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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. Kyushu Univ., Saga Univ. A. Recent studies (2008 – 2010). K. K. , H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B 662 (2008) 26.

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## Importance of imaginary chemical potential for QCD phase diagram in the PNJL model

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**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 Kyushu Univ., Saga Univ.A Recent studies (2008 – 2010) K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B 662 (2008) 26. 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. K. K., M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro, Phys. Rev. D 79 (2009) 076008. Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 79 (2009) 096001. K. K., M. Yahiro, H. Kouno, M. Matsuzaki and Y. Sakai, J. Phys. G. 36 (2009) 105001. H. Kouno, Y. Sakai, K. K., and M. Yahiro, J. Phys. G. 36 (2009) 115010. K. K, H. Kouno and M. Yahiro, Phys. Rev. D 80 (2009) 117901. 1/20**Introduction**In recent theoretical studies, novel scenarios for QCD phase diagram are suggested. ex.) Multi critical-endpoints generation 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. Quarkyonic phase 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 Lifshitz-point induced by the inhomogeneous phase D. Nickel, Phys. Rev. Lett. 103 (2009) 072301; Phys. Rev. D 80 (2009) 074025. and more … Qualitative understanding is now running well !! However, Quantitative understanding of the QCD phase diagram at finite mRis quite poor.**Introduction**Quantitative understanding of the QCD phase diagram is important. A schematic view To investigate these properties quantitatively, our present understanding of the QCD phase diagram is not enough. LHC RHIC Early universe SPS AGS JPARC GSI KEK-PS Compact star ρ0**Problem**However, first principle lattice QCD simulation have the sign problem at real chemical potential. A schematic view Effective models have some ambiguities. K. K,H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B 662 (2008) 26.**Imaginary chemical potential**Strategy We pay an attention to the imaginary chemical potential. Reason 1 : LQCD simulation can exactly calculated in the imaginary chemical potential, because there is no sign problem. Reason 2: Imaginary chemical potential region has almost all information of real one. In usual studies using information in the mI region, we can not reach the moderate and high mR region. Toexploit it more, We propose the new approach ; imaginary chemical potential matching approach. 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.**Strategy**QCD phase diagram at finite q Imaginary chemical potential matching approach Roberge Weiss (RW) phase transition line 1. We determine the strengths of interactions from LQCD data at mI . 0 4p/3 (Usual effective models have some ambiguityies caused by chemical potential effects) 2. We then apply the model to the mR region. (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 mI.) The imaginary chemical potential region have several and important information of real chemical potential. Through this method, we can explore the moderate and high chemical potential !**Imaginary chemical potential**Phase diagram at imaginary chemical potential Roberge Weiss (RW) phase transition line Dimensionless chemical potential 2p/3 q = mI/T Remains of the Z3symmetry in pure gauge limit. 4p/3 0 Properties: Period：2p/3 Thermodynamical quantities have the RW periodicity. New transition line appears. RW transition line A. Roberge and N. Weiss, Nucl. Phys. B 275 (1986) 735. These properties are directly obtained from QCD. These properties become strong constraint for the extended model.**PNJL model**Which model should be used in the imaginary chemical potential matching approach? Our model must have the RW periodicity. 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. We already know the model! The Lagrangian density of the PNJL model K. Fukushima, Phys. Lett. B 591 (2004) 277. Chiral phase transition Intuitively, （Approximately) Deconfinement phase transition The PNJL model can reproduce QCD properties at finite T and zero m. ex.) C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D 73 (2006) 014019. Effective gluon propagator S. Ro¨ßner, C. Ratti, and W. Weise, Phys. Rev. D 75 (2007) 034007.**K. K., H. Kouno, T. Sakaguchi, M. Matsuzaki and M. Yahiro,**Phys. Lett. B 647 (2007) 446. PNJL model K. K., M. Matsuzaki, H. Kouno, and M. Yahiro, Phys. Lett. B 657 (2007) 143. Importance of multi-leg interaction K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B662 (2008) 26. Usual NJL-type model ・・・scalar-type four-quark interaction only. 8-leg 4-leg Our model Other interactions are neglected. (vector-type four-quark, scalar-type eight-quark interactions…) (Vector-type) However, there are no reason that these are neglected. m0 Mass term + + Same order （1/Nc expansion） The scalar-type eight-quark interaction leads the T and m dependence to the effective coupling constant. G G4 (in Lagrangian density) Gs8**RW periodicity in chiral limit**K.K., Y. Sakai, H. Kouno, M. Matsuzaki and M. Yahiro, J. Phys. G36 (2009) . Modified Polyakov-loop These are RW periodic quantities. Lattice data: H. S. Chen and X. Q. Luo, Phys. Rev. D 72 (2005) 034504 In this region, different –order discontinuities can co-exist . This fact can be proofed by model independent analysis. M. D’ Elia and M. P. Lambard, Phys. Rev. D 67 (2003) 145005.**(Test fitting)**PNJL model in realistic case Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 79 (2009) 096001. Parameter set obtained by our approach and results Set C Lattice data: 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.**PNJL model**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 !! Phase diagram**Meson mass**Meson masses in PNJL model Meson masses are good quantities to determined strengths and kinds of interactions in the PNJL model. Meson masses do not depend on the renormalization point. Model parameters (NJL part) largely affect meson masses. The 2 flavor case: s and p meson The 2+1 flavor case: s, k, a0, f0 and p, K, h, h’ meson**Meson mass formula**We consider the scalar and pseudo-scalar meson masses. Meson masses in PNJL model Random phase approximation p p p4 +(m – iA4) p4 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. K. K, M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro, Phys. Rev. D 79 (2009) 076008.**PNJL results**K. K, M. Matsuzaki, H. Kouno, Y. Sakai and M. Yahiro, Phys. Rev. D79 (2009) 076008. 2 flavor Meson masses have the RW periodicity! T=160 MeV Opposite oscillation Mp**PNJL results**T Matsumoto, K. K, H. Kouno and M. Yahiro, in preparation. Next is the 2+1 flavor results We use the 2+1-flavor PNJL model. To consider the UA(1) anomaly effect, following determinant interaction is introduced. This K can have the T and m 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.**PNJL results**T Matsumoto, K. K, H. Kouno and M. Yahiro, in preparation. 2+1 flavor T=300 MeV Meson masses also have the RW periodicity! Result Chiral symmetry re-broken Qualitatively behavior is same as the 2-flavor results. Near q=p/3, the chiral symmetry is broken again. To investigate the UA(1) anomaly effects, we will vary the strength of K.**PNJL results**T Matsumoto, K. K, H. Kouno and M. Yahiro, in preparation. 2+1 flavor Qualitative difference arises near q=p/3. T=300 MeV T=300 MeV T=300 MeV Chiral symmetry re-broken**PNJL results**T Matsumoto, K. K, H. Kouno and M. Yahiro, in preparation. 2+1 flavor Qualitative difference arises near q=p/3. T=300 MeV h meson mass h’ meson mass**Summary**We investigate properties of the imaginary chemical potential by using the PNJL model. To quantitatively investigate the phase structure at finite real chemical potential, we propose the imaginary chemical potential matching approach. The imaginary chemical potential matching approach can be applied to the 2+1 flavor system. At the imaginary chemical potential, the strengths of the vector-type interaction and also determinant interaction can be determined. If we can refer LQCD data for several meson masses and thermodynamical quantities, we can quantitatively investigate the phase structure at finite mR.**Strange world**Imaginary chemical potential Phase diagram at imaginary chemical potential 2p/3 2p/3 periodicity Partition function of SU(N) gauged theory with imaginary chemical potential**Strange world**Imaginary chemical potential (Gauge) ZNtransformation Ｚ（θ）has the periodicity of 2πk/N !!**Strange world**Imaginary chemical potential We assume the smooth connection at m2=0. (4-flavor) M. P. Lombardo, PoSCPOD2006 (2006) 003, hep-lat/0612017. P[0,M] is corresponding the to Taylor N-th partial sum. Those results suggest that the physics at real m is deeply related to imaginary one. This strange world is useful space because there are many important information about real chemical potential.**Numerical extrapolation**Good method? ex.) Gross-Neveu model More simpler model than the NJL (QCD motivated) model Phase diagram Tri-critical point F. Karbstein and M. Thies, Phys. Rev. D 75 (2007) 025003. M. P. Lombardo, PoSCPOD2006 (2006) 003, hep-lat/0612017. But, 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. H. Kohyama, Phys. Rev. D 77 (2008) 045016.**Numerical extrapolation**In QCD Point of extrapolation Thermodynamical value become q-even or q-odd function at imaginary chemical potential. ex.) q-even ・・・: chiral condensate, real part of modified Polyakov-loop q-odd: ・・・ quark number density, imaginary part of modified Polyakov-loop Quantities are function of m2. CP invariance Relation between imaginary and real chemical potential Fourier representation: Fugacity expansion:**Result 3-3**Meson mass is good indicator of the consistency about current mass. T=160 MeV**Derivation of the NJL model from QCD 1**Lagrangian density of QCD**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.**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.**Interaction of the NJL model 1**1 gluon exchange interaction 4 quark interaction Gluon degree of freedom is integrated out Local approximation Color-singlet Attractive gluon Color symmetric Color anti-symmetric Attractive Repulsive In the ordinary NJL model, interactions are constructed by four-quark interaction only.**Formalism**Polyakov-loop potential RTW05: C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D73, 014019 (2006). RRW06: S. Ro¨ßner, C. Ratti and W. Weise. Phys. Rev. D75, 034007 (2007).**PNJL model**K. K., H. Kouno, T. Sakaguchi, M. Matsuzaki and M. Yahiro, Phys. Lett. B 647 (2007) 446. Parameter set K. K., M. Matsuzaki, H. Kouno, and M. Yahiro, Phys. Lett. B 657 (2007) 143. K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B662 (2008) 26. Parameters: Gs, Gv, Gs8 Our usual procedure Gs(scalar type 4-quark interaction) Gs8(scalar type 8-quark interaction) Λ(3 dimensional momentum cutoff ). or Gv is free parameter. Parameters: Gs, Gv, Gs8 Our new procedure All parameters are fitted in imaginary chemical potential region (and m=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.

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