Spectral action and the electroweak $\theta$-terms for the Standard Model without fermion doubling

TL;DR

This paper derives the spectral action for a noncommutative geometry model of the Standard Model without fermion doubling, explicitly showing the emergence of electroweak topological $ heta$-terms and CP-symmetry breaking effects.

Contribution

It introduces a chiral spectral triple with a non-product Dirac operator, enabling the calculation of the spectral action without fermion doubling and revealing electroweak $ heta$-terms.

Findings

Explicit derivation of the spectral action for a noncommutative geometry model

Demonstration of topological $ heta$-terms in the electroweak sector

Model allows for CP-symmetry breaking

Abstract

We compute the leading terms of the spectral action for a noncommutative geometry model that has no fermion doubling. The spectral triple describing it, which is chiral and allows for CP-symmetry breaking, has the Dirac operator that is not of the product type. Using Wick rotation we derive explicitly the Lagrangian of the model from the spectral action for a flat metric, demonstrating the appearance of the topological $\theta$-terms for the electroweak gauge fields.