Ghost condensation on the lattice

TL;DR

This study investigates ghost condensation in SU(2) lattice gauge theory, analyzing ghost propagators and their fluctuations, and provides insights into the behavior of ghost condensates and related gauge field components.

Contribution

It offers the first detailed numerical analysis of ghost condensation in the Overhauser channel on the lattice, highlighting the role of spontaneous symmetry breaking and providing bounds on the ghost condensate.

Findings

( p ) is zero within error bars but has large fluctuations.

|( p )| scales as L^{-2} p^{-z} with z pprox 4.

Estimated upper bound for ghost condensate v is about 0.058 GeV^2.

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: \beta = 2.2, 2.3, 2.4. Our data show that the quantity \phi^b(p) = \epsilon^{bcd} G^{cd}(p) / 2 is zero within error bars, being characterized by very large statistical fluctuations. On the contrary, |\phi^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 \phi^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 |\phi^b(p)| instead of \phi^b(p), to avoid a null average due to tunneling between different broken vacua. Also, we show that \phi^b(p) is proportional to the Fourier-transformed gluon field components {\widetilde A}_{\mu}^b(q). This explains the L^{-2} dependence of |\phi^b(p)|, as induced by the behavior of | {\widetilde A}_{\mu}^b(q) |. We fit our data for |\phi^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.