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]
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(q0,q)
ｃ) data from new analysis (preliminary)
5. Summary and Outlook
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 ?
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 (Z1 = 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
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 Z1 = Z2
4) Same results in the constant ξ gauges
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 Z1 = Z2, and
to get gauge “invariant” result (in the same sense at T=0 analysis)
2. Hard-Thermal-Loop Re-summed
Dyson-Schwinger Equations
PTP 107 (2002) 759
Real Time Formalism
e2
 i ( P )  
R
2


d 4K
 S ( K , K ) G


( K  P, P  K )
4
RR
,

RA
RAA
RAA
(2 )
   S ( K , K ) G
( K  P, P  K ) 
AAR RA,
RAA RR
Fermion :
S R ( P)  S RA ( P, P) 
1
P  i 0   R
S C ( P)  S RR ( P, P)  (1  2n F ( p 0 ))( S R ( P)  S A ( P))
 ( P)  (1 A( P)) pi i  B( P) 0  C ( P)
R
A(P), B(P), C(P) : Invariant complex functions
HTL resummed gauge boson propagator

G R ( K )  G RA
( K , K )
1
1




A

B

D
 TR  K 2  ik 0
 RL  K 2  ik 0
K 2  ik 0
~  ~

K
K
K  K ~






A  g  B  D ,B  
,D 
, K  ( k , k 0 k )
2
2
K
K

GC ( K )  G RR
( K , K )  (1  2n B (k 0 ))(G R ( K )  G A ( K ))

Improved Instantaneous Exchange Approximation
（ set k0 = 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)
HTL resummed vertex and the
point vertex approximation



ijk   ijk  ijk ,



 RAA   AAR   ,
otherwise 0
Then set
 0
(Improved Ladder Approximation)
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

4
d
K
2




C ( P )  e 
g
1

2
n
(
p

k
)
Im
G
B
0
0
R (P  K )
4 
(2 )
C(K )

 1  2nF (k0 )
2
2 2
2
k0  B( K )  i   A( K ) k  C ( K )



C(K )
 GR ( P  K ) Im
2
2 2
2
 k0  B( K )  i   A( K ) k  C ( K ) 


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
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 (   0 )
because
A(P)=1
Z1  Z 2 holds
for the point vertex
T. Maskawa and H. Nakajima, PTP 52,1326(1974)
PTP 54, 860(1975)
T≠0
Find the gauge  ( q 0 , q ) such that A(P)= 1 holds


Z1 = Z2 (= 1) holds
“Gauge invariant” results
4. Numerical calculation
• Cutoff at
1
1
0
1
   dk  dk 0 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
Momentum dependent ξ analysis
   (q0 , q) : function of momentum
Require
A( P )  1
integral equation for
   (q0 , q)
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
α=4.0 : ξ(q0,q) v.s. constant ξ
1.6
1.5
1.4
Re[A]
1.3
1.2
1.1
ξ(q0,q)
1
0.9
ξ= 0.05
●
ξ= 0.025
●
ξ= 0.0
●
ξ= -0.025
●
ξ= -0.05
●
0.8
0.7
0.6
0.12
Real ξ
Complex ξ
0.125
○
●
0.13
0.135
T/Λ
0.14
0.145
α=4.0 : ξ(q0,q) v.s. constant ξ
0.4
0.35
ξ(q0,q)
0.3
(Landau)
Re[M]
0.25
ξ= 0.0 ξ= -0.025
0.2
ξ= -0.05
ξ= 0.025
0.15
ξ= 0.05
0.1
0.05
0
0.12
0.125
0.13
0.135
T/Λ
0.14
0.145
0.15
Scaled data α=4.0 : ξ(q0,q) v.s. constant ξ
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
Real and complex ξ give the same solution
when the condition A(P)= 1 is properly imposed！
(fixed α analysis)
0.6
Real ξ
0.5
Re[M]
0.4
0.3
○
ν= 0.380
α= 4.0
Complex ξ ●
ν= 0.445
α= 3.5
ν= 0.378
α= 4.5
ν= 0.350
α= 5.0
ν= 0.423
α= 3.7
0.2
0.1 α= 3.2
ν=0.400～0.460
6E-16
0.105
0.11
0.115
0.12
-0.1
T/Λ
0.125
0.13
0.135
0.14
Phase Diagram ｉｎ (T,1/α)-plane
(Comparison with the Landau gauge analysis)
0.4
Symmetric Phase
0.35
1/αc
0.3
0.25
0.2
ξ(q0,q)
ξ=0
Broken Phase
0.15
0.1
0.08
0.1
0.12
Tc/Λ
0.14
0.16
Data from new analysis (preliminary)
1. Symmetry under p0 ⇄ -p0 (⇐ 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: αc➝αcT=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 ec agrees with T=0 analysis !
Site #-dependence
Phase Diagram (Landau gauge)
1.2
1
1/αc
0.8
0.6
0.4
0.2
0
0
0.05
0.1
0.15
Tc
0.2
0.25
Phase Diagram (Landau gauge)
linear fit [αc(Tc=0)=1.306]
0.06
0.05
(Tc)2
0.04
0.03
0.02
0.01
0
0
1
2
3
4
5
6
αc
7
8
9
10
11
12
Im[B]/T data (fixed coupling)
Im[B]/T vs charge e=sqr(4πα)
8.0
7.0
6.0
Im[B]/T
5.0
4.0
3.0
2.0
1.0
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.011.012.013.0
e
0.09
0.095
0.1
0.105
0.11
0.115
0.12
0.125
0.13
0.135
0.14
0.145
0.15
0.16
0.17
0.18
0.2
0.3
**
0.4
Im[B]/T vs coupling α= e2/4π
8.0
7.0
6.0
Im[B]/T
5.0
4.0
3.0
2.0
1.0
0.0
0.00
2.00
4.00
6.00
α
8.00
10.00
0.09
0.095
0.1
0.105
0.11
0.115
0.12
0.125
0.13
0.135
0.14
0.145
0.15
0.16
0.2
0.18
0.17
0.3
12.00
0.4
**
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 ] /Tc and Im[B/T] /(T/Tc)2
・ Analysis of critical exponents: underway
5. Gauge-independent solution
・ A(P) = 1 must hold ⇔ Z1 = Z2
・ No solution in gauges with constant ξ
⇒ must find a solution in nonlinear ξ gauges
Im[B] data (various fixed ξ gauges)
Scaled Im[B] data (various fixed ξ gauges)
Re[C] data (various fixed ξ gauges)
Scaled Re[C] data (various fixed ξgauges)
Scaled Re[C/A] data (various fixed ξ gauges)
Scaled Re[A] data (various fixed ξ gauges)
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 ξ(q0,q) is small
• Critical exponents:   0.395,   0.522
ν : depends on the coupling strength !?
η : independent of the temperature
Summary and Outlook (cont’d)
• Both the Real and Complex ξ(q0,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