Imaginary Chemical potential and Determination of QCD phase diagram

1. From the effective theory Imaginary Chemical potential and Determination of QCD phase diagram M. Yahiro (Kyushu Univ.) Collaborators: H. Kouno (Saga Univ.), K. Kashiwa, Y. Sakai(Kyushu Univ.) 2009/08/3 XQCD 2009

2. Our papers on imaginary chemical potential 2008-2009 • Polyakov loop extended NJL model with imaginary chemical potential,Phys. Rev. D77 (2008), 051901. • Phase diagram in the imaginary chemical potential region and extended Z(3) symmetry,Phys. Rev. D78(2008), 036001. • Vector-type four-quark interaction and its impact on QCD phase structure, Phys. Rev. D78(2008), 076007. • Meson mass at real and imaginary chemical potential, Phys. Rev. D 79, 076008 (2009). • Determination of QCD phase diagram from imaginary chemical potential region, Phys. Rev. D 79, 096001 (2009). • Correlations among discontinuities in QCD phase diagram, J. Phys. G to be published.

3. Prediction of QCD phase diagram First-principle lattice calculation is difficult at finite real chemical potential, because of sign problem. Where is it ? Lattice calculation is done with some approximation. Sign problem Where is the critical end point?

4. Imaginary chemical potential T ? • Motivation Lattice QCD has no sigh problem. • Lattice data P. de Forcrand and O. Philipsen, Nucl. Phys. B642, 290 (2002); P. de Forcrand and O. Philipsen, Nucl. Phys. B673, 170 (2003). M. D’Elia and M. P. Lombardo, Phys. Rev. D 67, 014505(2003); Phys. Rev. D 70, 074509 (2004); M. D’Elia, F. D. Renzo, and M. P. Lombardo, Phys. Rev. D 76, 114509(2007); H. S. Chen and X. Q. Luo, Phys. Rev. D72, 034504 (2005); arXiv:hep-lat/0702025 (2007). S. Kratochvila and P. de Forcrand, Phys. Rev. D 73, 114512 (2006) L. K. Wu, X. Q. Luo, and H. S. Chen, Phys. Rev. D76, 034505(2007). O.K. 0 μ2 effective model Real μ Imaginary μ

5. Roberge-Weiss periodicity Nucl. Phys. B275(1986) Dimensionless imaginary chemical potential: Temperature: QCD partition function

6. Z3 transformation where is an element of SU(3) with the boundary condition for any integer

7. RW periodicity and extended Z3 transformation under Z3 transformation. for integer Roberge-Weiss Periodicity Invariant under the extended Z3 transformation

8. QCD has the extended Z3 symmetry in addition to the chiral symmetry This is important to construct an effective model. The Polyakov-extended Nambu-Jona-Lasinio (PNJL) model Fukushima; PLB591

9. Polyakov-loop Nambu-Jona-Lasinio (PNJL)model Two-flavor Fukushima; PLB591 gluon potential quark part (Nambu-Jona-Lasinio type) It reproduces the lattice data in the pure gauge limit. , Ratti, Weise; PRD75

10. Mean-field Lagrangian in Euclidean space-time for Performing the path integration of the PNJL partition function the thermodynamic potential

11. Thermodynamic potential (1) where invariant under the extended Z3 transformation

12. Thermodynamic potential (2) Polyakov-loop is not invariant under the extended Z3 transformation; Modified Polyakov-loop Thermodynamic potential Extended Z3 invariant RWperiodicity:

13. Θ-evenness Stationary condition Invariant under charge conjugation Θ-even

14. Model parameters Gs This model reproduces the lattice data at μ=0. , Ratti, Weise; PRD75

15. Thermodynamic Potential low T=Tc high T=1.1Tc Kratochvila, Forcrand; PRD73 low T RW transition high T

16. Polyakov-loop susceptibility PNJL Lattice data: Wu, Luo, Chen, PRD76(07).

17. Phase of Polyakov loop PNJL Lattice data: Forcrand, Philipsen, NP B642(02), Wu, Luo, Chen, PRD76(07)

18. Phase diagram for deconfinement phase trans. Lattice data: Wu, Luo, Chen, PRD76(07) PNJL RW RW periodicity

19. Chiral condensate and quark number density Quark Number Chiral Condensate Θ-odd Θ-even Low T High T Lattice D’Elia, Lombardo(03)

20. Phase diagram for chiral phase transition PNJL RWline Chiral Deconfinement Forcrand,Philipsen,NP B642 Chiral Deconfinement Θ-even higher-order interaction

21. Zero chemical potential PNJL Lattice data: Karsch et al. (02)

22. Higher order correction 8-quark PNJL ＋ Θ-even in next-to-leading order ＋ Power counting rule based on mass dimension Lattice Karsch, et. al.(02)

23. Θ-even in next-to-leading order PNJL 8-quark ＋ RW Chiral Forcrand,Philipsen, NP B642 Deconfinement

24. Another correction 8-quark (Θ-even) PNJL ＋ RW difference Chiral Forcrand,PhilipsenNPB642 Deconfinement

25. Vector-type interaction 8-quark (Θ-even) Vector-type (Θ-odd) PNJL ＋ ＋ RW Chiral Forcrand,PhilipsenNPB642 Deconfinement

26. Phase diagram at real μ 8-quark Vector-type PNJL ＋ ＋ RW Lattice de-confined Chiral Deconfinement CEP CEP confined 1’ st order

27. Critical End Point StephanovLattice2006 Lattice Taylor Exp.(LTE) Reweighting(LR) Model Our result (784, 125)

28. Meson mass Mesonic correlation function Random phase approximation (Ring diagram approximation) One-loop polarization function H. Hansen, W. M. Alberico, A. Beraudo, A. Molinari, M. Nardi and C. Ratti, Phys. Rev. D 75 (2007) 065004. K. Kashiwa,M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro1, Phys. Rev. D 79, 076008 (2009).

29. Meson mass with RW periodicity T=160 MeV oscillation

30. Extrapolation T=160 MeV PNJL

31. Conclusion • QCD has a higher symmetry at imaginary μ, called the extended Z3 symmetry. • PNJL has this property. • PNJL well reproduces lattice data at imaginary μ. • PNJL predicts that the CEP survives, even if the vector interaction is taken into account. • Meson mass also has RW periodicity at imaginary μ.

32. Thank you

33. Higher order correction Θ-even in next-to-leading order PNJL 8-quark ＋ ＋ Mean field approx. 1/N expansion Kashiwa et al. PLB647(07),446; PLB662(08),26. Lattice Karsch, et. al.(02)