1. Importance of imaginary chemical potential for QCD phase diagram in the PNJL model Kouji Kashiwa H. Kouno A, Y. Sakai, T. Matsumoto and M. Yahiro Recent studies (2008 – 2010) Kyushu Univ., Saga Univ. A 1/20 Y. Sakai, K. K., H. Kouno and M. Yahiro, Phys. Rev. D 77 (2008) 051901(R). Y. Sakai, K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 78 (2008) 036001. Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 79 (2009) 096001. K. K., M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro, Phys. Rev. D 79 (2009) 076008. K. K., M. Yahiro, H. Kouno, M. Matsuzaki and Y. Sakai, J. Phys. G. 36 (2009) 105001. K. K, H. Kouno and M. Yahiro, Phys. Rev. D 80 (2009) 117901. K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B 662 (2008) 26. H. Kouno, Y. Sakai, K. K., and M. Yahiro, J. Phys. G. 36 (2009) 115010.

2. Introduction Quantitative understanding of the QCD phase diagram at finite  R is quite poor. In recent theoretical studies, novel scenarios for QCD phase diagram are suggested. Quarkyonic phase Multi critical-endpoints generationex.) M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Prog. Theor. Phys. 108 (2002) 929. T. Hatsuda, M. Tachibana, N. Yamamoto and G. Baym, Phys. Rev. Lett. 97 (2006) 122001. Z. Zhang, K. Fukushima and T. Kunihiro, Phys. Rev. D 79 (2009) 014004. M. Harada, C. Sasaki and S. Takemoto, arXiv:0908.1361. Qualitative understanding is now running well !! However, and more … Lifshitz-point induced by the inhomogeneous phase D. Nickel, Phys. Rev. Lett. 103 (2009) 072301; Phys. Rev. D 80 (2009) 074025. L. McLerran and R. D. Pisarski, Nucl. Phys. A 796 (2007) 83. Y. Hidaka, L. McLerran and R. D. Pisarski, Nucl. Phys. A 808 (2008) 117. K. Miura, T. Z. Nakano and A. Ohnishi. Prog. Theor. Phys. 122 (2009) 1045. L. McLerran, K. Redlich and C. Sasaki, hep-ph/0812.3585

3. Introduction A schematic view LHC RHIC GSI JPARC Early universe Compact star ρ0ρ0 AGS SPS KEK-PS Quantitative understanding of the QCD phase diagram is important. To investigate these properties quantitatively, our present understanding of the QCD phase diagram is not enough.

4. Problem A schematic view However, first principle lattice QCD simulation have the sign problem at real chemical potential. K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B 662 (2008) 26. Effective models have some ambiguities.

5. Imaginary chemical potential Reason 1 : Strategy LQCD simulation can exactly calculated in the imaginary chemical potential, because there is no sign problem. Imaginary chemical potential region has almost all information of real one. Reason 2: In usual studies using information in the  I region, we can not reach the moderate and high  R region. To exploit it more, K. K, M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro, Phys. Rev. D 79 (2009) 076008. M. P. Lombardo, PoSCPOD2006 (2006) 003, hep-lat/0612017. We pay an attention to the imaginary chemical potential.

6. Strategy QCD phase diagram at finite  0 4  /3 Imaginary chemical potential matching approach 1 1. We determine the strengths of interactions from LQCD data at  I. 2 2. We then apply the model to the  R region. (Usual effective models have some ambiguityies caused by chemical potential effects) (This extended model can well describe (real) chemical potential effects because the effects are taken into account through the comparison between model result and LQCD data at finite  I.) Through this method, we can explore the moderate and high chemical potential ! The imaginary chemical potential region have several and important information of real chemical potential. Roberge Weiss (RW) phase transition line

7. Imaginary chemical potential Phase diagram at imaginary chemical potential Properties: Thermodynamical quantities have the RW periodicity. New transition line appears. Period ： 2  /3 RW transition line These properties become strong constraint for the extended model. 2  /3 0 Remains of the Z 3 symmetry in pure gauge limit. A. Roberge and N. Weiss, Nucl. Phys. B 275 (1986) 735. 4  /3 These properties are directly obtained from QCD.  =  I /T Roberge Weiss (RW) phase transition line Dimensionless chemical potential

8. PNJL model The Lagrangian density of the PNJL model Chiral phase transition （ Approximately) Deconfinement phase transition K. Fukushima, Phys. Lett. B 591 (2004) 277. Which model should be used in the imaginary chemical potential matching approach? Our model must have the RW periodicity. We already know the model! Effective gluon propagator Y. Sakai, K. K, H. Kouno and M. Yahiro, Phys. Rev. D 77 (2008) 051901(R). Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 78 (2008) 036001. Intuitively, ex.) C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D 73 (2006) 014019. The PNJL model can reproduce QCD properties at finite T and zero . S. Ro¨ßner, C. Ratti, and W. Weise, Phys. Rev. D 75 (2007) 034007.

9. PNJL model Importance of multi-leg interaction Usual NJL-type model ・・・ scalar-type four-quark interaction only. Other interactions are neglected. (vector-type four-quark, scalar-type eight-quark interactions…) However, there are no reason that these are neglected. Mass term Same order （ 1/N c expansion ） The scalar-type eight-quark interaction leads the T and  dependence to the effective coupling constant. m0m0 GG4G4 G s8 K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B662 (2008) 26. K. K., H. Kouno, T. Sakaguchi, M. Matsuzaki and M. Yahiro, Phys. Lett. B 647 (2007) 446. K. K., M. Matsuzaki, H. Kouno, and M. Yahiro, Phys. Lett. B 657 (2007) 143. 4-leg 8-leg (in Lagrangian density) + + Our model (Vector-type)

10. K.K., Y. Sakai, H. Kouno, M. Matsuzaki and M. Yahiro, J. Phys. G36 (2009). RW periodicity in chiral limit Lattice data: H. S. Chen and X. Q. Luo, Phys. Rev. D 72 (2005) 034504 M. D’ Elia and M. P. Lambard, Phys. Rev. D 67 (2003) 145005. In this region, different –order discontinuities can co-exist. This fact can be proofed by model independent analysis. Modified Polyakov-loop These are RW periodic quantities.

11. Parameter set obtained by our approach and results PNJL model in realistic case P. de Forcrand and O. Philipsen, Nucl. Phys. B 642 (2002) 290. L. K. Wu, X. Q. Luo and H. S. Chen, Phys. Rev. D 76 (2007) 034505. Lattice data: (Test fitting) Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 79 (2009) 096001. Set C

12. PNJL model Phase diagram Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 79 (2009) 096001. In our extended PNJL model has one critical endpoint !!

13. Meson mass Meson masses in PNJL model Meson masses are good quantities to determined strengths and kinds of interactions in the PNJL model. The 2 flavor case:  and  meson The 2+1 flavor case:  a 0, f 0 and  ’ meson Meson masses do not depend on the renormalization point. Model parameters (NJL part) largely affect meson masses.

14. Meson mass formula Meson masses in PNJL model Random phase approximation At finite chemical potential in the PNJL model : H. Hansen, W. M. Alberico, A. Beraudo, A. Molinari, M. Nardi and C. Ratti, Phys. Rev. D 75 (2007) 065004. p4p4 p 4 +(  – iA 4 ) We consider the scalar and pseudo-scalar meson masses. K. K, M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro, Phys. Rev. D 79 (2009) 076008. p p

15. Meson masses have the RW periodicity! Opposite oscillation T=160 MeV PNJL results K. K, M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro, Phys. Rev. D79 (2009) 076008. MM

16. T Matsumoto, K. K, H. Kouno and M. Yahiro, in preparation. PNJL results We use the 2+1-flavor PNJL model. To consider the U A (1) anomaly effect, following determinant interaction is introduced. This K can have the T and  dependence because of the variation of instanton density. However, quantitative behavior is not known. Therefore, the 2+1-flavor system is very ambiguous at finite real chemical potential. ex.) J. -W. Chen, K. Fukushima, Hi. Kohyama, K.Ohnishi, U. Raha, Phys. Rev. D80 (2009) 054012.

17. Meson masses also have the RW periodicity! T=300 MeV T Matsumoto, K. K, H. Kouno and M. Yahiro, in preparation. PNJL results Chiral symmetry re-broken Qualitatively behavior is same as the 2-flavor results. Result Near  =  /3, the chiral symmetry is broken again. To investigate the U A (1) anomaly effects, we will vary the strength of K.

18. T Matsumoto, K. K, H. Kouno and M. Yahiro, in preparation. T=300 MeV PNJL results Chiral symmetry re-broken Qualitative difference arises near  =  /3. T=300 MeV

19. T Matsumoto, K. K, H. Kouno and M. Yahiro, in preparation. PNJL results Qualitative difference arises near  =  /3.  meson mass  ’ meson mass T=300 MeV

20. Summary We investigate properties of the imaginary chemical potential by using the PNJL model. At the imaginary chemical potential, the strengths of the vector-type interaction and also determinant interaction can be determined. To quantitatively investigate the phase structure at finite real chemical potential, we propose the imaginary chemical potential matching approach. If we can refer LQCD data for several meson masses and thermodynamical quantities, we can quantitatively investigate the phase structure at finite  R. The imaginary chemical potential matching approach can be applied to the 2+1 flavor system.

21. END

22. Rw periodicity

23. Strange world Imaginary chemical potential 2  /3 periodicity Phase diagram at imaginary chemical potential Partition function of SU(N) gauged theory with imaginary chemical potential 2  /3

24. Strange world Imaginary chemical potential Ｚ（ θ ） has the periodicity of 2πk/N !! Z N transformation (Gauge)

25. Relation between imaginary and real chemical potential

26. Strange world Imaginary chemical potential This strange world is useful space because there are many important information about real chemical potential. M. P. Lombardo, PoSCPOD2006 (2006) 003, hep-lat/0612017. We assume the smooth connection at  2 =0. Those results suggest that the physics at real  is deeply related to imaginary one. (4-flavor) P[0,M] is corresponding the to Taylor N-th partial sum.

27. Numerical extrapolation Good method? M. P. Lombardo, PoSCPOD2006 (2006) 003, hep-lat/0612017. ex.) Gross-Neveu model Tri-critical point H. Kohyama, Phys. Rev. D 77 (2008) 045016. if the CSC is take into account, the phase structure is modified. In this simple case, we can not reach the structure at high chemical potential region by using the extrapolation. But, More simpler model than the NJL (QCD motivated) model Phase diagram F. Karbstein and M. Thies, Phys. Rev. D 75 (2007) 025003.

28. In QCD Thermodynamical value become  -even or  -odd function at imaginary chemical potential. Point of extrapolation ex.) chiral condensate, real part of modified Polyakov-loop  -even ・・・ :  -odd: ・・・ quark number density, imaginary part of modified Polyakov-loop CP invariance Quantities are function of  2. Fugacity expansion: Relation between imaginary and real chemical potential Fourier representation: Numerical extrapolation

29. Current quark mass dependence

30. Meson mass is good indicator of the consistency about current mass. T=160 MeV Result 3-3

31. QCD to GCM to NJL model

32. Derivation of the NJL model from QCD 1 Lagrangian density of QCD

33. Derivation of the NJL model from QCD 2 Expansion in powers of quark current. The GCM consists in keeping only W (2). It can reproduce properties of the QCD such as the confinement, asymptotic freedom.

34. Derivation of the NJL model from QCD 3 Lagrangian density of GCM Effective gluon propagator Fierz transformation Local approximation Other method is using the field strength method. In this method, quadratic expansion around auxiliary field and limit of small momenta are used.

35. Interaction of the NJL model 1 1 gluon exchange interaction4 quark interaction gluon Color-singlet Attractive Gluon degree of freedom is integrated out Local approximation Color anti-symmetric Color symmetric AttractiveRepulsive In the ordinary NJL model, interactions are constructed by four-quark interaction only.

36. Parameter set

37. Formalism Polyakov-loop potential RTW05: RRW06: C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D73, 014019 (2006). S. Ro¨ßner, C. Ratti and W. Weise. Phys. Rev. D75, 034007 (2007).

38. PNJL model Parameter set K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B662 (2008) 26. K. K., H. Kouno, T. Sakaguchi, M. Matsuzaki and M. Yahiro, Phys. Lett. B 647 (2007) 446. K. K., M. Matsuzaki, H. Kouno, and M. Yahiro, Phys. Lett. B 657 (2007) 143. Gs (scalar type 4-quark interaction) G s8 (scalar type 8-quark interaction) Λ (3 dimensional momentum cutoff ). Parameters: G s, G v, G s8 or Our usual procedure Our new procedure Parameters: G s, G v, G s8 G v is free parameter. All parameters are fitted in imaginary chemical potential region (and  =0 region). Y. Sakai, K. K, H. Kouno and M. Yahiro, Phys. Rev. D 77 (2008) 051901(R). Y. Sakai, K. K, H. Kouno and M. Yahiro, Phys. Rev. D 78 (2008) 036001. Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 78 (2008) 076071.