A path-integral representation of the free one-flavour staggered-fermion determinant

A complete study of QCD on the lattice requires the numerical simulation of dynamical fermions. These Monte Carlo calculations are extremely computationally costly, since the effects of quarks must be included by first integrating out the fermion path integral, and then describing the resulting non-local dynamics of the fermion determinant. For a review of recent developments, see e.g. [1, 2]. With present techniques, the most costeffective means of performing these simulations is to use the staggered fermion formulation of Kogut and Susskind [3]. Recent calculations by the MILC collaboration [4] have demonstrated good agreement between experimentally known strong-interaction measurements and staggered fermion lattice QCD simulation. The formulation as it stands has a serious deficiency for dynamical simulations. In four dimensions, the staggered fermion determinant describes four flavours of fermion, not one. This means that while it is very simple to simulate four mass-degenerate fermions with the staggered method, the study of one or two flavours must use a fractional power of the fermion determinant. This raises difficult theoretical problems: what are the fermion fields, and what is the local action on these fields which reproduces this determinant? Without a path-integral representation of the fermion determinant, all the standard quantum field theory construction of propagators (which are the two-point functions of the underlying quark fields) is poorly defined. If no local action exists, an even more severe issue arises, since there is then no guarantee that the continuum limit of the lattice simulation is in the same universality class as QCD and the link with physics is lost. In this paper, we describe a numerical construction of an operator that defines a lattice quantum field theory equivalent to a single, free staggered fermion. Most of the construction is performed in two dimensions to ease the computations, but some suggestive results in four dimensions indicate the same construction works there too. Note that all the work in this paper is for the theory of