A lattice formulation of chiral gauge theories Page: 4 of 69
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fore the gauge-field cutoff, the violations of chiral symmetry vanish with at least one power
of the ratio of cutoffs. The use of this double limit in conjunction with the modification of
the magnitude of the fermion determinant has been emphasized previously in Refs. [6,7
Most of the analysis in this paper is couched in weak-coupling coupling perturbation
theory. However, we are able to show, by exploiting the finite radius of convergence the
perturbation expansion of the fermion determinant, that our method is also valid in the
presence of nonperturbative gauge-field configurations.
The remainder of this paper is organized as follows. In Section II we discuss, in general
terms, fermion doubling, its elimination through the use of a Wilson mass, and the breaking
and restoration of chiral symmetry. In Section III we introduce a lattice implementation of
a theory of left-handed fermions coupled to a non-Abelian gauge field. Although our specific
analyses in subsequent sections of the paper refer to this model, our methods generalize
immediately to models that contain right-handed as well as left-handed fermion fields and
to models that contain scalar particles. In Section IV we discuss the nature of the violations
of gauge invariance that arise from the introduction of a Wilson mass. Section V contains an
analysis of the chiral-symmetry properties of the fermion determinant in the presence of a
background gauge field. This analysis allows us to derive a modification of the determinant
that restores the chiral symmetry in the case of an anomaly-free theory. In Section VI
we discuss the difficulties that arise from dynamical gauge fields and present the double
limiting procedure for dealing with them. In Section VII we indicate how the methods used
in computing the fermion determinant can also be applied in computing matrix elements of
operators containing fermion fields. A proof of the validity of the methods for computing
the fermion determinant and operator matrix elements in the presence of nonperturbative
gauge fields is sketched in Section VIII. Finally, in Section IX, we summarize our results
and discuss various options for implementing our chiral-fermion method.
While this paper was in preparation, a paper by Hernandez and Sundrum  on the
same subject appeared. The methods that these authors propose for computing the fermion
determinant (but not the matrix elements of fermion operators) are essentially identical to
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Bodwin, G.T. A lattice formulation of chiral gauge theories, report, December 1, 1995; Illinois. (https://digital.library.unt.edu/ark:/67531/metadc693548/m1/4/: accessed May 27, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.