Fermion hierarchies in SU(5) grand unification from $$ {\Gamma}_6^{\prime } $$ modular flavor symmetry
Yoshihiko Abe, Tetsutaro Higaki, Junichiro Kawamura, Tatsuo Kobayashi
Abstract
A bstract We construct an SU(5) grand unified model in which the hierarchies of the quark and lepton masses and mixing are explained by the $$ {\Gamma}_6^{\prime } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Γ</mml:mi> <mml:mn>6</mml:mn> <mml:mo>′</mml:mo> </mml:msubsup> </mml:math> modular flavor symmetry. The hierarchies are realized by the Froggatt-Nielsen-like mechanism due to the residual $$ {Z}_6^T $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mn>6</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:math> symmetry, approximately unbroken at τ ~ i∞ . We argue that the $$ {\Gamma}_6^{\left({}^{\prime}\right)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Γ</mml:mi> <mml:mn>6</mml:mn> <mml:mfenced> <mml:msup> <mml:mrow /> <mml:mo>′</mml:mo> </mml:msup> </mml:mfenced> </mml:msubsup> </mml:math> symmetry is the minimal possibility to realize the up-type quark mass hierarchies, since the Yukawa matrix is symmetric. We find a combination of the representations and modular weights and then show numerical values of (1) coefficients for the realistic fermion hierarchies.