1. Color confinement mechanism and the type of the QCD vacuum Tsuneo Suzuki (Kanazawa Univ.) Collaborators: K.Ishiguro, Y.Mori, Y.Nakamura, T.Sekido （ M.Polikarpov, M.Chernodub, V.Zakharov)

2. １． Color confinement mechanism Color confinement could be understood as the dual Meissner effect. (’75 ‘tHooft & Mandelstam) How to find a magnetically charged particle from QCD? Perform a partial gauge-fixing called abelian projection. QCD is reduced to U(1) 2 theory with magnetic monopoles. If the monopoles condense, the dual Meissner effect occurs. (’80 ‘tHooft)

3. 2. Lattice QCD study of abelian projection When we perform a partial gauge-fixing called Maximally abelian (MA) gauge (’87 Kronfeld et al.), the ‘tHooft- Mandelstam conjecture seems to be correct. 1.Abelian dominance, monopole dominance 2. Monopole condensation occurs. Entropy dominates over energy. 3.The dual Meissner effect is observed clearly. (M.Chernodub and M.Polikarpov, hep-th/9710205; T.Suzuki,P.T.P.Suppl.131(‘98)633; R.W. Haymaker, P.R.315, 153 (‘99), H.Shiba and T.Suzuki, P.L. B 351, 519 (‘95), Y.Koma et al, KI & PRD68 (2003) 114504 and references therein)

4. Dual Meissner effect: Electric field flux tube Monopole current distribution Dual Meissner picture is seen very beautifully.

5. 3. gauge-independent? Ｉｓ the dual Meissner effect in the infrared region gauge- independent? (1)Polyakov gauge where Polyakov loops are diagonalized. Monopoles are always static. Do not contribute to the usual abelian Wilson loop. Monopole dominance is broken.(M.Chernodub ’00) (2)Landau gauge: Configurations are so smooth. No DeGrand-Toussaint monopoles. No?

6. Existence of the linear potential indicates the flux- squeezing occurs generally in QCD. Is it understood as the dual Meissner effect? What squeezes the electric field?

7. 1)Imroved Iwasaki gauge action: Nearer to the continuum limit 2) (Landau) Gauge fixing: study of electric and magnetic field flux directly 3) Comparison with MA gauge + U1 Landau ４． Study of flux in Landau gauge (T.Suzuki et al, Phys. Rev. Lett. 94, 132001 (2005))

8. Define abelian field strength MA

9. Non-abelian Wilson loops (a source of a Q-Qbar pair) Electric field Magnetic field

10. Results 1 Abelian electric field is squeezed also as in non- abelian one. Other components are almost zero. Quark-antiquark are located in z direction. R r

11. Results 2: Electric field is squeezed by abelian magnetic displacement current Magnetic solenoidal current

12. Solenoidal current observed !!! Landau gauge perpendicular plane

13. Landau gauge MA gauge Landau gauge MA Gauge （ Bali (’98))

14. Comments 1.The positions of the peak of the solenoidal current distribution in Landau and in MA are almost the same. 2. The dual Meissner effect is observed also in Landau gauge, where naive DT lattice monopoles do not exist. New monopole or essentially non-static? The dual Meissner effect must be a universal confinement mechanism.

15. 5. Type of the QCD vacuum in MA (Kanazawa + M.Chernodub, M.Polikarpov, V.Zakharov) The penetration length ( inverse of the dual photon mass) is fixed from the electric field flux. R=Distnce between Q and Qbar

16. How to determine the coherence length (inverse of the dual Higgs mass) Study monopole around QCD string in dual Ginzburg-Landau theory: (Suzuki 1988)

17. Behaviors of classical solutions

18. place the static quarks at the (spatial) infinities of the z axis in London limit

19. Real world: dominant

20. Expected behavior of monopole density around QCD string The coherence length can be derived from this behavior !!

21. Numerical data of the coherence length

22. Correlation of monopole density and D=2 gluon operator

23. Is minimized in MA

24. Various D=2 gluon operators in MA+U(1) LA

25. Border between the type 1 and the type 2 ( Cea et al,1995, Singh et al.1993, Matsubara et al.,1994, kato et al.,1998, Bali 1998, Koma 2003)

26. 6. The type of the QCD vacuum in Landau gauge

27. Border between type 1 and 2 ? Ｌａｎｄａｕ ｇａｕｇｅ

28. 7. Study of gauge dependence Penetration length Coherence length

29. 8. Summary and outlook The dual Meissner effect works good also in Landau gauge. How about in other general gauges? Expectation: It must work in general. Magnetic displacement current plays a role of monopole currents if we see naive abelian components. The vacuum is near the border of the type 1 and the type 2.

30. Existence of the dual Meissner effect suggests that something magnetic condenses in the vacuum. Gauge- invariant monopoles are working? Extraction of monopole need not always abelian projection. Or essentially non-static? 1. Gubarev’s monopoles may be important and should be studied extensively.(Gubarev, hep-lat/0204018, Gubarev- Zakharov hep-lat/0204017) 2. Violation of non-abelian Bianchi identity may be important? (Gubarev & Morozov, hep-lat/0503023)

31. D=2 gluon condensate operator important? （１） Consider A + A - in MA gauge: The effect of the off-diagonal gluons appears only in the screening of adjoint charged particles. Fundamental quark confinement is not affected. (T.Suzuki and M.Chernodub, P.L. B563 (2003) 183) （２）Ａ ３ Ａ ３ ： non-perturbative part is given by monopoles (3) In LA gauge, both effects are included in