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The QCD vacuum wave-functional and confinement in Coulomb gauge

This study delves into the concept of confinement in quantized non-abelian gauge theories, especially focusing on the QCD vacuum wave-functional in the Coulomb gauge. It explores the relationship between the vacuum wave-functional and confinement, offering an approximate form that shows promise in capturing the true ground state of the theory. Various Coulomb-gauge observables and their relation to the proposed wave-functional are discussed, highlighting the potential in replicating values obtained through standard lattice Monte Carlo techniques. The study also hints at future directions for research in this area, including exploring the wave-functional in numerical simulations and expanding into higher dimensions.

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The QCD vacuum wave-functional and confinement in Coulomb gauge

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  1. The QCD vacuum wave-functional and confinement in Coulomb gauge Jeff Greensite San Francisco State Univ. Štefan Olejník Institute of Physics, Slovak Acad. Sci. Bratislava, Slovakia TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA

  2. (Approximate) QCD vacuum wave-functional • Confinement is the property of the vacuum of quantized non-abelian gauge theories. In the hamiltonian formulation in D=d+1 dimensions and temporal gauge: Lattice 2010, Villasimius, Italy

  3. At large distance scales one expects: • Halpern (1979), Greensite (1979) • Greensite, Iwasaki (1989) • Kawamura, Maeda, Sakamoto (1997) • Karabali, Kim, Nair (1998) • Property of dimensional reduction: Computation of a spacelike loop in d+1 dimensions reduces to the calculation of a Wilson loop in Yang-Mills theory in d Euclidean dimensions. • The true vacuum wave-functional (VWF) cannot just be of the dimensional-reduction form - incorrect results at short distances/high frequencies. Lattice 2010, Villasimius, Italy

  4. Suggestion for an approximate vacuum wavefunctional • Greensite, ŠO, arXiv:0707.2860 [hep-lat]; Greensite, talk at Lattice 2007 Lattice 2010, Villasimius, Italy

  5. Arguments in favor of the proposed VWF In the free-field limit (g  0), Ψ0[A] becomes the well-known VWF of electrodynamics. The proposed form is a good approximation to the true vacuum also for strong fields constant in space and varying only in time. If we divide the magnetic field strength B(x) into “fast” and “slow” components, the part of the VWF that depends on Bslow takes on the dimensional-reduction form. The fundamental string tension is then easily computed as ¾F= 3m/4¯. If one takes the mass m in the wave-functional as a free variational parameter and computes (approximately) the expectation value of the YM hamiltonian, one finds that a non-zero (finite) value of m is energetically preferred. Lattice 2010, Villasimius, Italy

  6. Lattice evidence in favor of the proposed VWF “Recursion” lattices: ensemble of independent 2-d lattice configurations generated with the probability distribution given by the proposed VWF, with m fixed at given ¯ to get the correct value of the fundamental string tension. Monte Carlo lattices: ensemble of 2-d slices of configurations generated by MC simulations of 3-d euclidean SU(2) LGT with standard Wilson action; from each configuration, only one (random) slice at fixed euclidean time is taken. Lattice 2010, Villasimius, Italy

  7. Mass gap Lattice 2010, Villasimius, Italy

  8. Coulomb-gauge quantities • Why Coulomb gauge? • Low-lying spectrum of the Faddeev–Popov operator in Coulomb gauge probes properties of nonabelian gauge fields that are crucial for the confinement mechanism. • The ghost propagator in Coulomb gauge and the color-Coulomb potential are directly related to the inverse of the Faddeev–Popov operator, and play a role in various confinement scenarios. • In particular, the color-Coulomb potential represents an upper bound on the physical potential between a static quark and antiquark, which means that a confining color-Coulomb potential is a necessary condition to have a confining static quark potential. • Our aim was to see how well the proposed VWF can reproduce the values of Coulomb-gauge observables that can be obtained by standard lattice MC techniques. Lattice 2010, Villasimius, Italy

  9. From temporal to Coulomb gauge Classical Coulomb-gauge hamiltonian: Coulomb kernel: Lattice 2010, Villasimius, Italy

  10. From temporal to Coulomb gauge Lattice 2010, Villasimius, Italy • In the operator formalism, the minimal Coulomb gauge is a gauge fixing within the temporal gauge of the remnant local gauge invariance. The wave-functional in Coulomb gauge is the restriction of the WF in temporal gauge to transverse fields in FMR. • Greensite, ŠO, Zwanziger (2004)

  11. Coulomb-gauge ghost propagator Lattice 2010, Villasimius, Italy

  12. Color-Coulomb potential Lattice 2010, Villasimius, Italy

  13. Lattice 2010, Villasimius, Italy

  14. Lattice 2010, Villasimius, Italy

  15. Lattice 2010, Villasimius, Italy

  16. Conclusions • The proposed vacuum wave-functional for the temporal-gauge SU(2) Yang–Mills theory in 2+1 dimensions seems a fairly good approximation to the true ground state of the theory. • Two new pieces of evidence: • The ghost propagator in Coulomb gauge is practically identical in recursion and MC ensembles. • With the same statistics of “exceptional” configurations we expect also the color-Coulomb potential from recursion lattices to be close to that determined from MC lattices. • Still a long way to go: • Determination of the wave-functional in numerical simulations for “typical” field configurations. • Improvement of the variational estimate of the parameter m. • N-ality? Center vortices? • Generalization to 3+1 dimensions. Bianchi constraint. • ??? Thank you for your attention! Lattice 2010, Villasimius, Italy

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