The coe cients multiplying the counterterms required for O(a) improvement of the action and the isovector axial current in lattice QCD are computed non-perturbatively, in the quenched approximation and for bare gauge couplings g 0 in the range 0 g 0 1. A nite-size method based on the Schr odinger functional is employed, which enables us to perform all calculations at zero or nearly zero quark mass. As a by-product the critical hopping parameter c is obtained at all couplings considered.

The dominant cuto e ects in lattice QCD with Wilson quarks are proportional to the lattice spacing a. In particular, the isovector axial current satis es the PCAC relation only up to such e ects. Following a suggestion of Symanzik, they can be cancelled by adding local O(a) correction terms to the action and the axial current. We here address a number of theoretical issues in connection with the O(a) improvement of lattice QCD and then show that chiral symmetry can be used to x the coe cients multiplying the correction terms.

A finite-size technique is employed to compute the normalization constant Z A of the isovector axial current in lattice QCD. The calculation is carried out in the quenched approximation for values of the bare gauge coupling g 0 ranging from 0 to 1. In the lattice action and the lattice expression for the axial current we include the counterterms required for O(a) improvement, with non-perturbatively determined coefficients. With little additional work the normalization constant Z V of the improved isospin current is also obtained.

In a series of publications 1,2], L uscher et al. have demonstrated the usefulness of the Schr odinger functional in pure SU(2) and SU(3) gauge theory. In this paper, it is shown how their formalism can be extended to include fermions. In the framework of Wilson's lattice QCD, we de ne the Schr odinger functional by making use of the transfer matrix formalism. Boundary conditions for the fermions arise naturally. We then take the naive continuum limit of the action and show that no lattice peculiarities are left over. The corresponding free Dirac operator has a unique self-adjoint extension with purely discrete spectrum and no zero modes.

Starting from the Schr odinger functional, we give a non-perturbative de nition of the running coupling constant in QCD. The spatial boundary conditions for the quark elds are chosen such that the massless Dirac operator in the classical background eld has a large smallest eigenvalue. At one-loop order of perturbation theory, we determine the matching coe cient to the MS-scheme and discuss the quark mass e ects in the -function. To this order, we also compute the Symanzik improvement coe cient necessary to remove the O(a) lattice artefacts originating from the boundaries. For reasonable lattice resolutions and the standard Wilson action, lattice artefacts are found to be only weakly dependent on the lattice spacing a, while they vanish quickly with the improved action of Sheikholeslami and Wohlert.MPI{PhT/95-69