Logarithmic corrections to O(a) lattice artifacts

We compute logarithmic corrections to the O(a2) lattice artifacts for a class of lattice actions for the non-linear O(n) sigma-model in two dimensions. The generic leading artifacts are of the form a2[ln(a2)](n/(n−2)). We also compute the next-to-leading corrections and show that for the case n = 3 the resulting expressions describe well the lattice artifacts in the step scaling function, which are in a large range of the cutoff apparently of the form O(a). An analogous computation should, if technically possible, accompany any precision measurements in lattice QCD. 1. Most of our knowledge concerning renormalization of quantum field theories stems from perturbation theory. Although there are no rigorous proofs in general, many of the results are structural and hence considered to carry over to non-perturbative formulations. Indeed there is supporting evidence from various studies, e.g. of soluble models in 2 dimensions and of 1/n expansions of some theories. The same situation holds concerning cutoff artifacts in lattice regularized theories. It is generally accepted that these artifacts are summarized in Symanzik’s effective action. In this framework generic lattice artifacts are, in particular for asymptotically free (or trivial) theories, expected to be integer powers in the lattice spacing O(ap) , p = 1, 2, . . . up to possible multiplicative logarithmic corrections. This is an extremely important ansatz in the extrapolation of lattice data to the continuum limit, especially for present computations of lattice QCD where the lattice spacings are typically around 0.1fm. In this letter we will examine in more detail Symanzik’s theory for the 2dimensional non-linear O(n) σ-model. In such a simple bosonic model lattice artifacts are expected to be of the form O(a2). It was thus rather surprising that precision measurements [1], some years ago, of certain observables in the O(3) sigma model exhibited apparently linearly dependent O(a) artifacts for a rather large range of computable lattice spacings. The importance of finding the solution of this puzzle was emphasized by Hasenfratz in his lattice plenary talk in 2001 [2]. To define the measured quantity referred to above, one considers the model confined to a finite (1-dimensional) box of extension L (with periodic boundary conditions). The LWW coupling [3] is defined as u0 = Lm(L) , (1) where m(L) is the mass gap of the theory in finite volume. Next one measures u1, defined similarly with doubled box size. In the continuum limit u1 is a function of u0, called the step scaling function u1 = σ(2, u0). For the lattice regularized theory there are lattice artifacts and u1 = 2Lm(2L) = Σ(2, u0, a/L) . (2) The advantage of this measurement for the purpose of studying lattice artifacts is that there is no need to know the box size L or the mass gap m(L) in physical units. The results of the MC measurements are shown in Fig. 1. One can see that the lattice artifacts (cutoff effects) are very nearly linear as function of the lattice spacing a both for the case of the standard lattice action (ST) and for a modified action (MOD). Although the effects are in this case relatively very small, they seem not of the theoretically expected form. Note however the encouraging feature that computations with different lattice actions are consistent with the same continuum limit, supporting the crucial concept of universality.