Phase Structure of Thermal QCD/QED: A “Gauge Invariant” Analysis based on the HTL Improved Ladder Dyson-Schwinger Equation Hisao NAKKAGAWA Nara University in collaboration with Hiroshi YOKOTA and Koji YOSHIDA Nara University arXiv: [hep-ph] (to appear in proc. of sQGP’07, Nagoya, Feb. 2007) hep-ph/ (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]

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

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 (Z 1 = 1) 2) Improved ladder kernel (HTL resummed propagator) 3) Instantaneous exchange approximation to the longitudinal propagator transverse one: keep the full HTL resummed form

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 Z 1 = Z 2 4) Same results in the constant gauges [D] 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)

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

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)

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

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

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

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

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 that A(P)= 1 holds Z 1 = Z 2 (= 1) holds “Gauge invariant” results

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

depends on momentum Expand by a series of functions Minimize expansion coefficients (both real and complex studied) Require integral equation for Determine

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

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

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

Optimal Gauge

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

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

= α= 3.5 α= 4.0 α= 4.5 α= 5.0 α= 3.2 α= 3.7

T=0.100 T=0.105 T=0.110T=0.115 T=0.120 T=0.125 = Real ξ ○ Complex ξ ● Real and complex ξ give the same solution when the condition A(P)= 1 is properly imposed ！ (fixed T analysis)

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

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

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

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 is small Critical exponents ν : depends on the coupling strength !? η : independent of the temperature

Summary and Outlook (cont’d) Both the Real and Complex analyses : Give the same solution (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