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: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]
Transcript
Phase Structure of Thermal QCD/QED:A “Gauge Invariant” Analysis based on the
HTL Improved Ladder Dyson-Schwinger Equation
Hisao NAKKAGAWANara University
in collaboration with
Hiroshi YOKOTA and Koji YOSHIDANara University
arXiv: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]
Plan
1. Introduction2. HTL Resummed DS Equation a) Improved Ladder Approximation b) Improved Instantaneous Exchange Approximation
3. Consistency with the Ward-Takahashi Identity4. Numerical Calculation a) Landau gauge (constant gauges) b) nonlinear gauge : momentum dependent
5. Summary and Outlook
),( 0 qq
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
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)
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)
0 21 ZZ 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
Z1 = Z2 (= 1) holds
“Gauge invariant” results
0T ),( 0 qq
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”
1
1 0
1
0dkdk
PTP 107 (2002) 759 & 110 (2003) 777
depends on momentum
Expand by a series of functions
0124
mnC
AAPdA
)()(),( 00 qGqFCqq nmmn
mn
),( 0 qq
)(),( 0 qGqF nm),( 0 qq
Minimize
:mnC expansion coefficients (both real and complex studied)
1)( PARequire integral equation for ),( 0 qq
),( 0 qqDetermine
nakkfushimi
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
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
0.12 0.125 0.13 0.135 0.14 0.145
T/ Λ
Re[
A] ξ(q0,q
)
ξ= 0.05 ●
ξ= 0.025 ●
ξ= 0.0 ●
ξ= -0.025 ●
ξ= -0.05 ●
Real ξ ○ Complex ξ ●
α=4.0 : ξ(q0,q) v.s. constant ξ
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.12 0.125 0.13 0.135 0.14 0.145 0.15T/ Λ
Re[
M]
ξ(q0,q)
ξ= 0.05
ξ= 0.025
(Landau)
ξ= 0.0
ξ= -0.025 ξ= -
0.05
α=4.0 : ξ(q0,q) v.s. constant ξ
Optimal Gauge
Optimal Gauge
Real and complex ξ analyses give the same solution when 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.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14
T/Λ
Re[
M] α= 3.5
α= 4.0α= 4.5 α= 5.0
α= 3.2
ν= 0.445
ν= 0.380
α= 3.7ν= 0.423
ν= 0.378
ν= 0.350
ν=0.400 ～0.460
Real and complex ξ give the same solutionwhen the condition A(P)= 1 is properly imposed ！ (fixed α analysis)
Real ξ ○
Complex ξ ●
0
0.1
0.2
0.3
0.4
0.5
0.6
0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14T/ Λ
Re[
M]
<ν> = 0.395
α= 3.5
α= 4.0α= 4.5
α= 5.0
α= 3.2
α= 3.7
)(Re TTC c
0
0.1
0.2
0.3
0.4
0.5
0.6
2.7 2.9 3.1 3.3 3.5 3.7 3.9 4.1 4.3 4.5
α
Re[
M]
T=0.100
T=0.105 T=0.1
10T=0.115
T=0.120
T=0.125
<η> = 0.522
Real ξ ○
Complex ξ ●
Real and complex ξ give the same solutionwhen the condition A(P)= 1 is properly imposed ！ (fixed T analysis)
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.08 0.1 0.12 0.14 0.16Tc/ Λ
1/α
c
Symmetric Phase
Broken Phase
0
),( 0 qq
Phase Diagram ｉｎ (T,1/α)-plane(Comparison with the Landau gauge analysis)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.05 0.1 0.15Tc/ Λ
1/α
c
Symmetric Phase
Broken Phase
0),( 0 qq
Phase Diagram ｉｎ (T,1/α)-plane(Comparison with the Landau gauge analysis)
