# The fermion determinant and the chiral gauge theory on a lattice

## TL;DR

This paper examines how slight violations of gauge invariance in lattice chiral fermion determinants introduce a new, larger scale related to gauge variables, impacting the formulation of chiral gauge theories on a lattice.

## Contribution

It demonstrates that even minimal gauge invariance violations in lattice chiral fermion determinants necessitate an additional scale tied to gauge variables, affecting lattice gauge theory formulations.

## Key findings

- Violations of gauge invariance induce a new scale much larger than the lattice spacing.
- The new scale is associated with gauge variables and impacts the theory's consistency.
- Even mild violations have significant implications for lattice chiral gauge theories.

## Abstract

Considering as an example a simple lattice ansatz for the chiral fermion determinant, we demonstrate that even very mild violation of gauge invariance by the determinant at finite lattice spacing leads to the need for another scale in the full gauge theory. This new scale is much grater than the lattice spacing and is associated with the gauge variables.