Plan

1. Phase Structure of Thermal QCD/QED:A “Gauge Invariant” Analysis based on theHTL Improved Ladder Dyson-Schwinger Equation Hisao NAKKAGAWANara Universityin collaboration withHiroshi YOKOTA and Koji YOSHIDANara UniversityarXiv:0707.0929 [hep-ph] (to appear in proc. of sQGP’07, Nagoya, Feb. 2007) hep-ph/0703134 (to appear in proc. of SCGT’06, Nagoya, Nov. 2006)[An Isaac Newton Institute Workshop on Exploring QCD : Deconfinement, Extreme Environments and Holography, Cambridge, August 20-24, 2007]

2. Plan 1. Introduction 2. HTL Resummed DS Equation a) Improved Ladder Approximation b) Improved Instantaneous Exchange Approximation 3. Consistency with the Ward-Takahashi Identity 4. Numerical Calculation a) Landau gauge (constant gauges) b) nonlinear gauge : momentum dependent 5. Summary and Outlook

3. 1. Introduction [A] Why Dyson-Schwinger Equation (DSE)? 1) Rigorous FT eq. to study non-perturbative phenomena 2) Possibility of systematic improvement of the interaction kernel through analytic study inclusion of the dominant thermal effect (HTL), etc. [B] DSE with the HTL resummed interaction kernel Difficult to solve 1) Point vertex = ladder kernel (Z1 = 1) 2) Improved ladder kernel (HTL resummed propagator) 3) Instantaneous exchange approximation to the longitudinal propagator transverse one: keep the full HTL resummed form

4. Introduction (cont’d) [C] Landau gauge analysis 1) Importance of the HTL correction Large “correction” to the results from the free kernel 2) Large imaginary part: Real A, B, C rejected But ! 3) A(P) significantly larger than 1: A(P) ~1.4 or larger NB: A(P) = 1 required from the Ward-Takahashi Identity Z1 = Z2 4) Same results in the constant gauges [D] Nonlinear gauge inevitable to satisfy the Ward-Takahashi Identity Z1 = Z2, and to get gauge “invariant” result (in the same sense at T=0 analysis)

5. 2. Hard-Thermal-Loop Resummed Dyson-Schwinger EquationsPTP 107 (2002) 759 Real Time Formalism A(P), B(P), C(P) : Invariant complex functions

6. HTL resummed gauge boson propagator Improved Instantaneous Exchange Approximation （ set in the Longitudinal part ) To be got rid of at least in the Distribution Function Exact HTL resummed form for the Transverse part and for the Gauge part (gauge part: no HTL corrections)

7. HTL resummed vertex and thepoint vertex approximation (Improved Ladder Approximation)

8. HTL Resummed DS Equations for the Invariant Functions A, B, and C(A, BandC : functions with imaginary parts)PTP 107 (2002) 759 & 110 (2003) 777

11. 3. Consistency with the WT Identity Vacuum QED/QCD : In the Landau gauge A(P) = 1 guaranteed in the ladder SD equation where Z1 = 1 WT identity satisfied : “gauge independent” solution Finite Temperature/Density : Even in the Landau gauge A(P) ≠ 1 in the ladder SD equation where Z1 = 1 WT identity not satisfied : “gauge dependent” solution

12. To get a solution satisfying the WT identity through the ladder DSE at finite temperature: (1) Assume the nonlinear gauge such that the gauge parameter being a function of the momentum (2) In solving DSE iteratively, impose A(P) = 1 by constraint (for the input function at each step of the iteration) Can get a solution satisfying A(P) = 1 ?! thus, satisfying the Ward-Takahashi identity!! Same level of discussion possible as the vacuum QED/QCD

13. Gauge invariance (Ward-Takahashi Identity) T=0 Landau gauge ( ) holds because A(P)=1 for the point vertex T. Maskawa and H. Nakajima, PTP 52,1326(1974) PTP 54, 860(1975) Find the gauge such thatA(P)= 1 holds Z1 = Z2 (= 1) holds “Gauge invariant” results

14. 4. Numerical calculation • Cutoff at in unit of • A(P),B(P),C(P) at lattice sites are calculated by iteration procedure ★quantities at (0, 0.1) are shown in the figures corresponds to the “static limit” PTP 107 (2002) 759 & 110 (2003) 777

15. depends on momentum Requireintegral equationfor Expand by a series of functions expansion coefficients(both real and complex studied) Minimize Determine

16. Momentum dependent ξ analysis First, show the solution in comparison with those in the fixed gauge parameter • A(P) very close to 1 (imaginary part close to 0) • Optimal gauge ? complex ξ v.s. real ξ

17. α=4.0 : ξ(q0,q) v.s. constant ξ ξ= 0.05 ● ξ= 0.025● ξ= 0.0● ξ= -0.025 ● ξ= -0.05● ξ(q0,q) Realξ ○Complexξ●

18. α=4.0 : ξ(q0,q) v.s. constant ξ ξ(q0,q) (Landau) ξ= 0.0 ξ= -0.025 ξ= -0.05 ξ= 0.025 ξ= 0.05

19. Optimal Gauge

20. Optimal Gauge

21. Real and complex ξ analyses give the same solutionwhen the condition A(P)= 1 is properly imposed！ References: arXiv:0707.0929 [hep-ph], to appear in proc. of the Int’l Workshop on “Strongly Coupled QGP (sQGP’07)”, Nagoya, Feb. 2007. hep-ph/0703134, to appear in proc. of the Int’l Workshop on “Origin of Mass and Strong Coupling Gauge Theories (SCGT06)”, Nagoya, Nov. 2006.

22. Real and complex ξ give the same solution when the condition A(P)= 1 is properly imposed！ (fixed α analysis) ν= 0.378 Realξ ○ Complexξ ● ν= 0.350 ν= 0.380 α= 4.5 α= 5.0 α= 4.0 ν= 0.445 ν= 0.423 α= 3.5 α= 3.7 α= 3.2 ν=0.400～0.460

23. <ν> =0.395 α= 4.5 α= 5.0 α= 4.0 α= 3.5 α= 3.7 α= 3.2

24. Real and complex ξ give the same solutionwhen the condition A(P)= 1 is properly imposed！(fixed T analysis) <η> =0.522 T=0.120 T=0.105 T=0.110 T=0.115 T=0.125 T=0.100 Realξ ○ Complexξ ●

25. Phase Diagram ｉｎ (T,1/α)-plane(Comparison with the Landau gauge analysis) Symmetric Phase Broken Phase

26. Phase Diagram ｉｎ (T,1/α)-plane(Comparison with the Landau gauge analysis) Symmetric Phase Broken Phase

27. Phase Diagram ｉｎ (T,1/α)-plane(Landau gauge analysis)

28. 5. Summary and Outlook • DS equation at finite temperature is solved in the (“nonlinear”) gauge to make the WT identity hold • The solution satisfies A(P)=1, consistent with the WT identity Z1 = Z2 gauge “invariant” solution ! Very plausible!! • Significant discrepancy from the Landau gauge case, though is small • Critical exponents ν : depends on the coupling strength !? η : independent of the temperature

29. Summary and Outlook (cont’d) • Both the Real and Complex analyses : Give the samesolution(present result) ! gauge “invariant” solution ! stand the same starting level as the vacuum QED/QCD analysis • Application to QCD at finite T and density In future • Manifestly gauge invariant analysis • Analysis of the co-existing phases • Analytic solution