Testing the efficiency of different improvement programs

We study the finite-size behaviour of a tree-level on-shell improved action for the N-vector model. We present numerical results for N = 3 and analyti c results in the large-N limit for the mass gap. We also report a perturbat ive computation at one loop of the mass gap for states of spatial momentum p. We present a detailed comparison of the behaviour of this action with th at of other formulations, including the perfect action, and a critical disc ussion of the different approaches to the problem of action improvement. (C ) 1999 Elsevier Science B.V. All rights reserved.