# Beyond basis invariants

## TL;DR

This paper proposes using basis-covariant objects instead of basis invariants to analyze physical properties of models, simplifying the identification of key features like CP conservation in multi-Higgs-doublet models.

## Contribution

It introduces an alternative approach using basis-covariant objects for analyzing basis invariance, improving clarity and efficiency in complex models.

## Key findings

- Basis-covariant approach simplifies analysis of physical properties.
- New necessary and sufficient conditions for CP conservation.
- Enhanced understanding of basis invariance in multi-Higgs models.

## Abstract

Physical observables cannot depend on the basis one chooses to describe fields. Therefore, all physically relevant properties of a model are, in principle, expressible in terms of basis-invariant combinations of the parameters. However, in many cases it becomes prohibitively difficult to establish key physical features exclusively in terms of basis invariants. Here, we advocate an alternative route in such cases: the formulation of basis-invariant statements in terms of basis-covariant objects. We give several examples where the basis-covariant path is superior to the traditional approach in terms of basis invariants. In particular, this includes the formulation of necessary and sufficient basis-invariant conditions for various physically distinct forms of CP conservation in two- and three-Higgs-doublet models.