Flavor's Delight

Abstract

Discrete flavor symmetries provide a promising approach to understand the flavor sector of the standard model of particle physics. Top-down (TD) explanations from string theory reveal two different types of such flavor symmetries: traditional and modular flavor symmetries that combine to the eclectic flavor group. There have been many bottom-up (BU) constructions to fit experimental data within this scheme. We compare TD and BU constructions to identify the most promising groups and try to give a unified description. Although there is some progress in joining BU and TD approaches, we point out some gaps that have to be closed with future model building.

Keywords: eclectic symmetries; flavor; string compactifications.

Figure 1

The ${\mathbb{T}}^2/{\mathbb{Z}}_3$ orbifold (yellow shaded region) with three fixed points, $X,Y,Z$. Twisted states are localized at theses fixed points. Figure taken from ref. [54].

Figure 2

Fundamental domain of $\mathrm{SL}(2,\mathbb{Z})$ (dark shaded) and of its subgroup $\Gamma \left(3\right)\cong \mathrm{SL}(2,3\mathbb{Z})$ (light shaded). Figure taken from ref. [54].

Figure 3

Unbroken modular symmetries at special curves in moduli space, including the $\mathcal{CP}$-like generator $\mathrm{U}$, which maps $M\mapsto -\overline{M}$. Figure adapted from ref. [46].

Figure 4

Local flavor unification at special points and curves in moduli space. The traditional flavor symmetry $\Delta \left(54\right)$, valid at generic points, is enhanced to two (different) groups with GAP Id [108,17] at the vertical lines and semi-circles, including $\mathcal{CP}$-like transformations. At the intersections of curves, the flavor symmetry is further enhanced to [216,87] and [324,39], also with $\mathcal{CP}$-like transformations. Figure adapted from ref. [45].