Manifestly Gauge Covariant Treatment of Lattice Chiral Fermion

We propose a lattice formulation of the chiral fermion which maximally respects the gauge symmetry and simultaneously is free of the unwanted species doublers. This is achieved by directly dealing with the lattice fermion propagator and the composite operators, rather than the lattice action and the fermionic determinant. The latter is defined as a functional integral of the expectation value of the gauge current operator with respect to the background gauge field. The gauge anomaly is characterized as a non-integrability of this integration process and, the determinant is defined only for anomaly free cases. Gauge singlet operators on the other hand are always regularized gauge invariantly. Some perturbative check is performed to confirm the gauge covariance and the absence of the doublers. This formulation can be applied rather straightforwardly to numerical simulations in the quenched approximation. ⋆ e-mail: [email protected] The chiral fermion on the lattice has refused the manifestly gauge invariant formulation [1]. There even exists the no-go theorem [2] for such an endeavor. In the continuum counterpart, the chiral fermion exhibits a curious phenomenon called the quantum anomaly [3], or more definitely the gauge anomaly [4]. The difficulty in the lattice chiral gauge theory seems to be a rather natural consequence of the gauge anomaly. Suppose that we start from a well regularized partition function or a fermionic determinant defined by a manifestly gauge invariant lattice fermion action. In the continuum limit, the gauge anomaly is realized as a gauge variation of the partition function. Therefore if the fermion content is not free of the gauge anomaly, the partition function is not gauge invariant. But this contradicts with the very gauge invariance of the formulation. There are mainly two possible ways out: One is the appearance of the species doublers which cancels the gauge anomaly [5]. Another is some pathology in the continuum limit such as the non-Lorentz covariance [6]. The trouble of the lattice regularization is that it always regularizes the ultraviolet divergence, even when a gauge invariant regularization is impossible due to the gauge anomaly. From the above reasoning, the appearance of the unwanted doublers are quite natural. But the problem in the conventional approach is of course that the doublers appear even in the anomaly free cases. Presumably, the ideal lattice formulation of the chiral fermion will be the one which distinguishes the anomaly free gauge representations from the anomalous ones. † That unknown lattice action should have a structure that can be written down, for example, for the spinor rep. of SO(4) but not for the fundamental rep. of SU(3), because the latter is anomalous. Such an ideal formulation seems to require a deeper understanding on the origin of the quantum anomaly, which is not available at present. In this article, we simply abandon a direct gauge invariant definition of the † Besides the perturbative anomaly, it should also explain the topological global anomaly [7] because the partition function is ill-defined with this anomaly.