The eclectic flavor symmetry of the ℤ2 orbifold

Abstract

Modular symmetries naturally combine with traditional flavor symmetries and $$ \mathcal{CP} $$ CP , giving rise to the so-called eclectic flavor symmetry. We apply this scheme to the two-dimensional ℤ2 orbifold, which is equipped with two modular symmetries SL(2, ℤ)T and SL(2, ℤ)U associated with two moduli: the Kähler modulus T and the complex structure modulus U. The resulting finite modular group is ((S3× S3) ⋊ ℤ4) × ℤ2 including mirror symmetry (that exchanges T and U) and a generalized $$ \mathcal{CP} $$ CP -transformation. Together with the traditional flavor symmetry (D8× D8)/ℤ2, this leads to a huge eclectic flavor group with 4608 elements. At specific regions in moduli space we observe enhanced unified flavor symmetries with as many as 1152 elements for the tetrahedral shaped orbifold and $$ \left\langle T\right\rangle =\left\langle U\right\rangle =\exp \left(\frac{\pi \mathrm{i}}{3}\right) $$ T = U = exp π i 3 . This rich eclectic structure implies interesting (modular) flavor groups for particle physics models derived form string theory.

Completing the eclectic flavor scheme of the ℤ2 orbifold

“…In this paper we extend our previous discussion [1] of the T 2 /Z 2 orbifold. T 2 /Z 2 is the only two-dimensional orbifold with two unconstrained moduli T , U that transform under SL(2, Z) T × SL(2, Z) U and under mirror symmetry, which interchanges T and U .…”

“…Hence, it can serve as a building block for the discussion of six-dimensional orbifolds. 1 In our previous study, ref. [1], we had identified the traditional flavor symmetries and the finite modular symmetries Γ N for the T 2 /Z 2 orbifold.…”

“…1 In our previous study, ref. [1], we had identified the traditional flavor symmetries and the finite modular symmetries Γ N for the T 2 /Z 2 orbifold. The groups Γ N (for small N ) are isomorphic to groups like S 3 , A 4 , S 4 and A 5 that could be suitable for a description of discrete flavor symmetries in particle physics [7][8][9].…”

CP symmetry and symplectic modular invariance

“…If in addition the theory enjoys flavour modular invariance, the properties of the observed fermion spectrum such as masses, mixing angles and phases could be determined mostly by the vacuum, rather than by Lagrangian parameters. These features are parts of an appealing framework for the unification of flavor, CP and modular symmetries, advocated in recent works [39][40][41][42][43][44] in a top-down perspective. In a bottom-up approach we can hope to explore some aspect of this ideal framework.…”

“…A wealth of theoretical activity has in fact its focus on the study of Yukawa couplings in realistic string theory compactifications [7][8][9][10][11][12][13][14][15][16][17] and their modular properties . Moreover, in string theory finite modular invariance is in general only a component of a bigger Eclectic Flavour Group, which also involves CP and an ordinary flavour group leaving moduli invariant [39][40][41][42][43][44].…”