Order a improved renormalization constants

In Reference [1], we studied the improvement of the Wilson-Dirac theory by removing all lattice artifacts linear in the lattice spacing a in on-shell matrix elements. In general, such an improvement for the dimension 3 fermion bilinear operators requires one to tune the coefficient of the clover operator in the action, determine the coefficient of an extra term each in the vector, axialvector, and tensor currents, and obtain the mass dependence of all the renormalization constants. We showed that all these improvement constants, as well as the renormalization constants for the axial and vector currents, can be determined by imposing the axial and vector Ward identities. The remaining renormalization constants, those for the scalar, pseudoscalar and tensor operators, are scheme dependent and cannot be determined by this method. The scheme dependence is, however, the same for both the scalar and the pseudoscalar operators, and the corresponding ratio of renormalization constants can again be obtained.