The Standard Model on a finite bosonic lattice with continuum Weyl fermions

Abstract

We investigate a formulation of the Standard Model in which gauge and Higgs fields are regulated on a finite lattice, while quarks and leptons remain continuum Weyl fermions. This preserves the familiar continuum description of anomalies, spectral flow, and fermion zero modes, without requiring a finite-dimensional discretization of chiral fermions. The Standard Model's anomaly cancellation permits a gauge-invariant definition of the chiral fermion determinant. Yukawa interactions are included in the full fermion determinant using a gauge-covariant cutoff that acts on the interaction vertices rather than the fermion propagator. We show that removing this cutoff requires only the local subtractions expected from one-loop power counting, and construct a consistent complex fermion weight, including in topologically nontrivial backgrounds. At fixed bosonic lattice spacing and finite spacetime volume, the resulting bosonic functional integral is finite and exactly gauge invariant under the stated smoothness conditions. The construction provides a finite-regulator Standard Model formulation retaining the physical structure of continuum chiral fermions. Appendix F.5 establishes reflection positivity for a variant reconstruction at zero Yukawa coupling and zero theta. The bosonic continuum limit remains open.