1. Investigation of QCD phase structure from imaginary chemical potential Kouji Kashiwa RIKEN BNL Research Center 2014/02/06 BNL

2. Introduction : Quark and gluon Theme ： Phase structure of Quantum Chromodynamics at finite T and . Are there any different states? If those exist, where can they appear? Those are confined inside Hadrons. From experiments at Relativistic Heavy ion Collider (RHIC) and Large Hadron Collider (LHC), Some data are obtained which can not be understood from Hadronic state only. Question : Quarks and gluons can not be observed directly. What states? When and where those can be seen?

3. Introduction : Phase diagram Phase diagram ： quark-gluon system Recent conceptual drawing Several phases were predicted so far… There is no quantitative discussion at finite . K. Fukushima, T. Hatsuda, Rept. Prog. Phys. 74 (2011) 014001.

4. Introduction : Phase diagram LHC RHIC GSI JPARC Early universe Compact star ρ0ρ0 AGS SPS KEK-PS It is quite important for experiments and observation. Phase diagram ： quark-gluon system

5. Phase transition Deconfinement phase transition Chiral phase transition Chiral symmetry ： Polyakov-loop: Free energy for one quark excitation For example, L. D. McLerran and B. Svetitsky, Phys. Rev. D 24 (1981) 450. Symmetry under transformations of left- and right-handed components of quark independently. Order parameter : Chiral condensate Origin of the mass of proton, neutron, pion and so on. Z 3 symmetry (center of SU(3) ) ： It exists in pure gauge. （ twist at temporal boundary ） （ Zero quark mass ） Phase transition considered in this talk.

6. Phase transition In this talk, we assume there is the order-parameters for the deconfinement transition. There is the different clarification for confinement/deconfinement Topological order It was proposed in solid state physics (fractional quantum hall state) Order parameter Spontaneous symmetry breaking Example: chiral condensateChiral symmetry breaking There is no order parameter. Difference between those sates are characterized by the non-trivial degeneracy of the vacuum. We need the non-trivial topology. Masatoshi Sato, PRD 77 (2008) 0450013.

7. Problem? Lattice QCD simulation : first principle calculation of QCD Sig problem probability Probability Partition function Statistical dynamics Dirac operator : Probability can becomes complex (also minus) Several approaches to circumvent the sign problem: Taylor expansion Reweighting These can not reach very high . Analytic continuation Canonical approach

8. Problem? Ambiguity in effective models M. Stephanov, Prog. Theor. Phys. Suppl. 153 (2004) 139. M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Prog. Theor. Phys. 108 (2002) 929. NJL+CSC+G v case Ginzburg-Landau approach T. Hatsuda, M. Tachibana, N. Yamamoto and G. Baym, Phys. Rev. lett, 97 (2006) 122001. Multi critical endpoint ?

9. Sign problem free systems No sign problem ： Can we use these system? We construct the effective model by combining the LQCD data Imaginary chemical potential Iso-spin chemical potential (Baryon chemical potential = 0) Two color QCD Our approach: at imaginary chemical potential Because those reasons, we can not obtain reliable QCD phase diagram.

10. Z 3 symmetry Quark contribution (explicit center symmetry breaking) Pure gauge : Contour Plot Im  Re  Deconfined Confined Three degenerate minima are came from Z 3 symmetry Quark contribution breaks Z 3 symmetry explicitly. Two of them become metastable. What happen at finite  ?

11. Imaginary chemical potential QCD has characteristic properties at finite imaginary  ! ( It is similar to AB phase, but different ) Roberge Weiss (RW) phase transition line RW endpoint 2  /3 Phase diagram ： Imaginary chemical potential It is completely different from that at real . Non-trivial periodicity Roberge-Weiss (RW) periodicity First-order transition along T-axis RW transition A. Roberge and N. Weiss, Nucl. Phys. B275 (1986) 734.

12. Imaginary chemical potential Fugacity expansion: Fourier representation: 2  /3 A. Roberge and N. Weiss, Nucl. Phys. B275 (1986) 734. Even function has cusp. Odd function has gap. This coexistence: K.K., M. Yahiro, H. Kouno, M. Matsuzaki, Y. Sakai, J. Phys. G 36 (2009) 105001. Imaginary chemical potential

13. Fugacity expansion: Fourier representation: Imaginary chemical potential This relation means that the imaginary chemical potential has almost all information of the real chemical potential region. Actually, there are some method to use above relation in lattice QCD simulations. Analytic continuation method Canonical approach

14. Standard methods : Analytic continuation Analytic continuation Fig ： P. de Forcrand, S. Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 62. Data are collected at imaginary . Data are fitted by analytic functions. Example ： Based on Lattice QCD simulation only ：

15. Standard methods : Canonical approach Canonical approach Fig ： P. de Forcrand, S. Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 62. Check Maxwell contraction If there is first-order transition, S sharp structure is there （ in finite size system ） We should investigate (T,  ) where S sharp structure is vanished.

16. Problem? Convergence radius (Analytic continuation) Order of phase transition (Analytic continuation) Finite size system （ Canonical approach ） Color superconductivity （ Canonical approach, Analytic continuation ） We combine effective model and Lattice results. Dynamics of phase transition are included. Parameters can be determined at finite imaginary . Imaginary  has information of real  region.

17. Recent model development Fermion part What model should we use? Nambu—Jona-Lasinio (NJL) model If the gluonic contribution is not correctly introduced, the RW periodicity should be vanished. By using some approximations and ansatz, we can derive the NJL model from QCD. W (n) is the connected n-point function of gauge boson without quark loops. NJL model (This model only has 2  periodicity) For example: Quark color current :

18. Recent model development Fermion part What model should we use? Quark-meson model can be also used. (Basically it is almost equivalent with NJL model) Polyakov-loop extended Nambu—Jonal-Lasinio (PNJL) model Mean field approximation Thermodynamic potential If the gluonic contribution is not correctly introduced, the RW periodicity should be vanished. Gluonic contribution Nambu—Jona-Lasinio (NJL) model NJL model (This model only has 2  periodicity)

19. Gluon part Recent model development Polyakov-loop potential Meisinger-Miller- Ogilvie model Matrix model for deconfinement Effective potential from (Landau gauge) gluon and ghost propagator Strong coupling expansion Mass like parameter is introduced. Up to the second order term of high T expansion is included. Extension of MMO model. Gluon and ghost propagators in Landau gauge are used. To reproduce LQCD data in the pure gauge limit. P. N. Meisinger, T. R. Miller, M. C. Ogilvie, PRD 65 (2002) 034009. A. Dumitru, Y. Guo, Y. Hidaka, C. P. K. Altes, R. D. Pisarski, PRD 83 (2011) 034022. K. Fukushima, Phys. Lett. B 591 (2004) 277. + K. Fukushima, K.K., Phys. Lett. B 723 (2013) 360. RW periodicity can be reproduced by using following models. (RW periodicity is the remnant of the Z 3 symmetry) U U U U

20. Results : Model ambiguities Vector-type interaction It relates with ω 0 mode. Fermion part : Phase diagram If the vector-type interaction is sufficiently large, CEP should be vanished. For example, K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B662 (2008) 26. This behavior also appears in the NJL model.

21. Results : Vector interaction Vector-type interaction It relates with ω 0 mode. Vector-type interaction Set C Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 79 (2009) 096001. 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:

22. Results : Columbia plot Gluonic contribution Zero chemical potential RW endpoint Colombia plot Order of phase transition Gluonic part also has strong ambiguity even in perturbative regime of quark contribution Ambiguity appears even at large quark mass region. Larger ambiguity may be seen on the RW endpoint. Matrix There is the possibility that Region can be first order region. There is no phase boundary until 1 GeV in the case of Polyakov-Log. K.K., V. V. Skokov, R. D. Pisarski, Phys. Rev. D85 (2012) 114029. K.K., R. D. Pisarski, Phys. Rev. D87 (2013) 096009.

23. Related topic : Hosotani mechanism Imaginary chemical potential may be important for other topics. For the physics beyond the standard model  n f = 2  T (n + 1/2) +  I Matsubara frequency  n   = 2  T (n +  ) Matsubara frequency with arbitral boundary condition  n  = 2  T (n +  ) –  T + 2  T  Angle represents the arbitral boundary condition Imaginary  Boundary condition for temporal direction Fermion boundary condition is important for Hosotani mechanism. If the extra-dimension is not simply connected with the system, the gauge symmetry breaking vacuum expectation value can affect the system. Higgs can be understood as the fluctuation of extra-dimensional gauge boson component. Hosotani mechanism ： For example: Y. Hosotani, Phys. Lett. B 126 (1983) 309; Ann. Phys. 190 (1989) 233. 0 β, 1/L φ Compacted direction

24. Related topic : Hosotani mechanism q 1 = q 2 ≠ q 3 : SU(2) ×U(1), Gauge symmetry breaking is happen Eigen value For example, q 1 ≠ q 2 ≠ q 3 : U(1) ×U(1) Wilson loop in compacted direction Nth phases (q i ) for SU(N) Temporal direction is taken as compact dimension in following. Perturbative one-loop effective potential (free gas limit) Start from QCD Lagrangian density Decompose A 4 to “expectation value + fluctuation” Drop the interactions from the action Calculate the ln det (  2 n +p 2 ) Inverse of perturbative propagator. Divergences can be subtracted in 5D as same as 4D.

25. Free gas calculation Perturbative one-loop effective potential (free gas limit) = = ~ After summing up each integrations: 4D 5D5D +c D. Gross, R. Pisarski, L. Yaffe, Rev. Mod. Phys 53 (1981) 43. N. Weiss, Phys.Rev.D 24 (1981) 475.

26. Phase structure aPBC adjoint SU(3) SU(2)×U(1) U(1)×U(1) Large m Medium m Small m PBC adjoint Actual forms: Gauge boson Adjoint fermion Boundary angle Fermion : 1/2 Boson : 0,  Phase : Number of flavor : N a Fermion mass : m f, m a Arbitral dimensional form can be obtained similar form.

27. Phase diagram Phase Structure D : Deconfined phase S : Split (skewed) phase R : Re-confined phase C : Confined phase In previous studies for Hosotani mechanism, fermion mass effects were almost neglected. We use the perturbative one-loop potential. SU(3) U(1)×U(1) SU(2)×U(1) K.K., T. Misumi, JHEP 05 (2013) 042.

28. Lattice gauge results Phase Structure Scatter plot of Polyakov-loop Lattice setup: 2 flavor, 3 color and adjoint staggered fermion Lattice data : G. Cossu, M. D’Elia, JHEP 07(2009), 048.

29. Comparison Phase Structure We can understand it from Hosotani mechanism! SU(3) U(1)×U(1) SU(2)×U(1) K.K., T. Misumi, JHEP 05 (2013) 042.

30. Problem from confinement and U(1) ×U(1) phases Phase structure H. Nishimura, M. Ogilvie, Phys. Rev. D 81 (2010) 014018. K.K., T. Misumi, JHEP 05 (2013) 042. In their calculation, the confined and U(1)×U(1) phases are same...

31. Phase structure C Unknown In their calculation, the confined and U(1)×U(1) phases are same... Problem from confinement and U(1) ×U(1) phases H. Nishimura, M. Ogilvie, Phys. Rev. D 81 (2010) 014018. K.K., T. Misumi, JHEP 05 (2013) 042.

32. Chiral properties K.K., T. Misumi, JHEP 05 (2013) 042. With adjoint fermionWith adjoint and fundamental fermion To describe the chiral symmetry breaking and restoration, we use the Nambu—Jona-Lasinio type model.

33. 2+1+1 dimensional system QCD-like theory at finite temperature and one compactified spatial dimension is interesting. Standard local NJL model with moment cutoff can not be used. We use the nonlocal NJL model. K.K., T. Misumi, in preparation. This system may be useful to understood the system under the strong external magnetic field. The summation came from the Landau quantization appears as same as the Kaluza-Klein summation. Non-trivial chiral properties are obtained and almost all effective model can not explain it…

34. 2+1+1 dimensional system K. Farakov and P. Pasipoularides, Nucl. Phys. B 705 (2005) 92. M. Sakamoto and K. Takenaga, Phys. Rev. D76 (2007) 085016. Our results (these are still 3+1 dimensional system) Perturbative one-loop effective potential for massive particle Integral representation Poisson formula K.K., T. Misumi, in preparation. Distribution function

35. 5-dimensional SU(3) lattice gauge theory E. Itou, K.K., T. Nakamoto, in preparation. Investigation of the 5-dimensional system is important. However, phase structures of the 5-dimensional SU(3) lattice (pure) gauge theory is not well understood yet. We should know the critical  where bulk first order transition vanished. Small extra-dimensional systemMulti-(4-dimensional) layered system 4-dimensional layer large a 5

36. 5-dimensional SU(3) lattice gauge theory E. Itou, K.K., T. Nakamoto, in preparation.

37. Summary We study the QCD phase diagram from the imaginary chemical potential. Imaginary chemical potential has almost all information of real chemical potential. There is no sign problem and thus lattice QCD simulation is possible. We determined the vector-type interaction at the imaginary chemical potential and draw the phase diagram. To obtain more accurate diagram, we need more accurate data, We show the usefulness of the imaginary chemical potential to study it. Imaginary chemical potential can be converted to the boundary condition. It may be useful to understand the Hosotani mechanism. Physics beyond the standard model