Ghost condensation on the lattice

Abstract

We perform a numerical study of ghost condensation -- in the so-called Overhauser channel -- for SU(2) lattice gauge theory in minimal Landau gauge. The off-diagonal components of the momentum-space ghost propagator G^{cd}(p) are evaluated for lattice volumes V = 8^4, 12^4, 16^4, 20^4, 24^4 and for three values of the lattice coupling: β= 2.2, 2.3, 2.4. Our data show that the quantity ϕ^b(p) = ε^{bcd} G^{cd}(p) / 2 is zero within error bars, being characterized by very large statistical fluctuations. On the contrary, |ϕ^b(p)| has relatively small error bars and behaves at small momenta as L^{-2} p^{-z}, where L is the lattice side in physical units and z \approx 4. We argue that the large fluctuations for ϕ^b(p) come from spontaneous breaking of a global symmetry and are associated with ghost condensation. It may thus be necessary (in numerical simulations at finite volume) to consider |ϕ^b(p)| instead of ϕ^b(p), to avoid a null average due to tunneling between different broken vacua. Also, we show that ϕ^b(p) is proportional to the Fourier-transformed gluon field components {\widetilde A}_μ^b(q). This explains the L^{-2} dependence of |ϕ^b(p)|, as induced by the behavior of | {\widetilde A}_μ^b(q) |. We fit our data for |ϕ^b(p)| to the theoretical prediction (r / L^2 + v) / (p^4 + v^2), obtaining for the ghost condensate v an upper bound of about 0.058 GeV^2. In order to check if v is nonzero in the continuum limit, one probably needs numerical simulations at much larger physical volumes than the ones we consider. As a by-product of our analysis, we perform a careful study of the color structure of the inverse Faddeev-Popov matrix in momentum space.