On the fourth root prescription for dynamical staggered fermions

TL;DR

The paper tackles the fourth-root problem for dynamical staggered fermions by seeking a local single-flavour lattice Dirac operator $D$ satisfying $(\det D_{staggered})^{1/4}=\det D$. It derives a free-field candidate $D=\gamma^{\mu}\frac{1}{2a}\nabla_{\mu}+\frac{1}{2a}\sqrt{(2am)^2+\sum_{\nu}(\frac{1}{2}\Delta_{\nu})^2}$, obtained by adding a Wilson-type taste-mixing term, and proves this $D$ yields the SDR in the free-field theory while exhibiting exponential locality with localisation range $\sim 2\sqrt{2}\,\sqrt{a/m}$ as $a\to 0$. The work also analyzes the locality properties relative to previous approaches (e.g., Jansen) and discusses substantial challenges in extending the SDR to the interacting case, including gauging and the potential need for a lattice chiral symmetry such as a Ginsparg–Wilson relation. It argues that achieving a fully satisfactory interacting solution may require alternative viewpoints (e.g., RG decompositions like Shamir’s) or an exact chiral protection, and positions the free-field result as a valuable analytic prototype guiding future constructions. Overall, the paper provides a concrete, analytically tractable example of a local single-flavour operator compatible with the SDR in the free theory and clarifies the hurdles to a complete interacting implementation.

Abstract

With the aim of resolving theoretical issues associated with the fourth root prescription for dynamical staggered fermions in Lattice QCD simulations, we consider the problem of finding a viable lattice Dirac operator D such that (det D_{staggered})^{1/4} = det D. Working in the flavour field representation we show that in the free field case there is a simple and natural candidate D satisfying this relation, and we show that it has acceptable locality behavior: exponentially local with localisation range vanishing ~ (a/m)^{1/2} for lattice spacing a -> 0. Prospects for the interacting case are also discussed, although we do not solve this case here.

Figures (2)

• Figure 1: The "exclusion curves" $C_{\pm}$. The locations of the endpoints near 1 depend on $am$ and converge to $1$ for $a\to0$.
• Figure 2: The "exclusion curves" $C_{\pm}$ for the operator (\ref{['3.24']}). Compare with Fig. \ref{['frp1']} for the operator (\ref{['3.2']}).