Effective theory for quenched lattice QCD and the Aoki phase

Abstract

We discuss the symmetries of quenched QCD with Wilson fermions, starting from its lagrangian formulation, taking into account the constraints needed for convergence of the ghost-quark functional integral. We construct the corresponding chiral effective lagrangian, including terms linear and quadratic in the lattice spacing. This allows us to study the phase structure of the quenched theory, and compare it to that in the unquenched theory. In particular we study whether there may be an Aoki phase (with parity and flavor spontaneously broken) or a first order transition line (with no symmetry breaking but meson masses proportional to the lattice spacing), which are the two possibilities in the unquenched theory. The presence of such phase structure, and the concomitant long-range correlations, has important implications for numerical studies using both quenched and dynamical overlap and domain-wall fermions. We argue that the phase structure is qualitatively the same as in the unquenched theory, with the choice between the two possibilities depending on the sign of a parameter in the low-energy effective theory.Comment: 31 pages, a few minor correction

On the foundations of partially quenched chiral perturbation theory

“…[1], which do not take into account all the subtleties coming from the ghost sector, rather than the correct symmetries introduced in Refs. [17,19,23]. The subtleties do not affect the general argument.…”

“…M v,g,s are the masses of (flavor nondiagonal) valence, ghost, and sea pions; in 19 Negative squared masses would not imply imaginary masses, but rather signal a phase transition and the need to find a new vacuum state. We believe such transitions, driven by possible mass terms special to the partially quenched theory, cannot occur in the usual PQ case of degenerate valence and ghost masses, since the dynamics of such a theory are controlled by the sea masses.…”

Constraint on the low energy constants of Wilson chiral perturbation theory

“…This issue has been resolved, in the context of the quenched theory, in Ref. [33], and a simple generalization works here. The solution is to do an axial rotation in the τ 3 direction by angle π/4, such that the fermion operator becomes…”

“…[34], generalizing the methodology of Ref. [33]. 7 We could just as well add two isodoublets of valence quarks (and corresponding ghosts) and use q V 1 q V 2 (0)q V 2 q V 1 (n) .…”

“…Thus we work to leading order (LO) in the "large cut-off effects" or "Aoki" regime in which the power counting is m ∼ µ ∼ a 2 Λ 3 QCD , where m and µ are the physical errors [10], following a simple generalization of the analysis of Ref. [33]. Compared to the usual chiral Lagrangian, one has additional factors of ±iτ 3 in terms containing spurions coming from discretization errors.…”