The QCD vacuum wavefunctional and confinement in Coulomb

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 Lattice 2010, Villasimius, Sardinia, Italy, June 14 -19, 2010

(Approximate) QCD vacuum wave-functional l Confinementis the property of the vacuum of quantized non-abelian gauge theories. In the hamiltonian formulation in D=d+1 dimensions and temporal gauge: 18/06/2010 Lattice 2010, Villasimius, Italy 2

l At large distance scales one expects: l l Halpern (1979), Greensite (1979) l Greensite, Iwasaki (1989) Kawamura, Maeda, Sakamoto (1997) l Karabali, Kim, Nair (1998) l 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. l The true vacuum wave-functional (VWF) cannotjust be of the dimensionalreduction form - incorrect results at short distances/high frequencies. 18/06/2010 Lattice 2010, Villasimius, Italy 3

Suggestion for an approximate vacuum wavefunctional l 18/06/2010 Greensite, ŠO, ar. Xiv: 0707. 2860 [hep-lat]; Greensite, talk at Lattice 2007 Lattice 2010, Villasimius, Italy 4

Arguments in favor of the proposed VWF l l In the free-field limit(g 0), Ψ 0[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 varyingonly 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 = 3 m/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. 18/06/2010 Lattice 2010, Villasimius, Italy 5

Lattice evidence in favor of the proposed VWF l l “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. 18/06/2010 Lattice 2010, Villasimius, Italy 6

Mass gap 18/06/2010 Lattice 2010, Villasimius, Italy 7

Coulomb-gauge quantities l Why Coulomb gauge? l l 18/06/2010 Low-lying spectrum of the Faddeev–Popov operator in Coulomb gaug probes properties of nonabelian gauge fields that are crucial for the confinement mechanism. The ghost propagatorin Coulomb gauge and the color-Coulomb potentialare 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 8

From temporal to Coulomb gauge l Classical Coulomb-gauge hamiltonian: l Coulomb kernel: 18/06/2010 Lattice 2010, Villasimius, Italy 9

From temporal to Coulomb gauge l 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. l 18/06/2010 Lattice 2010, Villasimius, Italy Greensite, ŠO, Zwanziger (2004) 10

Coulomb-gauge ghost propagator 18/06/2010 Lattice 2010, Villasimius, Italy 11

Color-Coulomb potential 18/06/2010 Lattice 2010, Villasimius, Italy 12

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Conclusions l l 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 theory. Two new pieces of evidence: l l l 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: l l l 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! 18/06/2010 Lattice 2010, Villasimius, Italy 16