# Large Leptonic Dirac CP Phase from Broken Democracy with Random Perturbations

## TL;DR

This paper proposes that broken democratic mass matrices with random perturbations, guided by residual symmetries, naturally produce large leptonic CP phases and realistic fermion mixing patterns, improving upon anarchic models.

## Contribution

It introduces a framework where residual symmetries lead to democratic mass matrices with random perturbations, explaining large CP phases and realistic mixing angles.

## Key findings

- Leptonic Dirac CP phase peaks around ±π/2 with perturbations.
- Broken democracy models outperform anarchic models in predicting fermion properties.
- Framework links residual symmetries to realistic fermion mass and mixing predictions.

## Abstract

A large value of the leptonic Dirac CP phase can arise from broken democracy, where the mass matrices are democratic up to small random perturbations. Such perturbations are a natural consequence of broken residual $\mathbb S_3$ symmetries that dictate the democratic mass matrices at leading order. With random perturbations, the leptonic Dirac CP phase has a higher probability to attain a value around $\pm \pi/2$. Comparing with the anarchy model, broken democracy can benefit from residual $\mathbb S_3$ symmetries, and it can produce much better, realistic predictions for the mass hierarchy, mixing angles, and Dirac CP phase in both quark and lepton sectors. Our approach provides a general framework for a class of models in which a residual symmetry determines the general features at leading order, and where, in the absence of other fundamental principles, the symmetry breaking appears in the form of random perturbations.