Quark deconfinement and symmetry

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=Mc2. • 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. 1/18 Predicted QCD phase diagram (by Yuji Sakai)

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

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

13. Discrete mirror symmetry • Consider the potential which has mirror symmetry with respect to y-axis, • V(x)=x2 • V(x)=-2x2+x4 Ground state or vacuum is defined at the minimum of the potential V(x)

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

15. Continuous rotational symmetry r2=x2+y2 Consider rotational symmetric potential (1) v(r)=r2 • v(r)=-2r2+r4

16. Symmetry is preserved • < r>=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) < r>≠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 <x>≠0 broken phase T>Tc <x>=0 symmetric 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　（ＮＪＬ）

24. Ｍｅｓｏｎ　ｆｉｅｌｄ • If we identify as σand 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. <r>≠0⇒M≠0 Quark becomes heavy.

27. Restoration of chiral symmetry • At high temperature, becomes positive, chiral symmetry is restored. <r>=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 sinceF　～　-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 Z3 symmetry • Polyakov potential is invariant under discrete Z3 transformation where k is a integer.

32. Symmetry is preserved • Z3 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, Z3 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 <x>≠0 broken phase T<Tc <x>=0 symmetric phase

35. It should be remarked that • Different from Chiral symmetry, Z3 symmetry is preserved at low temperature and broken at high temperature. • Since Z3 symmetry is a discrete symmetry, there is no Nambu-Goldstone boson. • If the effects of quark-anitiquark pair creations are taken into account, Z3 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. 6/18 PNJL model PNJL　＝　NJL（chiral symmetry)　＋　Polyakov-loop（confinement） C. Ratti, et al. Phys. Rev. D73, 014019 (2006) O. Kaczmarek, et al., Phys. Lett. B 543 (2002) 41. ●Polyakov-loop ●Polyakov potential パラメータ

38. Gauge and direct interactions

39. 7/18 Thermodynamic potential

40. Z3 transformation • The PNJL thermodynamic potential is not invariant under the Z3 transformation

41. No phase transition • Since the Z3 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 Z3 transformation • However, the PNJL thermodynamic potential is invariant under the extended Z3 transformation with any integer k

43. It should be noted that • Since we change the external variable, chemical potential, the extended Z3 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 Z3 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 Z3 transformation is rewritten by

45. 7/18 Thermodynamic potential extended Z3 trans. 修正版Polyakovループ

46. 7/18 Thermodynamic potential extended Z3 trans. 修正版Polyakovループ Extended Z3 inv.

47. Roberge-Weiss periodicity • Since thermodynamic potential depends on the chemical potential only through the factor ei3θ, it is clear thatΩis invariant under extended Z3 transformation, and there is a Roberge-Weiss periodicity

48. 8/18 Extended Z3 symmetry RWeven RW odd periodicity Same symmetry

49. 9/18 TRW Thermodynamic potential TC Kratochvila, Forcrand PRD73,114512(2006)

50. 10/18 TRW Chiral condensate and quark density TC D’Elia, Lombardo PRD67, 014505 (2003) D’Elia, Lombardo PRD67, 014505 (2003)