Locality of the fourth root of the staggered-fermion determinant: Renormalization-group approach

Consistency of present day lattice QCD simulations with dynamical (“sea”) staggered fermions requires that the determinant of the staggered-fermion Dirac operator, $\mathrm{d}\mathrm{e}\mathrm{t}(D)$, be equal to ${\mathrm{d}\mathrm{e}\mathrm{t}}^4(D_{\mathrm{r}\mathrm{g}})\mathrm{d}\mathrm{e}\mathrm{t}(\mathcal{T})$ where $D_{\mathrm{r}\mathrm{g}}$ is a local one-flavor lattice Dirac operator, and $\mathcal{T}$ is a local operator containing only excitations with masses of the order of the cutoff. Using renormalization-group (RG) block transformations I show that, in the limit of infinitely many RG steps, the required decomposition exists for the free staggered operator in the “flavor representation.” The resulting one-flavor Dirac operator $D_{\mathrm{r}\mathrm{g}}$ satisfies the Ginsparg-Wilson relation in the massless case. I discuss the generalization of this result to the interacting theory.