=1: Propagating mode ( ) ・・・ unstable z=2: Overdamped mode ( ) ・・・ stable

20. Overdamped (diffusive) mode Anti-ferromagnet Rajagopal & Wilczek (1993) Particle (propagating) mode Hatsuda & Kunihiro (1985) The fate of meson mode near the chiral transition 4. Over-damping near the critical point Pion and sigma are not able to propagate and lose a particle-like nature

21. Ordered phase (Ferroelectric) Disordered phase Order parameter fluctuation ・・・ phonon mode Phonon mode near the ferroelectric transition 4. Over-damping near the critical point

22. Over-damping as a crossover between the two fixed points Universality of the propagating behavior Phonon mode is over-damped near the critical point Experimental fact Almairac et al. (1977) Softening with z=1 ・・・ Propagating fixed point Over-damping region (z=2) ・・・ Diffusive fixed point 4. Over-damping near the critical point

23.  Propagating mode in the O(N) model Meson mode at chiral transition Phonon mode at ferroelectric transition Canonical momentum is necessary as a slow variable  RG analysis of the propagating mode Meson mode near chiral transition is over-damped! Anti-ferromagnet (Rajagopal & Wilczek) Phonon mode near ferroelectric transition 5. Conclusion 2 fixed points for the propagating and diffusive modes Over-damping near the critical point