Weak coupling expansion of massless QCD with overlap Dirac operator and axial U(1) anomaly

We discuss the weak coupling expansion of massless QCD with the Dirac operator which is derived by Neuberger based on the overlap formalism and satisfies the Ginsparg-Wilson relation. The axial U(1) anomaly associated to the chiral transformation proposed by Lüscher is calculated as an application and is shown to have the correct form of the topological charge density for perturbative backgrounds. The coefficient of the anomaly is evaluated as a winding number related to a certain five-dimensional fermion propagator. e-mail address: [email protected] e-mail address: [email protected] Recently Dirac operators which may describe exactly massless fermion on a lattice are proposed by Neuberger [1] based on the overlap formalism [2, 3] and by Hasenfratz, Laliena and Niedermayer [4] based on the renormalization group method [5, 6]. These Dirac operators are very different from each other but both satisfy the Ginsparg-Wilson relation [7], which ensures that the fermion propagator anti-commutes with γ5 at nonzero distances and thus respects chiral symmetry in that sense. Subsequently Lüscher observed [8] that the Ginsparg-Wilson relation implies an exact symmetry of the fermion action and the anomalous behavior of the fermion partition function under its flavorsinglet transformation is expressed in terms of the index of the Dirac operator arised as the Jacobian factor of the path integral measure, providing a clear understanding of the exact index theorem on a lattice in Ref. [4]. Though it was pointed out in Ref. [7] that Dirac operators satisfying the GinspargWilson relation will necessarily exhibit non-local behavior in the presence of the dynamical gauge fields, a recent perturbative analysis of chiral gauge theory in the overlap formalism [9] suggests some of them will be well controllable as a field theory on a lattice. In this paper we consider the weak coupling expansion of massless QCD with the Dirac operator which is derived by Neuberger and satisfies the Ginsparg-Wilson relation. Then we apply our results to analyze the axial U(1) anomaly following the formulation of Ref. [8] and confirm that the Jacobian factor is given in the form of the topological charge density for slowly varing perturbative gauge fields, supplementing the original calculation of the same quantity of Ref. [7]. The covariant anomaly in the vacuum overlap formalism has been discussed by Neuberger and Narayanan [11], Randjbar-Daemi and Strathdee [10] and Neuberger [11]. Note also that the axial anomaly has been calculated in the context of the domain-wall QCD by Shamir [12]. We begin with a brief review of the symmetry argument of Ref. [8]. The partition function we consider is of the form Z = ∫ dψdψ̄e Σψ̄Dψ (1) 1 with a lattice Dirac operator D satisfying the Ginsparg-Wilson relation Dγ5 + γ5D = aDγ5D. (2) In Ref. [8] Lüscher pointed out that with the aid of the relation (2) the fermion action aΣψ̄Dψ is invariant under the infinitesimal transformation ψ → ψ + iǫT δψ, ψ̄ → ψ̄ + iǫδψ̄T where T is a generator of the rotation in the flavor space and δψ = γ5(1− 1 2 aD)ψ, δψ̄ = ψ̄(1− 1 2 aD)γ5. (3) The path integral measure yields the Jacobian factor −iǫatr{Tγ5D}, which does not vanish only for the flavor singlet chiral rotation, accounting for the index theorem of Ref. [4]. A Dirac operator satisfying the relation (2) was proposed by Neuberger in Ref. [1] and now we review its brief derivation starting from the domain wall fermion of the Shamir type [13] based on Ref. [14], trying to clarify the physics backgrounds of the Dirac operator. The domain wall fermion of this type is a Wilson fermion in five dimensions with the finite size N5 of the fifth space and is described by the action S = ∑