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The Polarization Tensor of the Massless Mode inYang-Mills Thermodynamics
Markus Schwarz
Karlsruhe Institute of Technology (KIT)
1st Symposium on Analysis of Quantum Field Theory9th International Conference of Numerical Analysis and Applied Mathematics
Haldiki, Greece, 19-25 September 2011
21 September 2011
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Table of Contents
Review of YMT
Polarization Tensor and Momentum Constraints
Approximate Calculation
Selfconsistent CalculationTransverse Dispersion RelationLongitudinal Dispersion Relation
Monopole Properties from a 2-Loop Correction to the Pressure
Summary and Outlook
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Table of Contents
Review of YMT
Polarization Tensor and Momentum Constraints
Approximate Calculation
Selfconsistent CalculationTransverse Dispersion RelationLongitudinal Dispersion Relation
Monopole Properties from a 2-Loop Correction to the Pressure
Summary and Outlook
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Review of Ralf’s talk
I After coarse-graining, nonperturbative YMT ground statedescribed by scalar field φ(T )[Herbst,Hofmann ’04, Hofmann ’05, Giacoa,Hofmann ’05]
I Effective action for top. trivial sector
SE [aµ] =
∫ 1T
0dx4
∫dx3Tr
(1
2GµνGµν + DµφDµφ+ Λ6φ−2
)I In deconfining phase (T > Λ): SU(2)→ U(1)
I Two tree-level massive modes (TLH) with massm2(T ) = 4e2(T )|φ(T )|2 = 4e2Λ3/2πT ,one tree-level massless mode (TLM)
I Eff. coupling e(T ) has plateau value eplateau ∼√
8π ≈ 8.8,no PT but loop expansion
I |φ| yields max. resolution
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Table of Contents
Review of YMT
Polarization Tensor and Momentum Constraints
Approximate Calculation
Selfconsistent CalculationTransverse Dispersion RelationLongitudinal Dispersion Relation
Monopole Properties from a 2-Loop Correction to the Pressure
Summary and Outlook
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Polarization Tensor, Motivation
I Effect TLH modes on propagation of TLM modes describedby polarization tensor Σµν .
I On one-loop level: Σµν = p p
k
p−k
A
+p p
kB
(one-loop sufficient, cmp. talk by Dariush)
I simplest radiative correction
I generic radiative correction
I interesting consequences for physics
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Polarization Tensor, Decomposition
U(1) gauge symmetry unbroken, ⇒ Σµν 4D transverse: pµΣµν = 0Decomposition into spatially transverse and longitudinal part:
Σµν = G (p4,p)PµνT + F (p4,p)PµνL
with
PµνL ≡ δµν − pµpν
p2− PµνT ,
projecting also onto p.
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Free Propagators
Free propagator for TLH and TLM modes in
I unitary gauge (particle content manifest)
I Coulomb gauge (∇A = 0)
DTLH,0µν,ab (k) = −δabD̃µν
1
k2 + m2, {a, b} ∈ {1, 2}
DTLM,0µν,ab (p) = −δa3δb3
(PTµν
1
p2+
uµuνp2
)uµ = δ4µ four-velocity of head bath.D̃µν projects out the component transverse to kPµνT projects out the component transverse to p.Gauge fixed completely ⇒ no ghost fields needed.
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Dressed Propagator
Propagator for interacting TLM mode (imaginary time)
DTLMµν,ab(p) = −δa3δb3
(PTµν
1
p2 + G+
p2
p2uµuνp2 + F
)
F (p4,p) =(
1− p24p2
)−1Σ44 describes propagation of longitudinal
mode A4
For p ‖ e3, G (p4,p) = Σ11 = Σ22 describes propagation oftransverse mode Ai
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Vertices and Momentum constraints
Γµνρ[3]abcΓµνρσ[4]abcd
I Momentum conservation at each vertex.I Effective vertex contains modes with
∣∣p2∣∣ > |φ|2:
p2 > |φ|2
I Exclude these modes in effective theory to avoid ”doublecounting” (already included in ag.s.µ ; cmp. talk by Ralf)
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Momentum constraints
|φ| yields maximum resolution in effective theory ⇒ constraints onmomentum transfer in vertex
p1
p2
p3
p1
p2
p34
p = + −
s-channel:∣∣(p1 + p2)2
∣∣ ≤ |φ|2t-channel:
∣∣(p1 − p3)2∣∣ ≤ |φ|2
u-channel:∣∣(p2 − p3)2
∣∣ ≤ |φ|2
RecallFinite temperature QFT defined in imaginary time x4 with fieldsbeing periodic in x4. Only discrete p4 momenta allowed(Matsubara sums).
ProblemMomentum constraints formulated in terms of physical, continuousfour momenta.
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Real time propagators
SolutionExpress Matsubara sums as integrals over continuous real time t.[Kapusta, LeBellac]
Free propagator for TLH and TLM modes in unitary Coulombgauge and real-time formalism
DTLM,0µν,ab (p) = δa3δb3
{PTµν
[−i
p2 + iε− 2πδ(p2) nB(|p0|/T )
]+ i
uµuνp2
}DTLH,0µν,ab (k) = −2πδabD̃µνδ(k2 −m2) nB(|k0|/T ) , a, b ∈ {1, 2}
No vacuum propagator for TLH modes (cmp. talk by Ralf):
+ +
(a)
+ +
p p p p p p
p p
p1
p2p p
p1
p2
p
(b)
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Modified dispersion relation of TLM
Dressed propagator of transverse and longitudinal TLM mode (realtime)
DTLMµν,ab(pt) = −δa3δb3PT
µν
[i
p2t − G+ 2πδ(p2t − G ) nB(|p0,t |/T )
]DTLMµν,ab(pl) = δa3δb3uµuν
[p2lp2l
i
p2l − F− 2πδ(p2l − F ) nB(|p0,l |/T )
]Poles yield dispersion relations (p0 = ω + iγ, assume γ � ω):
ω2t (pt) = p2t + ReG (ω(pt),pt)
γ(pt) = −ImG (ω(pt),pt)/2ω
ω2l (pl) = p2l + ReF (ωL(pl),pl)
γl(pl) = −ImF (ωl(pl),pl)/2ωl
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Diagrams for G and F
Chosing p ‖ e3:
G (p0,p) = Σ11 = Σ22
F (p0,p) =
(1− p20
p2
)−1Σ00
Σµν sum of two diagrams:
p p
k
p−k
A
Purely imaginary: ⇒ yields γ
p p
kB
Purely real: ⇒ yields dispersionrelation
One-loop level sufficient (see talk by Dariush)!
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Table of Contents
Review of YMT
Polarization Tensor and Momentum Constraints
Approximate Calculation
Selfconsistent CalculationTransverse Dispersion RelationLongitudinal Dispersion Relation
Monopole Properties from a 2-Loop Correction to the Pressure
Summary and Outlook
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Approximation p2 = 0
Applying Feynman rules to ReG = ReΣ11 =p p
kB
yields gap equation:
ReG (p0,p) = Σ11B (p) = 8πe2
∫|(p+k)2|≤|φ|2
[−(
3− k2
m2
)+
k1k1
m2
]×nB (|k0| /T )δ
(k2 −m2
) d4k(2π)4
∣∣∣∣p2=G
(1)
4 vertex imposes constraint∣∣(p + k)2
∣∣ ≤ |φ|2Difficulty
Equation (1) is a transzendental equation for G .
Approximation
Use p2 = 0 in constraint [Schwarz,Giacosa,Hofmann ’06].Valid if G � p2. Check later!
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Consequences of p2 = 0
1. For finite Σ00, F (p0,p) =(
1− p20p2
)−1Σ00 vanishes.
Hence, no propagation of longitudinal modes.
2. Diagram A vanishes:
p p
k
p−k
A
Momentum conservation at vertex vorbids TLM mode withp2 = 0 to split into two on-shell particles with mass m.
3. Diagram A = 0, hence no imaginary part of G , hence γ = 0and assumption γ � ω satisfied trivially.
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Calculation of diagram B, p2 = 0
With p2 = 0:
G (|p|,p) = 8πe2∫|2pk+k2|≤|φ|2
[g11
(3− k2
m2
)+
k1k1
m2
]× nB (|k0| /T ) δ
(k2 −m2
) d4k
(2π)4
Via δ-function, integration over k0 yields k0 → ±√k2 + m2.
Using p0 ≥ 0, p2 = 0, k2 = m2 = 4e2|φ|2, θ ≡ ∠(p, k), and
k0 = ±√
k2 + m2 constraint reads:∣∣∣∣2|p|(±√k2 + 4e2|φ|2 − |k| cos θ
)+ 4e2|φ|2
∣∣∣∣ ≤ |φ|2Integrand symmetric under k0 → −k0, but constraint is not!
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Dealing with constraints, + sign
Consider + sign and observe:∣∣∣∣2|p|(+
√k2 + 4e2|φ|2 − |k| cos θ
)+ 4e2|φ|2
∣∣∣∣ ≤ |φ|21. Term in parentheses always positive.
2. e &√
8π ∼ 8.8
Hence, constraint never satisfied.
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Dealing with constraints, – sign
Using X ≡ |p|/T , y ≡ k/|φ|, |φ|/T = 2πλ−3/2, and λ ≡ 2πT/Λconstraint reads
−1 ≤ −λ3/2Xπ
(√y2 + 4e2 + y3
)+ 4e2 ≤ 1
Convenient to use polar coordinates y1 = ρ cosϕ, y2 = ρ sinϕ.Constraint then reads
4e2 − 1
λ3/2π
X≤√ρ2 + y23 + 4e2 + y3 ≤
4e2 + 1
λ3/2π
X
Integration over ϕ not constraint!
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Dealing with constraints, – sign
4e2 − 1
λ3/2π
X≤√ρ2 + y23 + 4e2 + y3 ≤
4e2 + 1
λ3/2π
X
1. y3, ρ plane
2. y3 < ymax ≡ 4e2+1λ3/2
πX
3. ρ ≤ ρmax(y3) ≡√(πX
)2 (4e2+1)2
λ3− 2π
X4e2+1
λ3/2y3−4e2fory3 ≤ yM3 ≡ π
2X4e2+1λ3/2
− 2λ3/2Xπ
e2
4e2+1
4. ρ ≥ ρmin(y3) ≡√(πX
)2 (4e2−1)2λ3
− 2πX
4e2−1λ3/2y3−4e2
fory3 ≤ ym3 ≡ π
2X4e2−1λ3/2
− 2λ3/2Xπ
e2
4e2−1
0 20 40 60 80y3
20
40
60
80
Ρ
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Dealing with constraints, – sign
4e2 − 1
λ3/2π
X≤√ρ2 + y23 + 4e2 + y3 ≤
4e2 + 1
λ3/2π
X
1. y3, ρ plane
2. y3 < ymax ≡ 4e2+1λ3/2
πX
3. ρ ≤ ρmax(y3) ≡√(πX
)2 (4e2+1)2
λ3− 2π
X4e2+1
λ3/2y3−4e2fory3 ≤ yM3 ≡ π
2X4e2+1λ3/2
− 2λ3/2Xπ
e2
4e2+1
4. ρ ≥ ρmin(y3) ≡√(πX
)2 (4e2−1)2λ3
− 2πX
4e2−1λ3/2y3−4e2
fory3 ≤ ym3 ≡ π
2X4e2−1λ3/2
− 2λ3/2Xπ
e2
4e2−1
0 20 40 60 80y3
20
40
60
80
Ρ
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Dealing with constraints, – sign
4e2 − 1
λ3/2π
X≤√ρ2 + y23 + 4e2 + y3 ≤
4e2 + 1
λ3/2π
X
1. y3, ρ plane
2. y3 < ymax ≡ 4e2+1λ3/2
πX
3. ρ ≤ ρmax(y3) ≡√(πX
)2 (4e2+1)2
λ3− 2π
X4e2+1
λ3/2y3−4e2fory3 ≤ yM3 ≡ π
2X4e2+1λ3/2
− 2λ3/2Xπ
e2
4e2+1
4. ρ ≥ ρmin(y3) ≡√(πX
)2 (4e2−1)2λ3
− 2πX
4e2−1λ3/2y3−4e2
fory3 ≤ ym3 ≡ π
2X4e2−1λ3/2
− 2λ3/2Xπ
e2
4e2−1
0 20 40 60 80y3
20
40
60
80
Ρ
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Dealing with constraints, – sign
4e2 − 1
λ3/2π
X≤√ρ2 + y23 + 4e2 + y3 ≤
4e2 + 1
λ3/2π
X
1. y3, ρ plane
2. y3 < ymax ≡ 4e2+1λ3/2
πX
3. ρ ≤ ρmax(y3) ≡√(πX
)2 (4e2+1)2
λ3− 2π
X4e2+1
λ3/2y3−4e2fory3 ≤ yM3 ≡ π
2X4e2+1λ3/2
− 2λ3/2Xπ
e2
4e2+1
4. ρ ≥ ρmin(y3) ≡√(πX
)2 (4e2−1)2λ3
− 2πX
4e2−1λ3/2y3−4e2
fory3 ≤ ym3 ≡ π
2X4e2−1λ3/2
− 2λ3/2Xπ
e2
4e2−1
24.2 24.4 24.6 24.8 25.0 25.2 25.4y3
2
4
6
8
10
12
14
Ρ
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Expression for G , p2 = 0
Constraint in terms of boundaries for (y3, ρ) integration:
G (X ,T )
T 2=
[∫ ym3
−∞dy3
∫ ρmax
ρmin
dρ+
∫ yM3
ym3
dy3
∫ ρmax
0dρ
]
e2ρ
λ3
(ρ2
4e2− 4
) nB
(2πλ−3/2
√ρ2 + y23 + 4e2
)√ρ2 + y23 + 4e2
X = |p|/T momentum of external TLM mode in units oftemperature,λ ≡ 2πT/Λ temperature in units of YM scale Λ.Integration performed numerically via Gaussian quadrature (unlikeMonte-Carlo for 3-loop, comp. talk by Dariush).
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Result for G , p2 = 0
0.0 0.2 0.4 0.6 0.8 1.0X
10-3
10-2
10-1
ÈG�T2È
2Tc 3Tc 4Tc X2
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Result for G , p2 = 0
All results based on∣∣G/T 2
∣∣� X 2!
0.0 0.2 0.4 0.6 0.8 1.0X
10-3
10-2
10-1
ÈG�T2È
2Tc 3Tc 4Tc X2
I X & 0.2: G < 0(anti-screening)
I Dip: G = 0
I X . 0.2: G > 0(screening)
I G � p2 for X & 0.2
I G = 0 at X ∼ 0.2, dispersion relation solved selfconsistently
I G ≥ p2 for X . 0.1, approximation breaks down
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Table of Contents
Review of YMT
Polarization Tensor and Momentum Constraints
Approximate Calculation
Selfconsistent CalculationTransverse Dispersion RelationLongitudinal Dispersion Relation
Monopole Properties from a 2-Loop Correction to the Pressure
Summary and Outlook
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Table of Contents
Review of YMT
Polarization Tensor and Momentum Constraints
Approximate Calculation
Selfconsistent CalculationTransverse Dispersion RelationLongitudinal Dispersion Relation
Monopole Properties from a 2-Loop Correction to the Pressure
Summary and Outlook
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Full calculation of G
Gap equation:
ReG (p0,p) = 8πe2∫|(p+k)2|≤|φ|2
[−(
3− k2
m2
)+
k1k1
m2
]× nB (|k0| /T ) δ
(k2 −m2
) d4k
(2π)4
∣∣∣∣p2=G
≡ H(T ,p,G )
Via δ-function, integration over k0 yields k0 → ±√k2 + m2.
With p0 = ±√
p2 + G (p0,p) and p ‖ e3 constraint reads∣∣∣G + 2(±√p2 + G
√k2 + m2 − pk3
)+ m2
∣∣∣ ≤ |φ|2
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Full calculation of G
Strategy to solve ReG (p0,p) = H [T ,p,G (p0,p)][Ludescher,Hofmann ’08]:
1. fix T and p (integrand in H independent of p0 ⇒ G = G (p))
2. prescribe any value of G1
3. calculate H(T ,p,G1) using Monte-Carlo integration
4. repeat steps 2 and 3 to obtain H(T ,p,G )
5. solve G = H(T ,p,G ) numerically using Newton’s method
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Selfconsistent result for G , real part
æ æ æ æ æ æ æ æ ææ
ææ
æ
æ
æ
æ
æ
ææ
ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
ææ
à à à à à à à àà
àà
à
à
à
à
àà
àà à à à à à à à à à à à à à à à à à à à à à
ììììììììì
ì
ì
ìì
ì
ìì
ìììììììììììììììììììììììì
0.0 0.1 0.2 0.3 0.4 0.5 0.6X
10-3
10-2
10-1
ÈG�T 2È2Tc 3Tc 4Tc
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Selfconsistent result for G , real part
æ æ æ æ æ æ æ æ ææ
ææ
æ
æ
æ
æ
æ
ææ
ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
ææ
à à à à à à à àà
àà
à
à
à
à
àà
àà à à à à à à à à à à à à à à à à à à à à à
ììììììììì
ì
ì
ìì
ì
ìì
ìììììììììììììììììììììììì
0.0 0.1 0.2 0.3 0.4 0.5 0.6X
10-3
10-2
10-1
ÈG�T 2È2Tc 3Tc 4Tc
I X & 0.2: G < 0(anti-screening)
I Dip: G = 0
I X . 0.2: G > 0(screening)
Comparison with approximate result:
I Zeros of G agree (must be)
I For X & 0.2, approximate agrees with selfconsistent result(expected)
I Results different when G & X 2 (not suprising)
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Selfconsistent result for G , imaginary part
Imaginary part: ImG ∝ p p
k
p−k
A
At left vertex: particle with mass√G decaying into two on-shell
particles with mass m only possible if
G
T 2≥ 4
m2
T 2= 64π2
e2
λ3(2)
I G ≤ 0: condition (2) never satisfied
I G > 0:
G (X = 0,T )
T 2∝ 1
λ3� 64π2
e2
λ3∼ 5× 104/λ3
condition (2) never satisfied
Diagram A = 0, hence no imaginary part of G , hence γ = 0 andassumption γ � ω satisfied trivially.
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Table of Contents
Review of YMT
Polarization Tensor and Momentum Constraints
Approximate Calculation
Selfconsistent CalculationTransverse Dispersion RelationLongitudinal Dispersion Relation
Monopole Properties from a 2-Loop Correction to the Pressure
Summary and Outlook
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Full Calculation of F
Assume F ∈ R (turns out to be selfconsistent)
Apply Feynman rules to p2 = ReΣ00 =p p
kB
yields gap equation:
p2 = Σ00B (p) = 8πe2
∫|(p+k)2|≤|φ|2
[(3− k2
m2
)+
k0k0
m2
]× nB (|k0| /T ) δ
(k2 −m2
) d4k
(2π)4
∣∣∣∣p2=F
(3)
Strategy to find F similar to that of finding G[Falquez,Hofmann,Baumbach ’11].
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Selfconsitent Result for F
Yl ≡ ωl (pl ,T )T =
√F (p2l ,T )
T 2 +p2lT 2 , X ≡ |pl |/T
0.05 0.10 0.15 0.20 0.25 0.30X
0.25
0.50
0.75
1.00
1.25
1.50
YL
2Tc 3Tc 4Tc
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Selfconsitent Result for F
Yl ≡ ωl (pl ,T )T =
√F (p2l ,T )
T 2 +p2lT 2 , X ≡ |pl |/T
0.05 0.10 0.15 0.20 0.25 0.30X
0.25
0.50
0.75
1.00
1.25
1.50
YL
2Tc 3Tc 4Tc
I 3 branches
I YL defined only forX . 0.34
I superluminal groupvelocity
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Selfconsistent Result for F , Interpretation
Interpretation in terms of magnetic monopoles[Falquez,Hofmann,Baumbach ’11]
I longitudinal modes due to charge density wavesI light like propagation:
I stable (yet unresolved) monopoles released by large holonomycaloron dissociation [Diakonov et al. ’04]
I density disturbance can only be propagated by radiation field,which propagates at the speed of light
M M
I superluminal propagation:I unstable monopoles contained in small holonomy caloronI extended calorons provide instantaneous correlation between
monopoles, leading to superluminal propagation
MM
caloron
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Table of Contents
Review of YMT
Polarization Tensor and Momentum Constraints
Approximate Calculation
Selfconsistent CalculationTransverse Dispersion RelationLongitudinal Dispersion Relation
Monopole Properties from a 2-Loop Correction to the Pressure
Summary and Outlook
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2-Loop Corrections
I “Bubble diagrams” yield pressure (cmp. talk by Dariush)
I For T � Tc only relevant diagram:
50 100 150 200Λ
-0.0004
-0.0003
-0.0002
-0.0001
0.0000
0.0001
0.0002
DPHM�P1-LoopI ∆P ∝ −4× 10−4T 4
I Temperature of TLMgas reduced!
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Monopole Properties
Explanation
Energy used to break up calorons, creating monopoleanti-monopole pairs [Schwarz,Giacosa,Hofmann ’06].
Detailed analysis shows [Ludescher,Keller,Giacosa,Hofmann ’08]:
I average monopole-antimonopole distance d̄ < |φ|−1⇒ monopoles unresolved in effective theory
I screening length ls due to small-holonomy calorons: ls = 3.3d̄⇒ magnetic flux of monopole and antimonopole cancel (noarea law for spatial Wilson loop)
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Table of Contents
Review of YMT
Polarization Tensor and Momentum Constraints
Approximate Calculation
Selfconsistent CalculationTransverse Dispersion RelationLongitudinal Dispersion Relation
Monopole Properties from a 2-Loop Correction to the Pressure
Summary and Outlook
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Summary and Outlook
I Coarsegrained YMT ground state described by scalar field φ(T ),
SU(2)φ→ U(1)
I |φ| constrains loop momentaI Calculated polarization tensor Σµν for TLM mode on 1-loop levelI Approximation p2 = 0 in constraint
I Constraint solvable analyticallyI Dispersion relation for transverse mode;
no propagating longitudinal mode
I Selfconsistent calculationI Constraint implemented numericallyI Dispersion relation for transverse- and longitudinal mode
I In YMT monopoles are unresolvable and screened
QuestionIs 1-loop calculation sufficient?
AnswerSee talk by Dariush!
Thank you.
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References I
U. Herbst, R. Hofmann,Asymptotic freedom and compositeness[hep-th/0411214].
R. Hofmann“Nonperturbative approach to Yang-Mills thermodynamics”Int. J. Mod. Phys. A20, 4123-4216 (2005) [hep-th/0504064].
F. Giacosa, R. Hofmann,“Thermal ground state in deconfining Yang-Millsthermodynamics”Prog. Theor. Phys. 118, 759-767 (2007) [hep-th/0609172].
J. Kapusta, G. Charles“Finite-temperature field theory : principles and applications”Cambridge Univ. Press, 2.ed., 2006.
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References II
M. Le Bellac“Thermal field theory”Cambridge Univ. Press, 1996.
M. Schwarz, R. Hofmann, F. Giacosa,“Radiative corrections to the pressure and the one-looppolarization tensor of massless modes in SU(2) Yang-Millsthermodynamics”Int. J. Mod. Phys. A22, 1213-1238 (2007) [hep-th/0603078].
J. Ludescher, R. Hofmann,“Thermal photon dispersion law and modified black-bodyspectra”Annalen Phys. 18, 271-280 (2009) [arXiv:0806.0972 [hep-th]].
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References III
D. Diakonov, N. Gromov, V. Petrov, S. Slizovskiy,“Quantum weights of dyons and of instantons with nontrivialholonomy”Phys. Rev. D70, 036003 (2004) [hep-th/0404042].
C. Falquez, R. Hofmann, T. Baumbach,“Charge-density waves in deconfining SU(2) Yang-Millsthermodynamics”[arXiv:1106.1353 [hep-th]].
J. Ludescher, J. Keller, F. Giacosa, R. Hofmann,“Spatial Wilson loop in continuum, deconfining SU(2)Yang-Mills thermodynamics”Annalen Phys. 19, 102-120 (2010) [arXiv:0812.1858 [hep-th]].