Improved actions for the two-dimensional σ-model

In recent years there has been much work in improving lattice actions (see e.g. M. Lüscher’s and P. Hasenfratz’ talks at this conference). The idea behind all these attempts is to modify the lattice action with the addition of irrelevant operators in order to reduce lattice artifacts: the aim is to have scaling and finite-size-scaling at small values of the correlation length. Two different approaches have been used. The original idea of Symanzik [1] consisted in adding, on the basis of power conting, new operators to the action with coefficients such as to cancel corrections of order O(ga) in the correlation functions. It was later realized that it is possible to change the action so as to improve only on-shell quantities [2]. In this case the number of necessary operators is in general reduced. This approach can be applied order by order in perturbation theory in a straightforward (although computationally difficult) way. Recently the idea has been successfully implemented at a non-perturbative level [3] for the fermionic actions (here the corrections are of order a), obtaining actions which are full improved at order O(a). A different approach is the perfect action program of Ref. [4]: here the starting point is an action which is the fixed point of a renormalizationgroup transformation. The relation between the Symanzik approach and the fixed-point actions has been recently clarified in Ref. [5]. Fixedpoint actions are on-shell, tree-level improved to all orders in a (this last condition should not be very relevant — at least for the scaling of standard observables — as quantum corrections will introduce again terms of order a). Therefore these actions are particular Symanzik theories. Of course the main question is if this particular procedure chooses among all Symanzik-improved theories those which have a “better” behaviour. We do not know the answer to this question, but we can try to understand it by comparing the behaviour of fixed-point actions and other theories which are improved à la Symanzik using different criteria (for instance semplicity, locality .....). Here we will study this problem in the context of the two-dimensional σ-model focusing on the behaviour of the mass gap in a strip. Let us begin by discussing tree-level improvement and in particular the relation between onshell and off-shell improvement. In this case one can show that for generic actions which have only two-spin couplings on-shell and off-shell improvement at tree level are identical. More precisely, consider an action of the form