Lepton flavor symmetries

A general classification of flavor symmetries is provided according to their interplay with the proper Poincaré and gauge groups and to their linear or nonlinear action in field space. The focus is on the lepton sector, and the different types of symmetries describing neutrino masses and the lepton mixing matrix are reviewed. Several illustrative examples are presented for each type of symmetry, and specific strengths and limitations are discussed.

Figure 1

Diagrammatic representation of the three types of seesaw mechanisms. They all give rise to the Weinberg effective operator in Eq. (2.1) once the intermediate states are integrated out. The crosses denote lepton-number-violating mass insertions.

Figure 2

Space of invariants for $G=\mathrm{SU}(3)$ and $\phi$ in the real adjoint representation. The green region is the interior, which is defined by ${\gamma }_1^3-54{\gamma }_2^2>0$ and ${\gamma }_1>0$. The red point, where the full SU(3) symmetry is unbroken, is the intersection of the one-dimensional boundaries.

Figure 3

Two equivalent lattices with basis $(\tau ,1)$ and ($\tau +2,1$).

Figure 4

Fundamental region $\mathcal{F}$: a connected region of $\mathcal{H}$ such that each point of $\mathcal{H}$ can be mapped onto $\mathcal{F}$ by a $\overline{\mathrm{\Gamma }}$ transformation, but no two points in the interior of $\mathcal{F}$ are related under $\overline{\mathrm{\Gamma }}$.

Figure 5

Regions allowed in the $(m_{\text{lightest}},|m_{ee}|)$ plane for normal ordering (red) and inverted ordering (green), and predictions of modular invariant models at levels 3–5.