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)

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)

The Variation Principle M. Quandt (Uni Tübingen)

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)

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)

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)

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

Gaussian Trial Measure M. Quandt (Uni Tübingen)

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)

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 = + + M. Quandt (Uni Tübingen)

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)

Curvature Equation To given loop order = M. Quandt (Uni Tübingen) Gaussian Ansatz Curvature Equation To given loop order = M. Quandt (Uni Tübingen)

Counterterms Renormalization conditions (3 scales ) fix gluon field ghost field gluon mass Renormalization conditions (3 scales ) fix scaling/decoupling constitutent mass at M. Quandt (Uni Tübingen)

Propagators at T=0 M. Quandt (Uni Tübingen)

Scaling Solution IR exponents: sum rule violation: Propagators at T=0 Scaling Solution MQ, H. Reinhardt, J. Heffner, Phys. Rev. D89 035037 (2014) Lattice data from Bogolubsky et al., Phys. Lett. B676 69 (2009) IR exponents: sum rule violation: M. Quandt (Uni Tübingen)

Decoupling Solution gluon propagator ghost form factor Propagators at T=0 Decoupling Solution MQ, H. Reinhardt, J. Heffner, Phys. Rev. D89 035037 (2014) Lattice data from Bogolubsky et al., Phys. Lett. B676 69 (2009) gluon propagator ghost form factor M. Quandt (Uni Tübingen)

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

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)

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)

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)

longitudinal gluon transversal gluon ghost formfactor Finite Temperature MQ, H. Reinhardt, Phys. Rev. D92 025051 (2015) longitudinal gluon transversal gluon ghost formfactor M. Quandt (Uni Tübingen)

Polyakov Loop M. Quandt (Uni Tübingen)

Interpretation: free static quark energy Polyakov Loop Polyakov loop Interpretation: free static quark energy Center symmetry maps If unbroken [confinement] If broken [deconfinement] M. Quandt (Uni Tübingen)

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 M. Quandt (Uni Tübingen)

Effective potential of background field (Polyakov Loop) Weiß potential Similarly for G=SU(3) where M. Quandt (Uni Tübingen)

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)

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)

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)

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)

Sumary and Outlook M. Quandt (Uni Tübingen)

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)