Origin of family symmetries

TL;DR

This work investigates the origins of discrete family symmetries in UV-complete theories, emphasizing geometric origins from extra dimensions and explicit string constructions. It connects symmetry emergence to orbifold compactifications, space-group selection rules, and higher-dimensional Lorentz remnants, while clarifying the role of anomaly constraints and nonperturbative Green–Schwarz cancellation. The paper shows that non-Abelian discrete groups like $D_4$, $A_4$, and $ ext{SW}_4$ frequently arise in heterotic orbifolds, offering a framework to address fermion masses and mixing, with $R$-symmetries playing a special role in controlling the $ ext{mu}$ problem and proton decay operators. Overall, the findings underscore the interplay between geometry, string dynamics, and anomaly considerations in shaping viable flavor symmetries for beyond-Standard-Model physics.

Abstract

Discrete (family) symmetries might play an important role in models of elementary particle physics. We discuss the origin of such symmetries in the framework of consistent ultraviolet completions of the standard model in field and string theory. The symmetries can arise due to special geometrical properties of extra compact dimensions and the localization of fields in this geometrical landscape. We also comment on anomaly constraints for discrete symmetries.

Figures (4)

• Figure 1: Example for one extra compact dimension: $\mathbbm{S}^1/\mathbbm{Z}_2$ orbifold. Points which are related by a reflection on the dashed line are identified. The fundamental region of the orbifold is an interval with the fixed points sitting at the boundaries.
• Figure 2: Example for two extra compact dimensions: If the $\mathbbm{T}^2$ lattice vectors have equal length and enclose $120^\circ$, one can also fold the fundamental region of $\mathbbm{T}^2/\mathbbm{Z}_2$ (dark gray region in (a)) to the tetrahedron (b).
• Figure 3: Origin of symmetries in heterotic orbifold compactifications. By compactification of six dimensions and appropriate gauge embedding the 10D super Poincaré and $\mathrm{E}_{8}\times\mathrm{E}_{8}$ symmetries get broken to the 4D super Poincaré, a 4D gauge and various discrete $R$ and non--$R$ symmetries. The latter two get further broken to subgroups by non--trivial VEVs of certain charged fields.
• Figure 4: a) The four fixed points labeled by their constructing elements for a two--dimensional $\mathbbm{T}^2/\mathbbm{Z}_{2}$ example (i.e. with $\theta=-\mathbbm{1}_{2 \times 2}$). b) The orbifold of a) folded up to a pillow--like object with fixed points at the corners of the pillow.