1. Heavy quark potential and running coupling in QCD W. Schleifenbaum Advisor: H. Reinhardt University of Tübingen EUROGRADworkshop Todtmoos 2007

2. W. Schleifenbaum Outline Some basics of Yang-Mills theory Some basics of Yang-Mills theory Functional Schroedinger equation Functional Schroedinger equation Coulomb gauge Dyson-Schwinger equations Coulomb gauge Dyson-Schwinger equations Quark potential & confinement Quark potential & confinement Running coupling in Landau and Coulomb gauge Running coupling in Landau and Coulomb gauge

3. Todtmoos 2007 W. Schleifenbaum Yang-Mills theory Local gauge invariance of quark fields: Lagrangian acquires gauge field through QCD: nonabelian gauge group SU(3) Yang-Mills Lagrangian: dynamics of gauge fields nonabelian term

4. Todtmoos 2007 W. Schleifenbaum asymptotic freedom: running coupling: dimensional transmutation: → express dimensionless g in terms of  nonperturbative methods: End of perturbative methods  lattice gauge theory  continuum approach via integral equations k „The hamiltonian method for strong interaction is dead [...]“

5. Todtmoos 2007 W. Schleifenbaum Good gauge? Need unique solution infinitesimally: Faddeev-Popov determinant Gauge fixing task: separate gauge d.o.f. QED:.... easy: YM theory:.... hard! alternative method: fix the gauge CONFIGURATION SPACE “I am not smarter, I just think more.” same physics Gribov copy IR physics

6. Todtmoos 2007 W. Schleifenbaum Coulomb gauge Hamiltonian Canonical quantization: Gauß‘ law constraint: Weyl gauge Hamiltonian: Coulomb gauge: curved orbit space → gluon confinement heavy quark potential → quark confinement gauge invariance

7. Todtmoos 2007 W. Schleifenbaum Yang-Mills Schroedinger equation: ansatz for vacuum wave functional: minimizing the energy: mixing of modes: enhanced UV modes might spoil accuracy of IR modes IR modes are enhanced as well! Variational principle [Feuchter & Reinhardt (2004)] „It‘s no damn good at all!“ ?

8. Todtmoos 2007 W. Schleifenbaum Gap equation Initially, only one equation needs to be solved: Ghost propagator: Ghost Dyson-Schwinger equation: Gap equation: (infrared expansion) Cf. Landau gauge – [Alkofer & von Smekal (2001)]

9. Todtmoos 2007 W. Schleifenbaum Tree-level ghost-gluon vertex Non-renormalization: Tree-level approximation: Check by DSE/lattice studies (Landau gauge): [W.S. et al. (2005)] [Cucchieri et al. (2004)] crucial for IR behavior! renormalization constant:

10. Todtmoos 2007 W. Schleifenbaum Infrared analysis Propagators in the IR Infrared expansion of loop integrals [ Zwanziger (2004); W.S. & Leder & Reinhardt ( 20 06 )] Two solutions :

11. Todtmoos 2007 W. Schleifenbaum IR sector is dominated by Faddeev-Popov determinant In a stochastic vacuum, we have the following expectation values, and find the same equations: Horizon condition: [Zwanziger (1991)] Ghost dominance

12. Todtmoos 2007 W. Schleifenbaum If the ghost-loop dominates the IR, it better be transverse. In d spatial dimensions, there are two solution branches: Infrared transversality Only obeys transversality! supports Coulomb gauge: d=3

13. Todtmoos 2007 W. Schleifenbaum Full numerical solution for  =1/2 Excellent agreement with infrared analysis Excellent agreement with infrared analysis (in)dependence on renormalization scale (in)dependence on renormalization scale Confinement of gluons Confinement of gluons [D. Epple, H. Reinhardt, W.S., PRD 75 (2007)]

14. Todtmoos 2007 W. Schleifenbaum Heavy quark potential Two pointlike color charges, separated by r Approximation: (cf. ghost-gluon vertex) Solution with  =1/2 gives Coulomb string tension [D. Epple, H. Reinhardt, W.S., PRD 75 (2007)]

15. Todtmoos 2007 W. Schleifenbaum Perturbative tails & tales 1. Landau gauge In the ultraviolet, QCD is asymptotically free. Free theory: Interacting theory: (from renormalization group) Anomalous dimensions: (scaled by   

16. Todtmoos 2007 W. Schleifenbaum running coupling: nonperturbative UV-asymptotics: ghost DSE: sum rule gives correct 1/log behaviour setting gives correct  and  ! ghost and gluon DSEs: sophisticated truncation of gluon DSE necessary to reproduce nonperturbative IR-asymptotics: finite depends on renormalization prescription [WS & Leder & Reinhardt (2006)] [Fischer & Alkofer (2002)] [Lerche & von Smekal (2002)]

17. Todtmoos 2007 W. Schleifenbaum 2. Coulomb gauge: perturbation theory still subject to ongoing research Free theory: Interacting theory: ( ansatz) running coupling solution to gap equation: [Watson & Reinhardt, arXiv:0709.0140v1]

18. Todtmoos 2007 W. Schleifenbaum numerical result: [Epple & Reinhardt & WS (2007)] set the only scale: → very sensitive to accuracy of  (k) should-be result: set in ghost DSE:

19. Todtmoos 2007 W. Schleifenbaum Coulomb potential: over-confinement Heavy quark potential involved simple replacement Only upper bound for Wilson loop potential (→lattice) Lattice calculations: too large by a factor of 2-3. No order parameter for „deconfinement“. MISSING: ’ s knowledge of the quarks. [Zwanziger (1997)] („No confinement without Coulomb confinement“)

20. Todtmoos 2007 W. Schleifenbaum Summary and outlook minimized energy with Gaussian wave functional minimized energy with Gaussian wave functional gluon confinement gluon confinement quark confinement quark confinement computed running coupling, finite in the IR computed running coupling, finite in the IR need for improvement in the UV need for improvement in the UV calculation of Coulomb string tension calculation of Coulomb string tension