The Quenched Continuum Limit

Our increasing ability to carry out unquenched simulations means that the quenched approximation is no longer required. There is an outstanding issue associated with the quenched continuum limit, however, which we address here. At the same time we point out the ingredients that are necessary to obtain an accurate result in the continuum limit, since these ingredients will also be important for unquenched simulations. The outstanding issue is the question of whether all quark formulations give the same result in the quenched continuum limit. It is relatively trivial to demonstrate that they should, because this is the limit in which it is easy to analyse the formulations. At Lattice 2000, however, it seemed not to be the case [1]. A plot was made of the ratio of nucleon to vector meson mass at a fixed physical quark mass, as a function of the lattice spacing. The physical quark mass was chosen as the point at which the pseudoscalar meson mass divided by the vector meson mass was 0.5 (in those days quite a low mass to reach). The lattice spacing was given in units of the vector meson mass. Because of the instability of vector mesons this is not a plot that allows for a precision test against experimental results, even if it used a quark mass value from the real world and was not in the quenched approximation. However, it can be used as a comparison of different formalisms, provided the results have good statistical precision and are on large enough volumes. The original plot showed a 6σ disagreement between the results extrapolated to a = 0 from the Wilson formulation [2] and the unimproved staggered formulation [3]. The Wilson results were extrapolated linearly in a and the staggered results quadratically. Clover results were also included but their variation suggests a statistical error that means they are not particularly useful so we drop those results in this discussion. The disagreement, if true, would represent a major problem for lattice calculations and call the whole approach into question. With many large dynamical quark projects well underway and others planned, it is therefore time to readdress this issue, following on from [4]. In addition, there are more quark formulations in widespread use than there were in 2000, so the comparison of results can be significantly extended.