flow equations with momentum dependent vertex functions
TRANSCRIPT
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Jean-Paul Blaizot, IPhT CEA-CNRS
Flow equations with momentum dependent vertex functions
Collaborators: - R. Mendez-Galain, N. Wschebor - F. Benitez, H. Chate, B. Delamotte- A. Ipp- J. Pawlowski, U. Reinosa
ERG 2010Corfou Summer Institute
Corfou, Sept. 14, 2010
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Outline
- Truncation of the eRG flow equations (momentum dependent vertices)
- Application to perturbation theory at finite temperature
- ERG and 2PI relations
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The “exact” Renormalization Group
Basic strategy (ex scalar field theory)
Control infrared with “mass-like” regulator
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Exact flow equation
Theory at “scale” defined by the regulated action
or the “effective action”
Exact flow equation (Wetterich, 1993)
‘Initial conditions’
runs from ‘microscopic scale’ to zero (regulator vanishes)
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An infinite hierarchy of equations for n-point functions
Effective potential
2-point function
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Two observations
The vertex functions depend weakly on the loop momentum
The hierarchy can be closed by exploiting the dependence on the field
Beyond the local potential approximationJ.-P. B, R. Mendez-Galain, N. Wschebor (PLB, 2006)
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Structure of truncation
(LPA)
(‘BMW’-LO)
etc.
The equation for the 2-point function becomes a closed equation
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Applications
- Critical O(N) models (see B. Delamotte’s lecture)
- Bose-Einstein condensation
- Finite temperature field theory
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Non perturbative renormalization groupat finite temperature( J.-P B, A. Ipp, N. Wschebor, 2010)
Motivation : physics of the quark-gluon plasma
Naively: QCD asymptotic freedom implies that matter is simple at high temperature (weakly interacting gas of quarks and gluons)
Experiments (heavy ion collisions at RHIC) suggest that the quark-gluon plasma is ‘strongly coupled’
Technically: perturbation theory breaks down
A paradoxical situation :
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Perturbation theory is ill behaved at finite temperature
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Generic feature in (most) field theories, e.g. in scalar field theory
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Weakly AND strongly coupled …
Degrees of freedom with different wavelengths are differently coupled.
Expansion parameter
Dynamical scales
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Non perturbative renormalization groupat finite temperature
( J.-P B, A. Ipp, N. Wschebor, 2010 )
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weak coupling strong coupling
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weak coupling strong coupling
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RG techniques yield smooth extrapolation to strong coupling(scalar field theory)
(JPB, A. Ipp, N. Wschebor, 2010)
(high orders from J. O. Andersen et al, arXiv 0903.4596)
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Removing scheme dependence
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eRG and 2PI techniques
Gap equation
Phi-derivable approximation : choose a set of skeleton diagrams and solve the corresponding gap equation
( J.-P B, J. Pawlowski and U. Reinosa - see talk by Reinosa)
Luttinger-Ward expression for the thermodynamic potential
Equation for the four-point function
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eRG and 2PI truncation
The resulting flow is an exact derivative
Eq. for the 2-point function
Truncate with 2PI relation
Consequences : flow eq. ‘solves’ the gap equation
no residual dependence on ‘regulator’
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Conclusions
eRG is a nice toolwhy does it work so well ?