New Formulation for the Lattice-Fermion Derivative: Locality and Chirality without Spectrum Doubling

We present a formulation for the lattice fermion derivative which is both -local and explicitly chiral. Provided that the continuum and infinite volume limits are taken properly the formulation avoids spectrum doubling, and gives a satisfactory weak coupling perturbation theory for the case of Yukawa-coupled scalars. For gauge theories we find that a sum over long string paths must be introduced in order to maintain the correct weak-coupling perturbation theory in the continuum limit. -Submitted to Physical Review Letters * Work supported by the Department of Energy, contract DE AC03 76SF00515. .In this paper we present a local lattice theory of chiral fermions which is -_ free of~spectrum doubling. Our interest in this problem is motivated by a desire to develop methods which allow a non-perturbative study of truly chiral gauge theories (e.g., most grand-unified theories). We find that one can avoid the impli. cations of the Nielsen-Ninomiya theorem’ because, as with most no-go theorems, its proof requires an implicit assumption; namely that one should take the limit of infinite volume before taking the continuum limit. For a theory initially de_ fined in any finite volume, it is trivial to avoid spectrum doubling if one takes the in-finite volume and continuum limits carefully. We present a method which explicitly demonstrates these properties. Our formulation involves an additional parameter E which vanishes in the continuum limit. In an earlier paper’ we addressed a class of local free fermion derivatives which have no spectrum doubling. However, we found that these derivatives lead . . to non-covariant current-current correlation functions in the free theory and to non-covariant propagators in the case of the theory of a fermion interacting with a scalar field through a Yukawa interaction. We concluded that these problems could only be avoided by using a long-range derivative of the SLAC type. The purpose of this letter is to demonstrate that this In this paper we present a local chiral fermion theory which not only has an undoubled spectrum but which also produces covariant Green’s functions provided that the continuum limit is properly taken. Unfortunately, when this derivative is rendered gauge invariant by the usual technique of introducing only the shortest string-paths, the resulting Hamilto-nian does not correctly reproduce continuum weak-coupling perturbation theory for QED. The origin of this problem lies not in the derivative but in-the way the