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Investigation of QCD phase structure from imaginary chemical potential

Investigation of QCD phase structure from imaginary chemical potential. Kouji Kashiwa. RIKEN BNL Research Center. 2014/02/06 BNL. Introduction : Quark and gluon. Theme 　：　 Phase structure of Quantum Chromodynamics at finite T and m . Quarks and gluons can not be observed directly.

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Investigation of QCD phase structure from imaginary chemical potential

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  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 m. Quarks and gluons can not be observed directly. Those are confined inside Hadrons. Question : Are there any different states? If those exist, where can they appear? From experiments at Relativistic Heavy ion Collider (RHIC) andLarge Hadron Collider (LHC), Some data are obtained which can not be understood from Hadronic state only. When and where those can be seen? What states?

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

  4. Introduction : Phase diagram It is quite important for experiments and observation. Phase diagram：quark-gluon system LHC RHIC Early universe SPS AGS JPARC GSI KEK-PS Compact star ρ0

  5. Phase transition Phase transition considered in this talk. Chiral phase transition Chiral symmetry ： Symmetry under transformations of left- and right-handed components of quark independently. （Zero quark mass） Order parameter: Chiral condensate Origin of the mass of proton, neutron, pion and so on. Deconfinement phase transition Z3 symmetry(center of SU(3) ) ： It exists in pure gauge. （twist at temporal boundary） Free energy for one quark excitation Polyakov-loop: For example, L. D. McLerran and B. Svetitsky, Phys. Rev. D 24 (1981) 450.

  6. Phase transition In this talk, we assume there is the order-parameters for the deconfinement transition. Spontaneous symmetry breaking Order parameter Example: chiral condensate Chiral symmetry breaking There is the different clarification for confinement/deconfinement Masatoshi Sato, PRD 77 (2008) 0450013. Topological order It was proposed in solid state physics (fractional quantum hall state) 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.

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

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

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

  10. Z3 symmetry Z3 symmetry Contour Plot Pure gauge: Three degenerate minima are came from Z3symmetry Quark contribution breaks Z3symmetry explicitly. ImF Two of them become metastable. Confined Deconfined ReF What happen at finite m? Quark contribution (explicit center symmetry breaking)

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

  12. Imaginary chemical potential A. Roberge and N. Weiss, Nucl. Phys. B275 (1986) 734. Imaginary chemical potential 2p/3 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. Fourier representation: Fugacity expansion:

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

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

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

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

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

  19. Recent model development RW periodicity can be reproduced by using following models. (RW periodicity is the remnant of the Z3 symmetry) Gluon part To reproduce LQCD data in the pure gauge limit. Polyakov-loop potential K. Fukushima, Phys. Lett. B 591 (2004) 277. Strong coupling expansion Meisinger-Miller- Ogilvie model Mass like parameter is introduced. Up to the second order term of high T expansion is included. P. N. Meisinger, T. R. Miller, M. C. Ogilvie, PRD 65 (2002) 034009. U U U U Matrix model for deconfinement A. Dumitru, Y. Guo, Y. Hidaka, C. P. K. Altes, R. D. Pisarski, PRD 83 (2011) 034022. Extension of MMO model. + Effective potential from (Landau gauge) gluon and ghost propagator Gluon and ghost propagators in Landau gauge are used. K. Fukushima, K.K. , Phys. Lett. B 723 (2013) 360.

  20. Results : Model ambiguities Vector-type interaction Fermion part : Phase diagram It relates with ω0 mode. If the vector-type interaction is sufficiently large, CEP should be vanished. This behavior also appears in the NJL model. For example, K. K., H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Lett. B662 (2008) 26.

  21. Results : Vector interaction It relates with ω0 mode. Vector-type interaction Vector-type interaction Y. Sakai, K. K, H. Kouno, M. Matsuzaki and M. Yahiro, Phys. Rev. D 79 (2009) 096001. 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.

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

  23. Related topic : Hosotani mechanism Imaginary chemical potential may be important for other topics. For the physics beyond the standard model β, 1/L Matsubara frequency Imaginary m φ wnf = 2pT (n + 1/2) + mI Boundary condition for temporal direction Compacted direction Matsubara frequency with arbitral boundary condition wnf= 2pT (n + f) wnf= 2pT (n + 1/2) – pT + 2pTf Angle represents the arbitral boundary condition 0 Hosotani mechanism： For example: Y. Hosotani, Phys. Lett. B 126 (1983) 309; Ann. Phys. 190 (1989) 233. 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.

  24. Related topic : Hosotani mechanism Temporal direction is taken as compact dimension in following. Nth phases (qi) for SU(N) Wilson loop in compacted direction Eigen value Gauge symmetry breaking is happen q1 ≠ q2 ≠ q3 : q1 = q2 ≠ q3 : U(1)×U(1) SU(2)×U(1) , For example, Divergences can be subtracted in 5D as same as 4D. Perturbative one-loop effective potential (free gas limit) Start from QCD Lagrangian density Decompose A4 to “expectation value + fluctuation” Drop the interactions from the action Calculate the lndet (w2n+p2) Inverse of perturbative propagator.

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

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

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

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

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

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

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

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

  33. 2+1+1 dimensional system K.K., T. Misumi, in preparation. 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. 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.K., T. Misumi, in preparation. Perturbative one-loop effective potential for massive particle K. Farakov and P. Pasipoularides, Nucl. Phys. B 705 (2005) 92. M. Sakamoto and K. Takenaga, Phys. Rev. D76 (2007) 085016. Integral representation Poisson formula Our results (these are still 3+1 dimensional system) 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 b where bulk first order transition vanished. 4-dimensional layer Small extra-dimensional system Multi-(4-dimensional) layeredsystem large a5

  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

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