1. Phase Structure of Thermal QCD/QED through the HTL Improved Ladder Dyson-Schwinger Equation Hisao NAKKAGAWA Nara University in collaboration with Hiroshi YOKOTA and Koji YOSHIDA Nara University ・ Analysis underway (preliminary) ・ arXiv:0709.0323 ・ Talk at an Isaac Newton Institute Workshop on Exploring QCD : Deconfinement, Extreme Environments and Holography, Cambridge, August 20-24, 2007] ・ arXiv:0707.0929 [hep-ph] (in proc. of sQGP’07, Nagoya, Feb. 2007) ・ hep-ph/0703134 (in proc. of SCGT’06, Nagoya, Nov. 2006) [Seminar at the Institute of Physics, Academia Sinica, Taipei, Taiwan, March 13, 2009]

2. Plan 1. Introduction 2. HTL Re-summed 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 ξ gauges) b) nonlinear gauge : momentum dependent ξ(q 0,q) ｃ ) data from new analysis (preliminary) 5. Summary and Outlook

3. Pre-story 1. Existing QCD Phase diagram: ⊚ T ≠ 0, μ ≈ 0 : Lattice QCD simulation ⊚ Otherwise : Effective FT analyses (mostly, NJL model) 2. What does QCD itself really tell ?

4. 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 re-summed interaction kernel Difficult to solve 1) Point vertex = ladder kernel (Z 1 = 1) 2) Improved ladder kernel (HTL re-summed propagator) 3) Instantaneous exchange approximation to the longitudinal propagator transverse propagator: keep the full HTL re-summed form Bose-,Fermi-distribution function: exact form necessary for T ➝ 0

5. 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 deviates from 1 NB: A(P) = 1 required from the Ward-Takahashi Identity Z 1 = Z 2 4) Same results in the constant ξ gauges

6. Introduction (cont’d) [D] Gauge-dependence of the solution Really gauge dependent ? Further check necessary: to be reconfirmed ・ Error estimate: size of the systematic error ・ Determination of critical exponents ・ Analysis via invariant function B [E] Nonlinear gauge inevitable to satisfy the Ward-Takahashi Identity Z 1 = Z 2, and to get gauge “invariant” result (in the same sense at T=0 analysis)

7. 2. Hard-Thermal-Loop Re-summed Dyson-Schwinger Equations PTP 107 (2002) 759 Real Time Formalism A(P), B(P), C(P) : Invariant complex functions

8. HTL resummed gauge boson propagator Improved Instantaneous Exchange Approximation （ set k 0 = 0 in the Longitudinal part ) Should be got rid of at least in the Distribution Function Exact HTL re-summed form for the Transverse part and for the Gauge part (Gauge part: no HTL corrections)

9. HTL resummed vertex and the point vertex approximation (Improved Ladder Approximation)

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

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

14. 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

15. 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) T≠0 Find the gauge such that A(P)= 1 holds Z 1 = Z 2 (= 1) holds “Gauge invariant” results

16. 4. Numerical calculation Cutoff at in unit of A(P),B(P),C(P) at lattice sites are calculated by iteration procedure: check site #-dependence (New analysis underway ➩ systematic error estimate) ★ quantities at (0, 0.1) are shown in the figures corresponds to the “static limit” PTP 107 (2002) 759 & 110 (2003) 777

17. Momentum dependent ξ analysis : function of momentum Require integral equation for 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

18. ξ(q 0,q) ξ= 0.05 ● ξ= 0.025 ● ξ= 0.0 ● ξ= -0.025 ● ξ= -0.05 ● Real ξ ○ Complex ξ ● α=4.0 : ξ(q 0,q) v.s. constant ξ

19. ξ(q 0,q) ξ= 0.05 ξ= 0.025 (Landa u) ξ= 0.0 ξ= -0.025 ξ= -0.05 α=4.0 : ξ(q 0,q) v.s. constant ξ

20. Scaled data α=4.0 : ξ(q 0,q) v.s. constant ξ

21. Real and complex ξ analyses give the same solution when the condition A(P)= 1 is properly imposed ！ References: i) arXiv:0707.0929 [hep-ph], in proc. of the Int’l Workshop on “Strongly Coupled QGP (sQGP’07)”, Nagoya, Feb.’07. ii) hep-ph/0703134, in proc. of the Int’l Workshop on “Origin of Mass and Strong Coupling Gauge Theories (SCGT06)”, Nagoya, Nov.’06. iii) talk at an Isaac Newton Institute Workshop on “Exploring QCD: Deconfinement, Extreme Environments and Holography”, Cambridge, Aug. ‘07

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

23. Symmetric Phase Broken Phase Phase Diagram ｉｎ (T,1/α)-plane (Comparison with the Landau gauge analysis) ξ(q0,q)ξ(q0,q) ξ=0

24. Data from new analysis (preliminary) 1. Symmetry under p 0 ⇄ -p 0 ( ⇐ CC symmetry) ・ Re[A], Im[B], Re[C]: even ; Im[A], Re[B], Im[C]: odd 2. Site #-dependence: very small 3. Landau gauge ・ T ➝ 0 behavior of the critical coupling: α c ➝ α c T=0 =π/3 ・ Im[B] as a function of α (or e) and T : In symmetric phase, B ~ thermal mass Data shows Im[B] ~ eT !? ⊚ consistent also with αT, in the small range studied ⊚ in the region α:small and T:large : Im[B/T] ~α (in agreement with the HTL approximation ) ⊚ linear fit of Im[B/T] as function of e c agrees with T=0 analysis !

25. Site #-dependence

26. Phase Diagram (Landau gauge)

28. Im[B]/T data (fixed coupling)

29. Im[B]/T vs charge e=sqr(4πα)

30. Im[B]/T vs coupling α= e 2 /4π

31. 4. Gauge-dependence (from Landau to Feynman) ・ Can gauge-dependence be absorbed into “re-scaling” of the scale(cut-off)-parameter Λ ?! ξ-dependence never disappears ! see, scaled Im[B/T] data : Im[B ] /T c and Im[B/T] /(T/T c ) 2 ・ Analysis of critical exponents: underway 5. Gauge-independent solution ・ A(P) = 1 must hold ⇔ Z 1 = Z 2 ・ No solution in gauges with constant ξ ⇒ must find a solution in nonlinear ξ gauges

32. Im[B] data (various fixed ξ gauges)

33. Scaled Im[B] data (various fixed ξ gauges)

34. Re[C] data (various fixed ξ gauges)

35. Scaled Re[C] data (various fixed ξgauges)

36. Scaled Re[C/A] data (various fixed ξ gauges)

37. Scaled Re[A] data (various fixed ξ gauges)

38. 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 Z 1 = Z 2 gauge “invariant” solution ! Very plausible!! Significant discrepancy from the Landau gauge case, though ξ(q 0,q) is small Critical exponents: ν : depends on the coupling strength !? η : independent of the temperature

39. Summary and Outlook (cont’d) Both the Real and Complex ξ(q 0,q) analyses: Give the same solution (present result) ! ⇒ gauge “invariant” solution ! could stand the same starting level as the vacuum QED/QCD analysis Application to QCD at finite T and density Sys. Error estimate existence of gauge-dep. gauge “invariant” solutions In future Manifestly gauge invariant analysis: vertex correction, etc Tri-critical point phenomenology Analysis of the co-existing phases Analytic solution