Unification of flavor, [formula omitted], and modular symmetries

Alexander Baur,Hans Peter Nilles,Andreas Trautner,Patrick K.S Vaudrevange

doi:10.1016/j.physletb.2019.03.066

Abstract

Flavor symmetry plays a crucial role in the standard model of particle physics but its origin is still unknown. We develop a new method (based on outer automorphisms of the Narain space group) to determine flavor symmetries within compactified string theory. A picture emerges where traditional (discrete) flavor symmetries, CP -like symmetries and modular symmetries (like T-duality) of string theory combine to unified flavor symmetries. The groups depend on the geometry of compact space and the geographical location of fields in the extra dimensions. We observe a phenomenon of “local flavor groups” with potentially different flavor symmetries for the various sectors of quarks and leptons. This should allow interesting connections to existing bottom-up attempts in flavor model building.

Highlights

CP transformations and flavor symmetries were assumed to be of different origin
In the present paper we show that in string theory, where symmetries arise from the geometry of compactified extra dimensions and string selection rules [7,8,9], an even stronger link can be established: the CP and flavor transformations of the low-energy effective theory are unified in a common symmetry group
The main results of our paper originate from the discussion of the outer automorphisms of the “Narain space group” [37,38,39], where we find that non-Abelian flavor symmetries, modular symmetries of T -duality and CP have a common origin in string theory

Summary

Introduction

CP transformations and flavor symmetries were assumed to be of different origin. At a generic point in moduli space only the original flavor symmetry (possibly with CP as an outer automorphism) is present This allows a connection to the concept of “local grand unification” [17, 18], where the various fields of the standard model of particle physics (quarks, leptons, and Higgs bosons) live at different locations in compactified higher dimensions and feel different subgroups of the unified flavor group. A diversification of flavor symmetries to “local flavor groups” that depend on the location of fields in compactified extra dimensions (and could lead e.g. to different flavor groups for quarks and leptons) We shall apply this to our example of the two-dimensional Z3 orbifold and discuss the interplay of the original ∆(54) flavor symmetry and the relevant part of the modular transformation of T -duality, here given by Γ(3) (which is isomorphic to A4). We explore different regions in moduli space and construct the enhanced unified flavor groups

Outer automorphisms of the space group

Outer automorphisms of the Narain space group

Generic Point in T

Black Circles

Conclusions and Outlook

A Narain lattice