Lattice fermions with gauge noninvariant measure

Abstract

We define Weyl fermions on a finite lattice in such a way that in the path integral the action is gauge invariant but the functional measure is not. Two variants of such a formulation are tested in perturbative calculation of the fermion determinant in chiral Schwinger model. We find that one of these variants ensures restoring the gauge invariance of the nonanomalous part of the determinant in the continuum limit. A 'perfect' perturbative regularization of the chiral fermions is briefly discussed.

A lattice formulation of chiral gauge theories

“…3 This prescription has been discussed previously in the case of continuum theories [23] and in the case of lattice theories [20,24]; an equivalent formulation involving auxiliary fermion species has also been presented [18,19]. If one adopts 2 This property also holds if one drops the gauging of the Wilson term S W I on the right side of…”

“…Therefore, we conclude that our manipulations amount to the prescription that the magnitude of the chiral fermion determinant be replaced by the square root of the fermion determinant for a vector-like theory. 3 This prescription has been discussed previously in the case of continuum theories [23] and in the case of lattice theories [20,24]; an equivalent formulation involving auxiliary fermion species has also been presented [18,19]. If one adopts this prescription, then the magnitude of the fermion determinant and, correspondingly, the real part of the effective action have an exact gauge invariance.…”

“…A review of the present status of many of these lattice chiralfermion proposals has been given by Shamir [1]. Some proposals [2][3][4] have not yet been studied extensively. Others, such as the Eichten-Preskill model [5], the Smit-Swift model [6], and the staggered-fermion model [7,8], apparently fail to yield a chiral fermion spectrum because of the coupling of gauge degrees of freedom to the fermion [9,10].…”

Simple ansatz for the lattice fermion determinant

“…where D(A) is a lattice transcription of the chiral Dirac operator γ µ (∂ µ + igA µ P L ). The ansatz is based on the observation [7] that the gauge invariance of ℜΓ[A] without species doubling can be achieved if in the naive formulation the domain of integration over fermion loop momenta in all diagrams is narrowed down to D = (−π/(2a), π/(2a)) D , i.e. to the 1/2 D -th part of the fermion Brillouin zone B = (−π/a, π/a) D .…”

“…Note, that in the r.h.s. of (7) only local matrices are inverted, so the main difficulty is brought by the integration over t.…”