1. Second-order Relativistic Hydrodynamic Equations Compatible with Boltzmann Equation and Critical Opalescence around the QCD Critical Point Quark Matter 2009 Knoxville, March 30 --- April 4, 2009 Teiji Kunihiro (Kyoto) Yuki Minami (Kyoto) and Kyosuke Tsumura (Fuji Film co.)

2. Critical Opalescence around the QCD Critical Point and Second-order Relativistic Hydrodynamic Equations Compatible with Boltzmann Equation Quark Matter 2009 Knoxville, March 30 --- April 4, 2009 Teiji Kunihiro (Kyoto) Yuki Minami (Kyoto) and Kyosuke Tsumura (Fuji Film co.)

3. Contents I Density fluctuations around QCD critical point with rel. dissipative hydrodynamics; new possible signal for identifying the QCD critical point* II Derivation of Israel-Stewart type hydrodynamic equations on the basis of the (dynamical) renormalization group method** ; only brief exposition of the result * Yuki Minami and T.K., in preparation, ** Kyosuke Tsumura and T.K., in preparation

4. The same universality class; Z2 Density fluctuation is the soft mode of QCD critical point! The sigma mode is a slaving mode of the density. H. Fujii, PRD 67 (03) 094018;H. Fujii and M. Ohtani, Phys.Rev.D70(2004) Dam. T. Son and M. A. Stephanov, PRD70 (’04) 056001 T P Solid Liq. Classical Liq.-Gas c.f. The coupling of the density fluctuation with the scalar mode was discussed in, T.K. Phys. Lett. B271 (1991), 395 Critical point Triple.P gas

5. Spectral function of density fluctuations The density fluctuation depends on the transport as well as thermodynamic quantities which show an anomalous behavior around the critical point. Especially, the existence of the density-temperature coupling. For non-relativistic case with use of Navier-Stokes eq. L.D. Landau and G.Placzek(1934), L. P. Kadanoff and P.C. Martin(1963), R. D. Mountain, Rev. Mod. Phys. 38 (1966), 38 H.E. Stanley, `Intro. To Phase transitions and critical phenomena’ (Clarendon, 1971) We apply for the first time relativistic hydrodynamic equations to analyze the spectral properties of density fluctuations, and examine possible critical phenomena. Missing in the previous analyses.

6. Relativistic Hydrodynamics dissipative terms (1) Energy-frame (2) Particle frame ; Eckart(1940), unstable Tsumura-Kunihiro-Ohnishi, Phys.Lett.B646(2007) (3) Israel-Stewart

7. In the rest frame of the fluid, Inserting them into, and taking the linear approx. etc Linear approximation around the thermal equilibrium; Rel. effects ・ Linearized Landau equation (Lin. Hydro in the energy frame); Solving as an initial value problem using Laplace transformation, we obtain, in terms of the initial correlation. with

8. Spectral function of density fluctuations in the Landau frame In the long-wave length limit, k→0 Long. Dynamical ：： specific heat ratio ： sound velocity thermal expansion rate ： Rel. effects Rel. effects appear only in the width of the peaks. rate of isothermal exp. Notice: As approaching the critical point, the ratio of specific heats diverges! The strength of the sound modes vanishes out at the critical point. enthalpy sound modes thermal mode

9. Eq. of State of ideal Massless particles [MeV] [1/fm] [MeV] In the energy(Landau) frame, relativistic effects appear only in the peakheight and width of the Rayleigh peak. Rel.Non-rel. Rayleigh peak Brillouin peak sound mode thermal mode

10. Spectral function from I-S eq. [1/fm] IS Non-rel. (N-S) No contribution in the long-wave length limit k→0. For Conversely speaking, the first-order hydro. Equations have no problem to describe the hydrodynamic modes with long wave length, as it should.

11. [1/fm] Particle frame; the new equation Rel. case (TKO) Non-rel. (Navier-Stokes) Rel. effects appear in the Brillouin peaks (sound mode) but not in The Rayleigh peak. K. Tsumura, T.K. K. Ohnishi;Phys. Lett. B646 (2007) 134 TKO=

12. Critical behavior of the density-fluctuation spectral functions So, the divergence ofand the viscocities therein can not be observed, unfortunately. In the vicinity of CP, only the Rayleigh peak stay out, while the sound modes (Brillouin peaks) die out. Critical opalescence c.f.

13. Spectral function of density fluctuation at CP The sound mode (Brillouin) disappears Only an enhanced thermal mode remains. Suggesting interesting critical phenomena related to sound mode. Eg. Disappearance of Mach cone at CP! A hint for detecting CP! Spectral function at CP 0.4 The soft mode around QCD CP is thermally induced density fluctuations, but not the usual sound mode. Needs explicit examination.

14. Cf. STAR, arXiv:0805/0622, to be published in PRL R. B. Neufeld, B. Muller, and J. Ruppert,arXiv:0802.2254 gTgT gLgL  (z - ut)  

15. Why at all do sound modes die out at the Critical Point ? The correlation lengthThe wave length of sound mode The hydrodynamic regimel as t 0 However, around the critical point So the hydrodynamic sound modes can not be developed around CP! <<

16. II An RG derivation of 2 nd order hydrodynamic equations K.Tsumura and T.K., preliminary results was presented at JPS meeting Sep. 23, 2008, in preparation

17. Relativistic Boltzmann equation Conservation law of the particle number and the energy-momentum H-theorem. The collision invariants, the system is local equilibrium Maxwell distribution (N.R.) Juettner distribution (Rel.)

18. Geometrical image of reduction of dynamics X M O Invariant and attractive manifold ; distribution function in the phase space (infinite dimensions) ; the hydrodinamic quantities (5 dimensions), conserved quantities. eg.

19. RG/E equation Slow dynamics (Hydro dynamics + relaxation equations ） Energy frame Particle frame Relaxation equations (very long) (I) (II) T.K. Prog. Theor. Phys. 94 (’95), 503; 95(’97), 179 S.-I. Ei, K. Fujii and T.K., Ann. Phys. 280 (2000), 236 Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24 T.K. and K. Tsumura, J. Phys. A: Math. Gen. 39 (2006),

25. The viscocities are frame-independent, in accordance with Lin. Res. Theory. However, the relaxation times and legths are frame-dependent. The form is totally different from the previous ones like I-S’s, And contains many additional terms. contains a zero mode of the linearized collision operator. Conformal non-inv. gives the ambiguity.

26. Summary Density fluctuations is analyzed using rel. hydro. The second-order terms in IS theory do not affect the (long- wave length) hydrodynamic modes. As a dynamical critical phenomena, Brillouin peaks due to sound modes disappear. The disappearance of the sound mode suggests the suppresion or even disappearance of Mach cone, which can be a signal of the created matter hitting CP. Need further explicit calculation for confirmation, The (dynamical) RG method is applied to derive generic second-order hydrodynamic equations. There are so many terms in the relaxation terms which are absent in the previous works, especially due to the conformal non-invariance, which gives rise to an ambiguity in the separation in the first order and the second order terms (matching condition). A practical use of I-S level equations, however, may be problematic.

27. © University of Cambridge DoITPoMS, Department of Materials Science and Metallurgy, DoITPoMSDepartment of Materials Science and Metallurgy University of Cambridge University of Cambridge Information provided by doitpoms@msm.cam.ac.uk.doitpoms@msm.cam.ac.uk Critical Opalescence is a general phenomenon for the matter with 1 st order transition. Experiment at lab. HBT? http://www.msm.cam.ac.uk/doitpoms/tlplib/solid-solutions/videos/laser1.mov

28. Back Ups

29. for Definitions of critical exponents

30. Israel-Stewart eq. in particle frame ： relaxation times tends to coincide with the Eckart equation.

31. → diverges; unstable Eckart eq.(1 st order); I-S eq. → stable. Stability of I-S eq. → unstable! even with finite rel. time. Rel. time of thermal conductivity W.A.Hiscock and L.Lindblom Phys.Rev.D35(1987)

32. STAR data Away side shape modification STAR 2.5 < p T trig < 4 GeV/c 1< p T assoc < 2.5 GeV/c Technique: Measure 2- and 3- particle correlations on the away- side triggered by “high” p T hadron in central coll’s. Cone-shaped emission should show up in 3- particle correlations as signal on both sides of backward direction. Central Au+Au 0-12% (STAR) (  1 -  2 )/2 B. Muller@NFQCD08

33. The Mach cone  (z - ut)  Energy density Momentum density gTgT gLgL Unscreened source with  min/max cutoff B. Muller@NFQCD08 R. B. Neufeld, B. Muller, and J. Ruppert,arXiv:0802.2254

34. References on the RG/E method: T.K. Prog. Theor. Phys. 94 (’95), 503; 95(’97), 179 T.K.,Jpn. J. Ind. Appl. Math. 14 (’97), 51 T.K.,Phys. Rev. D57 (’98),R2035 T.K. and J. Matsukidaira, Phys. Rev. E57 (’98), 4817 S.-I. Ei, K. Fujii and T.K., Ann. Phys. 280 (2000), 236 Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24 T.K. and K. Tsumura, J. Phys. A: Math. Gen. 39 (2006), 8089 (hep-th/0512108) K. Tsumura, K. Ohnishi and T.K., Phys. Lett. B646 (2007), 134 L.Y.Chen, N. Goldenfeld and Y.Oono, PRL.72(’95),376; Phys. Rev. E54 (’96),376. C.f.