Flavor Symmetry and Galois Group of Elliptic Curves

TL;DR

This paper explores how the Galois group of elliptic curves with complex multiplication can serve as the origin of flavor symmetries in particle physics, linking number theory with fermion mass structures.

Contribution

It introduces a novel framework connecting elliptic curve Galois groups with flavor symmetries, including R-parity, in a brane-world model.

Findings

Galois group is an extension of Z_2 by an abelian group

A viable Galois group example reproduces fermion mass textures

The approach satisfies mixed-anomaly conditions

Abstract

A new approach to the generation structure of fermions is proposed. We consider a brane configuration in which the brane intersection yields a two-torus in the extra space. It is assumed that the two-torus is discretized and is given by the torsion points of the elliptic curve over Q . We direct our attention to the arithmetic structure of the elliptic curve with complex multiplication (CM). In our approach the flavor symmetry including the R-parity has its origin in the Galois group of elliptic curves with CM. We study the possible types of the Galois group. The Galois group is shown to be an extension of Z_2 by some abelian group. A phenomenologically viable example of the Galois group is presented, in which the characteristic texture of fermion masses and mixings is reproduced and the mixed-anomaly conditions are satisfied.