1. Gauge independence of Abelian dual Meissner effect in pure SU(2) QCD Katsuya Ishiguro Y.Mori, T.Sekido, T.Suzuki Kanazawa-univ & RIKEN Joint Meeting of Pacific Region Particle Physics Communities Sheraton Waikiki Hotel, Honolulu, Hawaii October 31, 2006 Collaborators Kanazawa-univ & RIKEN

2. Introduction Quark confinement (Y.Koma et al PRD68 (2003) 114504 and references therein, G.Bali, Talk at Confinement III, ’98 etc) Quarks are confined due to the dual Meissner effect. This idea seems to be correct when we perform Abelian projection in Maximally Abelian (MA) gauge. (‘tHooft & Mandelstam,’75) (‘tHooft,’80) Abelian magnetic currents are very important.

3. Motivations Abelian projection has infinite degrees of freedom of gauge-fixing. We know that Abelian projection scenario works good in some non-local gauges such as MA gauge-fixing. In another gauges Abelian dominance has not been observed in lattice simulation. Gauge dependence problem Previous problem (K. Bernstein et al PRD55(1997), S. Ito et al PRD67(2003) and references therein, etc.)

4. Motivations Abelian projection has infinite degrees of freedom of gauge-fixing. We know that Abelian projection scenario works good in some non-local gauges such as MA gauge-fixing. In another gauges Abelian dominance has not been observed in lattice simulation. Gauge dependence problem Previous problem (K. Bernstein et al PRD55(1997), S. Ito et al PRD67(2003) and references therein, etc.) Local unitary gauges are particularly bad gauges. e.g. F12 gauge, Polyakov gauge and etc. Good results have not been observed so far. The configurations with local unitary gauge-fixing have large fluctuations. High precision investigation with local unitary gauge.

5. The gauge-fixing matrix makes 1-2 plane plaquette diagonal. Local unitary gauge-fixing V is a gauge-fixing matrix. No Abelian dominance has been observed in F12 gauge on lattice. e.g. string tension, flux tube pictures This gauge-fixing condition is local.

6. The gauge-fixing matrix makes 1-2 plane plaquette diagonal. Local unitary gauge-fixing V is a gauge-fixing matrix. This gauge-fixing condition is local. There are large fluctuations in F12 gauge. If you want to know what happens actually, high precision investigation is needed. No Abelian dominance has been observed in F12 gauge on lattice. e.g. string tension, flux tube pictures

7. Method of noise reduction for local operators Luscher and Weisz, JHEP 09.010(2001),JHEP 07.049(2002) Multi level method is powerful noise reduction method. This method is effective only with respect to local operators. Ex) Polyakov loop correlation function 1 2 3 Lattice is divided into N_s sublattice. Sublattice average of operator is taken. The link variables are replaced using gauge update except for the spatial links at the boundary of the sublattice. (internal update) 4 Step 2,3 are repeated until stable values for sublattice average of operators are obtained. procedure We thank Y.Koma for giving us the code of the multi level.

8. Method of noise reduction for local operators Polyakov loop correlation function the boundary of the sublattice N_s=3 time space Average in sublattice

9. With gauge-fixing 1 2 3 Lattice is divided into N_s sublattice. Sublattice average of operator is taken. The link variables are replaced using gauge update except for the spatial links at the boundary of the sublattice. (internal update) 4 Step 2,3 are repeated until stable values for sublattice average of operators are obtained. procedure Local gauge-fixing and Abelian projection are implemented in each internal update. Operators are constructed by Abelian link fields. It is local with respect to time direction.

10. Parameters RIKEN-SX7 RCNP-SX5 Machine Parameters for multi level method Operator We calculate Abelian Polyakov loop correlation function using the multi level method. And we compare Abelian potential with non-Abelian potential.

11. Potential in F12 gauge Both string tensions are the same perfectly. This result shows Abelian dominance in F12 gauge. V-V0 Fitting function Potential in F12 gauge in comparison with non-Abelian potential. V-V 0

12. The gauge-fixing matrix makes 1-2-3 cube-like operator diagonal. Local unitary gauge-fixing V is a gauge-fixing matrix.

13. The gauge-fixing matrix makes spatial Polyakov line operator diagonal. Local unitary gauge-fixing V is a gauge-fixing matrix.

14. Potential in local unitary gauge V-V 0 These results show Abelian dominance in other local unitary gauges too. V-V 0 F123 gauge Spatial Polyakov line gauge

15. We measure the correlation between Wilson loop and Abelian field strength using the multi level method. These profiles are studied on a perpendicular plane at the midpoint between the two quarks. Spatial Polyakov line (SPL) gauge.

16. Abelian electric fields in SPL gauge Spatial Polyakov Line gauge W(R,T)=W(5,4) Abelian electric fields are squeezed in SPL gauge? Now calculating in detail. Fitting function small

17. No gauge-fixing In MA gauge Abelian projection works well without difficulty on lattice. In local unitary gauge Abelian projection also works well. But there are large fluctuations on lattice. These results suggest that Abelian confinement scenario is independent of gauge-fixing. Gauge-fixing may not be essential in Abelian confinement scenario. Our suggestion (A.Di Giacomo.et al,PRD61(2000). etc) A magnetically charged operator which is gauge independent has been observed as order parameter of confinement.

18. No gauge-fixing Each Abelian component in any color direction becomes dominant in the infrared region. Abelian dominance and the Abelian dual Meissner effect are seen in any color direction. A state which is Abelian neutral in any color direction can be a physical state. It is just a color-singlet state. Our suggestion First investigations Potential without gauge-fixing. Flux tube profile. (Correlations between electric fields and quark-antiquark pairs.) In this way, color confinement can be understood in the framework of Abelian scenario in gauge-invariant way.

19. Potential without gauge-fixing Fitting function Abelian potential without gauge-fixing in comparison with non-Abelian potential. This result shows Abelian dominance can be seen without gauge-fixing. Analytic results have been obtained. (M.Ogilvie, PRD59(1999)) (160000 conf. are used.) V-V 0

20. Correlations between non-Abelian Wilson loop and Abelian field strength without gauge-fixing are measured using connected operator. (Cea and Cosmai Phys.Rev.D52 (1995)) Similar simulation was done by Cea and Cosmai. Parameters Non-Abelian Abelian W L

21. Fitting function Abelian electric fields are squeezed. Penetration length is consistent with that in the MA gauge. R=5, T=5 Wilson loop Abelian electric fields without gauge-fixing

22. Width of electric field squeezing is not affected by the bridge part and. Fitting function Penetration lengths are almost the same in both cases. Connected and disconnected correlation functions in MA gauge R=5, T=7 W(5,3) W(5,5)W(5,7)

23. Abelian electric fields are squeezed with and without gauge-fixing. Penetration lengths are almost the same in these cases. Electric fields with and without gauge-fixing Non-Abelian Abelian (without gauge-fixing) Abelian (MA gauge) Abelian (F123 gauge)

24. Summary Gauge dependence problem In local unitary gauges Abelian dominance is confirmed using the multi level method. These results suggest that Abelian confinement scenario is independent of the choice of gauge-fixing condition.

25. Summary No gauge fixing Numerical results Abelian dominance is confirmed without gauge-fixing. Abelian electric fields are squeezed with and without gauge-fixing. Each Abelian component in any color direction becomes dominant in the infrared region. Abelian dominance and the Abelian dual Meissner effect are seen in any color direction. A state which is Abelian neutral in any color direction can be a physical state. It is just a color-singlet state. Our suggestion In this way, color confinement can be understood in the framework of Abelian scenario in gauge-invariant way.

26. Summary Future works What squeezes the Abelian electric fields? It is important to study flux-tube profiles of various operators in detail.. Why Abelian degrees of freedom are dominant in the infrared region ? Magnetic currents? Monopoles?