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I show that the conventional formulations of lattice domain-wall fermion with any finite N s (in the fifth dimension) do not preserve the chiral symmetry optimally and propose a new action which preserves the chiral symmetry optimally for any finite N s .

We test the convergence property of the chiral perturbation theory using a lattice QCD calculation of pion mass and decay constant with two dynamical quark flavors. The lattice calculation is performed using the overlap fermion formulation, which realizes exact chiral symmetry at finite lattice spacing. By comparing various expansion prescriptions, we find that the chiral expansion is well saturated at the next-to-leading order for pions lighter than approximately 450 MeV. Better convergence behavior is found, in particular, for a resummed expansion parameter xi, with which the lattice data in the pion mass region 290-750 MeV can be fitted well with the next-to-next-to-leading order formulas. We obtain the results in two-flavor QCD for the low energy constants l[over ]_{3} and l[over ]_{4} as well as the pion decay constant, the chiral condensate, and the average up and down quark mass.

Errata Erratum: Dragging effect on the inertial frame and the contribution of matter to the gravitational "constant" in a closed cosmological model of the Brans-Dicke theory [Phys. Rev. D 19,2861 (1979)l

We analyze the general solutions of the Ginsparg-Wilson relation for lattice Dirac operators and formulate a necessary condition for such operators to have a nonzero index in the topologically nontrivial background gauge fields. ͓S0556-2821͑99͒03405-0͔Recently there have been very interesting developments in the theoretical understanding of chiral symmetry on the lattice. The idea stems from the Ginsparg-Wilson ͑GW͒ relation ͓1͔ which was derived in 1981 as the remnant of chiral symmetry on the lattice after blocking a chirally symmetric theory with a chirality breaking local renormalization group transformation. The original GW relation iswhere D is lattice Dirac operator, R is a nonsingular Hermitian operator which is local in position space and trivial in Dirac space, and a is the lattice spacing which reminds us that D becomes chirally symmetric in the continuum limit a→0. According to the Nielsen-Ninomiya theorem ͓2͔ the chiral symmetry of a local Dirac operator defined on the regular lattice must be broken in order to avoid species doubling. The main advantage of the GW relation is that it introduces chiral symmetry breaking of D in the mildest way ͓1͔. Although it does not ensure the absence of the species doubling, it does incorporate two remarkable properties. The first is that the action Aϭ D has an exact symmetrywhere is a global parameter, which was discovered by Lüscher ͓3͔. The second is that any operator D satisfying the GW relation possesses a well defined integer index on a finite lattice ͓4,3͔where the left-hand side stands for the fermionic average of ␥ 5 calculated with the infinitesimal mass ⑀ added to the operator D, and n ϩ (n Ϫ ) are the number of the zero modes of D with positive ͑negative͒ chiralities. This is in contrast to the Wilson-Dirac operator for which the left-hand side ͑LHS͒ generally is not an integer on a finite lattice.It is essentially due to these two properties that such formulations of lattice QCD can possess the attractive features pointed out in Refs. ͓3-6͔. However, only the GW relation itself is not sufficient to guarantee that any D satisfying Eq. ͑1͒ must possess exact zero modes with definite chiralities, and reproduce the Atiyah-Singer index theorem on the lattice. In this paper, we analyze general solutions of GW relation and formulate a necessary condition for them to have nonzero indices in topologically nontrivial background gauge fields. We limit our consideration to the operators D satisfying the Hermiticity propertyFirst, we consider the case of nonsingular D which is relevant to topologically trivial gauge field background, except possibly some ''exceptional'' configurations. Then Eq. ͑1͒ is equivalent to the following equation linear in D Ϫ1 :and its general solution can be written in the formwhere D c is the chirally symmetric lattice Dirac operator, i.e.,Thus in the nonsingular case the problem of constructing explicit solutions of D reduces to finding a proper realization of the chirally symmetric operator D c . Note that by virtue of the condition ͑5...

We derive the topological susceptibility to the one-loop order in the chiral effective theory of QCD, for an arbitrary number of flavors.

The square root of the positive definite Hermitian operator D w † D w in Neuberger's proposal of exactly massless quarks on the lattice is implemented by the recursion formula Y kϩ1 ϭThe spectrum of the lattice Dirac operator for single massless fermion in two dimensional background U͑1͒ gauge fields is investigated. For smooth background gauge fields with nonzero topological charge, the exact zero modes with definite chirality are reproduced to a very high precision on a finite lattice and the index theorem is satisfied exactly. The fermionic determinants are also computed and they are in good agreement with the continuum exact solution. ͓S0556-2821͑98͒04519-6͔