# CP violation due to multi Froggatt-Nielsen fields

## TL;DR

This paper explores how to generate CP violation within the Froggatt-Nielsen framework by introducing multiple scalar fields and discrete symmetries, leading to predictions for CKM parameters and CP phases.

## Contribution

It proposes a novel mechanism involving at least two scalar fields with a discrete symmetry to produce CP violation in the FN model, linking CKM parameters to vacuum expectation values.

## Key findings

- CP violation arises from the relative phase between scalar VEVs.
- Predictions relate CKM angles and CP phase to VEV ratios and a CKM element.
- The simplest model's CP phase prediction does not match experimental data.

## Abstract

We study how to incorporate CP violation in the Froggatt--Nielsen (FN) mechanism. To this end, we introduce non-renormalizable interactions with a flavor democratic structure to the fermion mass generation sector. It is found that at least two iso-singlet scalar fields with imposed a discrete symmetry are necessary to generate CP violation due to the appearance of the relative phase between their vacuum expectation values. In the simplest model, ratios of quark masses and the Cabibbo-Kobayashi-Maskawa (CKM) matrix including the CP violating phase are determined by the CKM element |V_{us}| and the ratio of two vacuum expectation values R=|R|e^{i*alpha} (a magnitude and a phase). It is demonstrated how the angles phi_i (i=1--3) of the unitarity triangle and the CKM off-diagonal elements |V_{ub}| and |V_{cb}| are predicted as a function of |V_{us}|, |R| and \alpha. Although the predicted value of the CP violating phase does not agree with the experimental data within the simplest model, the basic idea of our scenario would be promising to construct a more realistic model of flavor and CP violation.