# Anomaly-free Model Building with Algebraic Geometry

## TL;DR

This paper introduces algebraic geometry techniques to construct anomaly-free gauged flavor models with rational charges, providing three viable models that explain the observed flavor structure beyond the Standard Model.

## Contribution

It applies algebraic geometry methods, such as Gr"obner bases and Mordell-Weil generators, to systematically build anomaly-free models with rational charges, expanding model-building tools.

## Key findings

- Three phenomenologically viable models identified
- Algebraic methods reduce charge constraints effectively
- Models include non-SUSY and supersymmetric scenarios

## Abstract

We present a method to find anomaly-free gauged Froggatt-Nielsen type models using results from algebraic geometry. These methods should be of general interest for model building beyond the Standard Model (SM) when rational charges are required. We consider models with a gauged $U(1)$ flavor symmetry with one flavon and two Higgs doublets and three right-handed SM singlets to provide three model examples based on different physical assumptions. The models we study are: anomaly-free with no SM neutral heavy chiral fermions, anomaly-free with SM neutral heavy chiral fermions, and supersymmetric with SM neutral heavy chiral fermions where the anomalies cancel via the Green-Schwarz mechanism. With these different models we show how algebraic methods may be used in model building; both to reduce the charge constraints by calculation of Gr\"obner bases, and to find rational solutions to cubic equations using Mordell-Weil generators. Using these tools we find three phenomenologically viable models explaining the observed flavor structure.