Exact renormalisation group at finite temperature
Transcript of Exact renormalisation group at finite temperature
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Exact renormalisation groupat finite temperature
Jean-Paul Blaizot, IPhT- Saclay
“Quarks, Hadrons, and the Phase Diagram of QCD”St Goar
2 Sept. 2009
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Outline
- weak coupling techniques and why they are useful to understand hot QCD
- insights from the exact Renormalization Group
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Hot QCD
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(from M. Bazavov et al, arXiv:0903.4379)
Thermodynamics of hot QCD
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Pressure for SU(3) YM theory at (very) high temperature
(from G. Endrodi et al, arXiv: 0710.4197)
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(SU(3) lattice gauge calculation from Karsch et al, hep-lat/0106019)
At T>3Tc Resummed Pert. Theory accounts for lattice results
Resummed Pert. Th.SB
Pressure
(resummed pert. th. from J.-P. B., E. Iancu, A. Rebhan: Nucl.Phys.A698:404-407,2002)
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Weak coupling techniques(non perturbative)
Effective theories- Dimensional reduction
Skeleton expansion (2PI formalism)
-not limited to perturbation theory; in fact weak coupling techniques can be used to study non perturbative phenomena(many degenerate degrees of freedom, strong fields)
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Perturbation theory is ill behaved
A two naive conclusion:
« weak coupling techniques are useless »
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Similar difficulty in scalar theory
The bad convergence of Pert. Th. is not related to non abelian features of QCD
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Effective theroyDimensional reduction
Integration over the hard modes
�
(gT ≤ ΛE ≤ T)
�
Di = ∂i − igEAi
�
gE ≈ g T
�
mE ≈ gT
�
λE ≈ g4TIn leading order
Non perturbative contribution
Integration over the soft modes
�
(g2T ≤ ΛM ≤ gT)
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Skeleton expansionPhi-derivable approximations
2PI formalism.......
• J. M. Luttinger and J. C. Ward, PR 118, 1417 (1960)• G. Baym, PR127,1391(1962)• J. M. Cornwall, R. Jackiw, and E. Tomboulis, PR D10, 2428 (1974)
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Pressure in terms of dressed propagators
Stationarity property
�
δPδG
= 0Entropy is simple!
�
S = dPdT
= dPdT G
�
P[G]
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THE TWO-LOOP ENTROPY
-effectively a one-loop expression; residual interactions start at order three-loop
-manifestly ultraviolet finite
-perturbatively correct up to order
�
g3
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State of the art
• J.-P. B., E. Iancu, A. Rebhan: Phys.Rev.D63:065003,2001• G. Boyd et al., Nucl. Phys. B469, 419 (1996).
from J.-P. B., E. Iancu, A. Rebhan: Nucl.Phys.A698:404-407,2002
pure-glue SU(3) Yang-Mills theory
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Large Nf
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Moore, JHEP 0210A.Ipp, Moore, Rebhan, JHEP 0301
Pressure at Large Nf
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Entropy in large Nf
Good agreement, even at large coupling
J.-P. B., A.Ipp, A. Rebhan, U. Reinosa, hep-ph/0509052
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Strong couplingComparison with SSYM
Hard-thermal-loop entropy of supersymmetric Yang-Mills theoriesJ.-P. B, E. Iancu, U. Kraemmer and A. Rebhan, hep-ph/0611393
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Entropy formula
Leading order correction
�
SS0
= 34
+ 4532
ς(3) 1λ3 / 2
At strong coupling (AdS/CFT)
�
λ ≡ g2Nc( )
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Insights from the functionalrenormalization group
(scalar theory)
J.-P. B, A. Ipp, R. Mendez-Galain, N. Wschebor (Nucl.Phys.A784:376-406,2007)
and work in progress
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Scales, fluctuations and degrees of freedom in the quark-gluon plasma
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Courtesy, A. Ipp
Non perturbative renormalization group
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Non perturbative renormalization group
(C. Wetterich 1993)
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Local potential approximation
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Beyond the local potential approximation
(J.-P. B, R. Mendez-Galain, N. Wschebor, Phys.Lett.B632:571-578,2006)
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(from F. Benitez, J.-P. Blaizot, H. Chate, B. Delamotte, R. Mendez-Galain, N. Wschebor, arXiv:0901.0128 [cond-mat.stat-mech])
Application to critical O(N) model
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29
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weak couplingstrong coupling
Flow of coupling constant(LPA)
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weak coupling strong coupling
Flow of pressure(LPA)
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Pressure versus m/T
2PI
LPA (Litim)
LPA (Exp.)
g^2
g^4g^3
2PI
LPA (Litim)
BMW (Exp.)
g^2
g^4g^3
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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.94
0.96
0.98
1.00
1.02
g^6
g^5
g^4
g^3
(g^6 from A. Gynther, et al ,hep-ph/0703307)
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Summary
•While strict perturbation theory is meaningless, weak coupling methods (involving resummations) provide an accurate description of the QGP for
• The functional renormalization group confirms results obtained with various resummations (at least in scalar case)