# Imaginary Chemical Potential, NJL-Type Model and Confinement–Deconfinement Transition

## Abstract

**:**

## 1. Introduction

- Sign problem free,
- Possible analytic continuation process to the real chemical potential,
- Relationship with the real chemical potential region via the canonical ensemble.

## 2. Roberge–Weiss Periodicity and Transition

**Roberge–Weiss (RW) periodicity:**Special $2\pi k/{N}_{\mathrm{C}}$ periodicity of the grand-canonical partition function ($\mathcal{Z}$) along the $\theta $-axis where $\theta :={\mu}_{\mathrm{I}}/T$ and $k\in \mathbb{N}$; $\mathcal{Z}(T,\theta )=\mathcal{Z}\left(T,\theta +\frac{2\pi k}{{N}_{c}}\right)$. See Section 2.1 for details.**Roberge–Weiss (RW) transition:**Special first-order transition which is characterized by the phase of the Polyakov loop and the quark number density appearing at $\theta =(2k-1)\pi /{N}_{\mathrm{c}}$ above the Roberge–Weiss endpoint temperature. See Section 2.2 for details.**Trivial and nontrivial ${\mathbb{Z}}_{{N}_{\mathrm{c}}}$ images:**Origin of the RW periodicity at high temperature. These are corresponding to minima of the thermodynamic potential characterized by the phase of the Polyakov loop. See Section 2.2.2 for details.**Spontaneous shift symmetry breaking:**Symmetry which characterizes the RW transition line. This symmetry is associated from the time reversal or the charge conjugation and ${\mathbb{Z}}_{{N}_{\mathrm{c}}}$ transformations via the semidirect product (It is first discussed by using the combination of the charge conjugation and ${\mathbb{Z}}_{{N}_{\mathrm{c}}}$ symmetries in Ref. [41].). In other words, the system symmetry at $\theta =(2k-1)\pi /{N}_{\mathrm{c}}$ is enhanced. The modified Polyakov-loop then becomes the order-parameter of the spontaneous breaking of this symmetry. See Section 2.4 for details.**Roberge–Weiss (RW) endpoint:**Endpoint of the first-order RW transition line. There are possibilities that the endpoint becomes the second-order (trivial scenario) or the first-order (nontrivial scenario) near the physical quark mass. The RW endpoint temperature is denoted by ${T}_{\mathrm{RW}}$. See Section 2.3 for details.

#### 2.1. Roberge–Weiss Periodicity

#### 2.2. Roberge–Weiss Transition

#### 2.2.1. RW Periodicity in the Confined Phase

#### 2.2.2. RW Periodicity in the Deconfined Phase

**Trivial**${\mathbb{Z}}_{3}$**-image:**$\varphi =0$ for $\theta =[-\pi /3,\pi /3]$,**Nontrivial**${\mathbb{Z}}_{3}$**-images:**$\varphi =2\pi /3$ for $\theta =[-\pi ,-\pi /3]$, $\varphi =-2\pi /3$ for $\theta =[\pi /3,\pi ]$,

#### 2.3. Roberge–Weiss Endpoint

#### 2.4. Shift Symmetry Breaking

## 3. Interplay of Imaginary Chemical Potential

- Prepare lattice QCD data for several observables at finite $\theta $.
- Prepare a suitable effective model which reproduces the RW periodicity and the transition.
- Set initial model parameters.
- Calculate observables by using the model and compare them.
- Reset model parameters.

#### 3.1. Analytic Continuation

#### 3.2. Boundary Condition of Fermion for the Temporal Direction

#### 3.3. Aharonov–Bohm Phase

- $U(1)$ flux insertion to holes of spatial closed loops.
- Exchanging of i-th and $i+1$-th quarks.
- Moving of the quark along loops.

## 4. NJL-Type Model at Finite Imaginary Chemical Potential

#### 4.1. Nambu–Jona–Lasinio Model

#### 4.2. Polyakov-Loop Extended Nambu–Jona–Lasinio Model

## 5. Application of Imaginary Chemical Potential to Explore the QCD Phase Diagram

#### 5.1. Analytic Continuation Method

#### 5.2. Canonical Ensemble Method

#### 5.2.1. Approach 1: Modified Polyakov-Loop Representation

#### 5.2.2. Approach 2: Trivial ${\mathbb{Z}}_{3}$-Image Restriction

#### 5.3. Lee–Yang Zero Analysis

## 6. Similarities Measurement

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- De Forcrand, P. Simulating QCD at finite density. PoS
**2009**, LAT2009, 010. [Google Scholar] - Parisi, G.; Wu, Y.S. Perturbation Theory Without Gauge Fixing. Sci. Sin.
**1981**, 24, 483. [Google Scholar] - Parisi, G. On complex probabilities. Phys. Lett.
**1983**, B131, 393–395. [Google Scholar] [CrossRef] - Witten, E. Analytic Continuation Of Chern-Simons Theory. AMS/IP Stud. Adv. Math.
**2011**, 50, 347–446. [Google Scholar] - Cristoforetti, M.; Di Renzo, F.; Scorzato, L. New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble. Phys. Rev.
**2012**, D86, 074506. [Google Scholar] [CrossRef] - Fujii, H.; Honda, D.; Kato, M.; Kikukawa, Y.; Komatsu, S.; Sano, T. Hybrid Monte Carlo on Lefschetz thimbles—A study of the residual sign problem. JHEP
**2013**, 1310, 147. [Google Scholar] [CrossRef] - Mori, Y.; Kashiwa, K.; Ohnishi, A. Toward solving the sign problem with path optimization method. Phys. Rev.
**2017**, D96, 111501. [Google Scholar] [CrossRef] - Mori, Y.; Kashiwa, K.; Ohnishi, A. Application of a neural network to the sign problem via the path optimization method. PTEP
**2018**, 2018, 023B04. [Google Scholar] [CrossRef] - Fukushima, K. Chiral effective model with the Polyakov loop. Phys. Lett.
**2004**, B591, 277–284. [Google Scholar] [CrossRef] - Haas, L.M.; Braun, J.; Pawlowski, J.M. On the QCD phase diagram at finite chemical potential. AIP Conf. Proc.
**2011**, 1343, 459–461. [Google Scholar] [CrossRef] - Biernat, E.P.; Gross, F.; Peña, T.; Stadler, A. Confinement, quark mass functions, and spontaneous chiral symmetry breaking in Minkowski space. Phys. Rev.
**2014**, D89, 016005. [Google Scholar] [CrossRef] - Fischer, C.S. QCD at finite temperature and chemical potential from Dyson–Schwinger equations. Prog. Part. Nucl. Phys.
**2019**, 105, 1–60. [Google Scholar] [CrossRef] - Biernat, E.P.; Gross, F.; Peña, M.T.; Stadler, A.; Leitão, S. Quark mass function from a one-gluon-exchange-type interaction in Minkowski space. Phys. Rev.
**2018**, D98, 114033. [Google Scholar] [CrossRef] - Miyahara, A.; Torigoe, Y.; Kouno, H.; Yahiro, M. Equation of state and transition temperatures in the quark-hadron hybrid model. Phys. Rev.
**2016**, D94, 016003. [Google Scholar] [CrossRef] - Gasser, J.; Leutwyler, H. Light Quarks at Low Temperatures. Phys. Lett.
**1987**, B184, 83–88. [Google Scholar] [CrossRef] - Allton, C.; Doring, M.; Ejiri, S.; Hands, S.; Kaczmarek, O.; Karsch, F.; Laermann, E.; Redlich, K. Thermodynamics of two flavor QCD to sixth order in quark chemical potential. Phys. Rev.
**2005**, D71, 054508. [Google Scholar] [CrossRef] - Borsanyi, S.; Fodor, Z.; Hoelbling, C.; Katz, S.D.; Krieg, S.; Ratti, C.; Szabo, K.K. Is there still any T
_{c}mystery in lattice QCD? Results with physical masses in the continuum limit III. JHEP**2010**, 9, 073. [Google Scholar] [CrossRef] - Borsanyi, S.; Fodor, Z.; Katz, S.D.; Krieg, S.; Ratti, C.; Szabo, K. Fluctuations of conserved charges at finite temperature from lattice QCD. JHEP
**2012**, 01, 138. [Google Scholar] [CrossRef] - Borsanyi, S.; Endrodi, G.; Fodor, Z.; Katz, S.D.; Krieg, S.; Ratti, C.; Szabo, K.K. QCD equation of state at nonzero chemical potential: Continuum results with physical quark masses at order mu
^{2}. JHEP**2012**, 8, 053. [Google Scholar] [CrossRef] - Hasenfratz, A.; Toussaint, D. Canonical ensembles and nonzero density quantum chromodynamics. Nucl. Phys.
**1992**, B371, 539–549. [Google Scholar] [CrossRef] - Alexandru, A.; Faber, M.; Horvath, I.; Liu, K.F. Lattice QCD at finite density via a new canonical approach. Phys. Rev.
**2005**, D72, 114513. [Google Scholar] [CrossRef] - Kratochvila, S.; de Forcrand, P. QCD at zero baryon density and the Polyakov loop paradox. Phys. Rev.
**2006**, D73, 114512. [Google Scholar] [CrossRef] - De Forcrand, P.; Kratochvila, S. Finite density QCD with a canonical approach. Nucl. Phys. Proc. Suppl.
**2006**, 153, 62–67. [Google Scholar] [CrossRef] [Green Version] - Li, A.; Alexandru, A.; Liu, K.F.; Meng, X. Finite density phase transition of QCD with N
_{f}= 4 and N_{f}= 2 using canonical ensemble method. Phys. Rev.**2010**, D82, 054502. [Google Scholar] [CrossRef] - Roberge, A.; Weiss, N. Gauge Theories With Imaginary Chemical Potential and the Phases of QCD. Nucl. Phys.
**1986**, B275, 734. [Google Scholar] [CrossRef] - Sato, M. Topological discrete algebra, ground state degeneracy, and quark confinement in QCD. Phys. Rev.
**2008**, D77, 045013. [Google Scholar] [CrossRef] - Wen, X.G. Topological Order in Rigid States. Int. J. Mod. Phys.
**1990**, B4, 239. [Google Scholar] [CrossRef] - Kashiwa, K.; Ohnishi, A. Topological feature and phase structure of QCD at complex chemical potential. Phys. Lett.
**2015**, B750, 282–286. [Google Scholar] [CrossRef] - Kashiwa, K.; Ohnishi, A. Quark number holonomy and confinement-deconfinement transition. Phys. Rev.
**2016**, D93, 116002. [Google Scholar] [CrossRef] - Kashiwa, K.; Ohnishi, A. Topological deconfinement transition in QCD at finite isospin density. Phys. Lett.
**2017**, B772, 669–674. [Google Scholar] [CrossRef] - D’Elia, M.; Lombardo, M.P. Finite density QCD via imaginary chemical potential. Phys. Rev.
**2003**, D67, 014505. [Google Scholar] [CrossRef] - De Forcrand, P.; Philipsen, O. The QCD phase diagram for small densities from imaginary chemical potential. Nucl. Phys.
**2002**, B642, 290–306. [Google Scholar] [CrossRef] - De Forcrand, P.; Philipsen, O. The QCD phase diagram for three degenerate flavors and small baryon density. Nucl. Phys.
**2003**, B673, 170–186. [Google Scholar] [CrossRef] - D’Elia, M.; Lombardo, M.P. QCD thermodynamics from an imaginary mu(B): Results on the four flavor lattice model. Phys. Rev.
**2004**, D70, 074509. [Google Scholar] [CrossRef] - Chen, H.S.; Luo, X.Q. Phase diagram of QCD at finite temperature and chemical potential from lattice simulations with dynamical Wilson quarks. Phys. Rev.
**2005**, D72, 034504. [Google Scholar] [CrossRef] - Bonati, C.; Cossu, G.; D’Elia, M.; Sanfilippo, F. The Roberge-Weiss endpoint in N
_{f}= 2 QCD. Phys. Rev.**2011**, D83, 054505. [Google Scholar] [CrossRef] - Nagata, K.; Nakamura, A. Imaginary Chemical Potential Approach for the Pseudo-Critical Line in the QCD Phase Diagram with Clover-Improved Wilson Fermions. Phys. Rev.
**2011**, D83, 114507. [Google Scholar] [CrossRef] - Bonati, C.; de Forcrand, P.; D’Elia, M.; Philipsen, O.; Sanfilippo, F. Chiral phase transition in two-flavor QCD from an imaginary chemical potential. Phys. Rev.
**2014**, D90, 074030. [Google Scholar] [CrossRef] - Takahashi, J.; Kouno, H.; Yahiro, M. Quark number densities at imaginary chemical potential in N
_{f}= 2 lattice QCD with Wilson fermions and its model analyses. Phys. Rev.**2015**, D91, 014501. [Google Scholar] [CrossRef] - Doi, T.M.; Kashiwa, K. Dirac-mode expansion of quark number density and its implications of the confinement-deconfinement transition. arXiv
**2017**, arXiv:1706.00614. [Google Scholar] - Kashiwa, K.; Sasaki, T.; Kouno, H.; Yahiro, M. Two-color QCD at imaginary chemical potential and its impact on real chemical potential. Phys. Rev.
**2013**, D87, 016015. [Google Scholar] [CrossRef] - Sakai, Y.; Kashiwa, K.; Kouno, H.; Yahiro, M. Polyakov loop extended NJL model with imaginary chemical potential. Phys. Rev.
**2008**, D77, 051901. [Google Scholar] [CrossRef] - Nishida, Y. Phase structures of strong coupling lattice QCD with finite baryon and isospin density. Phys. Rev.
**2004**, D69, 094501. [Google Scholar] [CrossRef] - Kawamoto, N.; Miura, K.; Ohnishi, A.; Ohnuma, T. Phase diagram at finite temperature and quark density in the strong coupling limit of lattice QCD for color SU(3). Phys. Rev.
**2007**, D75, 014502. [Google Scholar] [CrossRef] - Garcia Martin, R.; Pelaez, J. Chiral condensate thermal evolution at finite baryon chemical potential within Chiral Perturbation Theory. Phys. Rev.
**2006**, D74, 096003. [Google Scholar] [CrossRef] - Ayala, A.; Bashir, A.; Dominguez, C.; Gutierrez, E.; Loewe, M.; Raya, A. QCD phase diagram from finite energy sum rules. Phys. Rev.
**2011**, D84, 056004. [Google Scholar] [CrossRef] - Gross, D.J.; Pisarski, R.D.; Yaffe, L.G. QCD and Instantons at Finite Temperature. Rev. Mod. Phys.
**1981**, 53, 43. [Google Scholar] [CrossRef] - Weiss, N. The Effective Potential for the Order Parameter of Gauge Theories at Finite Temperature. Phys. Rev.
**1981**, D24, 475. [Google Scholar] [CrossRef] - Sakamoto, M.; Takenaga, K. On Gauge Symmetry Breaking via Euclidean Time Component of Gauge Fields. Phys. Rev.
**2007**, D76, 085016. [Google Scholar] [CrossRef] - Kouno, H.; Sakai, Y.; Kashiwa, K.; Yahiro, M. Roberge-Weiss phase transition and its endpoint. J. Phys.
**2009**, G36, 115010. [Google Scholar] [CrossRef] - Kashiwa, K.; Yahiro, M.; Kouno, H.; Matsuzaki, M.; Sakai, Y. Correlations among discontinuities in the QCD phase diagram. J. Phys.
**2009**, G36, 105001. [Google Scholar] [CrossRef] - De Forcrand, P.; Philipsen, O. Constraining the QCD phase diagram by tricritical lines at imaginary chemical potential. Phys. Rev. Lett.
**2010**, 105, 152001. [Google Scholar] [CrossRef] [PubMed] - D’Elia, M.; Sanfilippo, F. The Order of the Roberge-Weiss endpoint (finite size transition) in QCD. Phys. Rev.
**2009**, D80, 111501. [Google Scholar] [CrossRef] - Bonati, C.; D’Elia, M.; Mariti, M.; Mesiti, M.; Negro, F.; Sanfilippo, F. Roberge-Weiss endpoint at the physical point of N
_{f}= 2 + 1 QCD. Phys. Rev.**2016**, D93, 074504. [Google Scholar] [CrossRef] - Bonati, C.; Calore, E.; D’Elia, M.; Mesiti, M.; Negro, F.; Sanfilippo, F.; Schifano, S.F.; Silvi, G.; Tripiccione, R. Roberge-Weiss endpoint and chiral symmetry restoration in N
_{f}= 2 + 1 QCD. Phys. Rev.**2019**, D99, 014502. [Google Scholar] [CrossRef] - Goswami, J.; Karsch, F.; Lahiri, A.; Schmidt, C. QCD phase diagram for finite imaginary chemical potential with HISQ fermions. arXiv
**2018**, arXiv:1811.02494. [Google Scholar] - Shimizu, H.; Yonekura, K. Anomaly constraints on deconfinement and chiral phase transition. Phys. Rev.
**2018**, D97, 105011. [Google Scholar] [CrossRef] - Kikuchi, Y. ’t Hooft Anomaly, Global Inconsistency, and Some of Their Applications. Ph.D. Thesis, Kyoto University, Kyoto, Japan, 2018. [Google Scholar]
- Nishimura, H.; Tanizaki, Y. High-temperature domain walls of QCD with imaginary chemical potentials. arXiv
**2019**, arXiv:1903.04014. [Google Scholar] - Kashiwa, K.; Matsuzaki, M.; Kouno, H.; Sakai, Y.; Yahiro, M. Meson mass at real and imaginary chemical potentials. Phys. Rev.
**2009**, D79, 076008. [Google Scholar] [CrossRef] - Bilgici, E.; Bruckmann, F.; Gattringer, C.; Hagen, C. Dual quark condensate and dressed Polyakov loops. Phys. Rev.
**2008**, D77, 094007. [Google Scholar] [CrossRef] - Bilgici, E.; Bruckmann, F.; Danzer, J.; Gattringer, C.; Hagen, C.; Ilgenfritz, E.M.; Maas, A. Fermionic boundary conditions and the finite temperature transition of QCD. Few Body Syst.
**2010**, 47, 125–135. [Google Scholar] [CrossRef] - Bilgici, E. Signatures of Confinement and Chiral Symmetry Breaking In Spectral Quantities of Lattice Dirac Operators. Ph.D. Thesis, University of Graz, Graz, Austria, 2009. [Google Scholar]
- Bruckmann, F.; Endrodi, G. Dressed Wilson loops as dual condensates in response to magnetic and electric fields. Phys. Rev.
**2011**, D84, 074506. [Google Scholar] [CrossRef] - Fischer, C.S. Deconfinement phase transition and the quark condensate. Phys. Rev. Lett.
**2009**, 103, 052003. [Google Scholar] [CrossRef] - Fischer, C.S.; Mueller, J.A. Chiral and deconfinement transition from Dyson-Schwinger equations. Phys. Rev.
**2009**, D80, 074029. [Google Scholar] [CrossRef] - Kashiwa, K.; Kouno, H.; Yahiro, M. Dual quark condensate in the Polyakov-loop extended NJL model. Phys. Rev.
**2009**, D80, 117901. [Google Scholar] [CrossRef] - Gatto, R.; Ruggieri, M. Dressed Polyakov loop and phase diagram of hot quark matter under magnetic field. Phys. Rev.
**2010**, D82, 054027. [Google Scholar] [CrossRef] - Zhang, Z.; Miao, Q. Dual condensates at finite isospin chemical potential. Phys. Lett. B
**2015**, 753, 670–676. [Google Scholar] [CrossRef] - Zhang, Z.; Lu, H. Dual meson condensates in the Polyakov-loop extended linear sigma model. arXiv
**2017**, arXiv:1705.09953. [Google Scholar] - Xu, F.; Mao, H.; Mukherjee, T.K.; Huang, M. Dressed Polyakov loop and flavor dependent phase transitions. Phys. Rev.
**2011**, D84, 074009. [Google Scholar] [CrossRef] - Sasagawa, S.; Tanaka, H. The separation of the chiral and deconfinement phase transitions in the curved space-time. Prog. Theor. Phys.
**2012**, 128, 925–939. [Google Scholar] [CrossRef] - Flachi, A. Deconfinement transition and Black Holes. Phys. Rev.
**2013**, D88, 041501. [Google Scholar] [CrossRef] - Benič, S. Physical interpretation of the dressed Polyakov loop in the Nambu-Jona-Lasinio model. Phys. Rev.
**2013**, D88, 077501. [Google Scholar] [CrossRef] - Aharonov, Y.; Bohm, D. Significance of electromagnetic potentials in the quantum theory. Phys. Rev.
**1959**, 115, 485–491. [Google Scholar] [CrossRef] - Huang, S.; Schreiber, B. Statistical mechanics of relativistic anyon-like systems. Nucl. Phys. B
**1994**, 426, 644–660. [Google Scholar] [CrossRef] - Kondo, K.I. Toward a first-principle derivation of confinement and chiral-symmetry-breaking crossover transitions in QCD. Phys. Rev.
**2010**, D82, 065024. [Google Scholar] [CrossRef] - Kashiwa, K.; Hell, T.; Weise, W. Nonlocal Polyakov-Nambu-Jona-Lasinio model and imaginary chemical potential. Phys. Rev.
**2011**, D84, 056010. [Google Scholar] [CrossRef] - Kitazawa, M.; Koide, T.; Kunihiro, T.; Nemoto, Y. Chiral and color superconducting phase transitions with vector interaction in a simple model. Prog. Theor. Phys.
**2002**, 108, 929–951. [Google Scholar] [CrossRef] - Kashiwa, K.; Kouno, H.; Sakaguchi, T.; Matsuzaki, M.; Yahiro, M. Chiral phase transition in an extended NJL model with higher-order multi-quark interactions. Phys. Lett.
**2007**, B647, 446–451. [Google Scholar] [CrossRef] - Kashiwa, K.; Kouno, H.; Matsuzaki, M.; Yahiro, M. Critical endpoint in the Polyakov-loop extended NJL model. Phys. Lett.
**2008**, B662, 26–32. [Google Scholar] [CrossRef] - Sugano, J.; Takahashi, J.; Ishii, M.; Kouno, H.; Yahiro, M. Determination of the strength of the vector-type four-quark interaction in the entanglement Polyakov-loop extended Nambu-Jona-Lasinio model. Phys. Rev.
**2014**, D90, 037901. [Google Scholar] [CrossRef] - Braun, J.; Gies, H.; Pawlowski, J.M. Quark Confinement from Color Confinement. Phys. Lett.
**2010**, B684, 262–267. [Google Scholar] [CrossRef] - Roessner, S.; Ratti, C.; Weise, W. Polyakov loop, diquarks and the two-flavour phase diagram. Phys. Rev.
**2007**, D75, 034007. [Google Scholar] [CrossRef] - Schaefer, B.J.; Pawlowski, J.M.; Wambach, J. The Phase Structure of the Polyakov—Quark-Meson Model. Phys. Rev.
**2007**, D76, 074023. [Google Scholar] [CrossRef] - Haas, L.M.; Stiele, R.; Braun, J.; Pawlowski, J.M.; Schaffner-Bielich, J. Improved Polyakov-loop potential for effective models from functional calculations. Phys. Rev.
**2013**, D87, 076004. [Google Scholar] [CrossRef] - Meisinger, P.N.; Miller, T.R.; Ogilvie, M.C. Phenomenological equations of state for the quark gluon plasma. Phys. Rev.
**2002**, D65, 034009. [Google Scholar] [CrossRef] - Pisarski, R.D. Why the quark gluon plasma isn’t a plasma. In Proceedings of the Strong and Electroweak Matter Meeting (SEWM), Marseille, France, 13–17 June 2000; pp. 107–117. [Google Scholar] [CrossRef]
- Dumitru, A.; Pisarski, R.D. Degrees of freedom and the deconfining phase transition. Phys. Lett.
**2002**, B525, 95–100. [Google Scholar] [CrossRef] - Scavenius, O.; Dumitru, A.; Lenaghan, J.T. The K/pi ratio from condensed Polyakov loops. Phys. Rev.
**2002**, C66, 034903. [Google Scholar] [CrossRef] - Ratti, C.; Thaler, M.A.; Weise, W. Phases of QCD: Lattice thermodynamics and a field theoretical model. Phys. Rev.
**2006**, D73, 014019. [Google Scholar] [CrossRef] - Dumitru, A.; Guo, Y.; Hidaka, Y.; Altes, C.P.K.; Pisarski, R.D. How Wide is the Transition to Deconfinement? Phys. Rev.
**2011**, D83, 034022. [Google Scholar] [CrossRef] - Fukushima, K.; Kashiwa, K. Polyakov loop and QCD thermodynamics from the gluon and ghost propagators. Phys. Lett.
**2013**, B723, 360–364. [Google Scholar] [CrossRef] - Kashiwa, K.; Pisarski, R.D.; Skokov, V.V. Critical endpoint for deconfinement in matrix and other effective models. Phys. Rev.
**2012**, D85, 114029. [Google Scholar] [CrossRef] - Kashiwa, K.; Pisarski, R.D. Roberge-Weiss transition and ’t Hooft loops. Phys. Rev.
**2013**, D87, 096009. [Google Scholar] [CrossRef] - Bornyakov, V.G.; Boyda, D.L.; Goy, V.A.; Molochkov, A.V.; Nakamura, A.; Nikolaev, A.A.; Zakharov, V.I. New approach to canonical partition functions computation in N
_{f}= 2 lattice QCD at finite baryon density. Phys. Rev.**2017**, D95, 094506. [Google Scholar] [CrossRef] - Karbstein, F.; Thies, M. How to get from imaginary to real chemical potential. Phys. Rev. D
**2007**, 75, 025003. [Google Scholar] [CrossRef] - Kouji, K.; Hiroaki, K. Roberge-Weiss periodicity, canonical sector and modified Polyakov-loop. arXiv
**2019**, arXiv:1903.11737. [Google Scholar] - Yang, C.N.; Lee, T.D. Statistical theory of equations of state and phase transitions. 1. Theory of condensation. Phys. Rev.
**1952**, 87, 404–409. [Google Scholar] [CrossRef] - Nakamura, A.; Nagata, K. Probing QCD phase structure using baryon multiplicity distribution. PTEP
**2016**, 2016, 033D01. [Google Scholar] [CrossRef] - Nagata, K.; Kashiwa, K.; Nakamura, A.; Nishigaki, S.M. Lee-Yang zero distribution of high temperature QCD and the Roberge-Weiss phase transition. Phys. Rev.
**2015**, D91, 094507. [Google Scholar] [CrossRef] - Wakayama, M.; Borynakov, V.G.; Boyda, D.L.; Goy, V.A.; Iida, H.; Molochkov, A.V.; Nakamura, A.; Zakharov, V.I. Lee-Yang zeros in lattice QCD for searching phase transition points. arXiv
**2018**, arXiv:1802.02014. [Google Scholar] - Kashiwa, K.; Ohnishi, A. Investigation of confinement-deconfinement transition via probability distributions. arXiv
**2017**, arXiv:1712.06220. [Google Scholar] - Almasi, G.A.; Friman, B.; Morita, K.; Lo, P.M.; Redlich, K. Fourier coefficients of the net-baryon number density and chiral criticality. arXiv
**2018**, arXiv:1805.04441. [Google Scholar] - Ejiri, S.; Karsch, F.; Redlich, K. Hadronic fluctuations at the QCD phase transition. Phys. Lett.
**2006**, B633, 275–282. [Google Scholar] [CrossRef] - Karsch, F.; Redlich, K. Probing freeze-out conditions in heavy ion collisions with moments of charge fluctuations. Phys. Lett.
**2011**, B695, 136–142. [Google Scholar] [CrossRef] - Lin, J. Divergence measures based on the Shannon entropy. IEEE Trans. Inf. Theory
**1991**, 37, 145–151. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The $\theta $-dependence of $\sigma /{\sigma}_{0}$ and ${n}_{q}/{T}^{3}$. The dotted and solid lines represent the result at $T=100$ and 300 MeV, respectively.

**Figure 2.**The right and left panels show the $\tilde{D}{.}_{\mathrm{JS}}^{J}$ trajectory on the 2-simplex for the PNJL model and the lattice QCD data [39], respectively.

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Kashiwa, K.
Imaginary Chemical Potential, NJL-Type Model and Confinement–Deconfinement Transition. *Symmetry* **2019**, *11*, 562.
https://doi.org/10.3390/sym11040562

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Kashiwa K.
Imaginary Chemical Potential, NJL-Type Model and Confinement–Deconfinement Transition. *Symmetry*. 2019; 11(4):562.
https://doi.org/10.3390/sym11040562

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Kashiwa, Kouji.
2019. "Imaginary Chemical Potential, NJL-Type Model and Confinement–Deconfinement Transition" *Symmetry* 11, no. 4: 562.
https://doi.org/10.3390/sym11040562