Renormaliation-group blocking the fourth root of the staggered determinant

Abstract

Lattice QCD simulations with staggered fermions rely on the "fourth-root trick." The validity of this trick has been proved for free staggered fermions using renormalization-group block transformations. I review the elements of the construction and discuss how it might be generalized to the interacting case.

Observations on staggered fermions at nonzero lattice spacing

“…9 The transformation Q (0) is not unique; see Ref. [14] for details. ) is expected to be a local functional of the gauge field.…”

“…(3.14) for Ξ = ξ 5 . This overlap operator is "natural," because it has been constructed such that the difference between D ov,n and D taste,n (0) is expected to be of order a 2 0 /a 2 n = 1/2 2n [14]. The distinction is that, by construction, D ov,n has exact SU(4) taste symmetry (in fact a full chiral SU(4) L × SU(4) R ), while D taste,n (0) does not.…”

“…A basic hypothesis of the RG treatment of staggered fermions (with or without the fourth root) is that the taste-breaking terms of the RG-blocked operator D taste,n (m) tend to zero in the limit of infinitely many RG blocking steps [15,14]. The tastebreaking part ∆ n is given explicitly by writing…”

“…We refer to Ref. [14] for a discussion of the status of this assumption, as well as a more precise statement about the gauge fields for which it is expected to apply. Under this scaling hypothesis it is trivial that D taste,n (m) and D inv,n (m) have a common n → ∞ limit, for any m. Furthermore, by Eq.…”

“…(A.6) and (A.7) respectively. 14 Note that corrections of O(m 2 a) are absent in Eq. (A.8) because the difference between γ5 and γ 5 is O(pa), and ordinary chiral symmetry (as opposed to the GWL type) would forbid mass terms in D ov,χ (m).…”