The Schrödinger functional — a renormalizable probe for non-abelian gauge theories

Abstract

Following Symanzik we argue that the Schrödinger functional in lattice gauge theories without matter fields has a well-defined continuum limit. Due to gauge invariance no extra counter terms are required. The Schrödinger functional is, moreover, accessible to numerical simulations. It may hence be used to study the scaling properties of the theory and in particular the evolution of the renormalized gauge coupling from low to high energies. A concrete proposition along this line is made and the necessary perturbative analysis of the Schrödinger functional is carried through to 1-loop order.

Recent results on moments of parton distribution functions

“…In particular we use Schrödinger functional (SF) boundary conditions [6] for our computation, which allows the extraction of the matrix element from one large plateau around the middle of the time extent T of the lattice. The correlation function of the matrix element f M is obtained by inserting the O 44 operator between two SF nucleon states S and S ′ at the time boundaries t = 0 and T and suitable normalisation with the boundary-to-boundary corre-…”

Abelian monopole and vortex condensation in lattice gauge theories

“…(1.4) the sum over topological inequivalent classes. However, it turns out that [43][44][45][46] on the lattice such an average is not needed because the functional integral Eq. (1.4) is already invariant under arbitrary gauge transformations of A (i) and A (f ) .…”

“…It is possible to improve the lattice action S by modifying the weights W µν 's [43][44][45][46]. Note that, due to the fact that U (i) = U (f ) , one cannot impose periodic boundary conditions in the Euclidean time direction.…”

“…At zero temperature we employ a disorder parameter defined in terms of a gaugeinvariant lattice effective action introduced [40][41][42] using the lattice Schrödinger functional [43][44][45][46]. At finite temperature the disorder parameter has been defined [7] in terms of a thermal partition functional.…”

“…The lattice implementation [43][44][45][46] of the Schrödinger functional, Eq. (1.4), is now straightforward:…”