1. Quark deconfinement and symmetry Hiroaki Kouno Dept. of Phys., Saga Univ. Collaboration with K. Kashiwa, Y. Sakai, M. Yahiro （ Kyushu. Univ.) and M. Matsuzaki (Fukuoka Univ. of Education)

2. Confinement of quark At zero temperature and zero density, quarks are confined in hadrons (baryons, mesons) Color is also confined. Hadrons are white.

3. If you want to “cut” a meson,… you need “Energy” which creates a pair of quark and anti- quark, because of the Einstein’s famous relation E=Mc 2. You only get two mesons!! Not a isolated quark.

4. Quark-gluon plasma (QGP) However, at finite- temperature and/or finite density, it is expected that hadron will melt and quark- gluon plasma will be formed. This phenomenon is regarded as a phase transition.

5. Lattice QCD (LQCD) simulations LQCD is a computer simulation of QCD. Since we can not construct continuous space time in computer, we use discrete lattice space-time as an approximation. Quark and gluon live in this lattice space time.

6. Lattice QCD simulation At critical temperature, there is a jump in the energy density of the system just like a liquid-gas (vapor) phase transition. At finite density, it is difficult to do the Lattice QCD simulation because of the sign problem. However, the phenomenological model calculations predict the quark phase at high density.

7. Hadron-QGP transition

8. Chiral symmetry restoration There is also chiral phase transition at Tc, where the quark mass becomes small suddenly.

9. Predicted QCD phase diagram (by Yuji Sakai) 1/18

10. Introduction Interaction Result of NJL model Result of PNJL model Summary Recent development of the dense quark matter Problems of lattice QCD calculation Several approaches ? From NASA From RHIC in the finite chemical potential region (T>>μ), lattice QCD calculation is not feasible. Therefore, the low energy effective theory of QCD is often used in finite chemical potential.

11. Recent development of the dense QCD study Problems of lattice QCD calculation Several approaches Experiment Inside of compact star Relativistic Heavy Ion Collider (RHIC) Large Hadron Collider (LHC) ・ ・ ・ Many experimental evidences are obtained at RHIC. But there are no absolute evidence. Equations of state have many ambiguity in quark part. We do not know the method to calculate the dense QCD at moderate density region exactly !! Introduction Interaction Result of NJL model Result of PNJL model Summary

12. Phase transition and symmetry Phases are classified by symmetry and an order parameter φ. =0 ⇒ symmetry is preserved. Symmetric phase ≠0 ⇒ symmetry is spontaneously broken. Symmetry broken phase

13. Discrete mirror symmetry Consider the potential which has mirror symmetry with respect to y-axis, (1)V(x)=x 2 (2)V(x)=-2x 2 +x 4 Ground state or vacuum is defined at the minimum of the potential V(x)

14. Discrete mirror symmetry (1) =0 ⇒ symmetry is preserved. Symmetric phase (2) ≠0 ⇒ symmetry is spontaneously broken Broken phase The vacuum solution breaks the symmetry!!

15. Continuous rotational symmetry r 2 =x 2 +y 2 Consider rotational symmetric potential (1) v(r)=r 2 (2) v(r)=-2r 2 +r 4

16. Symmetry is preserved (1) =0 ⇒ symmetry is preserved. Symmetric phase Ground state solution is (x,y)=(0,0). Rotational symmetry around (0,0).

17. Symmetry is spontaneously broken (2) ≠0 ⇒ symmetry is spontaneously broken Broken phase The vacuum solution breaks the symmetry!! No rotational symmetry around (x,y)=(1,0).

18. Nambu-Goldstone bosn If the symmetry is broken and the vacuum solution is (x,y)=(a,0) Square of mass of the particle x is proportional to Square of mass of the particle y proportional to The particle y is a massless particle Nambu-Goldstone boson

19. Phase transition T>Tc T<Tc ≠0 =0 symmetric phase broken phase

20. It should be remarked that The degeneracy of the ground states induces the discontinuity between symmetric phase and symmetry broken phase. If degeneracy disappears, the discontinuity disappears and phase transition disappears.

21. Phenomenological models Since the QCD itself is very complicated and is hard to be solved nonpertubatively, we use phenomenological model. ・ For chiral phase transition, we use the linear sigma model. ・ For deconfinement transition, we use the Polyakov-Nambu-Jona-Lasino (PNJL) model.

22. Direct interaction of quark At low energy, effective direct quark interaction is induced by the gauge interaction at high energy.

23. Nambu-Jona-Lasinio model Consider the direct quark-quark interaction instead of gauge gluon-quark interaction. ⇒ Nambu-Jona-Lasinio model （ＮＪ Ｌ）

24. Ｍｅｓｏｎ ｆｉｅｌｄ If we identify as σ and π meson fields, we obtain the linear sigma model.

25. Linear sigma model Rotational invariance in σ–π plane ⇒ chiral symmetry

26. Spontaneous breaking At low temperature and/or low density is negative. Chiral symmetry is spontaneously broken. πmeson is a NG boson. ≠0 ⇒ M≠0 Quark becomes heavy.

27. Restoration of chiral symmetry At high temperature, becomes positive, chiral symmetry is restored. =0 ⇒ M=0 quark becomes massless.

28. Polyakov Loop Polyakov Loop is defined by

29. Polyakov loop and confinement The isolated quark free energy F is given by F ～ -log(Φ). Therefore, if Φ is zero, a quark is confined since F ～ -log(Φ)=-log(0)=∞. If Φis finite, quarks are deconfined since F ～ -log(Φ)=finite.

30. Polyakov potential Pure LQCD results gives the Polyakov loop potential as

31. Discrete Z 3 symmetry Polyakov potential is invariant under discrete Z 3 transformation where k is a integer.

32. Symmetry is preserved Z 3 symmetry is preserved at low temperature. This means F is ∞, since F ～ -log(Φ) =-log(0)=∞. Therefore, a quark is confined.

33. Symmetry is spontaneously broken At high temperature, Z 3 symmetry is spontaneously broken. There are three degenerate ground states. ・ This means F is finite, since F ～ -log(Φ)=finite. Therefore, quarks are deconfined.

34. Deconfinement phase transition T>Tc T<Tc ≠0 =0 symmetric phase broken phase

35. It should be remarked that Different from Chiral symmetry, Z 3 symmetry is preserved at low temperature and broken at high temperature. Since Z 3 symmetry is a discrete symmetry, there is no Nambu-Goldstone boson. If the effects of quark-anitiquark pair creations are taken into account, Z 3 symmetry is explicitly broken. Therefore Φ is not an exact order parameter any more.

36. PNJL model To include quantum effects of quarks, we use PNJL model, in which the Polyakov loop potential is included as well as the NJL Lagrangian. PNJL = NJL +Polyakov Loop pot. + gauge interaction

37. PNJL model ● Polyakov potential パラメータ C. Ratti, et al. Phys. Rev. D73, 014019 (2006) O. Kaczmarek, et al., Phys. Lett. B 543 (2002) 41. ● Polyakov-loop PNJL ＝ NJL （ chiral symmetry) ＋ Polyakov-loop （ confinement ） 6/18

38. Gauge and direct interactions

39. Thermodynamic potential 7/18

40. Z 3 transformation The PNJL thermodynamic potential is not invariant under the Z 3 transformation

41. No phase transition Since the Z 3 symmetry explicitly broken, even at high temperature, the ground state is not degenerate and there no discontinuity between the confined phase and deconfined phase!! The transition becomes crossover.

42. Extended Z 3 transformation However, the PNJL thermodynamic potential is invariant under the extended Z 3 transformation with any integer k

43. It should be noted that Since we change the external variable, chemical potential, the extended Z 3 symmetry is not an internal symmetry and the ground state is not degenerate even at high temperature. To see the physical meaning of the extended Z 3 symmetry, we consider the system with imaginary chemical potential. (Not a real world!!)

44. Welcome to Imaginary world!! Below we consider imaginary chemical potential. Extended Z 3 transformation is rewritten by

45. Thermodynamic potential 7/18 extended Z 3 trans. 修正版 Polyakov ループ

46. Thermodynamic potential 7/18 extended Z 3 trans. 修正版 Polyakov ループ Extended Z 3 inv.

47. Roberge-Weiss periodicity Since thermodynamic potential depends on the chemical potential only through the factor e i3θ, it is clear that Ω is invariant under extended Z 3 transformation, and there is a Roberge- Weiss periodicity

48. Extended Z 3 symmetry RW even RW odd periodicity Same symmetry 8/18

49. Thermodynamic potential T RW 9/18 TCTC Kratochvila, Forcrand PRD73,114512(2006)

50. Chiral condensate and quark density D’Elia, Lombardo PRD67, 014505 (2003) T RW TCTC 10/18 D’Elia, Lombardo PRD67, 014505 (2003)

51. 11/18 Chen, Luo PRD72, 034504 (2005) Wu, Luo, Chen PRD76,034505(2007) Polyakov Loop T RW TCTC

52. RW phase transition At high temperature (T>T RW ), there is discontinuity at θ=(2k+1)π/3. It is call Roberge-Weiss (RW) phase transition. ・ The point (θ,T)=(π/3,T RW ) is the end point of the RW phase transition.

53. Phase diagram with imaginary chemical potential Chiral 転移線 Polyakov 転移線 RW 転移線 14/18

54. Phase diagram with imaginary chemical potential Chiral trans. Polyakov trans. RW trans. 14/18 Comparison with LQCD M. D’Elia et al. PRD76, 114509 (2007)

55. Phase diagram Lattice QCD PNJL model 多項式近似による外挿 CEP (m=1,2,3,4) 15/18

56. Remnant of deconfinement phase transition The RW phase transition is a remnant of deconfinement phase transition. The RW endpoint seems to dominate the deconfinement (crossover) transition at zero and real chemical potential, although it does not exist in the real world. It is important to study the properties of RW endpoint.

57. Summary Phase transition can be classified by the symmetry and the order parameter. If the order parameter is zero, symmetry is preserved. The symmetry is spontaneously broken, if the order parameter is nonzero. There is no internal symmetry and order parameter for the quark deconfinement transition.

58. Summary However, if we were admitted to transform the external variable, the chemical potential, we have the exended Z 3 symmetry and the RW phase transition. To analyze the deconfinement transition in the real world, it is important to study the properties of the endpoint of RW phase transition. The work is in progress.

59. Vector-type interaction PNJL 16/18 カイラル凝縮とクォーク数を見ればベクター相互作用の強さを決めることができる。

60. Phase diagram with vector-type interaction Ｇ v=0 Gv の強さに CEP は敏感に反映される。 17/18 Ｇ v=0.25Gs PNJL

61. Three ground states in PNJL