A lattice definition of QCD based on chiral defect fermions is discussed in detail. This formulation involves (infinitely) many heavy regulator fields, realized through the introduction of an unphysical fifth dimension. It is proved that non-singlet axial symmetries become exact in the limit of an infinite fifth dimension, and before the continuum limit is taken.

We construct a model in which four dimensional chiral fermions arise on the boundaries of a five dimensional lattice with free boundary conditions in the fifth direction. The physical content is similar to Kaplan's model of domain wall fermions, yet the present construction has several technical advantages. We discuss some aspects of perturbation theory, as well as possible applications of the model both for lattice QCD and for the on-going attempts to construct a lattice chiral gauge theory. email: ftshamir@weizmann.bitnet

We show that the use of the fourth-root trick in lattice QCD with staggered fermions corresponds to a non-local theory at non-zero lattice spacing, but argue that the non-local behavior is likely to go away in the continuum limit. We give examples of this non-local behavior in the free theory, and for the case of a fixed topologically non-trivial background gauge field. In both special cases, the non-local behavior indeed disappears in the continuum limit. Our results invalidate a recent claim that at nonzero lattice spacing an additive mass renormalization is needed because of taste-symmetry breaking.1 One reason is that breaking the taste degeneracy requires additional hopping terms in the lattice action, which, for a generic choice, make the fermion determinant complex. Also, the existence of a partially-conserved continuous chiral symmetry depends on the choice of mass term.2 In the isospin limit, the up-down sector is represented by a square root of a staggered determinant with the common light quark mass.

In this paper, we examine the phase diagram of quenched QCD with two flavors of Wilson fermions, proposing the following microscopic picture. The super-critical regions inside and outside the Aoki phase are characterized by the existence of a density of near-zero modes of the (hermitian) Wilson-Dirac operator, and thus by a non-vanishing pion condensate. Inside the Aoki phase, this density is built up from extended near-zero modes, while outside the Aoki phase, there is a non-vanishing density of exponentially localized near-zero modes, which occur in "exceptional" gauge-field configurations. Nevertheless, no Goldstone excitations appear outside the Aoki phase, and the existence of Goldstone excitations may therefore be used to define the Aoki phase in both the quenched and unquenched theories. We show that the density of localized near-zero modes gives rise to a divergent pion two-point function, thus providing an alternative mechanism for satisfying the relevant Ward identity in the presence of a non-zero order parameter. This divergence occurs when we take a "twisted" quark mass to zero, and we conclude that quenched QCD with Wilson fermions is well-defined only with a non-vanishing twisted mass. We show that this peculiar behavior of the near-zero-mode density is special to the quenched theory by demonstrating that this density vanishes in the unquenched theory outside the Aoki phase. We discuss the implications for domain-wall and overlap fermions constructed from a Wilson-Dirac kernel. We argue that both methods work outside the Aoki phase, but fail inside because of problems with locality and/or chiral symmetry, in both the quenched and unquenched theories. π would go negative, signalling the breaking of a symmetry. A pionic condensate forms in some direction in flavor space, and the line m 0 = m ′ (g 0 ) determines the location of a second-order phase transition. The corresponding pion becomes massive again for m 0 < m ′ , while the other two pions become Goldstone bosons associated with the spontaneous breaking of the SU(2) flavor symmetry ("isospin") down to a U(1) symmetry. Since the condensate is pionic, it breaks parity symmetry as well. Microscopically, the condensate arises from near-zero modes of the Wilson-Dirac operator [10], and can thus only occur in the region −8 < am 0 < 0.This argument does not provide much information on the detailed form of the Aoki phase. Additional analytical evidence comes from several sources. The location of the critical points along the line g 0 = 0 is obtained from weak-coupling perturbation 3 theory, which, however, gives no information on the existence of a condensate. The existence of a condensate was discussed in the context of the pion effective action in Refs. [11,12]. Ref. [12] showed that, if the Aoki phase extends all the way to g 0 = 0, it does so as indicated by the "finger" structure 2 in Fig. 1, with the width of these fingers proportional to (aΛ QCD ) 3 , where Λ QCD is the QCD scale. In the strongcoupling limit, the location of the two critical point...