Renormalizability of a lattice chiral fermion in the overlap formulation

Renormalizability of a lattice chiral fermion is studied at one loop level in the overlap formulation in four dimensions. The fermion chirality is examined including the self-energy corrections due to gauge interactions. Divergent terms breaking the chiral symmetry do not appear and the chiral fermion is renormalized, preserving the correct chiral properties without adding new counter-terms or tuning the parameters involved. The divergent part of the wave function renormalization factor agrees with that of the continuum theory. The overlap formulation of a lattice chiral fermion has passed the important test, the renormalizability, at one loop level. On leave of absence from: Department of Physics, University of Tokyo, Tokyo, 113 Japan. Kaplan’s original idea of the domain wall fermion [1] inspired many studies of lattice formulations of a chiral fermion [2, 3]. The overlap formulation [3] has been shown to have promising analytic properties [4]-[8] and gives results in good agreement with continuum theories in the numerical simulations of two dimensional systems [9]. This formulation provides the possibility of investigating chiral gauge theories in the strong coupling region, as well as a clearer analysis of the phenomenon of chiral symmetry breaking in vector gauge theories like QCD. Moreover, it has been recently argued that this formulation can be applied in the study of supersymmetric gauge theories [10] and strongly correlated fermion systems in three (two plus one) dimensions [11]. However, one of the most important tests of the validity of that formulation, the renormalizability of a chiral fermion, has not yet been examined. Since the existence of the triangle anomaly means that it is impossible to regularize a chiral fermion in a chiral invariant way [12], even if a chiral fermion is obtained in the continuum limit in a (lattice) regularization at tree level [13], it is not evident that the chirality of the regularized fermion is preserved after including quantum corrections. The fact that the chiral anomaly is correctly reproduced in the overlap formulation [5] shows that the chirality of the regularized fermion is preserved in the triangle anomaly diagram, where the dynamics of gauge fields does not play any role. However, this fact does not guarantee that the chirality of the regularized fermion is preserved after including the radiative corrections by the dynamics of the gauge bosons [14]. In this letter, we study the fermion propagator in the overlap formulation regularized on a lattice, including self-energy corrections due to the SU(N) gauge interactions at one loop level, and directly examine the chirality of the fermion and its renormalizability. In the overlap formulation, the effective action of a chiral fermion in the presence of gauge fields is expressed by the overlap of the two vacua |A±〉 of the Hamiltonians H±(A), where H±(A) = ∫ p ψ(p)H±(p)ψ(p) + V(A), (1) H±(p) = γ5 [ 4