Discrete symmetries, roots of unity, and lepton mixing

TL;DR

This paper explores the group-theoretical foundations of lepton mixing, identifying specific finite groups that can produce a particular lepton mixing matrix column, and proposes a natural mechanism within renormalizable models.

Contribution

It identifies the finite group Z_q x S_4 as capable of enforcing a specific lepton mixing matrix column using residual symmetries and roots of unity.

Findings

Finite group Z_q x S_4 enforces the lepton mixing pattern.

A natural mechanism for achieving the mixing pattern is proposed.

Realization of the mechanism in renormalizable models is demonstrated.

Abstract

We investigate the possibility that the first column of the lepton mixing matrix U is given by u_1 = (2,-1,-1)^T/sqrt{6}. In a purely group-theoretical approach, based on residual symmetries in the charged-lepton and neutrino sectors and on a theorem on vanishing sums of roots of unity, we discuss the finite groups which can enforce this. Assuming that there is only one residual symmetry in the Majorana neutrino mass matrix, we find the almost unique solution Z_q x S_4 where the cyclic factor Z_q with q = 1,2,3,... is irrelevant for obtaining u_1 in U. Our discussion also provides a natural mechanism for achieving this goal. Finally, barring vacuum alignment, we realize this mechanism in a class of renormalizable models.