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Understanding Heavy Quark Potential and Running Coupling in Quantum Chromodynamics

This paper explores the fundamental aspects of Yang-Mills theory, focusing on the heavy quark potential and running coupling within Quantum Chromodynamics (QCD). It discusses the functional Schrödinger equation, the role of Coulomb gauge, and the challenges of non-abelian gauge invariance. Key topics include confinement of quarks and gluons, dimensional transmutation, and the impact of ghost and gluon Dyson-Schwinger equations. The analysis addresses both perturbative and non-perturbative methods in QCD, with an emphasis on lattice gauge theories and continuum approaches.

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Understanding Heavy Quark Potential and Running Coupling in Quantum Chromodynamics

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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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