ar X iv : h ep - l at / 9 31 10 16 v 1 1 6 N ov 1 99 3 1 Fixed Point Actions for Lattice Fermions

The fixed point actions for Wilson and staggered lattice fermions are determined by iterating renormalization group transformations. In both cases a line of fixed points is found. Some points have very local fixed point actions. They can be used to construct perfect lattice actions for asymptotically free fermionic theories like QCD or the Gross-Neveu model. The local fixed point actions for Wilson fermions break chiral symmetry, while in the staggered case the remnant U (1)e ⊗ U (1)o symmetry is preserved. In addition, for Wilson fermions a nonlocal fixed point is found that corresponds to free chiral fermions. The vicinity of this fixed point is studied in the Gross-Neveu model using perturbation theory. The continuum limit of a lattice field theory is defined at a fixed point of the renormaliza-tion group. The lattice models on a renormal-ized trajectory emanating from the fixed point are free of cutoff effects and hence have perfect lattice actions. Recently, Hasenfratz and Nieder-mayer realized that perfect actions can be constructed explicitly for asymptotically free theories [1]. In addition, in the 2-d nonlinear σ-model the renormalization group transformation can be optimized such that the fixed point action is extremely local. This is essential for numerical simulations. The question arises if fixed point actions for fermionic theories are local as well [2]. Since the fixed point of an asymptotically free theory is close to the Gaussian fixed point this question can be studied perturbatively, to lowest order even in the free theory. The corresponding calculation for a free scalar field was done long ago by Bell and Wilson [3]. Let us consider free Wilson fermion fields Ψ and Ψ with the action S[Ψ, Ψ] on a hypercubic lattice Λ, which is then blocked to a lattice Λ ′ of doubled lattice spacing. Then each point x ′ ∈ Λ ′ corresponds to a hypercubic block of 2 d points x ∈ Λ and each point x belongs to exactly one block x ′ (we denote this by x ∈ x ′). The block transfor-* Based on two talks presented by the authors • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • …