The eclectic flavor symmetry of the ℤ2 orbifold
Alexander Baur, Moritz Kade, Hans Peter Nilles, Saúl Ramos-Sánchez, Patrick K. S. Vaudrevange
Abstract
A bstract Modular symmetries naturally combine with traditional flavor symmetries and $$ \mathcal{CP} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>CP</mml:mi> </mml:math> , giving rise to the so-called eclectic flavor symmetry. We apply this scheme to the two-dimensional ℤ 2 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 (( S 3 × S 3 ) ⋊ ℤ 4 ) × ℤ 2 including mirror symmetry (that exchanges T and U ) and a generalized $$ \mathcal{CP} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>CP</mml:mi> </mml:math> -transformation. Together with the traditional flavor symmetry ( D 8 × D 8 )/ℤ 2 , 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 $$ \left\langle T\right\rangle =\left\langle U\right\rangle =\exp \left(\frac{\pi \mathrm{i}}{3}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mi>T</mml:mi> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mi>U</mml:mi> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mo>exp</mml:mo> <mml:mfenced> <mml:mfrac> <mml:mrow> <mml:mi>π</mml:mi> <mml:mi>i</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mfenced> </mml:math> . This rich eclectic structure implies interesting (modular) flavor groups for particle physics models derived form string theory.