A Covariant Variational Approach to Yang-Mills Theory

1. A Covariant Variational Approach to Yang-Mills Theory Institut für Theoretische Physik Eberhard-Karls-Universität Tübingen A Covariant Variation Principle

2. A Covariant Variation Principle Overview Overview • Variational Principle for Effective Action • Gaussian Ansatz & Gap Equation • Renormalization • Numerical Results • Extension to finite Temperature • Conclusion and Outlook A Covariant Variation Principle

3. Effective Action Principle A Covariant Variation Principle

4. A Covariant Variation Principle Effective Action Effective Action Principle • Variation principle for functional probabilitymeasure • Entropy = available space for quantum fluctuations • Free action A Covariant Variation Principle

5. A Covariant Variation Principle Effective Action • Variation principle I Gibbs measure Schwinger functions • Variation principle II (Quantum effective action) Note: Usually and proper functions A Covariant Variation Principle

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

7. A Covariant Variation PrincipleA covariant variational principle Effective Action Comparision • Pro • 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 ) • Cons • vertex corrections not necessarily reliable • Confinement ? • Strong phase transitions ? • BRST ? [similar to Coulomb gauge] A Covariant Variation Principle

8. Gaussian Trial Measure A Covariant Variation Principle

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

10. A Covariant Variation Principle Gaussian Trial Measure Curvature Approximation • trial measure curvature curvature gone, reappears in entropy (natural) A Covariant Variation Principle

11. A Covariant Variation Principle Gaussian Trial Measure Remark • 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 A Covariant Variation Principle

12. A Covariant Variation Principle Overview Free action • classical action: Wick‘s theorem involves lengthy expression • relative entropy A Covariant Variation Principle

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

14. A Covariant Variation Principle Overview Ghost sector Use resolvent identity on FP operator in terms of ghost form factor -1 -1 = + rainbow approx. A Covariant Variation Principle

15. A Covariant Variation Principle Overview Curvature Equation To given loop order = A Covariant Variation Principle

16. Renormalization A Covariant Variation Principle

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

18. A Covariant Variation Principle Renormalization Important for finite-T calculation • All countertems have definite values, e.g. similarly • Final system curvature: quadratic + subleading log. divergence subtracted ! A Covariant Variation Principle

19. Numerical Results A Covariant Variation Principle

20. A Covariant Variation Principle Numerical Results Infrared Analysis Assume power law in the deep IR IR exponents Non-renormalization of ghost-gluon vertex û ü • scaling solution: Gluon prop IR vanishing, constituent mass irrelevant • decoupling solution: ü Gluon prop. IR finite, constituent mass dominant û Gluon prop. IR diverging, constituent mass A Covariant Variation Principle

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

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

23. Finite Temperature A Covariant Variation Principle

24. A Covariant Variation Principle Finite Temperature Extension to Finite Temperature • imaginary time formalism compactify euclidean time periodic b.c. for gluons (up to center twists) periodic b.c. forghosts (even though fermions) Extension to T>0 straightforward A Covariant Variation Principle

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

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

27. A Covariant Variation Principle Finite Temperature Renormalization • counterterms must be fixed atT=0 counter term • example: ghost equation finite T correction Ren. T=0 profile finite T correction vanishes at T=0 A Covariant Variation Principle

28. A Covariant Variation Principle Overview Renormalized System at T>0 • All finite temperature corrections have similar structure • Iterative solution A Covariant Variation Principle

29. A Covariant Variation Principle Overview Preliminary Results A Covariant Variation Principle

30. Conclusion A Covariant Variation Principle

31. A Covariant Variation Principle Overview Conclusion and Outlook • 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? • Best applied where effective 1-gluon exchange is relevant • inclusion of fermions • chemical potential A Covariant Variation Principle