Linear heavy quark potential in Coulomb gauge

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Linear heavy quark potentialin Coulomb gauge

• W. Schleifenbaum
• Advisor H. Reinhardt
• University of Tübingen, Germany
• Trento 2007

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Outline

• Hamiltonian approach to Yang-Mills theory
• Vacuum wave functional
• Dyson-Schwinger equations
• Infrared analysis
• Linear confining heavy quark potential
• Coherent superposition of gluon excitations
• Lowering of the Coulomb gauge string tension

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Hamiltonian approach to Coulomb gauge Yang-Mills
theory

• Yang-Mills Hamiltonian
• Gauß law constraint
• Coulomb gauge
• Gauge-fixed Hamiltonian
• Faddeev-Popov operator

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Vacuum wave functional

• Yang-Mills Schrödinger equation
• Ansatz for wave functional
• 
  variational kernel w(p)
• Variational principle
• Non-perturbative calculation, e.g.

C. Feuchter, H. Reinhardt, PRD 70 (2004)
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Dyson-Schwinger equations

• Ghost equation
• Minimizing the energy yields gap equation (for p
  0)
• Ghost-gluon vertex approx. bare

Cf. Landau gauge R. Alkofer, L. von Smekal,
Phys. Rep. 353 (2001)
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Infrared analysis

• Power law ansätze for propagators in the IR
• Infrared expansion of loop integrals

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Full numerical solution for k1/2

• Excellent agreement with infrared analysis
• (in)dependence on renormalization scale
• Running couling with infrared fixed point
• Confinement of gluons

D. Epple, H. Reinhardt, W.S., PRD 75 (2007)
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Logarithms

• IR power laws unique?
• Further dressing of the IR power laws
• IR analysis
• If power laws are solution, then are
  also solutions,
• with unchanged!

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Heavy quark potential

• Two pointlike color charges, separated by r
• Solution with k1/2 gives
• However
• is gauge dependent!
• Gauge invariant potential

D. Epple, H. Reinhardt, W.S., PRD 75 (2007)
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Im not quite sure about this
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Response of vacuum to external charges

• Presented vacuum solution without quarks
• Now include heavy quarks
• and construct more general state
• with a variational field

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Gluon excitations

• Recall that vacuum wave functional is Gaussian
• Quasi-particle states
• Coherent state nonperturbative

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Coherent states

• Nobel prize 2005 (Glauber, Hall, Häntsch)
• Eigenstates of annihilation operator
• Over-complete basis
• Minimal uncertainty
• Poissonian distribution

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Calculation of the energy

• Property of coherent states
• Shift theorem
• (VEVs with
  background field)
• Energy expression to be minimized w.r.t. Z
• Functionals quadratic in ImZ
• Non-polynomial in ReZ, due to Faddeev-Popov terms
  

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Asymptotic estimate (preliminary)

• Faddeev-Popov operator with background field Q
• Infrared limit
• What is in the infrared ?

Gluon chain cf. J. Greensite, C. Thorn, JHEP
0202
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Chain link L(p)

• Assume
• For p 0 we get
• If a1, all terms contribute,

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• For SU(2)
• Lowering of the Coulomb string tension
• Saturation mechanism?
• WORK IN PROGRESS

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Summary

• Variational vacuum solution found
• IR analysis gives gluon and heavy quark
  confinement
• Logarithms in the IR?
• Gluonic excitations along the flux tube
• Lowering of Coulomb string tension

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IR fixed point of running coupling
Define RG invariant Analytically determined
fixed point
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Saturation?