Scale Invariance in the Spectral Action

TL;DR

This paper introduces a dynamical dilaton field into the spectral action to achieve scale invariance, analyzing low-energy terms and applying results to the standard model's spectral action, resulting in a nearly scale-invariant effective action.

Contribution

It demonstrates how to make the spectral action scale invariant by introducing a dilaton, aligning with models of inflation and the Randall-Sundrum scenario without fine tuning.

Findings

Effective matter couplings are scale invariant except for specific terms.

The resulting action matches models for inflation and the Randall-Sundrum model.

All desirable features are obtained with correct signs and no fine tuning.

Abstract

The arbitrary mass scale in the spectral action for the Dirac operator in the spectral action is made dynamical by introducing a dilaton field. We evaluate all the low-energy terms in the spectral action and determine the dilaton couplings. These results are applied to the spectral action of the noncommutative space defined by the standard model. We show that the effective action for all matter couplings is scale invariant, except for the dilaton kinetic term and Einstein-Hilbert term. The resulting action is almost identical to the one proposed for making the standard model scale invariant as well as the model for extended inflation and has the same low-energy limit as the Randall-Sundrum model. Remarkably, all desirable features with correct signs for the relevant terms are obtained uniquely and without any fine tuning.