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Heavy quark potential and running coupling in QCD

Heavy quark potential and running coupling in QCD. W. Schleifenbaum Advisor: H. Reinhardt University of Tübingen. EUROGRAD workshop Todtmoos 2007. Outline. Some basics of Yang-Mills theory Functional Schroedinger equation Coulomb gauge Dyson-Schwinger equations

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Heavy quark potential and running coupling in QCD

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  1. Heavy quark potentialand running couplingin QCD W. Schleifenbaum Advisor: H. Reinhardt University of Tübingen EUROGRAD workshop Todtmoos 2007

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

  3. nonabelian term 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 W. Schleifenbaum

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

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

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

  7. Variational principle 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! [Feuchter & Reinhardt (2004)] „It‘s no damn good at all!“ ? W. Schleifenbaum

  8. 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)] W. Schleifenbaum

  9. renormalization constant: 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! W. Schleifenbaum

  10. Two solutions : [Zwanziger (2004); W.S. & Leder & Reinhardt (2006)] Infrared analysis Propagators in the IR Infrared expansion of loop integrals W. Schleifenbaum

  11. Ghost dominance 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)] W. Schleifenbaum

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

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

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

  15. Perturbative tails & tales 1. Landau gauge In the ultraviolet, QCD is asymptotically free. Free theory: Interacting theory: (from renormalization group) Anomalous dimensions: (scaled by b0) W. Schleifenbaum

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

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

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

  19. MISSING: ’s knowledge of the quarks. 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“. [Zwanziger (1997)] („No confinement without Coulomb confinement“) W. Schleifenbaum

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

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