Hamiltonian Flow in Coulomb Gauge Yang-Mills theory

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Presentation transcript:

Hamiltonian Flow in Coulomb Gauge Yang-Mills theory H. Reinhardt Tübingen Collaborators: M. Leder, J. Pawlowski, A.Weber

Hamiltonian approach to YMT canonical quantization

Hamiltonian approach to YMT canonical quantization Coulomb gauge

Hamiltonian approach to YMT canonical quantization Coulomb gauge variational solution ansatz for vacuum wave functional DSE to calculate minimization of

Hamiltonian approach to YMT canonical quantization Coulomb gauge variational solution ansatz for vacuum wave functional DSE to calculate minimization of FRG flow equations

Hamiltonian approach to YMT canonical quantization Coulomb gauge variational solution ansatz for vacuum wave functional DSE to calculate minimization of FRG flow equations infinite tower of flow eqs. for static propagators indirect determination of truncation of flow equations Ansätze for propagators initial condition to the flow (UV-regime)

Variational approach trial ansatz : C.Feuchter & H. R. PRD70(2004) gluon propagator determined from variational kernel

D. Epple, H. Reinhardt, W.Schleifenbaum, PRD 75 (2007) Numerical results D. Epple, H. Reinhardt, W.Schleifenbaum, PRD 75 (2007) gluon energy

Static gluon propagator in D=3+1 G. Burgio, M.Quandt , H.R., PRL102(2009)

W. Schleifenbaum, M. Leder, H.R. PRD73(2006) running coupling W. Schleifenbaum, M. Leder, H.R. PRD73(2006)

Coulomb potential

D. Epple, H. Reinhardt, W.Schleifenbaum, PRD 75 (2007) ghost formfactor gost propagator ghost form factor D. Epple, H. Reinhardt, W.Schleifenbaum, PRD 75 (2007) Input:

ghost formfactor: lattice see talk by G. Burgio

D. Campagnari, H. R., A. Weber, Phys. Rev D(2009) Perturbation theory D. Campagnari, H. R., A. Weber, Phys. Rev D(2009) Rayleigh-Schrödinger PT vacuum (QED) ß-function

The color dielectric function of the QCD vacuum ghost propagator dielectric „constant“ H.Reinhardt,PRL101 (2008)

The color dielectric function of the QCD vacuum ghost propagator dielectric „constant“ horizon condition: : QCD vacuum-perfect color dia-electricum H.Reinhardt,PRL101 (2008)

The color dielectric function of the QCD vacuum ghost propagator dielectric „constant“ horizon condition: : QCD vacuum-perfect color dia-electricum QED: screening H.Reinhardt,PRL101 (2008)

no free color charges in the vacuum: confinement

Confinement scenarios Gribov-Zwanziger dual superconductor horizon condition perfect dia-electricum

Can we avoid the input of the horizon condition ? Yes we can : RG-flow equation: indirect test of our ansatz for the wave functional M. Leder, J. Pawlowski, H. R, A. Weber

Renormalization group flow equation

Renormalization group flow equation Wetterich

Effective action: 1PI-vertices

RG-flow equation Wetterich 1993

RG-flow equation propagator flow

Hamiltonian flow

Hamiltonian flow in Coulomb gauge YMT

RG- flow equation

Hamiltonian FRG flow equation no ansatz for indirect specification of truncation of flow equation form of the propagators&vertices assumed intial condition to propagators&vertices in the UV

Truncation of FRG flow equation gluon propagator ghost propagator ghost-gluon-vertex no tadpoles ghost dominance no gluon loops

RG- flow equation ghost dominance

Integrating the RG-flow equation

ghost form factor d(p)

gluon energy

FRG & DSE Replacement in loop integrals of FRG: FRG flow eq. DSE-variational approach

RG-Flow vs DSE: ghost form factor

RG-Flow vs DSE: gluon energy

RG-Flow vs DSE: running coupling

IR- exponents satisfy sum rule smaller than for DSE

Summary & Conclusion Hamiltonian FRG-flow equation of YMT in Coulomb gauge : Input: ghost dominance scaling of ghost in the IR output : horizon condition YM vacuum=perfect dual superconductor IR-exponents satisfy sum rule smaller than in variational approach outlook: Coulomb form factor inclusion of quarks

Summary & Conclusion Hamiltonian approach to YMT in Coulomb gauge variational solution of the YM Schrödinger eq. , input: gluon confinement quark confinement satisfactory agreement with lattice data Hamiltonian FRG-flow equation: horizon condition IR-exponents smaller than in variational approach dielectric function of the YM vacuum YM vacuum=perfect dual superconductor

Thanks for your attention

non-perturbative approaches to continuum Yang-Mills theory DSE FRG flow equations Variational Hamiltonian approch

Hamiltonian Flow in Coulomb Gauge Yang-Mills theory Introduction Hamilton approach to YMT FRG flow equation Numerical results Conclusions

Variational approach trial ansatz : C.Feuchter & H. R. PRD70(2004) gluon propagator determined from variational kernel