1. Institut für Theoretische Physik Eberhard-Karls-Universität Tübingen
Finite Temperature and the Polyakov Loop in the Covariant Variational Approach to Yang-Mills Theory
Institut für Theoretische Physik
Eberhard-Karls-Universität Tübingen
M. Quandt (Uni Tübingen)

2. Overview Variational principle for the effective action in YM theory
Propagators at T=0
Propagators at T > 0
Effective potential of the Polyakov loop
Summary and Outlook
M. Quandt (Uni Tübingen)

3. The Variation Principle
M. Quandt (Uni Tübingen)

4. Effective Action Principle
Variation principle for functional probability measure
Free action
euclidean action
entropy
Variation principle I
Gibbs measure
Schwinger functions
M. Quandt (Uni Tübingen)

5. Variation principle II
Effective Action Principle
Variation principle II
(Quantum effective action)
Note: Usually
and proper functions
Variation principle III
(Yang-Mills-Theory)
FP determinant
relative entropy
M. Quandt (Uni Tübingen)

6. Ansatz for trial measure:
Effective Action Principle
Ansatz for trial measure:
Determines kernels in
2. Minimal value is effective action (1PI)
M. Quandt (Uni Tübingen)

7. A Covariant Variation Principle
Effective Action Principle
Integral equation system like DSE or FRG
simple renormalization through counter terms
can discriminate competing solutions (lowest action wins)
easy access to thermodynamics
Strategy
study system with relatively few kernels in ansatz
need to optimize ansatz
Systematic improvement possible using DSE
Optimization of DSE truncation
M. Quandt (Uni Tübingen)
A Covariant Variation Principle

8. Gaussian Trial Measure
M. Quandt (Uni Tübingen)

9. Curvature Approximation
Gaussian Trial Measure
Gaussian ansatz
UV : gluons weakly interacting
IR : configurations near Gribov horizon dominant
self-interaction in such configs sub-dominant
Curvature Approximation
curvature
M. Quandt (Uni Tübingen)

10. Free action Gap Equation evaluation of : only Wick‘s theorem = + + -1
Gaussian Ansatz
Free action
evaluation of : only Wick‘s theorem
Gap Equation -1 -1 = + +
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11. Ghost sector Use resolvent identity on FP operator
Gaussian Ansatz
Ghost sector
Use resolvent identity on FP operator
in terms of ghost form factor -1 -1 = +
rainbow approx.
M. Quandt (Uni Tübingen)

12. Curvature Equation To given loop order = M. Quandt (Uni Tübingen)
Gaussian Ansatz
Curvature Equation
To given loop order =
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13. Counterterms Renormalization conditions (3 scales ) fix gluon field
ghost field
gluon mass
Renormalization conditions (3 scales )
fix
scaling/decoupling
constitutent mass at
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14. Propagators at T=0
M. Quandt (Uni Tübingen)

15. Scaling Solution IR exponents: sum rule violation:
Propagators at T=0
Scaling Solution
MQ, H. Reinhardt, J. Heffner, Phys. Rev. D (2014)
Lattice data from Bogolubsky et al., Phys. Lett. B (2009)
IR exponents:
sum rule violation:
M. Quandt (Uni Tübingen)

16. Decoupling Solution gluon propagator ghost form factor
Propagators at T=0
Decoupling Solution
MQ, H. Reinhardt, J. Heffner, Phys. Rev. D (2014)
Lattice data from Bogolubsky et al., Phys. Lett. B (2009)
gluon propagator
ghost form factor
M. Quandt (Uni Tübingen)

17. A Covariant Variation Principle
Finite Temperature
M. Quandt (Uni Tübingen)
A Covariant Variation Principle

18. Extension to Finite Temperature
imaginary time formalism
compactify euclidean time
periodic b.c. for gluons (up to center twists)
periodic b.c. for ghosts (even though fermions)
Extension to T>0 straightforward
M. Quandt (Uni Tübingen)

19. Lorentz structure of propagator
Finite Temperature
Lorentz structure of propagator
heat bath singles out restframe (1,0,0,0) breaks Lorentz invariance
two different 4-transversal projectors
3-transversal
3-longitudinal
Two Lorentz structures for kernel and curvature
M. Quandt (Uni Tübingen)

20. Gap Equations induced gluon masses now temperature-dependent
Finite Temperature
Gap Equations
induced gluon masses now temperature-dependent
renormalization by T=0 counter terms
M. Quandt (Uni Tübingen)

21. longitudinal gluon transversal gluon ghost formfactor
Finite Temperature
MQ, H. Reinhardt, Phys. Rev. D (2015)
longitudinal gluon
transversal gluon
ghost formfactor
M. Quandt (Uni Tübingen)

22. Polyakov Loop
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23. Interpretation: free static quark energy
Polyakov Loop
Polyakov loop
Interpretation: free static quark energy
Center symmetry
maps
If unbroken
[confinement]
If broken
[deconfinement]
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24. G=SU(2) Polyakov gauge [ ] Background gauge [ ]
Polyakov Loop
Alternative order parameter
G=SU(2) Polyakov gauge [ ] Background gauge [ ]
Background gauge
Transfer Landau -- Background
replace
in basis where rhs is diagonal
replace
are the simple roots
replace
sum over simple roots
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25. Effective potential of background field (Polyakov Loop)
Weiß potential
Similarly for G=SU(3)
where
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26. Phase transition for G=SU(2)
Polyakov Loop
Phase transition for G=SU(2)
MQ, H. Reinhardt, Phys. Rev. D, in press (2016)
2nd order transition
critical temperature
Eff. Potential for Polyakov loop
Lattice
Lucini, Teper, Wenger, JHEP 01 (2004) 061
M. Quandt (Uni Tübingen)

27. Phase transition for G=SU(3)
Polyakov Loop
Phase transition for G=SU(3)
MQ, H. Reinhardt, Phys. Rev. D, in press (2016)
slice of eff. Potential for
Polyakov loop
1st order transition
critical temperature
Lattice
Lucini, Teper, Wenger, JHEP 01 (2004) 061
M. Quandt (Uni Tübingen)

28. Effective potential for Polakov loop in G=SU(3)
Polyakov Loop
Effective potential for Polakov loop in G=SU(3)
MQ, H. Reinhardt, Phys. Rev. D, in press (2016)
Deconfined phase
Confined phase
V(x,y) maximal at center symmetrc points
V(x,y) minimal at center symmetrc points
M. Quandt (Uni Tübingen)

29. Effective potential for Polakov loop in G=SU(3)
Polyakov Loop
Effective potential for Polakov loop in G=SU(3)
MQ, H. Reinhardt, Phys. Rev. D, in press (2016)
Deconfined phase
Confined phase
V(x,y) maximal at center symmetrc points
V(x,y) minimal at center symmetrc points
M. Quandt (Uni Tübingen)

30. Sumary and Outlook
M. Quandt (Uni Tübingen)

31. Conclusions
Summary
Variational Principle for Effective Action + Gaussian Ansatz
Propagators at T=0 (fix all renorm. constants)
Propagators at T > 0
Polyakov loop and deconfinement, ghost dominance
Outlook
Simple access to thermodynamics
DSE
Inclusion of fermions
M. Quandt (Uni Tübingen)