1. Markus Quandt Quark Confinement and the Hadron Spectrum St. Petersburg September 9,2014 M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 1 A Covariant Variational Approach to Yang-Mills Theory Institut für Theoretische Physik Eberhard-Karls-Universität Tübingen

2. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 2 Overview 1.Variational Principle for Effective Action 2.Gaussian Ansatz & Gap Equation 3.Renormalization 4.Numerical Results 5.Extension to finite Temperature 6.Conclusion and Outlook A Covariant Variation Principle Overview

3. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 3 Effective Action Principle

4. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 4 Effective Action Principle A Covariant Variation Principle Effective Action probability measure Variation principle for functional probability measure Entropy = available space for quantum fluctuations Free action

5. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 5 Variation principle I Variation principle II Gibbs measure Schwinger functions (Quantum effective action) Note: Usually and proper functions A Covariant Variation Principle Effective Action

6. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 6 1.Take (restricted) measure space with parameters K gap equation 2.Minimize [gap equation] 3.Quantum effective action 4.Proper functions = derivatives of Modification for gf. Yang-Mills Theory A Covariant Variation PrincipleA Covariant variational principle Effective Action relative entropy replace entropy by relative entropy (available quantum fluctuations in curved field space)

7. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 7 - non-perturbative corrections to arbitrary proper functions - closed system of integral equations - renormalization through standard counter terms - can discriminate competing solutions - can be optimized to many physical questions (choice of ) Comparision A Covariant Variation PrincipleA covariant variational principle Effective Action Pro Cons - vertex corrections not necessarily reliable - Confinement ? - Strong phase transitions ? Coulomb gauge - BRST ? [similar to Coulomb gauge]

8. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 8 Gaussian Trial Measure

9. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 9 A Covariant Variation Principle Gaussian Trial Measure Physical picture in Landau gauge Gaussian trial measure for effective (constitutent) gluon - variational parameters - Lorentz structure in Landau gauge: - UV : gluons weakly interacting - IR : configurations near Gribov horizon dominant self-interaction in such configs sub-dominant

10. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 10 Curvature Approximation A Covariant Variation Principle Gaussian Trial Measure curvature trial measure curvatureentropy curvature gone, reappears in entropy (natural)

11. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 11 Remark A Covariant Variation Principle Gaussian Trial Measure put classical field if interested in propagator alternative: effective action for gluon propagator gap equation gives best gluon propagator that is available within the ansatz

12. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 12 Free action A Covariant Variation Principle Overview classical action: Wick‘s theorem relative entropy involves lengthy expression

13. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 13 Gap Equation A Covariant Variation Principle Overview Use O(4) symmetry to perform angular integrations constituent gluon mass (4-gluon vertex gap eq.) constutent gluon self energy (ghost contribution curvature) graph + + =

14. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 14 Ghost sector A Covariant Variation Principle Overview + = Use resolvent identity on FP operator in terms of ghost form factor rainbow approx.

15. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 15 Curvature Equation A Covariant Variation Principle Overview To given loop order =

16. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 16 Renormalization

17. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 17 Counterterms A Covariant Variation Principle Renormalization gluon field gluon mass ghost field not Added to exponent of trial measure, not Renormalization conditions Renormalization conditions (3 scales ) fix scaling/decoupling constitutent mass at

18. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 18 A Covariant Variation Principle Renormalization All countertems have definite values, e.g. Final system curvature: quadratic + subleading log. divergence subtracted ! Important for finite-T calculation similarly

19. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 19 Numerical Results

20. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 20 Gluon prop. IR diverging, constituent mass decoupling solution: Gluon prop. IR finite, constituent mass dominant Gluon prop IR vanishing, constituent mass irrelevant Infrared Analysis A Covariant Variation Principle Numerical Results Assume power law in the deep IR IR exponents Non-renormalization of ghost-gluon vertex scaling solution:  

21. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 21 Scaling Solution A Covariant Variation Principle Numerical Results Lattice data from Bogolubsky et al., Phys. Lett. B676 69 (2009) IR exponents: sum rule violation: lattice data:  H. Reinhardt, J. Heffner, M.Q., Phys. Rev. D89 065037 (2014)

22. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 22 Decoupling Solution A Covariant Variation Principle Numerical Results Lattice data from Bogolubsky et al., Phys. Lett. B676 69 (2009) lattice data: sum rule:  H. Reinhardt, J. Heffner, M.Q., Phys. Rev. D89 065037 (2014)

23. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 23 Finite Temperature

24. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 24 Extension to Finite Temperature A Covariant Variation Principle 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

25. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 25 A Covariant Variation PrincipleA covariant variational principle Finite Temperature Lorentz structure of propagator heat bath singles out restframe (1,0,0,0) breaks Lorentz invariance two different 4-transversal projectors Same Lorentz structure for Gaussian kernel and curvature 3-transversal 3-longitudinal

26. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 26 Gap Equations A Covariant Variation Principle Finite Temperature induced gluon masses now temperature-dependent must be finite (no counter term)

27. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 27 Renormalization A Covariant Variation Principle Finite Temperature counterterms must be fixed at T=0 example: ghost equation counter term finite T correction vanishes at T=0 Ren. T=0 profile

28. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 28 Renormalized System at T>0 A Covariant Variation Principle Overview All finite temperature corrections have similar structure Iterative solution

29. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 29 Preliminary Results A Covariant Variation Principle Overview

30. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 30 Conclusion

31. M. Quandt (Uni Tübingen) A Covariant Variation Principle Confinement XI 31 A Covariant Variation Principle Overview Variational Principle for Effective Action + Gaussian Ansatz - yields optimal truncation - conventionally renormalizable - very good agreement with lattice data - Easily extensible to finite temperatures - BRST? Confinement? Conclusion and Outlook Best applied where effective 1-gluon exchange is relevant - inclusion of fermions - chemical potential